Complete Theory

Theory and Background

E Copyright LMS International 2000

Table of Contents Part I

Signal processing

Chapter 1 1.1 1.2 1.3

1.4 1.5

Digital signal processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leakage and windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Window characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Window types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing window functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Window correction mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Window correction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 2.1 2.2 2.3

3.2

3.3

3.4 3.5

2 8 10 11 12 13 15 15 17 18 20

Structural dynamics testing

Signal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signature analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 3.1

Spectral processing

24 25 27

Functions

Time domain functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Record . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crosscorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency domain functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Autopower Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crosspower spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principal Component Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Response Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overall level (OA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Order sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octave sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rms calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32 32 32 33 34 34 34 36 36 37 38 39 41 42 44 46 46 46 48 48 49 50

Part II

Acoustics and Sound Quality

Chapter 4 4.1

4.2

4.3 4.4

Chapter 5 5.1

5.2 5.3 5.4 5.5

6.2 6.3

56 56 56 56 57 58 59 60 60 60 60 61 61 62 64

Acoustic measurements

Acoustic measurement functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound pressure level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Residual intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure residual intensity index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of acoustic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic measurement surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic ISO standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F1 Sound field temporal variability indicator . . . . . . . . . . . . . . . . . . . . . . F2 Surface pressure-intensity indicator . . . . . . . . . . . . . . . . . . . . . . . . . . F3 Negative partial power indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F4 Non-uniformity indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement mesh adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6 6.1

Terminology and definitions

Acoustic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound power (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound (Acoustic) intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic impedance (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dB scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound power level Lw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle velocity level Lv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound (Acoustic) intensity level LI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound pressure level LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octave bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 68 68 69 70 72 74 74 76 77 77 77 78 79 79 80

Sound quality

The basic concepts of Sound Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The perception of sounds by the human ear . . . . . . . . . . . . . . . . . . . . . . . Binaural hearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loudness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Masking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temporal effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sound quality analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of sound signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binaural recording and playback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reading list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 84 85 86 86 87 87 88 89 90 91 91 93 95

Chapter 7 7.1 7.2 7.3

7.4 7.5 7.6 7.7 7.8 7.9 7.10

Chapter 8 8.1 8.2

Sound metrics

Sound pressure level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time domain sound pressure level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalent sound pressure level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loudness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Stevens Mark VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Stevens Mark VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Loudness Zwicker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sharpness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuation strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Articulation index (AI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speech interference level (SIL, PSIL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impulsiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acoustic holography

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic holography concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temporal and spatial frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summation of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagating and evanescent waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Back) propagating to other planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Wiener filter and the AdHoc window . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of other acoustic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part III

100 100 101 102 103 104 104 107 109 110 111 112 114 115 118 119 119 121 122 124 125 126

Time data processing

Chapter 9

Statistical functions

Minimum, maximum, range and extremum . . . . . . . . . . . . . . . . . . . . . . . . Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Root mean square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crest factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Median . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Percentiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variance and standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean absolute deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Skewness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Markov regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130 130 130 131 131 131 132 133 133 134 134 134 135 136

Chapter 10 Time frequency analysis 10.1 10.2 10.3 10.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear time-frequency representations . . . . . . . . . . . . . . . . . . . . . . . . . . . The Short Time Fourier Transform (STFT) . . . . . . . . . . . . . . . . . . . . . . . . Wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quadratic time-frequency representations . . . . . . . . . . . . . . . . . . . . . . . . The Wigner-Ville distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 142 142 143 146 147 148 150

Chapter 11 11.1

11.2 11.3

Resampling

Fixed resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Integer downsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Integer upsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Fractional ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Arbitrary ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adaptive resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 153 154 156 157 159 159 162

Chapter 12 Digital filtering 12.1 12.2

12.3 12.4 12.5

Basic definitions relating to digital filtering . . . . . . . . . . . . . . . . . . . . . . . . FIR and IIR filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Filter design terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear phase filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filter types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Design of FIR filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design of an FIR window filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIR multi window Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIR Remez filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Design of IIR filters using analog prototypes . . . . . . . . . . . . . . . . . Step 1) Specify the filter characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 2) Compute the analog frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . Step 3) Select the suitable analog filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bessel filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Butterworth filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chebyshev (type I) filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inverse Chebyshev (type II) filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cauer (elliptical) filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step 4) Transform the prototype low pass filter . . . . . . . . . . . . . . . . . . . . Step 5) Apply a bilinear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . Determining the filter order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 IIR Inverse design filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applying filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 170 171 171 171 172 174 174 176 177 178 178 179 180 180 181 182 183 183 185 185 185 187 188 189 191

Chapter 13 Harmonic tracking 13.1 13.2

13.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditions for use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Determination of the Rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Waveform tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structural equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Data equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194 194 195 195 195 196 196 199

Chapter 14 Counting and histogramming 14.1 14.2

14.3

14.4

Part IV

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One dimensional counting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Peak count methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Level cross counting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Range counting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting of single ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Counting of range-pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-dimensional counting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 From-to-counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Range-mean counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 ``Range pair-range" or ``Rainflow'' method . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

204 206 206 207 208 208 209 211 211 212 213 217

Analysis and design

Chapter 15 Estimation of modal parameters 15.1 15.2

15.3

15.4 15.5

Estimation of modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A note about units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Single or multiple degree of freedom method . . . . . . . . . . . . . . . . 15.2.2 Local or global estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Multiple input analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.4 Time vs frequency domain implementation . . . . . . . . . . . . . . . . . . 15.2.5 Vibro-acoustic modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of a method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Peak picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Mode picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.3 Circle fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.4 Complex mode indicator function . . . . . . . . . . . . . . . . . . . . . . . . . . Cross checking and tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.5 Least squares complex exponential . . . . . . . . . . . . . . . . . . . . . . . . . Model for continuous data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model for sampled data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical implementation of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . Determining the optimum number of modes . . . . . . . . . . . . . . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model for sampled data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practical implementation of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.6 Least squares frequency domain . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.7 Frequency domain direct parameter identification . . . . . . . . . . . . Maximum likelihood method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Theoretical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of static compensation modes . . . . . . . . . . . . . . . . . . . . . . . . .

220 222 223 223 225 226 228 230 233 233 234 236 237 238 242 243 244 244 245 246 248 250 251 253 254 256 260 260 264

Chapter 16 Operational modal analysis 16.1 16.2

16.3

Why operational modal analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Stochastic substate identification methods . . . . . . . . . . . . . . . . . . . 16.2.2 Natural Excitation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Selection of the modal parameter identification method . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

268 270 270 275 277 279

Chapter 17 Running modes analysis 17.1 17.2

17.3 17.4

Running mode analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measuring running modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Transmissibility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Crosspower spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identification and scaling of running modes . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Scaling of running modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal Scale Factors and Modal Assurance Criterion . . . . . . . . . . . . . . . . Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

282 284 284 286 288 288 290 290 291

Chapter 18 Modal validation 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 18.10 18.11

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MSF and MAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reciprocity between inputs and outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized modal parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal phase collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode indicator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summation of FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthesis of FRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

294 295 297 298 300 302 303 304 305 307 308

Chapter 19 Rigid body modes 19.1

19.2 19.3

Calculation of rigid body properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Derivation of rigid body properties from measured FRFs . . . . . . . . . . . . Calculation of the rigid body properties . . . . . . . . . . . . . . . . . . . . . . . . . . . Rigid body mode analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Decomposition of measured modes into rigid body modes . . . . . . 19.2.2 Synthesis of rigid body modes based on geometrical data . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

310 310 311 316 317 318 320

Chapter 20 Design 20.1 20.2 20.3

20.4

Using the modal model for modal design . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Mathematical background to sensitivity analysis . . . . . . . . . . . . . . Modification prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.2 Implementation of Modification prediction . . . . . . . . . . . . . . . . . . 20.3.3 Definition of modifications to the model . . . . . . . . . . . . . . . . . . . . 20.3.4 Modification prediction calculation . . . . . . . . . . . . . . . . . . . . . . . . . 20.3.5 Units of scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of the application of a beam element . . . . . . . . . . . . . . . . . . . . . Static condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forced response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Mathematical background for forced response . . . . . . . . . . . . . . . .

322 325 325 329 329 338 339 347 348 349 351 354 354

Chapter 21 Geometry concepts 21.1 21.2

The geometry of a test structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

358 359 359 359

Theory and Background

Part I Signal processing

Chapter 1 Spectral processing . . . . . . . . . . . . . . . . . . . . .

1

Chapter 2 Structural dynamics testing . . . . . . . . . . . . . .

23

Chapter 3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

Chapter 1

Spectral processing

This chapter provides an overview of the terminology and techĆ niques used in general signal processing of vibrational and acoustic data. Digital signal processing Aliasing Leakage and windows Averaging This is by no means a comprehensive treatment of the subject and a reading list is given at the end.

1

Chapter 1

1.1

Spectral processing

Digital signal processing Time and frequency domains It is a property of all real waveforms that they can be made up of a number of sine waves of certain amplitudes and frequencies. Viewing these waves in the frequency domain rather than the time domain can be useful in that all the components are more readily revealed. amplitude

time

frequency

Each sine wave in the time domain is represented by one spectral line in the frequency domain. The series of lines describing a waveform is known as its frequency spectrum. Fourier transform The conversion of a time signal to the frequency domain (and its inverse) is achieved using the Fourier Transform as defined below. 

S x(f)


 x(t)āeā

jā2
āft

Ădt

Eqn. 1-1

 

x(t)


 S ā(f)eā x

Ăj2
ft

Ădf

Eqn. 1-2



This function is continuous and in order to use the Fourier Transform digitally a numerical integration must be performed between fixed limits. The Discrete Fourier Transform (DFT) The digital computation of the Fourier Transform is called the Discrete Fourier Transform. It calculates the values at discrete points (m
f) and performs a nuĆ merical integration as illustrated below between fixed limits (N samples).

2

The Lms Theory and Background Book

Spectral processing

x(t)āe j2ā
āmfāt

time


t

Since the waveform is being sampled at discrete intervals and during a finite observation time, we do not have an exact representation of it in either domain. This gives rise to shortcomings which are discussed later.

Hermitian symmetry The Fourier transform of a sinusoidal function would result in complex funcĆ tion made up of real and imaginary parts that are symmetrical. This is illusĆ trated below. In the majority of cases only the real part is taken into account and of this only the positive frequencies are shown. So the representation of the frequency spectrum of the sine wave shown below would become the area shaded in grey. S(f) imaj

X(t)

S(f) real A/2

A/2

A -f

0

+f

-f

0

+f

A/2

The Fast Fourier Transform (FFT) The Fast Fourier Transform is a dedicated algorithm to compute the DFT. It thus determines the spectral (frequency) contents of a sampled and discretized time signal. The resulting spectrum is also discrete. The reverse procedure is referred to as an inverse or backward FFT.

Part I

Signal processing

3

Chapter 1

Spectral processing

N samples

time

inverse

N/2 spectral lines

frequency

To achieve high calculation performance the FFT algorithm requires that the number of time samples (N) be a power of 2 (such as 2, 4, 8, ...., 512, 1024, 2048).

Blocksize Such a time record of N samples is referred to as a block of data with N being the blocksize. N samples in the time domain converts to N/2 spectral (frequency) lines. Each line contains information about both amplitude and phase.

Frequency range The time taken to collect the sample block is T. The lowest frequency that can be detected then is that which is the reciprocal of the time T.

T The frequency spacing between the spectral lines is therefore 1/T and the highĆ est frequency that can be determined is (N/2).(1/T).

4

The Lms Theory and Background Book

Spectral processing

N/2 spectral lines

frequency 1 T

2 T

3 T

f
1 T

N2 T

fmax
N . 1
N .f 2 T 2

The frequency range that can be covered is dependant on both the blocksize (N) and the sampling period (T). To cover high frequencies you need to sample at a fast rate which implies a short sample period.

Real time Bandwidth Remember that an FFT requires a complete block of data to be gathered before it can transform it. The time taken to gather a complete block of data depends on the blocksize and the frequency range but it is possible to be gathering a second time record while the first one is being transformed. If the computation time takes less than the measurement time, then it can be ignored and the process is said to be operating in real time. time record 1

time record 1

time record 2

time record3

time record 4

FFT 1

FFT 2

FFT 3

time record 2

time record3

time record 4

FFT 1

FFT 2

FFT 3

Real time operation

This is not the case if the computation time is taking longer than the measureĆ ment time or if the acquisition requires a trigger condition.

Overlap Overlap processing involves using time records that are not completely indeĆ pendent of each other as illustrated below.

Part I

Signal processing

5

Chapter 1

Spectral processing

time record 1 time record 2 time record3 time record 4

FFT 1

FFT 2

FFT 3

If the time data is not being weighted at all by the application of a window, then overlap processing does not include any new data and therefore makes no statistical improvement to the estimation procedure. When windows are being applied however, the overlap process can utilize data that would otherwise be ignored. The figure below shows data that is weighted with a Hanning window. In this case the first and last 20% of each sample period is practically lost and contribĆ utes hardly anything towards the averaging process.

Sampled data

Processed data with no overlap

6

The Lms Theory and Background Book

Spectral processing

Applying an overlap of at least 30% means that this data is once again included - as shown below. This not only speeds up the acquisition (for the same numĆ ber of averages) but also makes it statistically more reliable since a much higher proportion of the acquired data is being included in the averaging process.

Sampled data

Processed data with 30% overlap

Part I

Signal processing

7

Chapter 1

1.2

Spectral processing

Aliasing Sampling at too low a frequency can give rise to the problem of aliasing which can lead to erroneous results as illustrated below.

This problem can be overcome by implementing what is known as the Nyquist Criterion, which stipulates that the sampling frequency (fs ) should be greater than twice the highest frequency of the interest (fm ).

fs  2Ăfm The highest frequency that can be measured is fmax which is half the sampling frequency (fs ), and is also known as the Nyquist frequency (fn ).

f fmax
s Ă
fn 2

Ă

The problem of aliasing can also be illustrated in the frequency domain. measured frequency

fn f1 f1

f2 fn

f3

2 fn = fs

input frequency

f4 3 fn

4 fn

All multiples of the Nyquist frequency (fn ) act as `folding lines'. So f4 is folded back on f3 around line 3 fn , f3 is folded back on f2 around line 2 fn and f2 is folded back on f1 around line fn . Therefore all signals at f2 , f3 , f4 are all seen as signals at frequency f1 . The only sure way to avoid such problems is to apply an analog or digital antialiasing filter to limit the high frequency content of the signal. Filters are less than ideal however so the positioning of the cut off frequency of the filters must be made with respect to fmax and the roll off characteristics of the filter.

8

The Lms Theory and Background Book

Spectral processing

ideal filter

fmax

fmax

Part I

Signal processing

fs roll off characteristics of a real filter

fs

9

Chapter 1

1.3

Spectral processing

Leakage and windows A further problem associated with the discrete time sampling of the data is that of leakage. A continuous sine wave such as the one shown below should result in the single spectral line. continuous waveform

time

frequency Because the signals are measured over a sample period T, the DFT assumes that this is representative for all time. When the sine wave is not periodic in the sample time window, the result is a consequent leakage of energy from the original line spectrum due to the discontinuities at the edges. discretely sampled waveform

time

time

DFT assumed waveform

frequency The user should be aware that leakage is one of the most serious problems associated with digital signal processing. Whilst aliasing errors can be reduced by various techniques, leakage errors can never be eliminated. Leakage can be reĆ duced by using different excitation techniques and increasing the frequency resolution, or through the use of windows as described below.

10

The Lms Theory and Background Book

Spectral processing

1.3.1

Windows The problem of discontinuities at the edge can be alleviated either by ensuring that the signal and the sampling period are synchronous or by ensuring that the function is zero at the start and end of the sampling period. This latter situaĆ tion can be achieved by applying what is called a `window function' which normally takes the form of an amplitude modulated sine wave.

X

sample period T.

Frequency spectrum of a sine wave, periodic in the sample period T.

=

sample period T.

Frequency spectrum of a sine wave, not periodic with the sample period without a window.

Frequency spectrum of a sine wave that is not periodic with the sample period with a window.

The use of windows gives rise to errors itself of which the user should be aware and should be avoided if possible. The various types of windowing functions distribute the energy in different ways. The choice of window depends on the input function and on your area of interest.

Self windowing functions Self windowing functions are those that are periodic in the sample period T or transient signals. Transient signals are those where the function is naturally zero at the start and end of the sampling period such as impulse and burst sigĆ nals. Self windowing functions should be adopted whenever possible since the application of a window function presents problems of its own. A rectangular or uniform window can then be used since it does not affect the energy disĆ tribution. Note!

Part I

It should be noted that synchronizing the signal and the sampling time, or using a self windowing function is preferable to using a window.

Signal processing

11

Chapter 1

Spectral processing

Window characteristics The time windows provided take a number of forms - many of which are amĆ plitude modulated sine waves. There are all in effect filters and the properties of the various windows can be compared by examining their filter characterisĆ tics in the frequency domain where they can be characterized by the factors shown below. noise Bandwidth 0dB side lobe falloff highest side lobe

log
f The windows vary in the amount of energy squeezed in to the central lobe as compared to that in the side lobes. The choice of window depends on both the aim of the analysis and the type of signal you are using. In general, the broader the noise Bandwidth, the worse the frequency resolution, since it becomes more difficult to pick out adjacent frequencies with similar amplitudes. On the other hand, selectivity (i.e. the ability to pick out a small component next to a large on) is improved with side lobe falloff. It is typical that a window that scores well on Bandwidth is weak on side lobe fall off and the choice is therefore a trade off between the two. A summary of these characteristics of the windows provided is given in Table 1.1. Highest side lobe (dB)

Sidelobe falloff (dB/decade)

Noise BandĆ width (bins)

Max. Amp erĆ ror (dB)

Uniform

-13

-20

1.00

3.9

Hanning

-32

-60

1.5

1.4

Hamming

-43

-20

1.36

1.8

Kaiser-Bessel

-69

-20

1.8

1.0

Blackman

-92

-20

2.0

1.1

Flattop

-93

0

3.43

<0.01

Window type

Table 1.1 Properties of time windows

12

The Lms Theory and Background Book

Spectral processing

Window types

Uniform window This window is used when leakage is not a probĆ lem since it does not affect the energy distribuĆ tion. It is applied in the case of periodic sine waves, impulses, transients... where the function is naturally zero at the start and end of the samĆ pling period. The following windows Hanning, Hamming, Blackman, Kaiser-Bessel and Flattop all take the form of an amplitude modulated sine wave in the time domain. For a comparison of their frequency domain filter characteristics - see Table 1.1.

Hanning This window is most commonly applied for general purpose analysis of ranĆ dom signals with discrete frequency components. It has the effect of applying a round topped filter. The ability to distinguish between adjacent frequencies of similar amplitude is low so it is not suitable for accurate measurements of small signals.

Hamming This window has a higher side lobe than the Hanning but a lower fall off rate and is best used when the dynamic range is about 50dB.

Blackman This window is useful for detecting a weak component in the presence of a strong one.

Kaiser–Bessel The filter characteristics of this window provide good selectivity, and thus make it suitable for distinguishing multiple tone signals with widely different levels. It can cause more leakage than a Hanning window when used with ranĆ dom excitation.

Part I

Signal processing

13

Chapter 1

Spectral processing

Flattop This window's name derives from its low ripple characteristics in the filter pass band. This window should be used for accurate amplitude measurements of single tone frequencies and is best suited for calibration purposes.

Force window This type of window is used with a tranĆ sient signal in the case of impact testing. It is designed to eliminate stray noise in the excitation channel as illustrated here. It has a value of 1 during the impact periĆ od and 0 otherwise.

Exponential window This window is also used with a transient signal. It is designed to ensure that the sigĆ nal dies away sufficiently at the end of the sampling period as shown below. The form of the exponential window is deĆ scribed by the formula e -t . The `ExponenĆ tial decay' determines the % level at the end of the time window.

An exponential window is normally applied to the response (output) channels during impact testing. It is also the most appropriate window to be used with a burst excitation signal in which case it should be applied to all channels i.e. force(s) and response(s). It does however introduce artificial damping into the measurement data which should be carefully taken into account in further proĆ cessing in modal analysis.

14

The Lms Theory and Background Book

Spectral processing

Choosing window functions For the analysis of transient signals use : Uniform

for general purposes

Force

for short impulses and transients to improve the signal to noise ratio

Exponential

for transients which are longer than the sample period or which do not decay sufficiently within this period.

For the analysis of continuous signals use : Hanning

for general purposes

Blackman or Kaiser-Bessel

if selectivity is important and you need to distinguish beĆ tween harmonic signals with very different levels

Flattop

for calibration procedures and for those situations where the correct amplitude measurements are important.

Uniform

only when analyzing special sinusoids whose frequencies coincide with center frequencies of the analysis.

For system analysis i.e. measurement of FRFs use : Force

for the excitation (reference) signal when this is a hammer

Exponential

for the response signal of lightly damped systems with hammer excitation

Hanning

for reference and response channels when using random excitation signals

Uniform

for reference and response channels when using pseudo random excitation signals

Window correction mode Applying a window distorts the nature of the signal and correction factors have to be applied to compensate for this. This correction can be applied in one of two ways.

Part I

Amplitude

where the amplitude is corrected to the original value.

Energy

where the correction factor gives the correct signal energy for a particular frequency band. This is the only method that should be used for broad band analysis.

Signal processing

15

Chapter 1

Spectral processing

If a number of windows is applied to a function, the effect of the window may be squared or cubed, and this affects the correction factor required.

Amplitude correction Consider the example of a sine wave signal and a Hanning window. amplitude time

time

amplitude

unwindowed signal

amplitude

frequency

windowed signal

frequency

When the windowed signal (sine wave x Hanning window) is transformed to the frequency domain, then the amplitude of the resulting spectrum will be only half that of the equivalent unwindowed signal. Thus in order to correct for the effect of the Hanning window on the amplitude of the frequency specĆ trum, the resulting spectrum has to be multiplied by an amplitude correction factor of 2. Amplitude correction must be used for amplitude measurements of single tone frequencies if the analysis is to yield correct results.

Energy correction Windowing also affects broadband signals.

original signal

16

window function

windowed signal

The Lms Theory and Background Book

Spectral processing

In this case however it is the energy in the signal which it is usually important to maintain, and an energy correction factor will be applied to restore the enerĆ gy level of the windowed signal to that of the original signal. In the case of a Hanning window, the energy in the windowed signal is 61% of that the original signal. The windowed data needs to be multiplied by 1.63 therefore to correct the energy level.

Window correction factors The actual correction factor that is needed to compensate for the application of the time window depends on the window correction mode and the number of windows applied. Table 1.2 lists the values used. Window type

Amplitude mode

Energy mode

Uniform

1

1

Hanning x1

2

1.63

Hanning x2

2.67

1.91

Hanning x3

3.20

2.11

Blackman

2.80

1.97

Hamming

1.85

1.59

Kaiser-Bessel

2.49

1.86

Flattop

4.18

2.26

Table 1.2 Window correction factors

Part I

Signal processing

17

Chapter 1

1.4

Spectral processing

Averaging Signals in the real world are contaminated by noise -both random and bias. This contamination can be reduced by averaging a number of measurements in which the random noise signal will average to zero. Bias errors however, such as nonlinearities, leakage and mass loading are not reduced by the averaging process. A number of different techniques for averaging of measurements are provided.

Linear This produces a linearly weighted average in which all the individual measureĆ ments have the same influence on the final averaged value. If the average value of M consecutive measurement ensembles is x then M1

x
1  xm M m
0

Eqn 1-3

x
x a(n1)  x n The intermediate average is aān . The final averaging can be done at the end of the acquisition.

Stable In the case of stable averaging again all the individual measurements have the same influence on the final averaged value. In this case though, the intermediate averaging result is based on -





xn 1 xn
n  n Ăx n1  n

Eqn 1-4

The advantage of stable averaging is that the intermediate averaging results are always properly scaled. This scaling however makes the procedure slightly more time consuming.

Exponential Exponential averaging on the other hand yields an averaging result to which the newest measurement has the largest influence while the effect of the older ones is gradually diminished. In this case -





xn 1 Ăx xn
  n1   

18

Eqn 1-5

The Lms Theory and Background Book

Spectral processing

where  is a constant which acts as a weighting factor.

Peak level hold In this case a comparison has to be made between individual measurement enĆ sembles. When they contain complex data, comparison is done based on the amplitude information. For peak level hold averaging, the last measurement ensemble consisting of k individual samples, xn (k), (where k= 0...N-1 and N is the blocksize) is compared to the average of the n-1 previous steps, xn-1 (k). The new average xn (k), is then defined as x n(k)
x n(k)

if

x n(k)
x n1Ă(k)

|x n(k)| Ă |x n1(k)|

or

Eqn. 1-6

otherwise

In this way, the averaging result contains, for a specific k, the maximum value in an absolute sense of all the ensembles, considered during the averaging proĆ cess.

Peak reference hold In peak reference hold averaging, one channel determines the averaging proĆ cess. If x i is the ensemble for channel i and x r represents the reference channel, then the last measurement ensemble x rn (k) (where k= 0...N-1) is compared to the average of the n-1 previous steps, x rn-1 (k). The new average xn (k), is then defined as x i n(k)
x i n(k)

if

x i n(k)
x i n1Ă(k)

|x r n(k)| Ă |x r n1(k)|

or

Eqn 1-7

otherwise

This way, the averaging result contains all values that coincide with the maxiĆ mum values for the reference channel.

Part I

Signal processing

19

Chapter 1

1.5

Spectral processing

Reading list Signal and system theory J. S. Bendat and A.G. Piersol. Random Data : Analysis and Measurement Procedures Wiley - Interscience, 1971. J. S. Bendat and A.G. Piersol. Engineering Applications of Correlation and Spectral Analysis Wiley - Interscience, 1980. R.K. Otnes and L. Enochson. Applied Time Series Analysis John Wiley & Cons, 1978. J. Max Méthodes et Techniques de Traitement du Signal (2 Tomes) Masson, 1972, 1986. General literature in digital signal processing A.V. Oppenheimer and R.W. Schafer Digital Signal Processing Prentice Hall, Englewood Cliffs N.J., 1975. L.R. Rabiner and B. Gold Theory and Application of Digital Signal Processing Prentice Hall, Englewood Cliffs N.J., 1975. K.G. Beauchamp and C.K. Yueu Digital Methods for Signal Analysis George Allen & Unwin, London 1979. M. Bellanger Traitement Numérique du Signal Masson, Paris 1981. A. Peled and B. Liu Digital Signal Processing Theory, Design And Implementation John Wiley & Sons. Discrete Fourier Transform E.O. Brigham The Fast Fourier Transform Prentice Hall, Englewood Cliffs N.J., 1974.

20

The Lms Theory and Background Book

Spectral processing

R.W. Ramirez The FFT : Fundamentals and Concepts Prentice Hall, Englewood Cliffs N.J., 1985. C.S. Burrus and T.W. Parks DFT/FFT and Convolution Algorithms : Theory and Implementation John Wiley & Sons, 1985. H.J. Nussbaumer Fast Fourier Transform and Convolution Algorithms Springer Verlag, 1982. R.E. Blahut Fast Algorithms for Digital Signal Processing Addison Wesley, 1985. IEEE-ASSP Society Programs for Digital Signal Processing IEEE Press, New York, 1979.

Part I

Signal processing

21

Chapter 2

Structural dynamics testing

Understanding the structural dynamics of a structure is essential for both improving the performance of existing structures and the deĆ sign and development of new ones. This chapter provides an introduction to types of analysis used in exĆ amining the dynamic behavior of structures Signal analysis Signature analysis System analysis

23

Chapter 2

2.1

Structural dynamics testing

Signal analysis The dynamic analysis of a linear physical system can be achieved by measuring the response of the system (output) to a form of excitation. This excitation can be operational forces which, while typical, are not necessarily known. MeasurĆ ing the response to known excitation forces is discussed in section 2.2 In examining the vibrational behavior of a structure, there are a range of funcĆ tions that can be acquired which will provide information on the frequencies at which particular phenomena occur. These measurement functions are deĆ scribed in chapter 3. Noise levels are a common problem and specific information about acoustic measurement functions are given in a separate set of documentation on AcousĆ tics and sound quality. The examination of the behavior of a structure due to a changing environment, such as during an engine run up is termed signature analysis and this subject is discussed in section 2.3.

24

The Lms Theory and Background Book

Structural dynamics testing

2.2

System analysis System analysis refers to a method of examining the properties of a system, i.e. how a structure responds to a specific input. In the case of a linear system, this relationship between the input and the output is a fundamental characteristic of the system and can be used to predict the behavior of the system due to differĆ ent stimuli. output

output

output

output output

input

input

Modal analysis is a form of system analysis which results in a modal model of the system composed of a set of frequencies, damping values and mode shapes. The Frequency Response Function (FRF) is a frequency domain function exĆ pressing the ratio between a response (output) signal and a reference (input) signal. The position and direction of the measurements are termed Degrees Of Freedom DOFs. An FRF thus always depends on 2 DOFs, the response DOF (numerator) and the reference DOF (denominator). Input from reference DOF Xj

FRF H(f)

H(f)


Output from response DOF Xi

Xi Xj

For modal purposes the response signal is most commonly the acceleration at the response DOF due to a force input at another. In this case peaks in the FRF indicate that low input levels generate high response levels (resonances), while minima indicate low response levels, even for high inputs (anti-resonances).

Part I

Signal processing

25

Chapter 2

Structural dynamics testing

resonance log Amp

anti-resonance

frequency

Measurement points The number of acquisition channels determines the number of response and exĆ citation points that can be measured at any one time. Their position on the test system can be defined as part of the geometry of the structure. In order to visuĆ alize the response of each DOF, then their geometrical position must be defined.

Exciting the structure The input to the structure can be applied either from a hammer or a shaker. UsĆ ing a shaker will require a `Source' signal. The nature of this signal can take a number of forms. The choice of signal depends on the nature of the analysis. If the response is measured at several response DOFs and the system excited at a number of inputs then the resulting FRFs are termed Multiple Input Multiple Output. When a hammer is used to excite a mechanical structure the procedure is termed Impact testing. This type of testing can be done in one of two ways. Using the first method means measuring the response at a fixed point and apĆ plying the hammer at a number of excitation points. This case is termed `RovĆ ing hammer' The alternative is to apply the hammer to one point and to meaĆ sure the response at all the other points. This case is termed `Fixed hammer'.

26

The Lms Theory and Background Book

Structural dynamics testing

2.3

Signature analysis This involves analyzing a series of non-stationary signals that are varying over the analysis period. An example would be the vibrational/acoustical behavior of a structure as a function of rotational speed. Thus during `run-up' and/or `run-down' a series of signals are measured to determine the behavior of the structure and to determine the rotational speed. (the tacho signal). Spectral data are analyzed and plotted against the external parameter as illusĆ trated below. Such an arrangement is known as a waterfall or map of meaĆ sured functions. The functions that can be acquired during a run and placed in a waterfall are listed in sections 3.1 and 3.2. basic function tracking parameter

composite function As well as the waterfall of measured functions, signature analysis enables you to obtain so-called composite functions. These are two-dimensional functions that are directly related to the tracking parameter value. Such functions are overall levels and frequency sections and they are described in section 3.3. Measurements are taken during the acquisition but further analyses of the meaĆ sured functions in relation to the tracking parameters can be performed during post processing. Tracking The dominant parameter describing the change of a signal is termed the trackĆ ing parameter. This could be time, rpm, temperature or other. The rotational speed is a commonly used as a tracking parameter and for this a tacho signal is used to determine the rpm. A number of pulses per revolution are generated by the rotating shaft. The tacho channel uses a positive slope crossing of a trigger level to determine the time beĆ tween pulses and thus the rpm. t1

Part I

Signal processing

t2

t3

27

Chapter 2

Structural dynamics testing

While a number of channels can be used to measure tracking values, one must be used to control the acquisition, i.e. to determine when the measurements will be made. Parameters relating to signature analysis Sampling frequency f s
1 T

Sampling period

T
NĂ.ĂT

Number of revs P

M samplesā/ārev

M

M

M

Blocksize N = MP samples P= Number of revsā/ ā b ā lock =( Number of revs/sec) . ( Number of secs) rpm(Hz) P
rpmā(Hz)Ă.ĂT
f M = Number of samplesā/ārev = (Number of samplesā/sec) . (Number of secsā/rev) fs M
rpm(Hz) N= Number of samples = (Number of samples/rev) . (Number of revs) (data acquisition size) (blocksize) N
MĂ.ĂP Orders For rotating machinery most signal phenomena are related to the rotational speed and its harmonics. A rotational speed harmonic is called an order. It is the proportionality constant (O) between the rotational speed (rpm) and the frequency (f).

28

The Lms Theory and Background Book

Structural dynamics testing

f= O . rpm (Hz) For stationary signals the relevant analysis parameters are -

For rotational equipment the relevant analysis parameters are -

maximum frequency (fmax)

maximum order (Omax)

fmax= fs / 2

fmax= Omax . rpm

and frequency resolution (
f)
f = 1 / T

Omax= M / 2 and order resolution (
O)


f=
O . rpm


O = 1 / P

Fixed sampling This is another term for basic signature analysis, where signals are measured using the standard data acquisition techniques as described above i.e. with a fixed sampling frequency and sampling period. The rpm is measured but is used only for control of the acquisition, and annotation of the acquired blocks. In this case, the maximum order and the order resolution will vary with the roĆ tational speed (rpm).

Order tracking This involves measuring signals at different rotational speeds but in this case, the sampling frequency (fs ) and observation time (T) are dependent on the rpm. The data is sampled synchronously with the rotational speed (rpm). In this way the number of samples per revolution is kept constant. The signals are in fact sampled at constant shaft angle increments rather than time increments. This implies that the maximum order measured remains constant (Omax= M / 2). When order tracking, the number of revolutions /measurement (P) is indepenĆ dent of the rotational speed. Thus with a constant P, the order resolution is a constant (
O= 1 / P). The orders lie on spectral lines and leakage problems are avoided when an integer number of revolutions are measured.

Part I

Signal processing

29

Chapter 3

Functions

This chapter gives a brief description of the various functions that can be measured and their uses. Time domain measurements Frequency domain measurement functions This chapter does not deal with acoustic measurements which are dealt with in a separate set of documents Acoustics and sound qualĆ ity". It does describe the specific functions that are associated with signaĆ ture analysis and which are based on a tracking parameter. Composite functions In addition this chapter mentions the use of consistent units and how rms values are calculated for the various measurement functions. Units Calculation of rms values

Chapter 3

3.1

Functions

Time domain functions Time Record N instantaneous time samples x(n), are taken where N = the blocksize. The reĆ sult of a time record measurement x(n), is the ensemble average of a series of M instantaneous time records, where M= the number of averages and A desigĆ nates the averaging operator. xĂ(n)
A M1 m
0 Ă(x m(n))

Eqn 3-1

n
0 N  1

Averaging is useful in perceiving signals disguised by the presence of noise. The specification of the number of averages taken in the determination of a block of data as well as the various averaging methods used are described in secĆ tion 1.4. In the case of Signature Analysis, a map or waterfall is obtained of all the time measurements taken during the acquisition. Because this analysis deals with changing signals, averaging is only useful with signals that change slowly or in a stepwise fashion.

Autocorrelation Correlation is a measure of the similarity between two quantities. The autocorĆ relation function is found by taking a signal and comparing it with a time shifted version of itself. The time domain autocorrelation function Rxx () is thus acquired by multiplyĆ ing a signal by the same signal displaced by time () and integrating the prodĆ uct over all time.

R xxĂ()Ă
Ă

lim T 

 x(t)Ăx(t  )Ădt

Eqn 3-2

T

However this function is more commonly computed by using the correspondĆ ing frequency domain function. In this case the discrete auto correlation funcĆ tion Rxx (n) of a sampled signal x(n) is calculated as, R xxĂ(n)Ă
Ă F 1Ă ĂSxxĂ(k) ,ĂĂ

k
0...N  1

Eqn 3-3

n
0...N  1

32

The Lms Theory and Background Book

Functions

where F -1 is the inverse Fourier Transform and Sxx (k) is the discrete autopower spectrum. It can be seen that the greatest correlation will occur when    and the autoĆ correlation function will thus be a maximum at this point equal to the mean square value of x(t). Purely random signals will therefore exhibit just one peak at    Periodic signals however will exhibit another peak when the time shift equals a multiple of the period. The autocorrelation function of a periodic signal is also periodic and has the same period as the wave form itself. This property is useful in detecting sigĆ nals hidden by noise. The advantage of using the auto correlation function rather than linear averaging, is that no synchronizing trigger is required. CerĆ tain impulse type signals also show up better using the autocorrelation function rather than using a frequency domain function.

Crosscorrelation Cross correlation is a measure of the similarity between two different signals. It therefore requires multiple channels. In terms of the time domain it is defined as:

R xyĂ()Ă
Ă

lim T 

 x(t)Ăy(t  )Ădt

Eqn 3-4

T

As in the case of the autocorrelation function the discrete cross correlation funcĆ tion Rxy (n) between two sampled signals x(n) and y(n) is calculated as, R xyĂ(n)Ă
Ă F 1Ă ĂSxyĂ(k) ,ĂĂ

k
0...N  1 n
0...N  1

Eqn 3-5

with Sxy (k) being the discrete crosspower spectrum between the two signals. Cross correlation indicates the similarity between two signals as a function of the time shift. It is therefore useful in determining the time difference between such signals.

Part I

Signal processing

33

Chapter 3

Functions

Histogram The probability histogram q(j) describes the relative occurrence of specific sigĆ nal levels. Let the signal input range of a sampled signal x(n) be divided in J classes. Each class j,j = 0...J-1, can be characterized by an average value xj and a class increment x. 2

nr of classes

signal range

3

1 0 -1 -2

ÇÇ ÇÇ ÇÇ Ç ÇÇ ÇÇÇÇÇÇ Ç ÇÇ ÇÇ ÇÇ ÇÇÇÇÇ ÇÇ ÇÇ -3 -2 -1 0 1 2

-3

3

nr of classes

Figure 3-1 Histogram

The probability histogram of a sampled signal x(n) can then be defined as,

q(j)Ă
Ă 1 ĂĂ N

where

N1



Ă kĂ xĂ(n) Ă,ĂĂĂ jĂ
Ă 0...ĂJĂ Ă 1

Eqn 3-6

nĂ
Ă 0Ă kĂ x(n) Ă
Ă 1,Ă ifĂx jĂ Ă xĂ Ă xĂ(n)Ă Ă x jĂ Ă xĂĂ 2 2 kĂ x(n) Ă
Ă 0,Ă otherwiseĂĂ

The maximum value of J is either the number of time samples (Time data) or spectral lines in the block.

Probability Density The probability density p(j) is a normalized representation of the probability histogram q(j), p(j)Ă
Ă 100ĂĂ q(j),ĂĂ jĂ
Ă 0...JĂ Ă 1 x

Eqn 3-7

This function is expressed in percents per engineering unit.

Probability Distribution The probability distribution d(j) gives the probability (in percent) that the signal level is below a given value. This function is calculated from the probability histogram, q(t) given in equation 3-6.

34

The Lms Theory and Background Book

Functions

j

d(j)Ă
Ă

Ă q(i),ĂĂ jĂ
Ă 0...JĂ Ă 1

Eqn 3-8

i
0

Part I

Signal processing

35

Chapter 3

3.2

Functions

Frequency domain functions Spectrum The instantaneous discrete frequency spectrum X(k),is defined as the discrete Fourier transform of the instantaneous sampled time record. X(k)Ă
Ă F(ĂxāĂ(n)Ă),ĂĂĂ

Eqn 3-9

n
0...N  1 k
0...N  1

The result of a frequency spectrum measurement is the ensemble average of a series of M instantaneous discrete frequency spectra Xm (k), m = 0...M - 1,

_

XĂ(k)Ă
Ă A M1 m
0 Ă(X mĂ(k)),ĂĂ kĂ
Ă 0...ĂN  1

Eqn 3-10

Since only real valued time records are considered the frequency spectrum has a Hermitian symmetry. X(k)Ă
Ă X *Ă ( k)Ă
Ă X *Ă (N  k),ĂĂ kĂ
Ă 0..Ă NĂĂ 2

Eqn 3-11

where X * is the complex conjugate. The number of spectral lines is equal to half the number of time samples. The FFT algorithms produce a double sided Fourier transform which is corĆ rected to single-sided spectral quantities. Only the positive frequency values are considered. These are then adapted according to the format required. A Peak amplitude multiplies the result by a factor 2, so producing the amplitude of the time signal in case of a sine wave. Rms amplitude multiplies the result by
2. As with time record averaging, the non-synchronous signals will average out. This function is useful therefore in distinguishing a signal that is contaminated by noise. When a trigger signal is available the frequency spectrum has the adĆ vantage over autopower spectrum averaging in that the noise averages to zero, rather than to its mean square value.

36

The Lms Theory and Background Book

Functions

Autopower Spectrum The autopower spectrum is the squared magnitude of the frequency spectrum. The discrete autopower spectrum of a sampled time signal Sxx (k) is defined as the ensemble average of the squared magnitude of M instantaneous discrete frequency spectra Xm (k), * S xxĂ(k)Ă
ĂĂ A M1 m
0 Ă(XmĂ(k)ĂX mĂ(k)),ĂĂ kĂ
Ă 0...ĂN  1

Eqn 3-12

where X * is the complex conjugate. Thus if the frequency spectrum is complex you have phase information, while the autopower spectrum will be real and contain no phase information. Since only real valued time records are considered, the autopower spectrum is symmetric with respect to zero-frequency, S (xx)Ă(k)Ă
ĂĂ S xxĂ( k)Ă
Ă S xxĂ(N  k),ĂĂĂ kĂ
Ă 0Ă...Ă N 2 X

Sxx

Gxx

-f T

0

f

double sided frequency spectrum

-f

0

f

double sided autopower spectrum

signal Figure 3-2

A2/2

(A/2)2

A/2

A

Eqn 3-13

0

f

single sided (rms power) autopower spectrum

Autopower spectra

Of this double sided frequency spectrum, only the positive frequency values are considered. In order to obtain a time signal power estimate, a summation of the power spectra values at the positive and negative frequencies must be made, resulting in the so-called RMS Autopower spectra Gxx (k), G xxĂĂ(k)Ă
Ă S xx,ĂĂ whenĂkĂ
Ă 0 G xxĂĂ(k)Ă
Ă 2S xxĂ(k),ĂĂ whenĂkĂ
Ă 1... N Ă 1 2

Eqn 3-14

The power spectrum values correspond to the Fourier coefficients resulting from a double sided Fourier transform but these values are corrected to singlesided spectral quantities, expressed as RMS or as PEAK amplitude values.

Part I

Signal processing

37

Chapter 3

Functions

There are a number of formats in which autopower spectra are presented. The Power Spectral Density normalizes the level with respect to the frequency resolution. This overcomes differences that may arise from using a specific Bandwidth. This is the standard way of measuring stationary broadband sigĆ nals. For transient signals the Energy Spectral Density may be more interesting since this looks at the level of the energy rather than the average power over the total acquisition time and is obtained by multiplying the Power Spectral Density by the measurement period. The interrelationship of these autopower formats is shown in Table 3.1. The paĆ rameters A and T are as illustrated in Figure 3-2 , and
F is the frequency resoĆ lution. Examples of the different modes and units are shown below. Amplitude mode

Amplitude format

Value other than DC line

RMS

Power

A2/2

RMS

Linear

A/
2

RMS

PSD

A2/2
F

RMS

ESD

A2 T/2
F

Peak

Power

A2

Peak

Linear

A

Peak

PSD

A2/
F

Peak

ESD

A2 T/
F

Table 3.1 Autopower spectrum formats

Crosspower spectrum The cross power spectrum Sxy is a measure of the mutual power between two signals at each frequency in the analysis band. It is the dual of the cross corĆ relation function. It is defined as the following product -

*

S xyĂ(k)Ă
ĂĂ A M1 m
0 Ă X mĂ(k)Ă Ă Y mĂ(k) Ă,ĂĂ kĂ
Ă 0...ĂN  1 X*M (K)

38

Eqn 3-15

Is the complex conjugate of the instantaneous frequency spectrum of the one time signal X(n), and

The Lms Theory and Background Book

Functions

Ym (K)

Is the instantaneous frequency spectrum of a related time signal Y(n),

The crosspower spectrum contains information about both the magnitude and phase of the signals. Its phase at any frequency is the relative phase between the two signals and as such it is useful in analyzing phase relationships. Since it is a product, it will have a high value when the both signal levels are high, and a low value when both signal levels are low. It is therefore an indicaĆ tor of major signal levels on both the input and output. Its use in this respect should be treated with caution however since a high value can also arise from just the output level without indicating that the input is the cause. The interdeĆ pendence of input and output is revealed in the coherence function which is deĆ scribed in the following subsection. The cross power spectrum is used in the calculation of frequency response funcĆ tions. The Amplitude mode in which the crosspower spectrum is presented is as deĆ scribed in the previous section on Autopower spectrum. Rms and PEAK valĆ ues are considered.

Coherence There are three types of coherence functions; the ordinary coherence, partial coĆ herence and virtual coherence. Ordinary Coherence The (squared) ordinary coherence between a signal Xi (N) and Xj (N) is defined by,

 2 0 ijĂ(k)Ă
Ă

SijĂ(k) 2 S iiĂ(k)Ă  S jjĂ(k)

Eqn 3-16

where S ij(k) is the averaged crosspower. S ii(k) and S jj(k) are the averaged autoĆ powers. It is a ratio of the maximum energy in a combined output signal due to its variĆ ous components, and the total amount of energy in the output signal. CoherĆ ence can be used as a measure of the power in one channel that is caused by the power in the another channel. As such it is useful in assessing the accuracy of transfer function measurements. It does not however need to apply to input and output and can also be measured between shakers.

Part I

Signal processing

39

Chapter 3

Functions

The coherence function can take values that range between 0 and 1. A high valĆ ue (near 1) indicates that the output is due almost entirely to the input and you can feel confident in the frequency response function measurements. A low value (near 0) indicates problems such as extraneous input signals not being measured, noise, nonlinearities or time delays in the system. Multiple coherence (used in the calculation of the measurement function FRF) The multiple coherence function is the coefficient that describes, in the frequenĆ cy domain, the causal relationship between a single signal (an output spectrum) and a set of other signals (the considered input spectra) as a function of freĆ quency and all considered references. It is the ratio of the energy in an output signal, caused by several input signals to the total amount of energy in the outĆ put signal. It is used to verify the amount of noise on the measurements, as all responses should be related to the applied references (inputs). The multiple coherence function between a single response spectrum Y(k) and a set of reference spectra Xi (k) is calculated from

 2 y:xĂ(k)Ă
Ă 1 Ă

S yy.n!Ă(k)

Eqn 3-17

S yyĂ(k)

where Syy (k) is the autopower of response signal y(n) Syy.n! (k) is the part of autopower Syy (k) of which the contributions of all reference spectra Xi (k) have been eliminated The value of the multiple coherence is always between 0 and 1.

Partial Coherence The partial coherence is the ordinary coherence between conditioned signals. Conditioned signals are those where the causal effects of other signals are reĆ moved in a linear least squares sense. To define the partial coherence, consider the signals X1 ..., Xi , Xj ,... The partial coherence between Xi and Xj , after eliminating the signals X1 ... Xg is given by,

 2p ijĂgĂĂ(k)ĂĂ
Ă

 SijĂĂgĂ(k)Ă 2 SiiĂĂgĂ(k)Ă  S jjgĂ(k)

Eqn 3-18

with : Sii g (k) =

40

autopower of signal Xi without the influences of the signals X1 ...xg

The Lms Theory and Background Book

Functions

Sjjg (k) =

autopower of signal Xj without the influence of the signals X1 ...xg

Sijg (k) =

crosspower between signals Xi and Xj without the influences of the signals X1 ...xg .

The partial coherence can take values between 0 and 1. Virtual Coherence The Virtual coherence is an ordinary coherence between a signal and a princiĆ pal component which is discussed below. The virtual coherence is calculated from,  2 vijĂ(k)Ă
Ă

SijĂ(k) 2 S iiĂ(k)Ă  S jjĂ(k)

Eqn 3-19

with : S'ii (k)

autopower of principal component X'i

S'ij (k)

crosspower between signal xj and principal component X'i

The value of the virtual coherence is always between 0 and 1. The sum of the virtual coherences between any signal and all principal components is also in the range [0,1].

Principal Component Spectra Consider a set of signals, X... Xn . Now assume that a set of perfectly uncorreĆ lated signals can be determined such that, by linear combinations, they deĆ scribe the original set of signals. These signals (indicated by X'1 ... X'n .) are called the principal components of the signals in the original set. Note that the coherence between the principal components is exactly 0, as they are by definiĆ tion, perfectly uncorrelated. The principal components are in a sense the main independent mechanisms (sources) observable in the signal set. The Principal components can be calculated either on the sampled time data or on the corresponding spectra. The fundamental relations are,

 X(k) Ă
Ă [ U ] hĂ X(k) 

Eqn 3-20

 X(k) Ă
Ă [ U ]Ă X(k)  [ U ] h[ U ]Ă
Ă I [S xx]Ă
Ă [U ]hĂS xxĂ[ U ]

Part I

Signal processing

41

Chapter 3

Functions

where S'xx =

diagonal matrix with the autopower of the principal component spectra on the diagonal.

{X'(K)} =

an uncorrelated set of principal component signals.

[U] =

unitary transformation matrix.

The major application of the principal component spectra is in determining the number of uncorrelated mechanisms (sources) in a signal set. A well known example is the diagnosis of multiple input excitation for multiple input/multiĆ ple output FRF estimation.

Frequency Response Function The frequency response function (FRF) matrix [H(k) ] expresses the frequency domain relationship between the inputs and outputs of a linear time-invariant system. references

responses

Input Input

Output System

Input X(k)

Output Output

H(k)

Y(k)

If Ni be the number of system inputs and No the number of system outputs, let {X(N)} be a Ni -vector with the system input signals and {Y(N)} a No -vector with the system output signals. A frequency response function matrix [ H(k)] of size (No , Ni ) can then be defined such that,

 Y(k) Ă
Ă  H(k) Ă Ă  X(k) 

Eqn 3-21

The system described above is an ideal one where the output is related directly to the input and there is no contamination by noise. This is not the case in realĆ ity and various estimators are used to estimate [H(k)] from the measured input and output signals. The H1 Estimator The most commonly used one is the H1-estimator, which assumes that there is no noise on the input and consequently that all the X measurements are accuĆ rate.

42

The Lms Theory and Background Book

Functions

H

X

Y

Y

N Y = HX + N

X It minimizes the noise on the output in a least squares sense. In this case the transfer function is given by -

H 1(k) Ă
Ă

 Syx(k) 

Eqn 3-22

 Sxx(k) 

This estimator tends to give an underestimate of the FRF if there is noise on the input. H1 estimates the anti-resonances better than the resonances. Best results are obtained with this estimator when the inputs are uncorrelated.

The H2 Estimator Alternatively, the H2 estimator can be used. This assumes that there is no noise on the output and consequently that all the Y measurements are accurate. H

X

Y

Y

M Y = H(X - M) X It minimizes the noise on the input in a least squares sense and in this case the transfer function is given by -

H 2(k) Ă
Ă

 Syy(k)   Syx(k) 

Eqn 3-23

This estimator tends to give an overestimate of the FRF if there is noise on the output. this estimator estimates the resonances better than the anti-resonances.

Part I

Signal processing

43

Chapter 3

Functions

Note!

This estimator can only be implemented in the case of a single output

The Hv Estimator Finally with the Hv estimator, [ H(k)] is calculated from the eigenvector correĆ sponding to the smallest eigenvalue of a matrix [ Sxxy ]:

SxxyĂĂĂ
Ă





S xx S xy S yx S yy

Eqn 3-24

This estimator minimizes the global noise contribution in a total least squares sense. When using this estimator the partitioning of the noise over the input and output signals can be scaled. Y

H

X M

Y N

Y-N =H (X-M) X This estimator provides the best overall estimate of the frequency function. It approximates to the H2 estimator at the resonances and the H1 estimator at the anti-resonances. It does however require more computational time than the other two. Frequency response functions depend on there being at least one reference channel and one response channel.

Impulse Response The impulse response (IR) function matrix [h(t)] expresses the time domain relationship between the inputs and outputs of a linear system. This relationĆ ship takes the form of a convolution integral. y(t)Ă
Ă

44

 x()Ăx(t  )Ăd

Eqn 3-25

The Lms Theory and Background Book

Functions

[h(t)] is calculated using the inverse Fourier transform of the frequency reĆ sponse function as shown below -

 h(t) Ă
Ă F 1ĂĂ H(k) 

Eqn 3-26

Impulse response functions depend on there being at least one reference chanĆ nel and one response channel. The FRF estimators (H1, H2 and Hv) are as described above.

Part I

Signal processing

45

Chapter 3

3.3

Functions

Composite functions The functions described in this section represent functions that can be acquired or processed during a Signature analysis. Since this type of analysis is intended to examine the evolution of signals as a function of changing environment (e.g. rpm, time, ...), then there needs to be functions that express this evolution. These are called composite functions as they are derived from the `basic' meaĆ surement functions described in the previous section, for different environmenĆ tal conditions.

Overall level (OA) This function describes the evolution of the total energy in the measured signal. As such it is always expressed as a frequency spectrum rms value. It is availĆ able with all basic measurement functions. Energy correction is applied to this function.

ANSI 1.4 time based OA level calculation The time signal is exponentially averaged to calculate the Overall level over a t

particular bandwidth. An exponential weighting factor is used (e  ) where
t is the sample period of the signal and  is a time constant. The values of  deĆ pends on the type of signal and three standardized values are supplied.  = 35ms for impulse (peaky) signals  = 125 ms for fast changing signals  = 1000 ms for slow changing signals. When the signal contains spikes and is therefore defined as impulse" an addiĆ tional peak detector mechanism is implemented. In this case the signal is first averaged using the 35ms averaging time constant and then peaks are detected using a decay rate of 1500ms.

Frequency section This function describes the evolution of the energy of the measured signal over the rpm range in a specified frequency band. It is always expressed as an Rms frequency spectrum and is available only when the basic measurement function is a frequency domain function. The frequency section is calculated by integrating over a Bandwidth around the center frequency value.

46

The Lms Theory and Background Book

Functions

Bandwidth Lower bandvalue

Center frequency

Upper bandvalue

The center frequency is the frequency at which the section will be calculated and is specified by the Center parameter. The Lower bandvalue and the UpĆ per bandvalue are given by Center frequency +/- {BandwidthĂ/Ă2} The Bandwidth is determined by the Band mode parameter. Possible ways in which to express the Bandwidth are V

a fixed frequency range

V

a fixed number of spectral lines the lines closest to the exact frequency value are used.

V

a percentage of the selected center frequency

These options are illustrated below. rpm

f f Band mode=frequency Band mode=lines f=constant

Part I

Signal processing

rpm

f Band mode=%

f
constant fc

47

Chapter 3

Functions

Order sections This function describes the evolution of the energy of the measured signal in a specified `order' band. Orders are introduced chapter 2.3 in the chapter on types of testing. An `order' band is a frequency band whose center frequency changes as a function of the measurement environment or tracking parameter. It is necessary therefore that the tracking parameter be a `frequency' type of paĆ rameter (e.g. rotation speed in rpm). An order is nothing other than a multiple of this basic tracking parameter. The evolution of the energy in a specified orĆ der band is expressed as a function of the measured rpm. Through post procesĆ sing it is also possible to examine it in terms of measured time or frequency. Possible means of defining the span for integration are: V

a fixed frequency range

V

a fixed number of spectral lines the lines closest to the exact value are used.

V

a fixed order Bandwidth

V

a percentage of the selected order value

These three options are illustrated below. rpm

f f Band mode=frequency Band mode=lines f=constant

rpm

f f Band mode=order O=constant f=constant . rpm

rpm

f

f

Band mode= % O=constant Oorder i=Bandwidth (%) . i Oorder i+1=Bandwidth (%) . (i+1)

f=constant . rpm

Octave sections An octave section represents the summation of values over octave bands. The center frequencies of the bands are defined in the ISO norm 150 266. Possible octave bands are are 1/1, 1/2, 1/3, 1/12 and 1/24 octaves.

48

The Lms Theory and Background Book

Functions

3.4

Units To ensure consistency in the manipulation of data LMS software always operĆ ates with an internal set of reference units. The physical quantities with a caĆ nonical dimension of length, angle, mass, time, temperature, current, and light, each have a corresponding reference unit as listed below: Canonical dimension

Abbreviation

Reference unit

Abbreviation

length

le

meter

m

angle

an

radian

rad

mass

ma

kilogram

kg

time

ti

second

s

current

cu

Ampère

A

temperature

te

deg Kelvin

light

li

candela

0K

cd

Table 3.2 Reference units

This means that all data in either the internal data structures of the LMS softĆ ware or the database is stored in these units. A physical quantity with a dimenĆ sion that is a combination of the above canonical dimensions will be allocated a unit in the internal unit system that is a combination of the corresponding referĆ ence units. For example, a quantity with dimension of acceleration (length/time2) will have a unit that is the reference unit of length divided by the reference unit of time squared (m/s2).

Part I

Signal processing

49

Chapter 3

3.5

Functions

Rms calculations This section describes the ways in which rms calculations are performed for different measurement functions. RMS stands for Root Mean Square and is a measure of the energy in a signal. If the data is amplitude corrected, then it is automatically converted to energy correction for the calculations.

Time and Impulse records When dealing with time samples, then a certain number of sample must be analyzed in order to obtain a measure of the nature and the energy in the sigĆ nal. This is done by squaring values, summing them and then taking an averĆ age (mean) remove the influence of the number of samples. Then the square root of the mean is taken to arrive at the rms value. So for a range of samples starting at sample 0 and ending at sample k Rms




1 ā .ā k1

k



i
0

y 2i

yi Eqn. 3-27

i
0

Taking the example of a sine wave, of amĆ plitude A, then the rms value is A 2

i
k

rms A

Frequency spectra The frequency spectrum is first converted to a double sided amplitude spectrum Amplitude 2A

Amplitude A

frequency

50

-f

f=0

f

frequency

The Lms Theory and Background Book

Functions

The frequency range over which you want the rms value computed is defined by the upper and lower values of f1 and f2. All lines completely within the range will be included in the calculations (Ai) where i takes values of 1 to k-1. For the lines at the beginning and the end (A0 and A k), half of each value is taken. f2

f1 Ai -f

i=0

i=k

f

frequency

The rms value is then computed using the following formula Rms




A 2k A 20 k1  2 2Ă   A i   2  2 i
1

Eqn. 3-28

Autopower and crosspower spectra These spectra are first converted to a double sided power spectrum. The numĆ ber of lines (k) included in the calculations depends on the defined frequency span. As was the case for the frequency spectrum shown above, the value for the first and last sample (A0 and A k) are halved. The rms value is then comĆ puted using the following formula Rms




A 0 k1 A  2Ă   A i  k 2  2 i
1

Eqn. 3-29

FRF, Impedance, Transmissibility and Transmittance Rms values for these types of functions are not well defined. The Lms interĆ pretation for an FRF is to find the rms response when a force of amplitude 1 is applied. A force of amplitude 1 has an rms value Frms equal to F rms
1  k

Eqn. 3-30

where k is the number of samples in the range. The rms of the response, Xrms is derived from equation 3-28. The rms of the FRF therefore Hrms is

Part I

Signal processing

51

Chapter 3

Functions

H rms


H rms




X rms F rms

k1 2 A 2k 1 ĂA 0  2 Ai   2 k1 2



i
1

Eqn. 3-31

Sound power, sound intensity (active and reactive), SFTVI and SFUI SFTVI (sound field temporal uniformity indicator) and SFUI (sound field uniĆ formity indicator) are ISO defined functions for acoustic measurements and analysis. The rms computes the total energy in a band, so since these are alĆ ready a measure of energy, then the values of the spectral lines can simply be added. Rms


A0  2

 Ai  A2k

Eqn. 3-32

Particle velocity (active and reactive) Although the particle velocity is basically a frequency spectrum, since it is calĆ culated as a single sided spectrum it differs by a factor of 2 from equation 3-28. Rms


52



Ă

A 20 2



k1

2

 A2i  A2k

Eqn. 3-33

i
1

The Lms Theory and Background Book

Theory and Background

Part II Acoustics and Sound Quality

Chapter 4 Terminology and definitions . . . . . . . . . . . . . .

55

Chapter 5 Acoustic measurements . . . . . . . . . . . . . . . . .

67

Chapter 6 Sound quality . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Chapter 7 Sound metrics . . . . . . . . . . . . . . . . . . . . . . . . .

99

Chapter 8 Acoustic holography . . . . . . . . . . . . . . . . . . . .

53

117

Chapter 4

Terminology and definitions

This chapter contains definitions of basic terms associated with acoustics. Acoustic quantities Reference conditions Octave bands Acoustic weighting

55

Chapter 4

4.1

Terminology and definitions

Acoustic quantities

Sound power (P) The amount of noise emitted from a source depends on the sound power of that source. The sound power is a basic characteristic of a noise source, providing an absolute parameter that can be used for comparison. This differs from the sound pressure levels it gives rise to, which depend on a number of external factors. The total sound power PI of a source surrounded by N measurement surfaces is given by PI


N

 Pi

Eqn 4-1

i
1

The power of a sound source is expressed in Joules per second, or Watts. The sound power can also be represented by the letter W.

Sound pressure The effect of the sound power emanating from a source is the level of sound pressure. Sound pressure is what the ear detects as noise, the level of which deĆ pends to a great extent on the acoustic environment and the distance from the source. The sound pressure is defined as the difference between the actual and ambient pressure. This is a scalar quantity that can be derived from measured sound pressure spectra or autopower spectra either at one specific frequency (spectral line), or integrated over a certain frequency band. Sound pressure measurements can be obtained at each measurement point, and are independent of the measurement direction (X,Y, or Z). The units are Pascal (Pa) or N/m 2.

Sound (Acoustic) intensity An important quantity to be derived from the sound power is sound intensity. The sound intensity of a sound wave describes the direction and net flow of acoustic energy through an area.

56

The Lms Theory and Background Book

Terminology and definitions

TotalĂpowerĂP I


 I.dS



Eqn 4-2

I

S

Sound intensity is a vector, orientated in 3D-space with the fundamental units of W/m 2, (power transmitted per unit area). The area is represented as a vector in 3D space with a length equal to the amount of geometrical area, and a direction perpendicular to the measurement

surface. As such, the vector product ( I i.S i) represents the flow of acoustic enerĆ gy in a direction perpendicular to a surface. This is the usual direction in which intensity is measured. If the acoustic intensity vector lies within the surĆ face itself, the transmitted sound power equals zero. Intensity is also the time-averaged rate of energy flow per unit area. I
1 T

 I(t)dt T

Eqn 4-3

0

As such, if the energy is flowing back and forth resulting in zero net energy flow then there will be zero intensity. Normal sound intensity This is the component of the sound intensity vector normal to the measurement surface.

Free field This term refers to an idealized situation where the sound flows directly out from the source and both pressure and intensiĆ ty levels drop with increasing distance from the source according to the inverse square law.

Part II

Acoustics and Sound Quality

57

Chapter 4

Terminology and definitions

Diffuse field

In a diffuse field the sound is reflected many times such that the net intensity can be zero.

Particle velocity Pressure variations give rise to movements of the air particles. It is the product of pressure and particle velocity that results in the intensity. In a medium with mean flow therefore

I
pĂv

where

Eqn 4-4

p= sound pressure (Pa)

v = particle velocity (m/s)

The particle velocity of a medium is defined as the average velocity of a volĆ ume element of that medium. This volume element must be large enough to contain millions of molecules so that it may be thought of as a continuous fluid, yet small enough so that acoustic variables such as pressure, density and velocĆ ity may be considered to be constant throughout the volume element. Equation 4-4 can be used to compute the particle velocity, once the acoustic inĆ tensity and the sound pressure have been measured. Particle velocity is a vecĆ tor in 3D-space expressed in units of (m/s). In a diffuse field the pressure and velocity phase vary at random giving rise to a net intensity of zero. Under certain circumstances (i.e. plane progressive waves in a free field), the particle velocity can also be calculated from the pressure and the impedance of the medium (c). v
p eāāc

58

Eqn 4-5

The Lms Theory and Background Book

Terminology and definitions

where

pe = effective sound pressure (Pa) = mass density of the medium (kg/m 3) c= velocity of sound in the medium (m/s)

By combining equations 4-4 and 4-5 it can be seen that in a free field a relationĆ ship exists enabling the acoustic intensity to be determined from the effective pressure of a plane wave.

p 2e  I 
.c

Eqn 4-6

Acoustic impedance (Z) This is defined as the product of the mass density of a medium and the velocity of sound in that medium. Z
Ă.Ăc

Eqn 4-7

where  = mass density (kg/m 3) c = velocity of sound in the medium (m/s)

Part II

Acoustics and Sound Quality

59

Chapter 4

4.2

Terminology and definitions

Reference conditions It is a common practise to define standards for acoustic intensity, pressure, etc... at an air temperature of 20_C and a standard atmospheric pressure of 1023 hPa (1 bar). Under these conditions the density of air o = 1.21 (kg/m 3) the velocity of sound in air c = 343 (m/s) = 415 rayls (kg/m 2s) the acoustic impedance o .c

dB scale Since the range of pressure levels that can be detected is large and the ear reĆ sponds logarithmically to a stimulus, it is practical to express acoustic parameĆ ters as a logarithmic ratio of a measured value to a reference value. Hence the use of the decibel scales for which the reference values for intensity, pressure and power are defined below.

Sound power level Lw This is defined as the logarithmic measure of the absolute (unsigned) value of the sound power generated by a source. L W
10 log 10Ă



|P I| P0

Eqn 4-8

The reference sound power is P0 = 10-12 (W)

Particle velocity level Lv This is defined as the logarithmic measure of the particle velocity.



L v
20 log 10Ă vv 0

Eqn 4-9

The reference particle velocity is v0 = 50 10-9 (m/s)

60

The Lms Theory and Background Book

Terminology and definitions

Sound (Acoustic) intensity level LI This is the logarithmic measure of the absolute value of the intensity vector.



|I| L I
10 log 10Ă I0

Eqn 4-10

The commonly used reference standard intensity for airborne sounds is = 10-12 (W/m 2)

Normal acoustic intensity level (LI ) This is the logarithmic measure of the absolute value of the normal intensity vector.



|I n| L In
10 log 10Ă I0

dB

Eqn 4-11

Sound pressure level LP This is defined as



p L p
10 log 10Ă p 0
20 log 10

2



p p0

Eqn 4-12

p is the rms value of the acoustic pressure (in Pa) The above reference values for intensity and power correspond to an effective rms reference pressure of po = 0.00002 (Pa) = 20 Pa This sound pressure level of 20 Pa is known as the standardized normal hearĆ ing threshold and represents the quietest sound at 1000Hz that can be heard by the average person.

Part II

Acoustics and Sound Quality

61

Chapter 4

4.3

Terminology and definitions

Octave bands Complete (1/1) octave bands represent frequency bands where the center freĆ quency of one band is approximately twice (according to standardized values) that of the previous one.

fc, i

fc, i+1

fc, i+2 f c, i1
2. f c,i

Partial octave bands (1/3, 1/12 1/24 . . .) represent frequency bands where f c, i1
( 2 1x ). f c,i and where x = 3,12, 24 . . .

1/1 bands 1/3 bands

12x The Lower band limit of a 1/x octave band is f c.2 12x The Upper band limit of a 1/x octave band is f c.2

The bands defined by these formulas are termed the `natural' bands. The InterĆ national ISO norm 150266 defines normalized center frequencies for octave bands and the values for 1/1, 1/2 and 1/3 octave bands are listed in table 4.1. Natural frequencies are used for calculations but the normalized frequencies are used for annotation. Octave bands above or below the normalized values are annotated with the natural frequencies.

62

The Lms Theory and Background Book

Terminology and definitions

Normalized frequency 16

1/ 1/ 1/ 1 2 3 oct oct oct x

x

x

18 x

22.4

x

25

x x

x

x x

45

x

50

250

x

x x

x

x

x

x

71

x x

400

x

90

x

100

x

x

112

x x

x

x

x

140

800

x

x

x x

3150

x

4000

x

x

x

5000

x x

6300

x

8000

x

x

x

9000 x

x

x

1600

10000

x

11200

1250 1400

160

x

7100

1120 x

x

2250

5600

630

1000

x

4500

900 x

2000

3550 x

710

80

x

2800

560 x

1600

2240

315

500

1/ 1/ 1/ 1 2 3 oct oct oct 1800

450

56

Part II

200

355

40

Table 4.1

x

280

35.5

125

x

224

28

63

160 180

20

31.5

1/ 1/ 1/ 1 2 3 oct oct oct

x x

x

12500

x

14000 x

16000

x

x

x

Normalized frequencies (Hz)

Acoustics and Sound Quality

63

Chapter 4

4.4

Terminology and definitions

Acoustic weighting

Frequency weighting The human ear has nonlinear, frequency dependent characteristics, which means that the sensation of loudness cannot be perfectly described by the sound pressure level or its spectrum. To derive an experienced loudness level from the sound pressure signal, the frequency spectrum of the sound pressure signal is multiplied by a frequency weighting function. These weighting funcĆ tions are based on experimentally determined equal loudness contours which express the loudness sensation as a function of sound pressure level and freĆ quency. A number of equal loudness contours are shown in Figure 4-1. The loudness level is expressed in `Phons'. 1 kHz-tones are used as the reference, which means that for a 1000 Hz tone, the Phon value corresponds to the dB sound pressure level.

Figure 4-1

64

Equal loudness perception contours

The Lms Theory and Background Book

Terminology and definitions

A, B and C - weighting for acoustic signals. A-weighting modifies the freĆ quency response such that it follows approximately the equal loudness curve of 40 phons and is applied to signals with a sound pressure level of 40dB. The Aweighted sound level has been shown to correlate extremely well with subjecĆ tive responses. The B and C-weighting follow more or less the 70 and 100 phon contours respectively. These contours can be seen in Figure 4-2. The reĆ sulting value is then denoted by LA, LB,.... with unit dBA, dBB... Table 4.2 (overleaf) shows the relative response attenuations or amplifications of the 3 types of filters. In between the listed normal frequencies, these filter spectra are linearly interpolated on a log-log scale. Figure 4-2 shows the same information in a graphical form. Relative response (dB)

20 10 0

C

-10 -20

D

-30

B

-40 -50

A

-60

Frequency (Hz)

-70

10

Figure 4-2

Part II

2

5

102

2

5

103

2

5

104

2

Standardized weighting curves

Acoustics and Sound Quality

65

Chapter 4

Terminology and definitions

1/3 Octave band Center frequency Hz 16 20 25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 6300 8000 10,000 12,500 16,000 20,000 Table 4.2

66

A weighting dB

B weighting dB

C weighting dB

-56.7 -50.5 -44.7 -39.4 -34.6 -30.2 -26.2 -22.5 -19.1 -16.1 -13.4 -10.9 -8.6 -6.6 -4.8 -3.2 -1.9 -0.8 0 +0.6 +1.0 +1.2 +1.3 +1.2 +1.0 +0.5 -0.1 -1.1 -2.5 -4.3 -6.6 -9.3

-28.5 -24.2 -20.4 -17.1 -14.2 -11.6 -9.3 -7.4 -5.6 -4.2 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.1 0 0 0 0 -0.1 -0.2 -0.4 -0.7 -1.2 -1.9 -2.9 -4.3 -6.1 -8.4 -11.1

-8.5 -6.2 -4.4 -3.0 -2.0 -1.3 -0.8 -0.5 -0.3 -0.2 -0.1 0 0 0 0 0 0 0 0 0 -0.1 -0.2 -0.3 -0.5 -0.8 -1.3 -2.0 -3.0 -4.4 -6.2 -8.5 -11.2

Weighting of acoustic signals

The Lms Theory and Background Book

Chapter 5

Acoustic measurements

This chapter discusses the measurement of acoustic quantities. Measured acoustic functions In addition it describes the calculation of acoustic quantities based on measured ones and other parameters used in these calculations Calculation of acoustic quantities Acoustic measurement surfaces Frequency bands Field indicators

67

Chapter 5

5.1

Acoustic measurements

Acoustic measurement functions This section describes the acoustic quantities that can be measured. From meaĆ sured quantities it is possible to derive further quantities as described in section 5.2.

Sound pressure level This is defined by equation 4-12 and can be measured using a single channel. It will result in an averaged pressure or autopower spectrum. For measurements in the free field, and in the direction of propagation, the norĆ mal sound intensity level will be equal to the sound pressure level. In practice, when not working under free field conditions, the sound intensity level will be lower than the sound pressure level.

Sound Intensity The sound intensity in a specified direction at a point is the average rate of sound energy transmitted in the specified direction through a unit area normal to this direction at the point considered. In most situations it is the component of the sound intensity vector normal to

the measurement surface, I n , which is measured. In order to determine sound intensity you can measure both the instantaneous pressure and the corresponding particle velocity simultaneously. In practice, the sound pressure can be obtained directly using a microphone. The instantaĆ neous particle velocity can be calculated from the pressure gradient between two closely spaced microphones. A sound intensity probe can therefore consist of two closely spaced pressure microphones which measure both the sound pressure and the pressure gradient between the microphones. For frequency domain calculations, it can be shown that the sound intensity can be calculated from the imaginary part of the crosspower between the two miĆ crophone signals. The following formula is used I
ImagĂ

S 1,2

2
fĂĂd

Eqn 5-1

Where S1,2 is the double sided crosspower between the two microphone sigĆ nals, f is the signal frequency, d is the microphone distance and  is the air denĆ sity.

68

The Lms Theory and Background Book

Acoustic measurements

For this function, all channels are processed as channel pairs, each pair consistĆ ing of two consecutive channels. It therefore requires that an even number of channels is defined. The reactive sound intensity (non propagating energy) is calculated as I reactiveĂ
Ă

S 1,Ă1Ă  S 2,Ă2 2
ĂfĂĂd

Eqn. 5-2

For the idealized case of measurements in the free field (free space without reĆ flections) and in the direction of propagation, the reactive intensity is zero.

Residual intensity This is defined as

RI
L p   pĂI 0

Eqn 5-3

where L p is the measured sound pressure level and  pĂI0 is the pressure residual intensity index. To calculate the residual intensity therefore it is necessary to have the pressure residual intensity index available. This is described below. Intensity measurements can be made in a sound field where the sound intensity level is in the range

L p   pIo  L I  L p

Eqn 5-4

Lp is defined in equation 4-12, and LI in equation 4-10. In a free field the presĆ sure and intensity levels are the same, whereas in all other cases, the measured intensity will be less than the pressure. The residual intensity ( L p   pĂI 0) repĆ resents the lowest intensity level which can be detected by the system for the given sound pressure level.

Part II

Acoustics and Sound Quality

69

Chapter 5

Acoustic measurements

Pressure residual intensity index For the calculation of the pressure residual intensity index of a sound intensity probe, it is required to place the intensity probe in a sound field such that the sound pressure is uniform over the volume. In these conditions there will be no difference between the two signals at both microphones, and hence the meaĆ sured intensity should be zero. However, the phase mismatch between the two measuring channels causes a small difference between the two signals making it appear as if there is some intensity. The intensity detected can be likened to a noise floor below which measurements cannot be made. This intensity lower limit is not fixed but varies with the pressure level. What is fixed, is the differĆ ence between the pressure and the intensity level when the same signal is fed to both channels. It is this which is defined as the pressure residual intensity inĆ dex. Mathematically therefore the pressure residual intensity index is

 pIo
(ĂL p  L InĂ) dB

Eqn 5-5

where Lp is the sound pressure level and LIn is the normal sound intensity levĆ el.

Dynamic capability index In order to ensure a particular level of accuracy for the measurements it is necĆ essary to increase the measurement floor defined by the residual intensity level by an amount termed the `bias error factor' ( )

L p   pIo Ă ĂdB  LI  Lp

Eqn 5-6

LI dB Lp Ld pIo

residual intensity level frequency

Figure 5-1

70

Dynamic capability index Ld

The Lms Theory and Background Book

Acoustic measurements

The `bias error factor' ( ) is selected according to the grade of accuracy reĆ quired from the table below. Grade of accuracy

Bias error factor dB

Precision

(class 1)

10

Engineering

(class 2)

10

Survey

(class 3)

7

Table 5.1

Bias error factor ( )

The difference between the residual pressure-intensity index and therefore represents the range in which the probe should be operating and is termed the `dynamic capability index' (Ld ) for the probe.

L d
(Ă pIo  Ă) dB

Part II

Acoustics and Sound Quality

Eqn 5-7

71

Chapter 5

5.2

Acoustic measurements

Calculation of acoustic quantities Acoustic functions can be derived from ones that have been measured. This section describes these analysis functions and Table 5.2 gives an overview of them and the measured quantities required for their derivation. Calculations will be made over specific frequency bands This subject is disĆ cussed in section 5.4. Some functions are computed over a known area. The subject of defining surfaces (meshes) for acoustic functions is discussed in secĆ tion 5.3.

Effective sound pressure The effective sound pressure pe or prms may be computed from a measured sound pressure spectrum or from its autopower spectrum. f2

p 2e
2

  p(f)  df 2

f1 f2


2

 A (f)df

Eqn 5-8

p

f1

Acoustic intensity This as a vector quantity calculated directly from measured acoustic intensity functions. f2

I


 I(f)df

Eqn 5-9

f1

When intensity measurements are not available but sound pressure measureĆ ments are available, then the magnitude of the acoustic intensity can be comĆ puted from the effective sound pressure p and the acoustic impedance  .c p2e I
Ă.Ăc 0

72

Eqn 5-10

The Lms Theory and Background Book

Acoustic measurements

but only under the assumption of plane progressive waves in a free field. Sound power This is calculated from the geometrical area S and the acoustic intensity compoĆ nent perpendicular to a surface P
I nĂ.ĂS

Eqn 5-11

Under certain circumstances, intensity can be assumed to be proportional to efĆ fective sound pressure, and then p2 P
eĂc Ă.ĂS

Eqn 5-12

0

Particle velocities These can be calculated when both acoustic intensity and sound pressure data are available

v
pI

Eqn 5-13

All the possible analysis functions are summarized in Table 5.2. (These are based on the assumption of plane progressive waves in a free field.) Acoustic quantity Symbol Effective (RMS) sound pressure p

Intensity

Sound power p

P

I

Particle velocity Table 5.2

Part II

pe

v

Required data

Formula

MKS units

sound pressure spectrum p

2   p 

2

Pa or N/m2

pressure autopower A

2   A 

2

intensity

i

i

I

W/m2

i

Intensity and area



Sound pressure spectrum and area

p 2e

0Ăc .ĂS

(1)

pressure autopower and area

p 2e

0Ăc .ĂS

(1)

intensity and sound pressure

W

IĂ.ĂS

I p

m/s

Overview of analysis functions for acoustic signals

Acoustics and Sound Quality

73

Chapter 5

5.3

Acoustic measurements

Acoustic measurement surfaces Acoustic measurements differ from other types of signals in that they are meaĆ sured some distance away from the object rather than on the test structure itĆ self. The measurement points are termed associated nodes, that are surrounded by a hypothetical measurement surface. An organized collection of measureĆ ment surfaces and nodes are termed a measurement mesh and there are ISO standards that define such meshes for particular measurement types. acoustic measurement nodes

source

ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ reflecting plane

Figure 5-2 Sound source, acoustic measurement mesh and nodes

Acoustic measurement meshes can be parallelepiped, cylindrical or spherical in shape. Associated nodes on measurement meshes have a nodal orientation. This is alĆ ways Cartesian, and the orientation of the +Z nodal coordinate system for a measurement defines the measurement direction.

Acoustic ISO standards The ISO-3744 and ISO-3745 standards describe sound pressure measureĆ ments. The microphone positions are defined on a (hemi-) spherical or a paralĆ lelepiped measurement mesh. The possible dimensions of the measurement mesh depend on the characteristic distance of the reference surface. This referĆ ence surface is defined as the smallest rectangular box that encloses the noise source.

74

The Lms Theory and Background Book

Acoustic measurements

ISO-3744

Acoustics - Determination of sound power levels of noise sources - Engineering methods for free-field conditions over a reflecting plane.

ISO-3745

Acoustics - Determination of sound power levels of noise sources - Precision methods for anechoic and semi-anechoic rooms

The ISO-9614-1 standard describe sound intensity measurements. In this case the microphone positions of the measurement meshes are not defined. The quality of the mesh has to be judged during measurements. It describes a number of field indicators that allow a judgment of the accuracy of the measurements and the mesh.

Part II

Acoustics and Sound Quality

75

Chapter 5

5.4

Acoustic measurements

Frequency bands Whenever an acoustic quantity is integrated over a certain frequency band, the following formula applies

 a(f)df f2

a


Eqn 5-14

f1

The integration of a continuous function a(f) is replaced by a finite sum over the corresponding discrete samples:

a
1 a1  2

where

 a  12 a i

i

a1 a2 fā1 < ā fāiā < fā2

2

Eqn 5-15

= a(f1 ) = a(f2 )

This integration takes into account the full value of all data samples between the two limits, and 50 % of the first and last sample. It can be obtained between any two measured frequency limits. It is good practise to maintain the type of frequency band that was used in the acquisition of the data for the calculation. In fact data acquired in octave bands must remain in those bands for the analysis. The calculation of the field indicaĆ tors also makes little sense unless the analysis bands correspond with the meaĆ surement bands.

76

The Lms Theory and Background Book

Acoustic measurements

5.5

Field indicators When attempting to analyze the sound power being radiated from a noise source in situ, the international standard ISO 9614-1 lays out a number of meaĆ surement conditions which must be adhered to if the results are to be considĆ ered acceptable for this purpose. A number of criteria must be satisfied, based on the values of particular indicator functions, to ensure the requisite adequacy of the measurements and meshes. This section describes both the field indicaĆ tors themselves and the criteria used to assess the results.

F1 Sound field temporal variability indicator This gives the measure of temporal (or time) variability of the field. It is deĆ fined as follows F1
1 In



1 M1

M

 (ĂInk  InĂ)2

Eqn 5-16

k
1

Where I n is the mean value of M short time averages of Ink defined in the folĆ lowing equation.

In
1 M

M

 Ink

Eqn 5-17

k
1

F2 Surface pressure–intensity indicator In a free field where sound is only radiating out from a source, the pressure and intensity levels are equal in magnitude. In a diffuse or reactive field however, intensity can be low when the pressure is high. A lower measured intensity can also arise if the sound wave is incident at an angle to the probe since this also affects the phase change detected across the probe. The pressure-intensity indicator examines the difference between the pressure and the absolute values of intensity. This function can be determined on a point to point basis during the acquisition, but the function F2 described here represents the value averĆ aged over all the measured surfaces.

Part II

Acoustics and Sound Quality

77

Chapter 5

Acoustic measurements

F 2
L p  L |I n|

Eqn 5-18

L p is the surface sound pressure level defined as N p i 2!  1 ā ā  p  L p
10 log 10 o N  i
1

Eqn 5-19

where i indicates the measurement surface and N is the total number of surĆ faces (of the local component).

L |I n|

is the surface normal unsigned acoustic intensity level defined as

1 N |Ini|! L |I |
10 log 10ā ā Io  N  i
1

Eqn 5-20

n

where |I ni| is the absolute (unsigned) value of the normal intensity vector. Note!

A large difference between intensity and pressure suggests that the probe is not well aligned or that you are operating in diffuse field. In order to calculate F2 it is necessary to have both intensity and autopower (or pressure) measurements for all points on the mesh.

F3 Negative partial power indicator This indicator also examines the difference between measured intensity and pressure, but in this case the direction of the intensities is taken into account. Thus this function expresses the variation between intensities arising from the source under investigation (positive) and those being generated by extraneous sources (negative).

F3
L p  LIn Lp

78

Eqn 5-21

is the surface sound pressure level defined above.

The Lms Theory and Background Book

Acoustic measurements

L In is the surface normal signed acoustic intensity level defined as



ā Ă1 ā L In
10 log 10 N

 IInio Ă N

i
1

Note!

If the quantity

 II

ni



Eqn 5-22

is negative, then the effect of extraneous sources is

0

too great and the set of measurements do not satisfy the ISO requirements. In order to calculate F3 it is necessary to have both intensity and autopower (or pressure) measurements for all points on the mesh.

F4 Non–uniformity indicator This indicates the measure of spatial (or positional) variability that exists in the field. It can be compared with the statistical parameter standard deviation. F4
1 In



1 N1

N

(ĂIni  InĂ)2

Eqn 5-23

i
1

Where i indicates the measurement surface and N is the total number of surĆ faces. I n is the mean of the normal acoustic intensity vectors taken over the N surfaces. In
1 ā N

N

 Ini

Eqn 5-24

i
1

In order to calculate F4 , only intensity measurements are required.

5.5.1

The criteria Three criterion can be evaluated in verifying the results of an acoustic intensity analysis.

Part II

Acoustics and Sound Quality

79

Chapter 5

Acoustic measurements

Ld – F2

Measurement chain accuracy

If a measurement array is to be considered suitable for determining the sound power level of a noise source according to ISO 9614-1, then the dynamic capaĆ bility index (Ld ) must be greater than the indicator F2 for each frequency band. Ld  F2  0

Criterion 1

Ld is dependent on the measurement equipment and is defined in equation NO TAG. F2 is defined in equation 5-18. Ld is derived from the pressure residĆ ual intensity index which must be computed during the measurement phase. If this criterion is not satisfied then it is an indication that the levels being meaĆ sured are too low for the source and that it is necessary to reduce the average distance between the measurement surface and the source.

F3 – F2

Extraneous noise sources

If the difference between field indicators F2 and F3 is significant (greater than 3dB), it is a strong indication of the presence of a directional extraneous noise source in the vicinity of the noise source under test. If the difference between these two indicators is greater than 3 dB, then the situĆ ation can be improved by reducing the average distance between the measureĆ ment surface and the source, shielding measurement sources from the extraneĆ ous noises or reducing some reflections towards the source under investigation.

Measurement mesh adequacy A check on the adequacy of the measurement positions (mesh) can be made usĆ ing the following criterion. N  CĂ.ĂF 4ā 2

Criterion 2

where N is the number of measurement (probe) positions F4 is the indicator defined in equation 5-23 C is a factor selected from table 5.3 depending on the accuracy reĆ quired. Where the same mesh is used for a number of bands then the maximum value of C.F4 2 will be considered when evaluating the criterion.

80

The Lms Theory and Background Book

Acoustic measurements

Center frequencies (Hz)

C

Octave band

1/3 Octave band

Precision class 1

Engineering class 2

63-125

50-160

19

11

250-500

200-630

9

19

1000-4000

800-5000

57

29

6300

19

14

A weighted (63 - 4k or 50 - 6.3k) Hz Table 5.3

Part II

Survey class 3

8

Values of factor C for measurement mesh accuracy

Acoustics and Sound Quality

81

Chapter 6

Sound quality

The purpose of this chapter is to introduce you to the fundamentals of sound quality. Basic theory relating to sound quality Sound quality analysis An extensive reading list is included at the end of the chapter for more detailed information.

83

Chapter 6

6.1

Sound quality

The basic concepts of Sound Quality Sound signals The characteristics of a sound as it is perceived are not exactly the same as the characteristics of sound being emitted. The discussion starts with definitions which describe the actual sound signals, and then discusses the physical and psychological effects that influence the perception of a particular signal. Sound power and sound pressure The amount of noise emitted from a source depends on the sound power of that source. The effect of the sound power emanating from a source is the level of sound (or acoustic) pressure. Sound pressure is what the eardrum detects - the level of which depends to a great extent on the acoustic environment and the distance from the source. Sound pressure is what is measured by microphones and the majority of data used in the a sound quality analysis would have the dimension pressure and thus be referred to as a sound signal. This is not an absolute condition however and vibrational data too can be analyzed. Sound pressure level The basic descriptor of a sound signal is the sound pressure level (SPL) denoted by L and described in equation 4-12. The sound pressure level of 20 Pa is known as the standardized normal hearing threshold and represents the quietĆ est sound at 1000Hz that can be heard by the average person. Since the range of pressure levels that can be detected is large and the ear reĆ sponds logarithmically to a stimulus, it is practical to express acoustic parameĆ ters as a logarithmic ratio of a measured value to a reference value. Hence the use of the decibel scales. Hearing frequency range The threshold frequency for human hearing is around 20kHz. Signals with a frequency content below this value are referred to as audio signals. Sampling of audio signals therefore requires a sampling rate at least twice the maximum that can be detected by the ear in order to avoid aliasing problems. You will find therefore that CD recorders use a sampling rate of 44.1KHz and DAT reĆ corders 48kHz. Loudness and pitch A sound can be characterized by its loudness (related to the SPL) and its freĆ quency content. The common term for describing the frequency content of a sound (or tone) is its `pitch'. However pitch is very much a perceived frequency sensation and depends on its frequency and the sound pressure level. Both loudness and pitch are discussed further below.

84

The Lms Theory and Background Book

Sound quality

The perception of sounds by the human ear An important element in explaining why two sounds with an equal dB level may have a totally different subjective quality is related to the physics of the human hearing process. The human ear is a complex, nonlinear device, with specific frequency dependent transmission characteristics. In addition, the fact that hearing usually involves two ears (is `binaural') has a considerable influĆ ence on sound perception. The correct understanding of the hearing processes will lead to a better appreciation of why a sound has its specific quality, which in turn will result in improved models and quantitative analysis procedures. Physics alone, however, are not sufficient to explain all aspects of sound perĆ ception. It is also influenced by psychological factors such as attitude, backĆ ground, expectations, environment, context, etc. As a consequence, there is no better `judge' of sound quality than the human listener, despite all efforts at quantification and modelling. The purpose of this section is merely to highlight the salient points of this subĆ ject. For a more thorough understanding of this topic you should refer to the reading list at the end of the chapter. Specific references to items in this list are contained within brackets {1}thus.

The hearing process Before reaching the eardrum, an incident acoustic signal is considerably modiĆ fied by the spectral and spatial filtering characteristics of the human body and the ear. The human torso itself acts as a directional filter through diffraction, resulting in the fact that very significant interaural differences in sound presĆ sure level occur depending on the direction of the source,{2}. Figure 6-1 shows the various parts of the ear (from {5}). The outer ear consists of the pinna and the ear canal. Diffraction effects at the pinna and direction inĆ dependent effects within the ear canal result in the human ear being most sensiĆ tive in the frequency range 1 to 10 kHz. The middle ear links the eardrum to the cochlea, which is the actual sound receptor. The final link between an acoustic signal and a neural response takes place in the cochlea, which is in the inner ear.

Part II

Acoustics and Sound Quality

85

Chapter 6

Sound quality

Stirrup Oval window Hammer Anvil Semicircular canal

Nerve fibers Cochlea

Scala vestibuli

Pinna

Scala tympani Eustachian tube

Ear canal

Ear drum Outer ear

Figure 6-1

Round window Middle ear

Inner ear

The main parts of the ear

Binaural hearing Another essential characteristic of human hearing is that it is binaural in nature. The sound signals received by the left and right ear show a relative time delay as well as a spectral difference dependent on the direction of the sound. Below about 1500 Hz, the phase difference between the two signals will be the main contribution to localization, while above this frequency the interaural level difĆ ference and difference in spectrum will be the principal factors. Processing in the human brain not only allows the sound to be spatially localĆ ized, but also to suppress unwanted sounds and to concentrate on a sound coming from a specific direction {2 6}. This is the well known `party' effect where it is possible to focus one's hearing on an individual a certain distance away in the presence of significant background noise.

Sound perception The body, head and outer ear effects consist mainly of a spatial and spectral filĆ tering that is applied to the acoustic stimulus. Consequently, just looking at the frequency spectrum of a free positioned microphone does not necessarily lead to a correct assessment of the human response. In other words, there is no simĆ ple relationship between the measured physical sound pressure level and the human perception of the same sound.

86

The Lms Theory and Background Book

Sound quality

The effects of the inner ear are many, but the most important are its nonlinear characteristics. This means that the auditory impression of sound strength, which is referred to by the term `loudness' is not linearly related to the sound pressure level. In addition, the perceived loudness of a pure tone of constant sound pressure level varies with its frequency. Also the auditory impression of frequency, which is referred to by the term `pitch' is not linearly related to the frequency itself. These and other effects are described below.

Loudness The sound pressure level is not linearly related to the auditory impression of sound strength (or loudness). Together with the frequency dependencies disĆ cussed above, this means that the sensation of loudness cannot be correctly deĆ scribed by the acoustic pressure level or its spectrum. Figure 6-2 {5} shows a number of curves representing levels of perceived equal loudness (for sinusoidal tones) across a frequency range as a function of acoustic pressure level.

Figure 6-2

Equal loudness perception contours {5}

Pitch The perceived `frequency sensation', referred to as `pitch', is not directly related to the frequency itself {6}.

Part II

Acoustics and Sound Quality

87

Chapter 6

Sound quality

The pitch of a pure tone varies with both the frequency and the sound pressure level, and this relationship is itself dependent on the frequency of the tone. Pure tones can be used though to determine how pitch is perceived. One possiĆ bility is to measure the sensation of `half pitch'. In this case the subject is asked to listen to one pure tone, and then adjust the frequency of a second one such that it produces half the pitch of the first one. At low frequencies, the halving of the pitch sensation corresponds to a ratio of 2:1 in frequency. At high freĆ quencies however this does not occur and the corresponding frequency ratio is larger than 2:1. For example a pure tone of 8kHz produces a `half pitch' of only 1300Hz. So although the ratio between pitches can be determined from experiments, to obtain absolute values, it is necessary to determine a reference for the sensation `ratio pitch'. A reference frequency of 125 Hz was chosen so that at low freĆ quencies, the numerical value of the frequency is identical to the numerical valĆ ue of the ratio pitch. Because ratio pitch determined in this way is related to our sensation of melodies, it was assigned the dimension `mel'. Therefore a pure tone of 125 Hz has a ratio pitch of 125 mel, and the tuning standard, 440 Hz, shows a ratio pitch with almost the same numerical value. At high frequencies, the numerical value of frequency and that of the ratio pitch deviate substantially from another. The experimental finding that a pure tone of 8kHz has a `half pitch' of 1300Hz, is reflected in the numerical values of the corresponding ratio pitch. The frequency of 8 kHz corresponds to a ratio pitch of 2100 mel and the frequency of 1300 Hz corresponds to a ratio pitch of 1050 mel, which are half of 2100 mel.

Critical bands The inner ear can be considered to act as a set of overlapping constant percentĆ age Bandwidth filters. The noise Bandwidths concerned are approximately constant with a Bandwidth of around 110 Hz, for frequencies below 500 Hz, evolving to a constant percentage value (about 23 %) at higher frequencies. This corresponds perfectly with the nonlinear frequency-distance characterisĆ tics of the cochlea. These Bandwidths are often referred to as `critical BandĆ widths' and a `Bark' scale is associated with them as shown in Table 6.1.

88

The Lms Theory and Background Book

Sound quality

Critical Band (Bark) Center Frequency (Hz) Bandwidth (Hz)

1 50 100

2 150 100

3 250 100

4 350 100

5 450 110

6 570 120

7 700 140

8 840 150

Critical Band (Bark) Center Frequency (Hz) Bandwidth (Hz)

9 1000 160

10 1170 190

11 1370 210

12 1600 240

13 1850 280

14 2150 320

15 2500 380

16 290 450

Critical Band (Bark) Center Frequency (Hz) Bandwidth (Hz)

17 3400 550

18 4000 700

19 4800 900

20 5800 1100

21 7000 1300

22 23 24 8500 10500 13500 1800 2500 3500

Table 6.1

Table of critical bands

Masking The critical bands described above, have important implications for sounds composed of multiple components. For example, narrow band random sounds falling within one such filter Bandwidth will add up to the global sensation of loudness at the center frequency of the filter. On the other hand, a high level sound component may `mask' another lower level sound which is too close in frequency. An example of masking is shown below {5}. A 50 dB, 4 kHz tone (marked +) can be heard in the presence of narrow-band noise, centered around 1200 Hz, up to a level of 90 dB. If the noise level rises to 100 dB, the tone is not heard.

Sound pressure level (dB)

Level of masking noise

Threshold of hearing

Frequency

Figure 6-3

Part II

Masking effects of narrow band noise [5]

Acoustics and Sound Quality

89

Chapter 6

Sound quality

The higher the level of the masking sound, the wider the frequency band over which masking occurs. Again, it turns out that multiple sound components falĆ ling within one of the ear filter Bandwidths add up to the masking level, while when they are wider apart each can be considered as a separate sound with its own masking properties.

Temporal effects Finally, a number of temporal effects are associated with the hearing process. Sounds must `build up' before causing a neural reaction, the reaction time howĆ ever is dependent on the sound level. This has an effect on the perceived loudĆ ness since the loudness of a tone burst decreases for durations smaller than about 200 ms. For larger durations, the loudness is almost independent of duration. This also has its consequences for masking : - Short sounds preceding a second loud sound can be reduced in loudness or even masked. The time intervals for this temporal `pre-masking' phenomeĆ non are in the order of tens of milliseconds. - A similar effect may occur after switching off a loud sound. During a time interval up to 200 ms (dependent on masking and tone level), short tone bursts may be masked (post-masking). - In the presence of a given continuous sound, tone bursts with levels exceedĆ ing that of the first signal, might be obscured, depending on their length. This is called `simultaneous masking'. A detailed discussion of these temporal effects can be found in {6}.

90

The Lms Theory and Background Book

Sound quality

6.2

Sound quality analysis One of the fundamental problems with sound quality is that `what-you-hearis-not-what-you-get'. Nonlinear physical characteristics of the human ear mean that the sound perceived is not directly related to the sound level being generated. Furthermore `what-you-like-is-not-what-you-hear' since the apĆ preciation or non-appreciation of a sound depends to a great extent on the situĆ ation and the attitude of the listener. An appreciation of the physical and psyĆ cho-acoustic aspects of human hearing is essential to the understanding of sound quality and to this end a short summary of the significant points and terms used is given in section 6.1. In the majority of problems or studies related to acoustics, the issue at hand is acoustic comfort, and not hearing damage or structural integrity. In order to properly describe this acoustic comfort, it has long since become clear that the acoustic pressure level is by no means sufficient or even adequate to correctly represent the actual hearing sensations. This is due to the very complex nature of the auditory impressions of acoustic signals (or `sounds'), leading to the use of concepts such as the `quality' of the sound. Auditory impressions can be annoying, in which case the sound is unwanted and is often referred to as `noise'. Typical examples are irritating engine, road or wind noise in a car, aircraft noise, machine or fan noise in the working enviĆ ronment. Examples of vehicle noises which while being annoying do not contribute sigĆ nificantly to the sound pressure level, are wiper noise, fuel pump noise, alterĆ nator whine, dashboard squeaks. To express this negative quality or annoyĆ ance, a multitude of qualitative concepts like whine, rattle, boom, rumble, hiss, beat, squeak, speech interference, harshness, sharpness, roughness, fluctuation, strength... are used. But not everything you hear is either bad or unwanted. A sound can be an imĆ portant messenger of information in which, it conveys a positive feeling. ExĆ amples are the solidity of a door-slam, the feeling of sportiveness of a car enĆ gine (or exhaust) during acceleration, the smoothness of a limousine engine, the `catching' of a door lock, or a seat belt.... In these cases, the noise does not need to be removed, but it has to sound `right'.

Analysis of sound signals Having identified a problem the aim is to measure, evaluate and modify sounds and a prerequisite for this is a high quality recording of the sound.

Part II

Acoustics and Sound Quality

91

Chapter 6

Sound quality

Digital Spectral processing

Digital

Filtering

output

Replay

input Comfort analysis

ÄÄ ÄÄ Figure 6-4

Reporting

Sound quality analysis

ÎÎÎ ÎÎÎ ÎÎÎ Analysis

Measurements Sound quality measurements are acoustic measurements made with microĆ phones. These can be digitally recorded and imported into the computer sysĆ tem, but in order to successfully evaluate a sound it is absolutely essential that it is both recorded and replayed in the most accurate and representative way possible. Binaural recording is a technique whereby microphones are mounted inside the ear in an artificial head to represent the sensation of human hearing as closely as possible. Evaluation The next step in dealing with a sound quality issue, is to gain a proper underĆ standing of the quality of the sound. In order to evaluate sound quality characĆ teristics, different (non-exclusive) approaches may be followed. (a) The acoustic signal can be evaluated subjectively by a specialist or jury of listeners. This can be achieved by replaying the signal either digitally via a recorder or directly via an analog output to headphones or speakers. When using direct replay, cyclic repetition of a particular segment can be perĆ formed and techniques are provided to suppress the `click' at the start and end of a segment as well as on-line notch filtering. This latter facility can give a very fast assessment of the critical spectral characteristics of a sound. (b) The acoustic pressure signal is processed in such a way that perception-releĆ vant quantitative values can be obtained through the use of adequate sound quality metrics. Such metrics form part of the comfort analysis. Modification Important information on the nature of a sound can be obtained by modifying the sound signal and comparing its perceived quality with the original. This modification can be imposed in the time, frequency or order domains.

92

The Lms Theory and Background Book

Sound quality

An important consequence of sound modification, is that it can also serve to generate the `target' sounds which become the specifications for the subsequent product modifications.

Binaural recording and playback The ultimate goal of a sound quality analysis must be to record, analyze, possiĆ bly modify and then playback a sound in such a way as to reproduce exactly what the listener would have experienced if he had listened to the original sound. The purpose of this section is to give an overview of this whole process and to introduce the factors that are involved in it. It also serves as a means of clarifying the terminology used in such a process. Free or Diffuse field

Artificial head

DAT recorder

Recording equalization

Figure 6-5

Calibration

Listener

Computer

Sound Quality Analysis

de-equalization Equalization

Binaural recording and playback

Recording The first stage in this process is to make an exact recording of a sound. A single microphone situated in free space is insufficient for this since at least four miĆ crophones would be necessary to correctly capture the 3D nature of the sound. It has been demonstrated in the previous section that the pressure experienced by the eardrum will be greatly influenced by the presence of the head and torso of the listener and is further affected by the non-linear operating characteristics of the ear itself. As a consequence of this, one of the most accurate ways to reĆ cord a sound is to mimic the function of the ears themselves and place two miĆ crophones inside the ear canals. Such a technique is known as binaural recordĆ ing, which involves two inputs representing what the left and the right ears would hear. Although it is possible to place the earphones inside the head, it is more comĆ mon to use an artificial head which provides similar spatial filtering to that of an actual head shoulders and torso. Equalization You may wish to reconstruct this recording as if it were the original sound and not as it is heard inside the head. In this case, you will need to `undo' the modĆ ifications that were caused by the presence of the head. The sound can be reĆ constructed as if it were in a free field or a diffuse field.

Part II

Acoustics and Sound Quality

93

Chapter 6

Sound quality

A free field refers to an idealized situation where the sound flows directly out from the source and the pressure levels drop with increasing distance from the source. A diffuse field occupies a smaller space and the sound is reflected many times. Thus, when you are recording a sound you can determine the type of field you wish to reconstruct it in and the appropriate compensation or equalization will be applied. If you only wish to replay the sound through headphones, then you do not need equalization and so you can either select to have a non-equalĆ ized recording or you will have to de-equalize it before it is replayed through headphones. Transfer to computer The recording on the DAT recorder is held in a 16 bit audio format. When this is transferred to a computer system, it will be then converted to a 32 bit floating point format. To achieve this conversion a calibration factor is required. Replay When you need to replay the signal on the headphones, then de-equalization may be necessary if free-field or diffuse field equalization has been applied to the original recording. In addition, compensation is required to take account of the transfer function associated with the particular set of headphones to be used.

94

The Lms Theory and Background Book

Sound quality

6.3

Part II

Reading list 1

D.LUBMAN, Noise Quality, Toward a Larger Vision of Noise Control Engineering, JourĆ nal of Noise Control Engineering, ....

2

J.BLAUERT, Spatial Hearing, MIT Press, Cambridge (MA), 1983.

3

W.BRAY ET AL, Development and Use of Binaural Measurement Technique, Proc. Noise Con. `91, Tarytown (NY), July 14-16, 1991, pp 443-450.

4

D.HAMMERSHOI, H.MOLLER, Binaural Auralisation : Head-Related Transfer FuncĆ tion Measured on Human Subjects, Proceedings 93rd AES Convention, Vienna (A), March, 24-27, 1992, 7pp.

5

J.HASSAL, K.ZAVERI, Acoustic Noise Measurements, Bruel & Kjaer, DK2850 NaerĆ um, Denmark, 1988

6

E.ZWICKER,H.FASTL, Psychoacoustics, Facts and Models, Springer Verlag, Berlin (Germany), 1990.

7

J.HOLMES, Speech Synthesis and Recognition, Van Nostrand Reinhold, Wokingham, Berkshire (UK), 1988.

8

M.HUSSAIN, J.GOELLES, Statistical Evaluation of an Annoyance Index for Engine Noise Recordings, SAE Paper 911080, Proc. SAE Noise and Vibration Conference, TraĆ verse City (MI), May 16-18 1991 pp 359-368.

9

H.SHIFFBAENKER ET AL, Development and Application of an Evaluation Technique to Assess the Subjective Character of Engine Noise, SAE paper 911081, Proc. SAE Noise and Vibration Conference, Traverse City (MI), May 16-18 1991, pp 369-379.

10

K.TAKANAMI ET AL, Improving Interior Noise Produced During Acceleration, SAE paper 911078, Proc. SAE Noise and Vibration Conference, Traverse City (MI), May 16-18 1991, pp 339-348.

11

G.IRATO, G.RUSPA, Influence of the Experimental Setting on the Evaluation of SubjecĆ tive Noise Quality, Proceeding of the second International Conference on Vehicle ComĆ fort, Oct 14-16, 1992, Bologna (Italy), pp. 1033-1044.

12

INTERNATIONAL ORGANIZATION FOR STANDARDIZATION, Method for Calculating Loudness Level, ISO-532-1975 (E)

13

E.ZWICKER ET AL, Program for Calculating Loudness According to DIN45631 (ISO532B), Journal Acoustic Society Jpn (E), Vol. 12, Nr.1, 1991.

14

S.J.STEVENS, Procedure for Calculating Loudness : Mark VII, J. Acoust. Soc. Am., Vol. 33, Nr.11, pp.1577-1585, 1961.

15

S.J.STEVENS, Perceived Level of Noise by Mark VII and Decibel, J. Acoust. Soc. Am., Vol.511, Nr.2, pp. 575-601, 1971.

16

E.ZWICKER, Procedure for Calculating Loudness of Temporally Variable Sounds, J. Acoust. Soc. Am., Vol. 62, Nr. 3, pp 675-681, 1977.

17

L.L.BERANEK, Criteria for Noise and Vibration in Communities, Buildings and Vehicles in Noise and Vibration Control, revised edition, McGraw-Hill Inc., 1988.

18

W.AURES, Berechnungsverfahren fü r den Sensorischen Wohlklang beliebigen SchallsigĆ nale, Acustica, Vol.59, pp. 130-141, 1985

Acoustics and Sound Quality

95

Chapter 6

96

Sound quality

19

M.ZOLLNER, Psychoacoustic Roughness. A New Quality Criterion, Cortex Electronic, 1992.

20

W.AURES, Ein Berechnungsverfahren der Rauhigkeit, Acustica, Vol.58, pp. 268-280, 1985.

21

M.F.RUSSEL, What Price Noise Quality Indices, Proc. Engineering Integrity Society Symposium on NVH Challenges - Problem Solutions, Oct.21, 1992.

22

M.F.RUSSEL ET AL. Subjective Assessment of Diesel Vehicle Noise, IMechE paper 925187, Ref. C389/044, FISITA Conference Engineering for the customer, pp.37-42, 1992.

23

D.G.FISH, Vehicle Noise Quality - Towards Improving the Correlation of Objective MeaĆ surements with Subjective Rating, I. Mech. E. - paper 925186, Ref. C389/468 FISATAconference, Engineering for the customer, pp. 29-36, 1992.

24

G.TOWNSEND, A New Approach to the Analysis of Impulsiveness in the Noise of Motor Vehicles, Proc. Autotech `89, paper 7/26.

25

MOTOR INDUSTRY RESEARCH ASSOCIATION, Improving Correlation of ObjecĆ tive Measurements with Subjective Rating of Vehicle Noise, MIRA research report K3866326.

26

F.K.BRANDL ET AL, A Concept for Definition of Subjective Noise Character - A Basis for More Efficient Vehicle Noise Reduction Strategies, - Proceedings Internoise-89, Newport Beach (CA), Dec. 4-6, 1989, pp.1279-1282.

27

R.S.THOMAS, A Development Process to Improve Vehicle Sound Quality, SAE paper 911079, Proc, SAE Noise and Vibration Conference, Traverse City (MI), May 13-16 1991, pp. 349-358.

28

G.R.BIENVENUE, M.A.NOBILE, The Prominence Ratio Technique in Characterizing Perception of Noise Signals Containing Discrete Tones, Proc. Internoise `92, Toronto, Canada, July 20-22, 1992, pp. 1115-1118.

29

K.TSUGE ET AL, A Study of Noise in Vehicle Passenger Compartment during AcceleraĆ tion, SAE paper 8509665, Proceedings SAE Noise and Vibration Conference, Traverse City (MI), May 15-17, 1985, pp. 27-34.

30

T.WAKITA ET AL, Objective Rating of Rumble in Vehicle Passenger Compartment during Acceleration, SAE paper 891155, Proceedings SAE Noise and Vibration Conference, Traverse City (MI), May 16-18, 1989, pp. 305-312.

31

W.YAGISHASHI, Analysis of Car Interior Noise during Acceleration Taking into AcĆ count Auditory Impressions, JSAE Review (E), Vol. 12, nr.4, Oct. 1991, pp. 58-61.

32

K.FUJITA ET AL, Research on Sound Quality Evaluation Methods for Exhaust Noise, JSAE Review (E), Vol. 9, Nr. 2, April 1988, pp. 28-33.

33

American National Standard, S.3.14-1977 (R886), Rating Noise with Respect to Speech Interference, Acoustical Society of America.

34

H.STEENEKEN, T.HOUTGAST, RASTI, A Tool for Evaluating Auditoria, Bruel & Kjaer Technical Review, nr.3-1985, pp. 13-30.

35

M.NAKAMURA, T.YAMASHITA, Sound Evaluation in Cars by RASTI Method, JSAE Review, Vol.11, Nr.4, Oct 1990, pp.38-41.

36

H.MOLLER, Fundamentals of Binaural Technology, Applied Acoustics, Vol. 36, 1992, pp. 171-218.

The Lms Theory and Background Book

Sound quality

Part II

37

K.GENUIT, M.BURKHARD, Artificial Head Measurement System for Subjective EvalĆ uation of Sound Quality, Sound and Vibration, March 1992, pp. 18-23.

38

G.MICHEL, G.EBBIT, Binaural Measurements of Loudness as a Parameter in the EvalĆ uation of Sound Quality in Automobiles, Proc. Noise Con. `91, Tarytown (NY), July 14-16, 1991, pp. 483-490.

39

G.THEILE, The Importance of Diffuse Field Equalisation for Stereophonic Recording and Reproduction, Proc. 13-th Tonmeistertagung, 1984.

40

D.S.MANDIC, P.R.DONOVAN, An Evaluation of Binaural Measurement Systems as Acoustic Transducers, Proc. Noise Con 91, Tarytown (NY), July 14-16, 1991, pp. 459-466.

41

H.HAMMERSHOI, H.MOLLER, Artificial Head for Free Field Recording ; How Well Do They Simulate Real Heads ?, Proc. 14th ICA, Beijing, 1992, Paper H6-7 (2pp).

42

K.GENUIT, H.GIERLICH, Investigation between Objective Noise Measurement and Subjective Classification, SAE Paper 891154, Proceedings SAE Noise and Vibration Conference, Traverse City (MI), May 16-18 1989, pp 295-303.

43

H.MOLLER ET AL, Transfer Characteristics of Headphones, Proc. 92nd AES ConvenĆ tion, Vienna (A), March 24-27, 1992, 28 pp.

44

Y.OKAMOTO ET AL, Evaluation of Vehicle SOunds Through Synthesized Sounds that Respond to Driving Operation, JSAE Review (E), Vol.12, Nr.4, Oct.1991,pp.52-57.

45

S.M.HUTCHINS ET AL, Noise, Vibration and Harshness from the customer's Point of View, IMechE paper 925181, Ref. C389/049, Proc. FISATA-92 Conf, Engineering for the Customer.

46

H.AOKI ET AL, Effects of Power Plant Vibration on Sound Quality in the Passenger Compartment During Acceleration, SAE paper 870955, Proc. SAE Noise and Vibration Conf., Traverse City (MI), Apr. 28-30, 1987, pp.53-62.

47

K.C. PARSONS, M.J. GRIFFIN, Methods for predicting Passenger Vibration DiscomĆ fort, Society of Automotive Engineers Technical Paper Series 831921

48

M.J. GRIFFIN, Handbook of Human Vibration, Academic Press Ltd. 0-12-03040-4

49

J.D. LEATHERWOOD, L.M BARKER, A User-Oriented and Computerized Model for Estimating Vehicle Ride Quality, NASA Technical Paper 2299 (1984)

50

International Standard, Ref. No. ISO 2631/1 - 1985 (E)

51

International Standard, Ref. No. ISO 5349 - 1986 (E)

52

British Standards Institution, Measurement and evaluation of human exposure to wholebody mechanical vibration and repeated shock Ref. No. BS 6841 - 1987

53

American National Standard, S3.14 - 1977 (R-1986), Rating Noise with Respect to Speech Interference, order from the Acoustical Society of America.

54

ANSI S3.5, Calculation of the Articulation index, American National Standards InstiĆ tute, Inc., 1430 Broadway, New York, New York 10018 USA, 1969

55

International Standard, Ref. No. ISO 532 - 1975 (E)

Acoustics and Sound Quality

ISBN

97

Chapter 7

Sound metrics

It may be said that the best way to evaluate the quality of a sound is to listen to it and express an opinion about it, but in a lot of cases there is also a strong interest in correlating the results from these subjective evaluations with measurable parameters. Therefore a number of sound quality metrics exist where perception-relevant quantitative values are calculated from the acoustic pressure signal. Sound pressure levels Loudness metrics Sharpness Roughness Fluctuation strength Pitch Articulation index Speech interference levels Impulsiveness The references are listed in chapter 6

99

Chapter 7

7.1

Sound metrics

Sound pressure level The basic descriptor of a sound signal is sound pressure level (SPL) denoted by L. and described in equation 4-12. The stimulus of the sound pressure level needs to be interpreted as a hearing sensation and one approach consists of multiplying the frequency spectrum of the acoustic pressure signal with a weighting function before calculating the RMS level. Several weighting functions have been defined, of which the AB-C and D weightings are the most widely used. They are based on experiĆ mentally determined equal loudness contours which express the loudness sensation of single tones as a function of sound pressure level and frequency.

Time domain sound pressure level This function calculates the frequency and time weighted sound pressure level according to the IEC 651 and ANSI SI.4-1983 standards. Frequency weighting can be applied to the time signal using the A, B or C weightings described above. The time signal is then exponentially averaged to arrive at the sound pressure level. An exponential weighting factor is used t

(e  ) where
t is the sample period of the signal and  is the time constant. The values of  depends on the type of signal (mode) and three default (standardĆ ized) values are supplied.  = 35ms for impulse (peaky) signals  = 125 ms for fast changing signals  = 1000 ms for slow changing signals. By selecting the type of signal (mode) then the appropriate time constant is apĆ plied. When the signal contains spikes and is therefore defined by the mode imĆ pulse" an additional peak detector mechanism is implemented. In this case when an increase in the averaged signal is detected, then the signal is followed exactly. When the signal is decreasing, then exponential averaging is used with a long time constant, set by default to 1500 ms. The time constant used in this situation is termed the decay time constant.

100

The Lms Theory and Background Book

Sound metrics

7.2

Equivalent sound pressure level The iso standards: ISO1996/1-1982 and ISO1999:1990 provide a definition for the `equivalent A-weighted sound pressure level in decibels' identified as LAeq,T. This function gives the value of the A-weighted sound pressure level of a conĆ tinuous, steady sound that, within a specified time interval T, has the same mean square sound pressure as the sound under consideration whose level varĆ ies with time. This leads to the expression: t   p 2A(t) 1 L Aeq,T
10 logt  t Ă 2 Ă dt p0 1 2 t  



2

Eqn 7-1

1

where LAeq,T is the equivalent continuous A-weighted sound pressure level, in deciĆ bels, determined over a time interval T starting at t1 and ending at t2 p0

is the reference sound pressure (20mPa);

pA(t)

is the instantaneous A-weighted sound pressure of the sound signal.

In practice with sampled data the equivalent sound pressure level is computed by a summation of the sampled values of the pressure level, in dB over the number of samples required. As a generalization, you can apply the same formula to a non-A-weighted sound pressure signal p(t) to obtain Leq,T.

Part II

Acoustics and Sound Quality

101

Chapter 7

7.3

Sound metrics

Loudness The equal loudness contours shown in Figure 6-2 in the document Sound quality" are the result of large numbers of psycho-acoustical experiments and are in principle only valid for the specific sound types involved in the test. These curves are valid for pure tones and depict the actual experienced loudĆ ness for a tone of given frequency and sound pressure level when compared to a reference tone. The resulting value is called the `loudness level'. The loudness level itself is expressed in Phons. 1 kHz-tones are used as the refĆ erence, which means that for a 1 kHz tone, the Phon value corresponds to the dB sound pressure level. The equal loudness contours for free field pure tones and diffuse field narrow-band random noise are standardized as ISO 226-1987 (E). A linear unit derived from the (logarithmic) Phon values is the Sone (S), which is related to the Phon (P) in the following way :

S
2 (P40)10

Eqn 7-2

The Sone scale's linear relationship to the experienced loudness makes it easier to interpret. A loudness of 1 Sone corresponds to a loudness level of 40 Phons. A tone which is twice as loud, will have double the loudness (Sone) value, and a loudness level which is 10 Phons higher. When broadband or multi-tone sounds are being considered, the frequency spectrum of the loudness is made in terms of critical bands instead of the total value. Critical bands and barks are described in Table 6.1 in the chapter onSound quality". In this case the terminology `specific loudness' is used, exĆ pressed in Sones/Bark. For steady state sounds, standardized calculation procedures have been deĆ fined by Zwicker and Stevens and are accepted as ISO standards {12, 13, 14}. A more recent procedure by Stevens {15} has not yet been accepted as an ISO stanĆ dard. They are both based on : V

a convention for the relation between octave band sound pressure levĆ els and octave band partial (specific) loudness descriptions

V

a convention to combine the specific loudness values into a global loudĆ ness, taking into account masking effects.

For temporally varying sounds, Zwicker has also proposed an approach taking into account temporal effects {16}, which is not yet accepted as an ISO standard.

102

The Lms Theory and Background Book

Sound metrics

7.3.1

Stevens Mark VI The Stevens (Mark VI) method, standardized as ISO 532-A-1975 and ANSI S3.4-1980, starts from octave band sound pressure levels. Their loudness is compared to that of a critical band noise at 1 kHz. It is only defined for diffuse sound fields with relatively smooth, broadband spectra. Through a set of stanĆ dardized curves, each octave band level is converted into a partial loudness inĆ dex (s) see Figure 7-1. The partial loudness values are then combined into a total loudness (in Sones), using equation 7-3.

st
s m  FĂ(Ăs  smĂ) where sāt = sām = s= F=

the total loudness in Sones the greatest of the loudness indices, in Sones the sum of the loudness indices of all bands, in Sones fractional loudness contribution factor, reflecting masking effects. It depends on the type of octave measurement (0.3 1/1 octaves,0.15 for 1/3 octaves).

for

Figure 7-1

Part II

Eqn 7-3

Loudness (Mark VI)

Acoustics and Sound Quality

103

Chapter 7

7.3.2

Sound metrics

Stevens Mark VII A more recent calculation scheme is Stevens Mark VII {15, 17}, which uses a more refined partial loudness calculation ( see Figure 7-2 ), as well as a level dependent calculation for F in equation 7-3. The reference frequency is 3150 Hz. Apart from the loudness (in Sones), the logarithmic unit `perceived loudĆ ness level' (PLdB) is used here, which is 32 dB for a loudness of 1 Sone at 3150 Hz. PLdB values will be about 8 dB lower than the loudness level in Phones. Examples are discussed in {5} and {17}.

Figure 7-2

7.3.3

Loudness (Mark VII)

Loudness Zwicker Loudness assessment using the Zwicker method (standardized as ISO 532B) starts from 1/3 octave band sound pressure level data, which can originate from either a free or diffuse sound field. It is capable of dealing with complex broadband noises, which may include pure tones.

104

The Lms Theory and Background Book

Sound metrics

The method takes masking effects into account. Masking effects are important for sounds composed of multiple components. A high level sound component may `mask' another lower level sound which is too close in frequency. An exĆ ample of masking is shown below {5}. A 50 dB, 4 kHz tone (marked +) can be heard in the presence of narrow-band noise, centered around 1200 Hz, up to a level of 90 dB. If the noise level rises to 100 dB, the tone is not heard.

Sound pressure level (dB)

Level of masking noise

Threshold of hearing

Frequency

Figure 7-3

Masking effects of narrow band noise (5)

The method uses different sets of graphs for diffuse and free fields that relate loudness level to sound pressure level and that take the masking into account by a sloping-edge filter characteristic for each octave band. This way, domiĆ nant and hence masking frequency bands will show their influence over a large frequency range and prevent masked sounds contributing to the total level. Figure 7-4 shows an example of the Zwicker method. The 1/3 octave band data are transferred to the appropriate Zwicker diagram.

Part II

Acoustics and Sound Quality

105

Sound metrics

Sound pressure level (dB)

Chapter 7

Frequency

Figure 7-4

Example loudness calculation according to Zwicker's method {5}

The partial loudness contours are computed for each defined segment (global evaluation) or frame (tracked evaluation) using a classical Zwicker loudness calculation. The frame or segment size should be selected to ensure that the spectral resolution needed for the FFT-based octave band analysis can be achieved. The frame size can be used to restrict the analysis to time periods over which time-varying signals can be regarded as stationary. The Zwicker loudness analysis allows you to distinguish between unmasked and masked contours thus allowing you to see that certain levels are either wholly or completely masked by previous ones. The total loudness is calculated as the surface under the enveloping partial loudĆ ness contours and can be expressed in Sones, or as loudness level in Phones as a function of time. This is presented as a single value in the global evaluation and a trace of values for the tracked evaluation.

106

The Lms Theory and Background Book

Sound metrics

7.4

Sharpness A sensation which is relevant to the pleasantness of a sound is its `sharpness', allowing you to classify sounds as shrill (sharp) or `dull'. The sharpness sensaĆ tion is strongly related to the spectral content and center frequency of narrowband sounds and is not dependent on loudness level or the detailed spectral content of the sound. Roughly, it corresponds to the first spectral moment of the specific loudness, with a pre-emphasis for higher frequencies. A quantitative procedure has been proposed, expressing the sharpness in the unit `acum'. The reference sound of 1 acum is a narrow-band noise, one critical band wide, and at a center frequenĆ cy of 1 kHz and having a level of 60 dB. The dependency of sharpness on the center frequency and bandwidth of the noise is shown in Figure 7-5 {6}. The middle curve represents a noise of one critical bandwidth as a function of center frequency, the upper and lower curves representing the sharpness of noises with respect to fixed upper (10 kHz) or lower (0.2 kHz) cut-off frequency as a function of the other cut-off valĆ ue. Higher frequency noises produce higher sharpness.

Figure 7-5

Part II

Loudness of bandlimited noise

Acoustics and Sound Quality

107

Chapter 7

Sound metrics

The specific sharpness calculation (S '(z) ) is made according to:

S (z)


0.11ĂN (z)Ăg(z)Ăz 24āBark



N(z)Ăz

Eqn 7-4

0āBark

where

N'(z) is the specific Zwicker loudness g(z) a weighting function that pre-stresses higher frequency compoĆ nents (Figure 7-6). g(z) has unit value below 16 Bark and rises expoĆ nentially as

g(z)
0.066e 0.171z

Figure 7-6

Eqn 7-5

Sharpness calculation weighting function

The total sharpness S expressed in `acums' is obtained by integrating the specific sharpness. 24āBark

S




S (z)Ăz

Eqn 7-6

0āBark

108

The Lms Theory and Background Book

Sound metrics

7.5

Roughness The roughness or harshness of a sound is a quality associated with amplitude modulations of tones. When this modulation frequency is very low (15 Hz), the actual time varying loudness fluctuations can be perceived. This fluctuation sensation is discussed in section 7.6. At high modulation frequencies (above 150-300 Hz), three separate tones can be heard. In the intermediate frequency range (15-300 Hz), the sensation is of a stationary, but rough tone, which renders it rather unpleasant. This sensation is often associated with engine noise, where fractional orders can cause the moduĆ lation effects. Roughness increases with degree of modulation and with modulation frequenĆ cy, and is less sensitive to the base frequency. The unit used to describe roughĆ ness is the asper"; 1 asper being produced by a 100%, 70 Hz modulated 1 kHz tone of 60 dB. The dependency relationship between modulation depth and frequency is howĆ ever not straightforward. An important element is that the temporal variations of the loudness can cause masking effects, and a temporal masking depth (L) is introduced, representing the difference between maximum and minimum in the actually perceived time dependent loudness pattern. Due to post masking, this masking depth is smaller than the modulation depth, with the difference becoming greater at higher frequencies. The roughness (R) of an amplitude modulated sound can then be approximated as

R " f mod L

Eqn 7-7

Quantitative procedures to calculate roughness have been proposed. They inĆ volve the calculation of partial or specific roughness" in each critical band, based on modulation and depth, including masking effects and integrating them to obtain total roughness.

Part II

Acoustics and Sound Quality

109

Chapter 7

7.6

Sound metrics

Fluctuation strength When the sound functions have modulation frequencies below 20 Hz, they are perceived as changes in the sound volume over time. Typically, fluctuation sigĆ nal sound louder (and more annoying) than steady state signals of the same rms amplitude. In this case, the intensity of the sensation is referred to as Fluctuation strength" with the unit vacil". A reference sound of 1 vacil correĆ sponds to a 1 KHz tone of 60 dB with a 100 % amplitude modulation of 4Hz. The ear is most sensitive to fluctuations at 4 Hz. Quantitative models have been proposed for the fluctuation strength {6} which take into account the temĆ poral masking effects due to the sound fluctuation. The dependency of the fluctuation strength (F) on the modulation frequency (fmod) and masking depth (L) is then the following

F#

110

L (f mod4Hz)  (4Hzf mod)

Eqn 7-8

The Lms Theory and Background Book

Sound metrics

7.7

Pitch Pitch is a sound attribute that classifies sounds on a scale from low to high. For pure tones, pitch depends largely on the frequency of the tone, but it is also inĆ fluenced by its level. In a complex tone, consisting of many spectral components, one or more pitches can be perceived. These pitches also depend to a large extent on the frequenĆ cies of the constituent components, but also masking effects can occur, making some pitches more prominent than others. Pitches, both for pure and complex tones, which can be derived from the specĆ tral content of the signals, are called spectral pitches. It has been observed that in a complex tone, consisting of a fundamental freĆ quency and a number of its harmonics, a pitch corresponding to the fundamenĆ tal frequency is perceived, even when that fundamental frequency is filtered out of the signal. In this case, the perceived pitch does not relate anymore to a component actually present in the signal but relates to the difference between the higher harmonics. This type of pitch is called residue pitch or virtual pitch. The pitch calculation is implemented according the method developed by Terhardt (J. Acoust. Soc. Am. Vol 71, pp 679-688, 1982). Both spectral and virtual pitches can be derived as well as the weight of each calculated pitch. These indicate how prominently the pitches are perceived. If, in the calculation the effect of the tone level on the pitch is taken into acĆ count, the calculated pitch is called true pitch. If the influence of level on the tone is neglected, it is called nominal pitch.

Part II

Acoustics and Sound Quality

111

Chapter 7

7.8

Sound metrics

Articulation index (AI) The Articulation Index is a parameter developed with a view to assuring speech privacy. Speech privacy can be defined as the lack of intrusion of recogĆ nizable speech into an area when background sound or noise then provides a positive quality of privacy. The measure of interference caused by noise to the masking of speech can be calculated by weighting the noise spectrum (in 1/3 octave bands) according to its importance to the understanding of speech. From this weighted spectrum, the Articulation Index is derived.

A graphical equivalent of the calculation is given in Figure 7-7 (from {17}). The 1/3 octave bands relevant to speech are weighted by a number of dots. When the sound pressure level is plotted on this graph, the AI can be derived as the numĆ ber of dots above the spectrum divided by the total number. Practical calculaĆ tions are of course based on tables. Figure 7-7

Graphical representation of the Articulation Index

This index can then be related to a perĆ centage of syllables understood (see FigĆ ure 7-8 from {17}) For complete privacy, an AI of 0.05 is the limit, for semi-privacy to discuss non-confidential matters, an AI of 0.1 is acceptable {17}.

Figure 7-8 Intelligibility of sentences as a function of articulation index

There are two methods available. V

112

Standard The calculation is based on the work of Beranek as set out in The deĆ sign of speech communication systems", Proceedings of the IRE, Vol 45, 880-884, 1947. The results of this method will lie in the range 0-100%

The Lms Theory and Background Book

Sound metrics

V

Part II

Modified These calculations are based upon the AIM method which has been deĆ scribed in the work mentioned above, but which opens up the internal floating range of 30dB to a fixed range of 80dB between the limits of 20 and 100dB. The results of this method will lie in the range-107% to alĆ most 160%

Acoustics and Sound Quality

113

Chapter 7

7.9

Sound metrics

Speech interference level (SIL, PSIL) When the comprehension of speech is the goal, background sound or noise has the negative quality of interference. It can cause annoyance, and even be hazĆ ardous in a working environment where instructions need to be correctly unĆ derstood. Therefore, a noise rating called `Speech Interference Level' (SIL) was developed. Beranek originally defined it as the arithmetic average of the sound pressure levels in the bands 600-1200, 1200-2400 and 2400-4800 Hz. Since the definition of the new preferred octave band limits, this definition was changed to the PreĆ ferred Speech Interference Level' or PSIL, defined as the average sound pressure level in the 500, 1000 and 2000 Hz octave bands {5,17}. In 1977, the Speech Interference level was standardized as ANSI S3.14-1977(R-1986) {33}, which also included the 4 kHz octave band. This is in accordance with an ISO suggestion, described in ISO Technical Report TR 3352-1974. On average, the ANSI-SIL is about 1 dB higher than the original (Beranek) and about 2.5 dB lower than the PSIL {17}. The application of the SIL to the actual understanding of speech is presented in several graphs and tables {see 5,17}. These papers show the relationship beĆ tween SIL and the conditions under which speech can be understood. As an example, Figure 7-9 shows the relationship between ease of face-to-face conĆ versation with ambient noise level in PSIL, and separation distance in meters {5}.

Figure 7-9

114

Communication limits in the presence of background noise (after WebĆ ster)

The Lms Theory and Background Book

Sound metrics

7.10

Impulsiveness This metric is used to quantify the impulsive nature of a signal. It is used for instance in the quantification of the diesel engine noise. The algorithm for calculating impulsiveness is based on the signal envelop, and results in a number of output values; the mean impulse peak level, mean imĆ pulse rise rate and mean impulse duration. Each of these parameters is deĆ scribed in the Figure below. In addition the mean impulse rate (occurrence) is determined. Peak level

signal envelope

rise rate

threshold center position

threshold offset rms level

rise time

fall time

A certain threshold is used to determine the occurrence of an impulsive event. That threshold is the sum of the overall segment RMS value (in the global comĆ putation) and the RMS of the frame (in the case of tracked calculation) and the a user-defined threshold offset. The start of the impulse is defined by the minimum which occurs before the crossing of the threshold. The rise time is the time between the impulse start, and the moment at which the impulse peak level is reached. The peak level is expressed in dB, and is the difference between the impulse peak and the threshĆ old level. The end of the peak is defined by the first minimum which occurs after the threshold level. The duration of the peak duration is the sum of rise time and fall time. The rise rate is the maximum rise rate occurring between the impulse start and the impulse peak.

Part II

Acoustics and Sound Quality

115

Chapter 8

Acoustic holography

This chapter describes the background to acoustic holography.

117

Chapter 8

8.1

Acoustic holography

Introduction Acoustic holography allows you to accurately localize noise sources. It thereĆ fore helps in both the reduction of unwanted vibro-acoustic noise and optiĆ mization of noise levels. It : V

estimates the acoustic power and the spectral content emitted by the object under examination.

V

maps sound pressure, velocity and intensity on the measurement plane and on all parallel planes. The mapping of these acoustical quantities outside the measurement plane is done through acoustical holography (near field - far field).

V

estimates the acoustic level of the principal sources, including contribuĆ tion analysis.

This document describes the principles of taking acoustic measurements and the subsequent analysis of acoustic holography data, for both stationary and transient measurements.

Basic principles In performing acoustic holography, you need to measure cross spectra between a set of reference transducers and the hologram microphones. From these meaĆ surements you can derive sound intensity, particle velocity and sound power values. A basic assumption is that you are operating in free field conditions and that the energy flow is coming directly from the source. Measurements need to be taken close to the source. It provides you with an accurate 3D characterization of the sound field and the source with a higher spatial resolution than is possible with conventional intenĆ sity measurements.

118

The Lms Theory and Background Book

Acoustic holography

8.2

Acoustic holography concepts The principle of acoustic holography is to decompose the measured pressure field in plane waves, by using a spatial Fourier transform. With the frequency being fixed, we can calculate how each of these plane waves propagates, and by adding them we can find the pressure field on any plane which is parallel to the measurement plane. Consider an acoustic wave. Measuring the pressure on a plane means cutting the wavefronts by the measurement plane : measurement plane

ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ

The goal is to determine the whole acoustic wavefront from the known pressure on the measurement plane. Each microphone in the array measures the comĆ plex pressure (amplitude and phase).

Temporal and spatial frequency In considering how to do this we will compare the time and the spatial domain. Time domain When considering measurements in the time domain, then the position from the sound source (m) is fixed and we obtain a measure of the pressure variation as a function of time.

ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ

pressure T time

m

f=1/T

=cĂ/Ăf

The transformation from the time to the frequency domain is achieved using the Fourier Transform given below

Part II

Acoustics and Sound Quality

119

Chapter 8

Acoustic holography



F(āā)


 Ă f (ātā)e

Eqn 8-1

jāātĂdt



Spatial domain If we now consider measurements where time is fixed and pressure varies as a function of distance, we can obtain a measure of energy flow.

ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ ÄÄÄÄÄ

m

pressure distance

The spatial frequency of this function or wavenumber (k0) is defined as :

k0 = 2  f = = 2  c

c



where c is the speed of sound and f is the temporal frequency

If we fix the temporal frequency, this means that the acoustic wavelength is fixed too. The complex pressure as a function of the space is called the pressure image at the specified frequency. Conversion from the spatial domain is also done using a Fourier transform. In Acoustic holography pressure is measured in two dimensions (x and y for example), so a 2-dimensional transformation is performed.

Sā(k x, k y)


 Ă P

measuredĂ(āxā,ā y)Ăe

j(k xxk yy)

ādxā.ādy

Eqn 8-2

S

where S (kx , ky) is the spatial transform of the measured pressure field to the wavenumber (kx and ky ) domain resulting in the 2-D hologram pressure field.

120

The Lms Theory and Background Book

Acoustic holography

Pmeasured

spatial domain x,y

wavenumber domain kx, ky

A measured pressure (sound) wave with a particular temporal frequency can propagate in a number of directions, so the wavenumber vector (k) will have a number of components. The appearance of these vectors depends on the plane on which you are looking at them. The aim is to find the components of these vectors in the 2 dimensions that define the plane and to do this projections of the vectors in the plane are made. pressure levels

kx ky

wavenumber domain kx ky

Summation of plane waves The spatial Fourier transform implies that a measured pressure field can be considered as a sum of sinusoidal functions.

Each of these sinusoidal functions can be understood as the result of cutting the wavefronts of a plane wave by the measurement plane.

Part II

Acoustics and Sound Quality

121

Chapter 8

Acoustic holography

measurement plane

spatial periodicity

wavelength = c/f

wavelength = c/f

There is a coincidence between the nodes of the sinusoidal function and the waĆ vefronts. In effect, decomposing the pressure field into a sum of sinusoidal functions means decomposing the real acoustic wave into a sum of plane waves. Whatever the angle of incidence, the spatial periodicity must be greater than the wavelength .

Propagating and evanescent waves There are two kinds of plane waves : propagating waves whose level remains the same as they propagate but who undergo a phase shift.

evanescent waves whose level decreases as they propagate.

Propagating waves represent the sound field that is propagated away from the near towards the far field. Evanescent waves describe the complex sound field in the near field of the source. To understand why we must take evanescent plane waves into account, let us consider our decomposition of the pressure field into sinusoidal functions. If the spatial periodicity of a sinusoidal function is shorter than the wavelength, it cannot be the result of cutting a propagating plane wave by the measurement plane :

122

The Lms Theory and Background Book

Acoustic holography

spatial periodicity

measurement plane

Whatever the direction of the propagating plane wave may be, there is no posĆ sible coincidence between the nodes of the sinusoidal function and the waveĆ fronts. Therefore, this sinusoidal function must be understood as the intersecĆ tion between an evanescent wave (which can have a smaller spatial periodicity than propagating waves) and the measurement plane. A mathematical interpretation of the evanescent waves is based on the value of kz which is the component perpendicular to the measurement directions in the wave number domain. kz

ky

measurement plane

kx kz can be determined from the wave number k0 and the known values of kx and ky from the transformation. k 0
k 2x  k 2y  k 2z
 c kz


 c  (k  k ) 2

2 x

Eqn 8-3

2 y

 2 2 2 kz is real when k x  k y  ( c ) (the spatial periodicity is greater than the waveĆ length). This means that the waves lie in the circle defined by the radius /c in the wave number domain. kz is imaginary outside of this region.

Part II

Acoustics and Sound Quality

123

Chapter 8

Acoustic holography

ky k 2x  k 2y  k 20 k z is imaginary evanescent waves

k0
 c

k 2x  k 2y  k 0

2

kx

k z is real propagating waves

When kz is imaginary, the propagation factor becomes a damped exponential function ( e jk zĂz) meaning that a propagated wave undergoes an amplitude modification while the phase is not changed.

(Back) propagating to other planes Pressure levels at other planes can be found using Raleigh's integral Equation with Dirichlet's Green function : P(r)


 P(r)G ā(r  r)ādxādy

Eqn 8-4

d

where the Green function Gd can be thought of as the transformation function to transform the sound pressure field from one plane to another. We can use wave domain properties (k) to predict the pressure at a different spatial position (z). The practical computation of Raleigh's equation is for z > z'

S(k z, k y, z)
S(k z, k y, z)āg dā(k z, k y, z  z)

for 0 < z < z'

Sā(k x,ā k y,ā z)
Sā(k x,ā k y,ā z)

1 g d(k x,ā ky,ā z  z)

Eqn 8-5 Eqn 8-6

where z' is the measurement plane and z is the position of the required plane. The green function is given by g d
eā jākzāz and kz can be found from equation 8-3.

124

The Lms Theory and Background Book

Acoustic holography

The final step is to perform an inverse transformation back to the temporal doĆ main.

The Wiener filter and the AdHoc window As mentioned above, evanescent waves undergo a change in amplitude when propagating. Propagating towards the source implies an amplification of the signal that is a function of kz. Evanescent waves that lie far away from the unit circle have a large kz, therefore their amplitude is amplified significantly when propagating to the source. The contribution of these evanescent waves results in an increase of spatial resolution. Note that the inclusion of evanescent waves is only appropriate when propagating towards the source. Propagating away from the source, the evanescent waves decrease so rapidly in amplitude that their contribution to the spatial resolution becomes negligible. However the further away a wave is located from the circle, the less accurate the amplitude estimate becomes so that at a certain point noise is propagated and at that point the propagated image starts to blur. propagating waves evanescent waves

ÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏ ÏÏÏÏÏÏÏÏ

ky

kx

radius = k0 When propagating towards the source, a Wiener filter can be used to include a certain number of evanescent waves to improve the resolution. Taking a higher number of waves taken into account may result in the amplification becoming unstable. This depends on a parameter of the Wiener filter known as the Signal to Noise Ratio (SNR). When the SNR value is greater than 15dB, then the amĆ plification will become unstable as the number of evanescent waves included increases. Using an low SNR value (5dB for example) means that the evanesĆ cent waves are taken into account but they are so attenuated that the improveĆ ment in resolution is negligible. The default value of 15dB provides the best compromise in terms of resolution and amplification. When the Wiener filter is used, the pressure image needs to be multiplied by a two-dimensional window. As is the case with a single FFT, the observed presĆ sure must be `periodic' within the observed hologram. If this is not the case, then truncation errors occur as with a single FFT. These truncation errors maniĆ fest themselves as ghost sources at the borders of the observed area. Two windows are used

Part II

Acoustics and Sound Quality

125

Chapter 8

Acoustic holography

The rectangular window, which does not modify the pressure image. In case of a rectangular window, only propagating waves are included in the calculations resulting in a resoluĆ tion equivalent to an intensity measurement. The so-called Ad Hoc window For a time signal, the FFT algorithm takes the time signal and duplicates it from minus to plus infinity. If the amplitude of the measured time signal differs beĆ tween the start and the end of the window, a discontinuity occurs during this multiplication introducing an error in the FFT algorithm. This can be corrected using a Hanning window. Holography used a double FFT so the AdHoc winĆ dow is used, which is basically a two dimensional Hanning window thus reĆ moving discontinuities in the both the x and y directions. The one-dimension Ad Hoc window (W) would be: When

N  1 (1  )  I  N  1 (1  ) 2 2 Wā[I]
1

When

I  N  1 (1  ) 2 W[I]
0.5  0.5 cos

When

I  N  1 (1  ) 2 W[I]
0.5  0.5 cos

2
I  N 2 1 (1  )  N  .N 

2
I  N 2 1 (1  )  N  .N 

Derivation of other acoustic quantities If we know how the plane waves propagate, we can calculate the pressure field in any parallel plane, by adding the contributions of all plane waves. This will be correct only if all acoustic sources are on the same side of both planes : correct calculation plane measurement plane

ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ ÄÄÄÄÄÄ 126

correct calculation plane

incorrect

The Lms Theory and Background Book

Acoustic holography

Knowing the pressure field on the parallel plane, it is possible to calculate the particle velocity and eventually the intensity on this plane. The particle velocity (V) will be known if the pressure differential can be deterĆ mined -which is the case with Acoustic holography since the pressure can be measured at r and (r + r) Pā(r)
fā(P(r), P(r  ādr))

V


Eqn 8-7

j P(r)

ck

Once the pressure and the velocity are known then the intensity is just the product of the two. I
PāV

Part II

Acoustics and Sound Quality

Eqn 8-8

127

Theory and Background

Part III Time data processing Chapter 9 Statistical functions . . . . . . . . . . . . . . . . . . . . .

129

Chapter 10 Time frequency analysis . . . . . . . . . . . . . . . . .

139

Chapter 11 Resampling . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

Chapter 12 Digital filtering . . . . . . . . . . . . . . . . . . . . . . . . .

163

Chapter 13 Harmonic tracking . . . . . . . . . . . . . . . . . . . . . .

193

Chapter 14 Counting and histogramming . . . . . . . . . . . . .

128

203

Chapter 9

Statistical functions

Descriptive statistics provide information that characterizes sets of data. This chapter gives a very brief summary of a variety of some statistical functions.

129

Chapter 9

Statistical functions

Minimum, maximum, range and extremum These functions are shown in Figure 9-1 and described below. x

maximum + extremum

range

t minimum

real value

Figure 9-1

N
t

absolute value

Minimum, maximum, range and extreme of a function

Minimum This is defined as the lowest value contained within the specified range of valĆ ues. Maximum This is defined as the highest value contained within the specified range of valĆ ues. Range The range is the difference between the minimum and maximum values. Extremum The extreme is the highest absolute value contained within the specified range. It is equal to the maximum when the absolute value of the maximum is greater than the absolute value of the minimum, and is equal to the minimum value otherwise.

Sum This is the summation of all the (N) values within the frame Sum


N1

 Ă xj

Eqn. 9-1

j
0

Integration This is the area under the curve of values, found by multiplying the half of the sum of the values by the time increment.

130

The Lms Theory and Background Book

Statistical functions

x j1 xj

area


N2

 ĂĂ xj 2xj1Ă t

Eqn. 9-2

j
0

t

Root mean square The root mean square, also called effective value, is given by

RMS




Ă1 N

N1

 Ă x2j

Eqn 9-3

j
0

where N is the number of samples. Its energy content is equivalent to that of the original time series.

Crest factor The crest factor is given by |max  min| 2ĂRMS

Eqn 9-4

The crest factor provides a measure of the ``spikeness'' in the data. A sine sigĆ nal has a crest factor of 1.4. A random signal has a crest factor of about 3 or 4. A short spike will yield a high crest factor.

Mean The mean of a set of data values (x) estimates the central value contained withĆ in the set. It is defined as 

x
1 N

Part III Time data processing

N1

 Ă xj

Eqn 9-5

j
0

131

Chapter 9

Statistical functions

where N is the number of samples. The mean is not the only parameter which characterizes the central value of a distribution. An alternative is the median. The mean and the median both provide information on the average or central value of the data. The choice of the most suitable one to use depends on the skewness described on page 134.

Median The median of a probability function p(x) is the value for which larger and smaller values of x are equally probable: x med



 Ă p(x)ĂdxĂ
 Ă p(x)ĂdxĂ
12



Eqn 9-6

xmed

For discrete data, the median is defined as the middle value of the data samples when they are arranged in increasing (or decreasing) order. When N is odd, the median is

Eqn 9-7

x med
x N1 2

Thus half the values are numerically greater than the median and half are smaller. When N is even, the median is estimated as the mean of the two unique central values.

x med


x N1  x N 2

2

2

Eqn 9-8

The mean and median both provide information on the average or central valĆ ue of a set of data. Which is the most suitable one to use in a particular circumĆ stance depends on the skewness of the data. Skewness is illustrated in Figure 9-2.

132

The Lms Theory and Background Book

Statistical functions

p(x)

p(x)

p(x)

x (a) mean = median

Figure 9-2

Negative skewness

Positive skewness

Symmetrical data no skew

x (b) mean > median

x (c) mean < median

Symmetrical and skew data distributions.

Skewness refers to the shape of the distribution about the central value. PerĆ fectly symmetrical data has no skew. Data distributions where there is a small number of extremely high values are said to exhibit positive skew. Those with a few extremely low values show negative skew. The mean is more influenced by such extreme values than the median, but can be used with confidence if the skewness lies within the range -1 to 1. For the calculation of skewness see Equation 9-13 below.

Percentiles The median can also be expressed as the 50th percentile since it represents the value where 50% of all the values in the data set are below it and 50% lie above it. It is also possible to compute the 10th, 25th, 75th and 90th percentiles. The nth percentile of a probability function p(x) is the value at which n% of the values in the set are smaller then the percentile value. So 10% of the values are smaller than the 10th percentile and 90% are larger.

Variance and standard deviation Further information on the range of values in a distribution can be obtained by determining how much the data values vary from the mean value. So the variĆ ance is given by varĂ(x 0, ..., x N1)
1 N1

N1

 xj  x 2

Eqn 9-9

j
0

and as such can also be regarded as the second order moment of a distribution. The standard deviation is defined as the square root of the variance:

Part III Time data processing

133

Chapter 9

Statistical functions

Ă(x 0, ..., x N1)
ĂvarĂ(x 0, ..., x N1)

Eqn 9-10

The standard deviation is in the same units as the original measurement.

Mean absolute deviation It is not uncommon, in real life, to be dealing with a distribution whose second order moment does not exist (i.e. is infinite). In this case, the variance or stanĆ dard deviation is useless as a measure of the data width around its central valĆ ue. This can occur even when the width of the peak looks perfectly finite to the eye. A more robust estimator of the width is the average deviation or mean absolute deviation, defined by: ADevĂ(x 0, ..., x N1)
1 N

N1

 Ăxj  x Ă

Eqn 9-11

j
0

Extreme deviation The extreme deviation is given by maxĂ(max  mean,Ă mean  min) 

Eqn 9-12

The extreme deviation is similar to the crest factor, except that it is referenced to the mean and will therefore follow data which drifts away from zero.

Skewness Skewness was illustrated in Figure 9-2. It characterizes the degree of asymmeĆ try of the distribution around its central value. It is defined as  3

x j  x ! skewĂ(x 0, ..., x N1)
1  Ă   N  j
0 N1

Eqn 9-13

The skewness is a unitless parameter known as the third order moment of a distribution.

134

The Lms Theory and Background Book

Statistical functions

Even if the estimated skewness is other than zero, it does not necessarily mean that the data is in fact skewed. You can have confidence in the skewness only when the estimated skewness is larger than the standard deviation on this estiĆ mated parameter (Eqn 9-13). For the idealized case of a normal (Gaussian) disĆ tribution, the standard deviation on the estimated skewness is approximately 6N . In real life it is good practice to place confidence in skewness only when the estimated value is several times as large as this.

Kurtosis One further characteristic of a distribution can be obtained from the kurtosis of a function. This is also a unitless parameter that measures the relative sharpĆ ness or flatness of a distribution relative to a normal or Gaussian one. This is illustrated in Figure 9-3. p(x)

p(x)

normal distribution

Figure 9-3

x

p(x)

positive kurtosis

x

negative kurtosis

x

Distributions with positive and negative kurtosis compared to a normal distribution.

The kurtosis is defined as  4  N1 x  x  !! j  1  kurtĂ(x 0, ..., x N1)Ă
Ă $Ă Ă Ă   Ă %3  N j
0  

Eqn 9-14

The term -3 is necessary, so that a Gaussian distribution has a kurtosis of zero. The kurtosis is the fourth order moment of a distribution and is a unitless paĆ rameter. A positive value indicates that the distribution has longer tails than the Gaussian distribution, while a negative value indicates that the distribution has shorter tails. The standard deviation of (Eqn 9-14) is 24N, for the idealized case of a norĆ mal (Gaussian) distribution. However, the kurtosis depends on such a high moment, that there are many real-life distributions for which the standard deviation of equation 9-14 is effectively infinite.

Part III Time data processing

135

Chapter 9

Statistical functions

Note!

Higher order moments (skewness and kurtosis), are often less robust than lower order moments which are based on linear sums. (It is possible that the calculation of the skewness or kurtosis generates an overflow.) They must be used with caution.

Markov regression This function provides you with a measure of the likelihood of one data value within a set being similar to another. It is based on the circular autocorrelation R(.) of a set of data. This calculates the correlation between one particular value and a value displaced by a certain lag, as illustrated below.

lag

lag

The circular correlation takes the last shifted value and wraps it to the start. The circular correlation for a lag of 1 data sample is given by

 N2  xjĂxj1Ă x0ĂxN1  j
0 

R(1)
Ă

Eqn 9-15

The circular correlation for a lag of (0) is given by

R(0)


N1

 x2j

Eqn 9-16

j
0

The Markov regression coefficient is the ratio of these two quantities MarkovĂregressionĂcoefficient


136

R(1) R(0)

Eqn 9-17

The Lms Theory and Background Book

Statistical functions

This function can therefore take values between 0 (very low correlation) and 1 (high similarity). It approaches 1 for a narrow or filtered band and 0 for broadĆ band signals. It provides an indication therefor of how much a broadband sigĆ nal has been filtered.

Part III Time data processing

137

Chapter 10

Time frequency analysis

The objective of a time-frequency analysis is to examine the spectral (frequency) contents of a signal when this is varying in time. This chapter provides very brief account of the background theory reĆ lated to this type of analysis. Introduction to the theory Linear representations Quadratic representations

139

Chapter 10 Time frequency analysis

10.1

Introduction A great deal of physical signals are non-stationary. Fourier analysis establishes a one-to-one relationship between the time and the frequency domain, but proĆ vides no time localization of a signal's frequency components. Whilst an overĆ all representation of all frequencies that appeared during the observation periĆ od is presented, there is no indication as to exactly at what time which frequencies were present. Time-frequency analysis methods describe a signal jointly in terms of both time and frequency. The aim is to find a distribution that determines the portion of the signal's energy which lies in a particular time and/or frequency range. In addition these distributions might or might not satisfy some other interesting mathematical properties, such as the marginal equations". The instantaneous power of a signal at time t is given by |s(t)| 2

= Energy or intensity per unit time at time t

The intensity per unit frequency is given by the squared modulus of the Fourier transform S() |S()| 2

= Energy or intensity per unit frequency at frequency 

The joint function P(,t) should represent the energy per unit time and per unit frequency P(, t)

=

Energy or intensity per unit frequency (at frequency ) per unit time (at time t )

Ideally summing this energy distribution over all frequencies should give the instantaneous power 

 P(, t)ād
|s(t)|

2

Eqn 10-1



and summing over all time should give the energy density spectrum. 

 P(, t)ādt
|S()|

2

Eqn 10-2



140

The Lms Theory and Background Book

Time frequency analysis

Equations 10-1 and 10-2 are known as the `marginal' equations and in addition the total energy, E 

E


 P(, t)ādtād

Eqn 10-3



should be equal to the total energy in the signal while satisfying the marginals. There are a number of distributions which satisfy equations 10-1 and 10-2 but which demonstrate very dissimilar behavior. In general there are two main classes of time-frequency analysis methods V

linear techniques discussed in section 10.2

V

quadratic techniques discussed in section 10.3.

Part III Time data processing

141

Chapter 10 Time frequency analysis

10.2

Linear time–frequency representations These are representations that satisfy the linearity principle. If x1 , and x2 are signals, then T(t,f) is a linear time-frequency representation if x 1(t) & T x1(t, f) x 2(t) & T x2(t, f) x(t)
c 1x 1(t)  c 2x 2(t) & T x(t, f)
c 1T x1(t, f)  c 2T x2(t, f) Two linear techniques are discussed V

The Short Time Fourier Transform

V

Wavelet analysis

The Short Time Fourier Transform (STFT) A standard method used to investigate time-varying signals is the so-called Short Time Fourier Transform (STFT). This involves selecting a relatively narĆ row observation period, applying a time window and then computing the freĆ quencies in that range. The observation window then slides along the entire time signal to obtain a series of spectra shown as vertical bands in Figure 10-1. time window g(t) t

sliding frame

time t



frequency Figure 10-1



The Short Time Fourier Transform

For a time signal s(t) multiplied by a window function g(t), the Short Time Fourier Transform located at time  is given by

142

The Lms Theory and Background Book

Time frequency analysis



STFT(, )
1 2


e

jtĂs(t)Ăg *(t  )ādt

Eqn 10-4



This is a useful technique if it is possible to select the observation period so that the signal can be regarded as being stationary within that period. There are a whole range of signals however where the frequency contents change so rapidĆ ly that the time period required would be unacceptably small. This technique suffers from a further disadvantage in that the same time winĆ dow is used throughout the analysis and it is this that determines the frequency resolution (
f= 1/T). This fixed relationship means that there has to be a trade off between frequency resolution and time resolution. So, if you have a signal composed of short bursts interspersed with long quasi stationary periods, then each type of signal component can be analyzed with either good time resoluĆ tion or good frequency resolution but not both. An alternative view of the STFT is gained if it is expressed in terms of the Fourier transforms of the signal S() and the window function G(). Equation 10-4 then becomes 

STFT(, )
1 2


 eĄ

jĂS()āG *(  )Ăād

Eqn 10-5



By analogy with the previous discussion this reflects the behavior around the frequency  ``for all times'' as illustrated by the horizontal bands in Figure 10-1. These bands can be regarded as a bank of bandpass filters which have impulse responses corresponding to the window function.

Wavelet analysis A method that provides an alternative for the analysis of non-stationary sigĆ nals, where it becomes difficult to find the right compromise between time and frequency resolution for the analysis window of the STFT is the Wavelet analyĆ sis. In effect, the Fourier transform decomposes the signal using a set of basis funcĆ tions, which in this case are sine waves. The Wavelet transform also decomĆ poses the signal, but it uses another set of basis functions, called wavelets. These basis functions are concentrated in time, which results in a higher time localization of the signal's energy. One prototype basis function is defined, and a scaling factor is then used to dilate or contract this prototype function to arĆ rive at the series of basis functions needed for the analysis.

Part III Time data processing

143

Chapter 10 Time frequency analysis

This brings us to the definition of the Continuous Wavelet transform. If h(t) is the prototype function (basic wavelet) localized in time t0 and frequency 0 then the scaled versions (wavelets) are given by

h a(t)


1 Ă h t

|a| a

Eqn 10-6

where a is the scale factor given by 0 / The Continuous Wavelet Transform CWT is given by 

CWT(a, t)


1

|a|

 s()āh  a t ād

Eqn 10-7



where  is the time localization. A disadvantage of the STFT is that it uses a single analysis window of constant width. The result is that there is a fixed relationship between the frequency and time resolutions. Improving one could only be achieved at the cost of the other. Mapping this onto the time/frequency plane results in a fixed grid as shown on Figure 10-2(a). The use of the scaling factor to dilate or contract the basic wavelet results in an analysis window that is narrow at high frequencies and wide at low frequenĆ cies. Figure 10-1 likens the STFT to a series of constant width bandpass filters. Using this concept again, the wavelet transform can be considered as a bank of constant relative Bandwidth filters. f
c f where c is a constant. This is illustrated in Figure 10-2(b) where by allowing both the frequency and time resolutions to vary, a multi-resolution analysis is possible.

144

The Lms Theory and Background Book

Time frequency analysis

time

time

(a) STFT frequency Figure 10-2

(b) Wavelet analysis frequency

Mapping of the time/frequency plane

This is in fact a very natural way to analyse a signal. Low frequencies are pheĆ nomena that change slowly with time so requiring a low resolution in this doĆ main. In this situation, a good time resolution can be sacrificed for a high freĆ quency resolution. High frequency phenomena vary rapidly with time which then becomes the important dimension, so under these conditions wavelet analĆ ysis increases the time resolution at the cost of frequency. This type of analysis is also very closely related to the human hearing process, since the human ear seems to analyse sounds in terms of octave bands.

Part III Time data processing

145

Chapter 10 Time frequency analysis

10.3

Quadratic time–frequency representations Whilst linearity is a desirable property, in many cases, it is more interesting to interpret a time-frequency representation as a time-frequency energy distribuĆ tion which is a quadratic signal representation. This type of time-frequency representations will exhibit many desirable mathematical properties, but it is important to investigate the consequences of the bilinearity principle. x(t) & T x(t, f) y(t) & T y(t, f) Eqn. 10-8 z(t)
c 1x(t)  c 2y(t) & T z(t, f)
|c1| 2T x(t, f)  |c 2| 2T y(t, f)  c 1c 2 * T xy(t, f)  c 2c 1 * T yx(t, f) The first two terms in this result can be seen as signal terms", and the last two terms as the interference terms". These interference terms are necessary to satĆ isfy mathematically desirable properties like the marginal equations", but they often make interpretation of the results difficult. The interference terms can be recognized by their oscillatory nature, and differĆ ent so -called smoothing" techniques can be used to reduce their effect. This, however, leads us to a new tradeoff; that of a reduction of interference terms against time-frequency localization. The spectral smearing effect of the smoothing windows will disperse the signal's energy in the time-frequency plane, thereby reducing the time-frequency localization of all signal compoĆ nents. Two examples of quadratic time-frequency representations are the spectrogram and the scalogram. spectrogram = |STFT|2 scalogram = |WT|2 which are the two energy-counterparts of the Short-Time Fourier Transform (STFT) and the Wavelet Transform (WT) respectively. The interference terms for these representations only exist where different signal components overlap. Hence if the signal components are sufficiently far apart in the time-frequency plane, the interference terms will be essentially zero. While neither of these representations satisfies the marginal equations, this is not of great concern for a qualitative energy localization assessment. For an adequate interpretation of time-frequency analysis results, it is often good practice to use several techniques (STFT or WT together with a quadratic method), which makes it possible to distinguish the signal components" from the interference terms".

146

The Lms Theory and Background Book

Time frequency analysis

The Wigner–Ville distribution The Wigner-Ville distribution is 

W(, t)
1 2


 s * t  2ā Ăe

jĂs



t  2ā ād

Eqn 10-9

where  is the local time. In terms of the spectrum it is 

W(, t)
1 2


 S *   2ā Ăe



  2ā ād

jtĂS

Eqn 10-10

where  is the local frequency. This distribution satisfies the marginals and is real. In addition, time and freĆ quency shifts in the signal cause corresponding shifts in the distribution. Many of its characteristics can be understood by considering the fact that in equation 10-9 at any point (t) a section of data prior to this point is being multiĆ plied with a section following this point and the results summed. This can be visualized by imagining that the segment to the left is folded over on top of the segment to the right. Where there is an overlap there will be a product and therefor a value for the distribution. The diagram below demonstrates that for a signal only starting a time (tstart ), all points to its left have value zero resulting in a distribution with the same value. The same will apply at the end point (tend ), tstart

tend

Thus one characteristic of the Wigner Ville distribution is that for a signal of finite duration the distribution is zero up to the start and beyond the end. The same can be said when considering the frequency version which means that for a band limĆ ited signal, the Wigner Ville distribution will be zero outside of that range. The same manoeuvre can be used to see why the reverse is true if at some point the signal level drops to zero. Consider the situation illustrated below.

Part III Time data processing

147

Chapter 10 Time frequency analysis

t0

At a point where the signal itself is zero (t0 ), multiplying the section to the left by the section to the right results in a non-zero value. In general it can be said that the Wigner distribution is not zero when the signal is. This unwelcome characĆ teristic makes it difficult to interpret, especially when analyzing signals with many components. The same mechanism accounts for noisiness that can be seen in the distribution in places where it is not present in the signal as shown below. t1

t2

When evaluating the distribution at point (t1 ) the overlapping sections will not include the noise, but even at point (t2 ) where there is no noise in the signal, it will already influence the distribution. Noise will be spread over a wider periĆ od than occurred in the actual signal therefore. The same reasoning can be used to explain the appearance of the interference terms along the frequency axis. This is especially so when a signal contains multiple frequency components at the same moment in time, which will result in interference terms at a frequency mid way between the frequencies of the different components. As mentioned above, these terms can easily be recogĆ nized by their oscillatory nature and smoothing techniques can reduce their efĆ fect. Some possible smoothing techniques are discussed below.

Generalization A generalization of the Wigner-Ville distribution leads to a whole class of timefrequency representations, with as main desirable mathematical property their invariance against operations like time shift, frequency shift or time/frequency scaling. This means that a shift in time or frequency of the signal leads to an equivalent shift of the time-frequency representation of that signal, or that scalĆ ing the signal leads to a corresponding scaling of the time-frequency represenĆ tation.

148

The Lms Theory and Background Book

Time frequency analysis

This more general class of time/frequency representations are defined as folĆ lows T x(t, f)


   (t  t, f  f)W ā(t, f)Ădtdf T

x

Eqn. 10-11

t f

where Wx (t',f') is the Wigner-Ville distribution of the signal x(t), and where ΨT is the kernel function". It is the choice of this kernel function that determines the basic properties of each specific time-frequency representation derived from this general definition. The kernel function can also be seen as a smoothĆ ing function applied to the Wigner-Ville distribution. Typical examples of techniques that can be defined in this framework are Spectrogram where the kernel = Wigner distribution of the analysis window. Smoothed Pseudo-Wigner Distribution (SPWD) where the kernel =separable smoothing function with independent smoothĆ ing spread in time- and frequency domain. Pseudo-Wigner Distribution (PWD) the same as the SPWD, but with no smoothing along the frequency axis. This can also be considered as short-time Wigner distribution". Choi-Williams Distribution (CWD) where the kernel = exponential smoothing function. The class of shift-invariant representations (time- and frequency shifting) is also called Cohen's class, and examples of representations belonging to that class are the spectrogram, the Wigner-Ville distribution, PWD, SPWD, ... The class of time shift/time scale invariant representations is also known as the Affine class, and examples of representations belonging to this class are the scaĆ logram, the Wigner-Ville distribution, CWD, ...

Part III Time data processing

149

Chapter 10 Time frequency analysis

10.4

References

Books Time-frequency analysis : Leon Cohen - Prentice Hall - 1995 - 299 pp. - ISBN 0-13-594532-1

Papers Linear and Quadratic Time-frequency Signal Representations : F. Hlawatsch, G.F.Boudreaux-Bartels (IEEE SP Magazine, April 1992) Time-frequency distributions - A review : Leon Cohen (Proc. of IEEE, July 1989) Wavelets and signal processing : O. Rioul, M. Vetterli (IEEE SP Magazine, October 1991) Time-frequency analysis applied to door slam sound quality problems. : H. Van der Auweraer, K. Wyckaert, W. Hendrickx (Journal de physique IV, May 1994)

150

The Lms Theory and Background Book

Chapter 11

Resampling

This chapter is concerned with both fixed resampling and adaptive (or synchronous) resampling. It discusses the general principles inĆ volved in both of these processes and contains a reading list for furĆ ther information. Fixed resampling Adaptive resampling

151

Chapter 11 Resampling

11.1

Fixed resampling The process of converting a signal that has been sampled at a particular rate to one that is sampled at a different rate is known as resampling. Resampling may be necessary for a number of reasons. A DAT recorder, for exĆ ample, samples a signal at a rate of 48000 samples per second. If the signal has a Bandwidth of only 200Hz then 500 samples a second would be adequate and as a consequence far more data than is needed to describe the signal exists. In this situation the sample rate could be decreased, a process which is referred to as decimation or downsampling. On the other hand, while a critically sampled signal may contain all the inforĆ mation to adequately describe the frequency contents of the signal, it may not look good, or be easy to interpret, in the time domain.

Increasing the sampling rate will generate a signal which has identical spectral contents but a much better defined time waveform. When the resampling inĆ volves an increase in the sampling rate it is referred to as interpolation or upsamĆ pling.

A further instance of when a specific sampling rate is required, is when it is reĆ quired to replay a signal through a D/A convertor. It may well be that the sigĆ nal must possess the very specific sampling rate supported by the D/A converĆ tor. This section considers the theoretical background to the process of digital reĆ sampling and the factors that must be taken into account to realize resampling and achieve the required accuracy of results. It should be noted however that the contents of this document are by no means a comprehensive treatment of this subject. For a more thorough understanding of this subject you should reĆ fer to the reading list given at the end of the section and in particular to referĆ ences [3] and [4].

152

The Lms Theory and Background Book

Resampling

11.1.1

Integer downsampling Integer downsampling by a factor n effectively means retaining every nth point of the source data. However it is necessary to take measures to avoid aliasing problems when doing this. The example below shows the effects of downsamĆ pling by a factor of 13, when the original number of samples per period was 16. Sampling a signal at a rate lower than 2 points per period of the highest freĆ quency in the signal will give rise to erroneous results.

To avoid aliasing, due to the resampling process, it is necessary to ensure that the signal does not contain frequencies that are any higher than can be deĆ scribed by the reduced sample rate. The use of a low pass filter will achieve this. To illustrate this, consider the example of downsampling by a factor of 5 described below. The original signal was sampled at 1kHz, implying a Bandwidth of 500Hz. It contains 2 spectral components, one at 8Hz and another at 325 Hz. 1.0 0.5 8Hz

325Hz

500Hz Bandwidth

Downsampling by a factor of 5 will reduce the sample rate to 200Hz and the Bandwidth to 100Hz. It is first necessary to apply a low pass filter to limit the spectral content of the data to the 100 Hz Bandwidth. This will remove the higher frequency component leaving a time domain signal containing 125 points per period for the remaining 8 Hz component. 1.0 0.5 8

100 Bandwidth

Part III Time data processing

É É É É É

325

frequency (Hz)

125 points per period

153

Chapter 11 Resampling

The downsampling by a factor 5 is performed by taking every 5th point. 1.0 0.5 8

100 Bandwidth

frequency (Hz)

Not applying the filter would result in the following. The 325 Hz component will fold to 75Hz in the 100Hz Bandwidth and as a consequence the result is heavily distorted. 1.0 0.5

8Hz

11.1.2

75Hz 100Hz Bandwidth

325Hz

Integer upsampling Integer upsampling by a factor n involves inserting (n-1) data points between the original measured ones. Normally the inserted points will have a value of zero, and it is then necessary to apply an appropriate filter to remove the harĆ monics introduced by the process.

The trace shown here is upsampled by a factor 4, which means that 3 zeros are added between each of the existing data points. The result is that in the time doĆ main the signal looks highly distorted.

It can be proven that the spectrum of the upsampled signal consists of the origiĆ nal one plus a mirrored version of it at all higher frequencies.

154

The Lms Theory and Background Book

Resampling

low pass filter

The `distortion' introduced by inserting zeroes, can therefore be filtered out by a properly designed low pass filter which will retain just the spectral contents of the original signal Bandwidth. The improvement in the time domain representation of a signal by upsampling is illustrated below for the case of a critically sampled sine wave. The sample rate is just greater than 2 points per period so the Nyquist criterion is satisfied. Although the time domain representation is poor, there is enough information for an accurate representation in the frequency domain. Upsampling by a facĆ tor 10 will make the time domain description of the signal more accurate.

f bw

f

bw

The resulting signal has identical spectral contents to the original. The inĆ creased number of points per cycle provides a much improved time domain deĆ scription of the waveform.

Part III Time data processing

155

Chapter 11 Resampling

11.1.3

Fractional ratios Resampling by a non-integer ratio can be realized by a combination of upsamĆ pling and downsampling. So downsampling by a factor of 2.5 can be achieved by first upsampling by a factor 2 then downsampling by a factor 5. The order in which these two processes are done is very important if the original signal content of interest is to be preserved. Consider a signal sampled at 2kHz and which contains signals up to 300 Hz. A new sampling rate of 800Hz is required, representing a downsampling by a facĆ tor of 2.5

If the signal is first downsampled by a factor of 5, a filter at 200Hz is required. As a result, all the signal content beĆ tween 200 and 300Hz will be eliminated, and the subsequent upsampling will not of course be capable of restoring this.

5

2

The correct procedure is to first upsample to 4kHz (a Bandwidth of 2kHz). A lowpass filter set at 1kHz will retain the original spectral content. The next stage is to downsample by a factor of 5 with a low pass filter at 400 Hz thus maintaining the original frequencies of up to 300Hz.

2

5

When a non-integer resampling factor is required, the software determines the optimum ratios and the sequence of resample operations required to achieve the desired sample rate conversion.

156

The Lms Theory and Background Book

Resampling

11.1.4

Arbitrary ratios Some resampling requirements can not be easily realized by a simple combinaĆ tion of an upsampling and a downsampling. For some ratios, even though they can be expressed as a fraction one needs an extremely high intermediate upĆ sampling ratio. The process imposes a heavy computational load and the result is numerically not well conditioned. Consider for instance a measurement at 8192 samples per second that is to be resampled to 8000 Hz for replay on digital audio hardware. This can theoretiĆ cally be realized by upsampling by a factor 125 followed by a downsampling with a factor 128, but this is computationally extremely costly. In this situation another strategy is used. Consider the signal shown below which was originally sampled at a rate indicated by the white circles. The reĆ quired sample rate is indicates by the filled circles. The new sample rate is not an integer ratio to the original.

The first stage is to upsample by a relatively high factor (a). This factor is known as the `Upsampling factor before interpolation' parameter and the deĆ fault value used is 15. The resulting sample rate is indicated by the squares. The second stage then involves performing a linear inĆ terpolation on the upsampled signal to arrive at a new sample rate that is an integer multiple (b) of the target frequency. This introduces an error  which will be small as long the source trace is upsampled at a high enough ratio. The maximum distortion that can occur with the upsampling factor is indicated by the softĆ ware.



This error is indicated in the form of the `SDR' (Signal to Distortion Ratio). It depends on the `Upsampling factor before interpolation' parameters and the filter 's cut-off frequency as shown below: SDR=10log10 (80*(100 R / ( <cut-off in percent>  ))) where R = Upsampling factor before interpolation cut-off in % = the cut off frequency as a % of the Nyquist frequency.

Part III Time data processing

157

Chapter 11 Resampling

The final stage in this process is to downsample by this integer factor (b) to the required rate. It is also possible that the downsampling is achieved directly by the interpolation process itself as long as the downsampling rate being perĆ formed is lower that the preceding upsampling rate (a).

158

The Lms Theory and Background Book

Resampling

11.2

Adaptive resampling Adaptive or synchronous resampling enables you to resample a signal such that its characteristics can be examined in a different domain. A well known mechanical application is the extraction of ``order-related" phenomena of enĆ gine vibrations based on the measurement of the rotation speed of one of its components. Phenomena which are very difficult to analyze or interpret in one domain, become clear and obvious in another. For synchronous averaging, for example, it is essential that repetitive phenomeĆ na occur at the very same instant for the different signal sections that are averĆ aged. Using the synchronous resampling technique, the data can be transĆ formed into that particular domain in which the phenomena are indeed repetitive. In the same way as the Fourier transform presents the contents of time domain data in the frequency domain, it converts angle domain data to the order doĆ main. Just as something that happens twice every second has a frequency of 2 Hz, something that occurs twice every cycle is related to order 2. Consider the example of measurements taken on an engine at a supposedly constant rpm. Even very slight variations in rpm will result in a frequency domain represenĆ tation where the related spectral components are sharp for the low orders, but become smeared out for higher frequencies. The small RPM variations, lead to leakage errors in the frequency domain. For applications where there is a need to investigate higher order phenomena (such as gear box analysis for example), such smearing makes it very difficult to discriminate order from resonance components. Transforming such data to the order domain, will result in all orders being clearly shown, but any resoĆ nance phenomenon present will be smeared out. The frequency and order doĆ main representations are therefore complementary to one another and useful information can be obtained in the domain most suited for analysis. Adaptive resampling facility enables you to convert from one domain to another.

Implementation example This example below illustrates the procedure involved in converting from the time to the angle domain. The principle can be used to convert between any two domains. Your original time signal must be measured in conjunction with a tracking sigĆ nal. This is most likely to be a tacho signal; a pulse train, that can be converted to an rpm /time function and then integrated to obtain an angle/time function.

Part III Time data processing

159

Chapter 11 Resampling

angle Ordinate time

Abscissa

In the case of a transformation from the time domain into the angle domain, the required (constant) resolution in the angle domain (
) defines the time interĆ vals at which data samples of the vibration measurement should be available.


angle measured points required points
t1


t2


t3

time

The most appropriate resolution (
) is based on the minimum slew rate which must be coped with. When sampling in the time domain the time increment is the reciprocal of the sampling frequency. 1
T Fs So according to the Nyquist criterion, information is available up to Fs/2. Adaptive resampling conforms to the same rules: so if you do not have enough samples then information is lost, while if you use too many samples then the processing effort is unnecessarily increased. It is necessary to determine the
angle which corresponds to the required Fs in the time/frequency domain. Adaptive resampling uses a varying time increĆ ment if the angle/time relationship is non linear. Data loss will occur first at the lowest rpm values (slew rate) and the aim is to determine the threshold angle
 between over and under sampling.




d

dt

min

Fs




rpm min Fs

So for example if the minimum slew rate (d/dt) is 500 rpm and the sample frequency is 2000Hz, then the threshold angle will be

160

The Lms Theory and Background Book

Resampling


 = 500/2000 = 0.25{ x (360/60) } = 1.5 degrees Using an angle increment less than this value will yield more data points in the angle domain without any gain in information thus representing excessive processing. Using a higher increment value will result in a loss of information in the lower rpm ranges which will not be recovered if the data is transformed back to the original domain. From the required point in the angle doĆ main α1, the α(t) function is consulted to find the corresponding time instant (t1). The value of the measured time signal at that instant (y'1) must then be determined as the value for the angle position. This is repeated for every value in the angle doĆ main. Depending on the resolution of the origiĆ nal signal and the relation between both domains, interpolation may be required as illustrated here. In order to maintain the dynamic nature of the signal, it is esĆ sential to preserve its spectral contents, so the signal is first upsampled before interĆ polation. The last interpolation ratio (and thus the corresponding upsampling factor) is govĆ erned by the actual local distance beĆ tween the available and required data samples. As a final stage the constructed angle-domain signal needs to be reĆ sampled (usually down-sampled) to match the desired angle resolution.

α α1 t

t1 y(t)

t1

t, ∆t

t1

t, ∆t’

y’(t)

y’1 y'(')

y’1



y''()

y’’ 1




downsampling



α


Preservation of the spectral characteristics during the up-sampling, interpolaĆ tion, and downsampling steps indicated above, requires the correct application of these procedures as well as a low-pass finite impulse response (FIR) filters with enough suppression in the stop-band, a low ripple in the pass-band, and yet optimal speed performance for acceptable computing times. The principles of resampling are discussed in section 11.1.4.

Part III Time data processing

161

Chapter 11 Resampling

11.3

162

References [1]

A. V. Oppenheimer and R.W. Schafer Digital Signal Processing Prentice Hall 1975

[2]

L.R. Rabiner and B. Gold Theory and Application of Digital Signal Processing Prentice Hall 1975

[3]

R.E. Crochiere and L.R. Rabiner Multirate Digital Signal Processing Prentice Hall 1983

[4]

J.G. Proakis and D.G. Manolakis Digital Signal Processing: Principles Algorithms and Applications MacMillan Publishing 1992

The Lms Theory and Background Book

Chapter 12

Digital filtering

Filtering is most often used to enhance signals by removing unĆ wanted components. This chapter describes the theoretical basis used in the design of digital filters. Basic definitions related to digital filtering Types of filters and their design Analysis of filters Application of filters This is by no means a comprehensive text and aims just to give some insight into the subject. A reading list is appended at the end of the chapter.

163

Chapter 12 Digital filtering

12.1

Basic definitions relating to digital filtering A linear time–invariant system Discrete time signals are defined for discrete values of time i.e. when t= n T. A general way of describing a sequence of discrete pulses of amplitude a(n) as ilĆ lustrated below is given in equation 12-1. a(n)

n

0





{a(n)}


m


a(m)Ău 0Ă(n  m)

Eqn 12-1

where u0 is the unit impulse. A discrete-time system is an algorithm for conĆ verting one sequence into another as represented below. In this case the input x(n) is related to the output y(n) by the specific system . y(n)

x(n)

yā(n)
Ă[xā(n)]

Eqn 12-2

A linear system implies that applying the input ax1 +bx2 will result in the output ay1 +by2 where a and b are arbitrary constants. A time-invariant system implies that the input sequence x(n-n0 ) will result in the output y(n-n0 ) for all n0 . From equation 12-1 the input x(n) to a system can be expressed as x(n)






m


164

x(m)Ău 0Ă(n  m)

Eqn 12-3

The Lms Theory and Background Book

Digital filtering

If h(n) is defined as the impulse response of a system which is the response to the sequence u0 (n), then by time invariance h(n-m) is the response to u0 (n-m). By linearity, the response to sequence x(m)u0 (n-m) must be x(m)h(n-m). Thus the response to x(n) is given by 

y(n)




x(m)ĂhĂ(n  m)

m









h(m)ĂxĂ(n  m)

Eqn 12-4

m


Equation 12-4 is known as the convolution sum and y(n) is known as the conĆ volution of x(n) and h(n), designated by x(n) * h(n). Thus for a linear time inĆ variant (LTI) system a relation exists between the input and output that is comĆ pletely characterized by the impulse function h(n) of the system. LTI system

x(n)

h(n)

y(n)

Stability and Causality The constraints of stability and causality define a more restricted class of linear time-invariant systems which have important practical applications. A stable system is one for which every bounded input results in a bounded outĆ put. The necessary and sufficient condition for stability is 



|h(n)|  

Eqn 12-5

n


A causal system is one for which the output for any n=n0 depends only on the input for n  n0. A linear time-invariant system is causal if and only if the unit sample response is zero for n<0, in which case it may be referred to as a causal sequence. Difference equations Some linear time-invariant systems have input and output sequences that are related by a constant coefficient linear difference equation. Representing such systems in this way, can provide means of making them realizable and the apĆ propriate difference equation reveals useful information on the characteristics of the system under investigation such as the natural frequencies, their multiĆ plicity, the order of the system, frequencies for which there is zero transmission ...

Part III Time data processing

165

Chapter 12 Digital filtering

The general form of an Mth order linear constant coefficient difference equation is given in equation 12-6. M

M

i
0

i
1

y(n)
 b iĂx(n  i)ĂĄ   a iĂy(n  i)Ă

Eqn 12-6

An example of a first order difference equation is given by y(n)
 a 1y(n  1)  b 0x(n)  b 1x(n  1)

Eqn 12-7

which can be realized as follows. x(n–1)

Delay

b1 y(n)

x(n) b0 y(n–1) –a1

Delay

Delay

represents a one sample delay. A realization such as this where sepaĆ rate delays are used for both input and output is known as Direct form 1. More detailed information on filter realizations can be obtained from the references listed at the end of this chapter. The z transform The z transform of a sequence x(n) is given by X(z)






x(n)Ăz n

Eqn 12-8

n


where z is a complex variable. The z transform is a useful technique for repreĆ senting and manipulating sequences. The information contained in the z transform can be displayed in terms of (poles) and zeros. If the poles of the function X(z) fall within a radius R1 where R1  1 then the system is stable.

166

The Lms Theory and Background Book

Digital filtering

In the z plane, the overall representation of a linear time invariant system is given by H(z)


Y(z) X(z)

Eqn 12-9

and H(z) can again be expressed in the general form of difference equations H(z)


a 0  a 1z 1  a 2z 2.Ă.Ă.a Mz M 1  b 1z 1  b 2z 2.Ă.Ă.b Nz N

Eqn 12-10

The frequency response of filters Consider the case when the input to a filter is x(n)= e j n (equivalent to a sampled sinusoid of frequency  ). From equation 12-4 

y(n)




h(m)Ăe Ăj0(nm)

m



e

j 0n





h(m)Ăe j0m

m



x(n)ĂHĂ(ej0)

Eqns 12-11

The quantity H(eā j) is the frequency response function of the filter, which gives the transmission of the system for every value of . This is in fact the z transform of the impulse response function with z=eāj. H(z)| z
ej
H(e j)






h(n)Ăe jn

Eqn 12-12

n


which means that the frequency response of a filter is an important indicator of a system's response to any input sequence that can be represented as a continuĆ ous superposition of input sequences x(n). Relationship between the frequency response and the Fourier transform of a filter The frequency response of a linear time invariant system can be viewed as the Fourier series representation of H(e j) .

Part III Time data processing

167

Chapter 12 Digital filtering

j



H(eā )




h(n)āe jn

Eqn 12-13

n



h(n)
1 2


ā H(eā

j)āe jn




where the impulse response coefficients are also the Fourier series coefficients. Since the above relationships are valid for any sequence that can be summed, the same can apply to x(n) and y(n) and it can be shown that Y(eā j)
X(eā j)ĂH(eā j)

Eqn 12-14

and so the convolution in the time domain has been converted to multiplication in the frequency domain. Discrete Fourier Transform For a periodic sequence of N samples, the Discrete Fourier Transform is given as

H p(k)


N1

 hp(k)āejā(2
N)ānk

Eqn 12-15

n
0

and the DFT coefficients are identical to the z transform of that same sequence evaluated at N equally spaced points around the unit circle. The DFT coeffiĆ cients are therefore a unique representation of a sequence of finite duration. The continuous frequency response can be obtained from the DFT coefficients, by artificially increasing the number of points equally spaced around the unit circle. So by augmenting a finite duration sequence with additional equally spaced zero valued samples the Fourier transform can be calculated with arbiĆ trary resolution.

Finite and Infinite Impulse Response Filters When an impulse response h(n) is made up of a sequence of finite pulses beĆ tween the limits N1 < n < N2 (as shown below) and is zero outside these limits then the system is called a finite impulse response (FIR) filter or system.

168

The Lms Theory and Background Book

Digital filtering

h(n)

N1

N2

n

Such filters are always stable and can be realized by delaying the impulse reĆ sponse by an appropriate amount. The design of FIR filters is described in secĆ tion 12.2.2. A filter (system) whose impulse response extends to either - or + (or both) is termed an infinite impulse response (IIR) filter or system. Design of these filters is discussed in sections 12.2.3 and 12.2.4.

Use of digital filters Digital filters can be used in a range of applications such as anti-aliasing, smoothing, elimination of noise, compensation (equalization), modification of fatigue/damage characteristics. They have some important advantages compared to analog filters high accuracy, consistent behavior and characteristics, few physical constraints, independent of hardware, the signals can be easily used by different processing algorithms.

Part III Time data processing

169

Chapter 12 Digital filtering

12.2

FIR and IIR filter design Filters fall into two distinct categories -the Finite Impulse Response (FIR) filĆ ters and the Infinite Impulse Response (IIR) filters. A comparison of the two categories of filters is given below. Characteristic

FIR

IIR

Stability

these filters are always stable (poles=0)

will be stable if |poles|<1

Phase

linear (important in applications nonlinear such as speech processing)

Efficiency

low the length (nr of taps) must be relatively large to produce an adequately sharp cut off

better lower order required

Round off error sensitivity

low

high

Start up transients finite duration

infinite duration

Adaptive filtering

easy

difficult

Realization

straightforward (direct form)

more critical (direct or cascaded)

There are nine basic designs of filters that are described in this chapter as listed below. FIR Window

see page 174

FIR Multi window

see page 176

FIR Remez

see page 177

IIR Bessel

see page 180

IIR Butterworth

see page 181

IIR Chebyshev

see page 182

IIR Inverse Chebyshev

see page 183

IIR Cauer

see page 183

IIR Inverse design

see page 187

This section begins with an introduction to the terminology used in filter deĆ sign. The following subsections deal with the processes and parameters inĆ volved in each sort of filter mentioned above.

170

The Lms Theory and Background Book

Digital filtering

12.2.1

Filter design terminology Filter characteristics The nomenclature used in describing a (low pass) filter is illustrated in Figure 12-1. pass band ripple

H 

attenuation

stop band ripple pass band

Figure 12-1

stop band transition band

Filter characteristics

The filter design functions operate with normalized frequencies with a unit freĆ quency equal to the sampling frequency. Normalized frequency = and thus lies in the range 0 to 0.5

frequency (Hz) sampling frequency

Angular frequency on the unit circle = Normalized frequency x 2


Linear phase filters The frequency response of a filter has an amplitude and a phase H(eā j)
|H(eā j)|.eā j() For a linear phase, () = - where -
  
. It can be shown that a necessary condition for this is that the impulse response function is symmetric,

Part III Time data processing

171

Chapter 12 Digital filtering

h(n)
h(N  1  n) and in this case = (N-1)/2. This means that for each value of N there is only one value of  for which exactĆ ly linear phase will be obtained. Figure 12-2 shows the type of symmetry reĆ quired when N is odd and even. center of symmetry

center of symmetry

N =11 =5

N =12 =5.5

N odd, even symmetry

N even, odd symmetry

Figure 12-2

Symmetrical impulses for odd and even N

Filter types Several types of filter are provided (some of which are illustrated below) as well as multipoint filters where the required response can be of an arbitrary shape. H 

H  low pass

band pass



H 

H 

high pass

Figure 12-3

band stop

Filter types

In addition it is also possible to design a Differentiator filter and a Hilbert transformer. These can both be designed using the Remez exchange algorithm and they are briefly described here. Differentiator filter Such a filter takes the derivative of a signal and an ideal differentiator has a deĆ sired frequency response of

172

The Lms Theory and Background Book

Digital filtering

H dā()
jĂĂĂĂĂĂ 
  


Eqn 12-16

The unit sample response is


h(n)
1 2


ā H ()āe d

jnd

Eqn 12-17






1 2


ā jāe

jnd





cosn
n Which is an anti symmetric unit sample response. In practice however the ideal case is not required and a pass band will be specified as shown here.

H 



stop band ripple

pass band stop band transition band

Figure 12-4

Characteristics of a differentiator filter

Hilbert transformer This filter imparts a 900 phase shift to the input. The ideal Hilbert transformer has a desired frequency response of H dā()
 jĂĂĂĂĂĂ0   

ĂĂĂ jĂĂĂ 
   0

Eqn 12-18

The unit sample response is


h(n)
1 2


ā H ()āe d

jnd

Eqn 12-19




 1
 2


0






ā jāe jnd 




 0

!  

ā jāe jnd

2 2 Ą sin (ā
n2)

n

Part III Time data processing

173

Chapter 12 Digital filtering

In practice however the ideal case is not required and the desired frequency reĆ sponse of a Hilbert transformer can be specified as Hd () = 1 between the limits l <<u as shown below. H 

l

Figure 12-5

12.2.2

u



Characteristics of a Hilbert transformer

Design of FIR filters Design of an FIR window filter The frequency response of a filter can be expanded into the Fourier series. H(eā j)






h(n)āe jn

Eqn 12-20

n



h(n)
1 2


ā H(eā

j)āe jn




The coefficients of the Fourier series are identical to the impulse response of the filter. Such a filter is not realizable however since it begins at - and is infiĆ nitely long. It needs to be both truncated to make it finite and shifted to make it realizable. Direct truncation is possible but leads to the Gibbs phenomenon of overshoot and ripple illustrated below.



Figure 12-6

174

Gibbs phenomenon due to truncation of the Fourier series

The Lms Theory and Background Book

Digital filtering

A solution to this is to truncate the Fourier series with a window function. This is a finite weighting sequence which will modify the Fourier coefficients to conĆ trol the convergence of the series. Then ^

h(n)
h(n)āw(n)

Eqn 12-21 ^

where w(n) is the window function sequence and h(n) gives the required imĆ pulse response. The desirable characteristics of a window function are d

a narrow main lobe containing as much energy as possible

d

side lobes that decrease in energy rapidly as  tends to
.

The windows supported are listed below. Rectangular This is equivalent to direct truncation. W(n)
1ĂwhenĂ

 (N  1) (N  1) n 2 2

W(n)

= 0 elsewhere -(N-1)/2

(N-1)/2

Hanning This type of window trades off transition width for ripple cancellation. In this case



 (N  1) (N  1) W(n)
  (1  ) cos 2
n ĂĂwhenĂĂ n 2 2 N

= 0 elsewhere  = 0.5 Hamming This has similar properties to the Hanning window described above. The forĆ mula is the same but in this case =0.54. Kaiser The Kaiser window function is a simplified approximation of a prolate spheroiĆ dal wave function which exhibits the desirable qualities of being a time-limited function whose Fourier transform approximates a band-limited function. It displays minimum energy outside a selected frequency band and is described by the following formula

Part III Time data processing

175

Chapter 12 Digital filtering

W(n)




I 0  1  [2n(N  1)] 2 I 0ā



ĂĂĂĂĂĂĂ when

 (N  1) (N  1) n 2 2

Where I0 is the zeroth order Bessel function and  is a constant representing a frequency trade-off between the height of the side lobe ripple and the width of the main lobe. Chebyshev This is another example of an essentially optimum window like the Kaiser winĆ dow, in the sense that it is a finite duration sequence that has the minimum spectral energy beyond the specified limits. The window function is derived from the Chebyshev polynomial which is described below. The Chebyshev polynomial of degree r in x where -1 x  1 is denoted by TrĂ Ă Tr (x). T rā(x) ( cos(r. cos 1ā(x)) and And so

T r1(x)
2.x.T r(x)  T r1(x) T0
1 T r(1)
1 T rā( 1)
( 1) r T 2rā(0)
( 1) r T 2r1ā(0)
0

The window function W(n) is obtained from the inverse DFT of the Chebyshev polynomial evaluated at N equally spaced points around the unit circle.

FIR multi window Filter This allows you to design a filter of arbitrary shape and is suited for narrow band selective filters. It uses the design technique known as frequency samĆ pling. It will be recalled from equations 12-15 that a filter can be defined by its DFT coefficients and that the DFT coefficients can be regarded as samples of the z transform of the function evaluated at N points around the unit circle.

176

The Lms Theory and Background Book

Digital filtering

H(k)


N1

 h(n)āej(2
N)ānk

n
0

h(n)
1 N

N1

 ā H(k)āej(2
N)nk

k
0

H(k)
H(z)| z
āej(2
N)k From these relationships and since e j2
k =1, it can be shown that H(z)


(1  zN) Ą N

N1

 [Ă1  z1H(k) eā j(2
N)ānkĂ]

Eqn. 12-22

k
0

The desired filter specification can be sampled in frequency at n equidistant points around the unit circle, to give the desired frequency response H(k). The continuous frequency response can be obtained by interpolation of these sampled values around the unit circle. The filter coefficients are obtained after applying an inverse FFT on the interpoĆ lated response. The coefficients are tapered smoothly to zero at the ends by multiplying the impulse response by the specified window function.

FIR Remez filter This uses the remez exchange algorithm and the Chebyshev approximation theory to arrive at filters that optimally fit the desired and the actual frequency responses, in the sense that the error between them is minimized. The ParksMcClellan algorithm employed enables you to design an equi-ripple optimal FIR filter. The desired frequency response is expressed as a gabarit which contains a numĆ ber of frequency bands. These bands are interpolated onto a dense grid in a similar way to that described for the multipoint FIR filter design using a winĆ dow described above. The weighted approximation error between the desired frequency response and the actual response is spread evenly across the passbands and the stopbands and the maximum error is minimized by linear optimization techniques. The approximation errors in both the pass and stop bands for a low pass filter are illustrated in Figure 12-7.

Part III Time data processing

177

Chapter 12 Digital filtering

1 1 1

 

2 2

Figure 12-7

0



Approximation errors

The filter coefficients are obtained after applying an inverse DFT on the optiĆ mum frequency response. Weighting For each frequency band the approximation errors can be weighted. This is done by specifying a weighting function W(). Applying a weighting function of 1 (unity) in all bands implies an even distribution of the errors over the whole frequency band. To reduce the ripple in one particular band it is necesĆ sary to change the relative weighting across the bands and in this case to ensure that the band of interest has a relatively high weighting. It is convenient to normalize W() in the stopband to unity and to set it to the ratio of the approxiĆ mation errors (2/1) in the passband.

12.2.3

Design of IIR filters using analog prototypes The steps involved in this design process are described in the following subsecĆ tions. References for further reading on filters can be found on page 191.

Step 1) Specify the filter characteristics The required filter characteristics are described in Figure 12-8. These will of course depend on the type of filter required.

178

The Lms Theory and Background Book

Digital filtering

 

maximum ripple in the pass band (dB) attenuation (dB)

u upper cutoff

lower cutoff l

Figure 12-8

Filter specification for IIR filters

Step 2) Compute the analog frequencies A prototype low pass filter will be designed based on the required digital cutĆ off frequency c . First however the digital frequency d must be converted to an analog one a . This is achieved through a bilinear transformation from the digital (z) plane to the analog (s) plane where s and z are related by



1 s
2 1  z 1 T 1z



Eqn 12-23

When z= e jT (the unit circle) and s=ja jT s
2 1  e jT
2 ā jā tan( dT2) T 1e T





Eqn 12-24

 a
2 ā tan( dT2) T

Eqn 12-25

The analog  axis is mapped onto one revolution the of the unit circle, but in a non-linear fashion. It is necessary to compensate for this nonlinearity (warpĆ ing) as shown below

Part III Time data processing

179

Chapter 12 Digital filtering

a

a

d computed analog frequencies

c d defined digital frequencies

Figure 12-9

Conversion from digital to analog frequencies

Step 3) Select the suitable analog filter It is now necessary to select a suitable low pass analog prototype filter that will produce the required characteristics. The selection can be made from the folĆ lowing types of filter. Bessel filters Butterworth filters Chebyshev type I filters Inverse Chebyshev (type II) filters Cauer (elliptical) filters

Bessel filters The goal of the Bessel approximation for filter design is to obtain a flat delay characteristic in the passband. The delay characteristics of the Bessel approxiĆ mation are far superior to those of the Butterworth and the Chebyshev approxiĆ mations, however, the flat delay is achieved at the expense of the stopband atĆ tenuation which is even lower than that for the Butterworth. The poor stopband characteristics of the Bessel approximation make it impractical for most filtering applications !

180

The Lms Theory and Background Book

Digital filtering

Bessel filters have sloping pass and stop bands and a wide transition width reĆ sulting in a cutoff frequency that is not well defined. The transfer function is given by H(s)


d0 B n(s)

Eqn 12-26

where Bn (s) is the nth order Bessel polynomial B n(s)
(2n  1)B n1(s)  s 2B n2(s)

Eqn 12-27

and d0 is a normalizing constant. d0


(2n)! 2 nn!

Eqn 12-28

Butterworth filters These are characterized by the response being maximally flat in the pass band and monotonic in the pass band and stop band. Maximally flat means as many derivatives as possible are zero at the origin. The squared magnitude response of a Butterworth filter is |H(s)| 2


1 1  (ssc) 2n

Eqn 12-29

where n is the order of the filter. The transfer function of this filter can be deĆ termined by evaluating equation 12-29 at s=j |H(j)| 2
H(s)H( s)


1 s2 n 1  ( j 2)

Eqn 12-30

c

Butterworth filters are all-pole filters i.e. the zeros of H(s) are all at s=. They have magnitude (1/2 ) when / c =1 i.e. the magnitude response is down 3dB at the cutoff frequency.

Part III Time data processing

181

Chapter 12 Digital filtering

|H  |2

3dB

n=4 n=10 c



Figure 12-10 Characteristics of a Butterworth filter

A means of determining the optimum order is described on page 185.

Chebyshev (type I) filters These are all pole filters that have equi-ripple pass bands and monotone stop bands. The formula is |H()| 2


1 1   2C 2n()

Eqn 12-31

where Cn () are the Chebyshev polynomials and  is the parameter related to the ripple in the pass band as shown below for n odd and even.  1 1  2

 1 1  2

n odd

n even

For the same loss requirements, the Chebyshev approximation usually requires a lower order than the Butterworth approximation, but at the expense of an equi-ripple passband. Therefore, the transition width of a Chebyshev filter is narrower than for a Butterworth filter of the same order. The increased stopband attenuation is achieved by changing the approximation conditions in that band thus minimizing the maximum deviation from the ideal flat characteristics. The stopband loss keeps increasing at the maximum posĆ sible rate of 6* dB/Octave.

182

The Lms Theory and Background Book

Digital filtering

Chebyshev filters show a non-uniform group delay and substantially non-linĆ ear phase. A means of determining the optimum order is described on page 185.

Inverse Chebyshev (type II) filters These contain poles and zeros and have equi-ripple stop bands with maximally flat pass bands. In this case |H()| 2




1



C n(r) 1   2 C ( n r)

2

Eqn 12-32

where Cn () are the Chebyshev polynomials,  is the pass band ripple parameĆ ter and r is the lowest frequency where the stop band loss attains a specified value. These parameters are illustrated below for n odd and even.  1 1  2

 1 1  2

n odd

r

n even r

...

For the same loss requirements, the Inverse Chebyshev approximation usually requires a lower order than the Butterworth approximation, but at the expense of an equi-ripple stopband. The increased passband flatness is achieved by changing the approximation conditions in that band thus minimizing the maximum deviation from the ideal flat characteristics.

Cauer (elliptical) filter These filters are optimum in the sense that for a given filter order and ripple specifications, they achieve the fastest transition between the pass and the stop band (i.e. the narrowest transition band). They have equi-ripple stop bands and pass bands.

Part III Time data processing

183

Chapter 12 Digital filtering

n odd

n even

The transfer function is given by |H()| 2


1 1   2R 2n(L)

Eqn 12-33

where Rn (L) is called a Chebyshev rational function and L is a parameter deĆ scribing the ripple properties of Rn (L). The determination of Rn (L) involves the use of the Jacobi elliptic function.  is a parameter related to the passband ripple. This group of filters is characterized by the property that the group delay is maximally flat at the origin of the s plane. However this characteristic is not normally preserved by the bilinear transformation and it has poor stop band characteristics. For a given requirement, this approximation will in general require a lower orĆ der than the Butterworth or the Chebyshev ones. The Cauer approximation will thus lead to the least costly filter realization, but at the expense of the worst delay characteristics. In the Chebyshev and Butterworth approximations, the stopband loss keeps inĆ creasing at the maximum possible rate of 6* dB/Octave. Therefore these approximations provide increasingly more loss than a certain wanted flat attenuation that is really needed above the edge of the stopband. This source of inefficiency for both approximations is remedied by the Cauer or elliptic approximation.

184

The Lms Theory and Background Book

Digital filtering

Step 4) Transform the prototype low pass filter At this point we have selected a suitable low pass filter prototype with a normalized cutoff frequency c =1. The next stage is to transform this low pass filter into the type of analog filter required with the desired cutoff frequencies. To achieve this the following transformations are applied. Transform

Frequency response

Replace s by s s

Low pass to low pass

c

c

 s sc

Low pass to high pass

c

Low pass to band pass

Low pass to band stop

s

s

s2   u l sā( u   l) l

u

l

u

sā( u   l) s 2   u l

Step 5) Apply a bilinear transformation The final stage in this design process is to apply a bilinear transformation to map the (s) plane to the (z) plane to obtain the desired digital filter. H(z)
H(s)|

s
T2



1z 1 1z 1



Eqn 12-34

The final result is a set of filter coefficients a and b, stored in vectors of length n+1,where n is the order of the filter. A facility, described below, enables you to determine the optimum order of a filter required for a particular design.

Determining the filter order You can determine the filter order and the cutoff frequency for a given set of design parameters that are shown in Figure 12-11.

Part III Time data processing

185

Chapter 12 Digital filtering

1 passband ripple 1-1 attenuation 2 p

s v

Figure 12-11 Specifications required to determine filter order

Ripple passband

This determines the ripple parameter 1. It is exĆ pressed in dB

Attenuation

When this is defined, the ripple parameter 2 is deĆ termined. It is expressed in dB.

Lower frequency Upper frequency

These are the two edge frequencies p (end of the pass band) and s (start of the stop band) of a low pass or high pass filter. Band pass and band stop filters will require a second pair of frequencies to be defined.

Sampling frequency

This is the sampling frequency at which the filter must operate.

The filter can be any one of the types mentioned above and the prototype can be either a Butterworth, Chebyshev type I or type II or a Cauer filter. This proĆ cess does not apply to the Bessel filter because of the particular condition perĆ taining to these filters in that the filter order affects the cutoff frequency. The minimum filter order required is determined from a set of functions deĆ scribed below. One function relates the pass band and stop band ripple specifications to a filter design parameter  where 
2

12

(1   1) (1   21)   22

Another parameter relates the pass band cut off frequency  p , the transition width v and the low pass filter transition ratio k where p tan  p2 k
ĄĄ  ĄĄ
ĄĄ s tan  s2 analog

186

digital

The Lms Theory and Background Book

Digital filtering

A final function relates the filter order n, the low pass filter transition ratio k and the filter design parameter  This relationship depends on the type of proĆ totype analog filter. n


n


n


 k

Butterworth

cosh1(ā 1) 11k 2 ln k





K(k)āK (1   2) K()āK(1  k 2)

Chebyshev

Elliptic

where K( .) is the complete elliptical integral of the first kind.

12.2.4

IIR Inverse design filter The `filter inverse design' command uses a direct digital design technique rathĆ er than the digitization of existing analog filters as described in section 12.2.3. An iterative procedure is used to perform a least squares error fit between the actual frequency response and the specified desired response. The required response is obtained from a specified gabarit that contains the necessary frequency and magnitude break points which are mapped onto a grid. The outcome is a set of filter coefficients.

Part III Time data processing

187

Chapter 12 Digital filtering

12.3

Analysis This section describes the functions that provide information on the characterisĆ tics of filters.

Frequency response of filters The magnitude and phase of the frequency response H(eā j) of the filter defined by the coefficients a and b in equation 12-10

Group delay The group delay of a set of filters provides a measure of the average delay of a filter as a function of frequency. The frequency response of a filter is given by H(z)| z
ej
H(eā j)
|H(eā j)|.eā j() The phase delay is defined as ā()  p()
 

Eqn 12-35

and the group delay is defined as the first derivative of the phase  g()


dā() d

Eqn 12-36

If the wave form is not to be distorted then the group delay should be constant over the frequency bands being passed by the filter. For a linear delay, () = - where -
  
then  is both the phase delay and the group delay.

188

The Lms Theory and Background Book

Digital filtering

12.4

Applying filters This section describes how filters can be applied to data. Direct trace filtering Implementing this method basically filters the data x according to the filter deĆ fined by coefficients a and b to produce the filtered data y. Zero phase filtering This option also filters the data using the filter defined by the coefficients a and b, but in such a way as to produce no phase distortion. In the case of FIR filters an exact linear phase distortion is possible since the output is simply delayed by a fixed number of samples, but with IIR filters the distortion is very non-linĆ ear. If the data has been recorded however and the whole sequence can be replayed, then this problem can be overcome by using the concept of `time reverĆ sal'. In effect the data is filtered twice, once in the forwards direction, then in the reverse direction which removes all the phase distortion but results in the magnitude effect of the filter being squared. If x(n)=0 when n<0, then the z transform of the time reversed sequence is 0

Z{āxā( n)}




Eqn 12-37

x( n)Ăz n

n


which if -n=u






 x(u)Ă(z1)u 0

X(z)
Z{x(n)}

So if

Z{x( n)}
X(z 1)

then

Time reversal filtering can be realized using the method shown in Figure 12-12. x(n)

a(n)=x(-n)

f(n)

Time reversal

b(n)=f(-n)

y(n)

Time reversal

Figure 12-12 Realization of zero phase filters

Part III Time data processing

189

Chapter 12 Digital filtering

In this case it can be seen that Aā(z)
Xā(z 1) Fā(z)
Aā(z)ĂHā(z)
Hā(z)ĂXā(z 1) Bā(z)
Fā(z 1)
Hā(z 1)ĂXā(z) Yā(z)
Hā(z)ĂBā(z)
Hā(z)ĂHā(z 1)ĂXā(z) So the `equivalent' filter for the input data is H eq(z)
Hā(z)ĂHā(z 1) with z
e j H eq(z)
Hā(e j)ĂHā(e j)
|H(e j)| 2 i.e. zero phase and squared magnitude. Using this filtering method results in starting and end transients, which in this implementation are minimized by carefully matching the initial conditions.

190

The Lms Theory and Background Book

Digital filtering

12.5

References [1]

A. V. Oppenheimer and R.W. Schafer Digital Signal Processing Prentice Hall 1975

[2]

L.R. Rabiner and B. Gold Theory and Application of Digital Signal Processing Prentice Hall 1975

[3]

R.E. Crochiere and L.R. Rabiner Multirate Digital Signal Processing Prentice Hall 1983

[4]

J.G. Proakis and D.G. Manolakis Digital Signal Processing: Principles Algorithms and Applications MacMillan Publishing 1992

Part III Time data processing

191

Chapter 13

Harmonic tracking

This chapter describes the concepts involved in Harmonic tracking using a Kalman filter. Theoretical background Practical considerations

193

Chapter 13 Harmonic tracking

13.1

Introduction There are a number of circumstances when it is necessary to track periodic comĆ ponents (orders) when the signal of interest is buried in noise, or the rotational speed is changing rapidly. Indeed some effects only manifest themselves when the rate of change of frequency is high. In these situations, real time analog and digital filters have limited of resolution due to transients and excessive procesĆ sing requirements. The Kalman filter however is able to accurately track sigĆ nals of a known structure concealed in a confusion of noise and other periodic components of unknown structure. An important characteristic of the Kalman filter is that it is non-stationary. It functions well at high slew rates, because the system model used does not preĆ sume either fixed time of frequency content, but adapts itself automatically as the system itself is changing. This ability to derive the system model for each time sample in the recording (within certain user-defined constraints) frees it from the usual time/frequency resolution constraint encountered with the traditional frequency transformations.

Conditions for use Some important capabilities of the Kalman filter are V

the ability to track an order with arbitrary fractional order resolution from signals sampled at a constant rate,

V

fine spectral resolution of the orders (i.e. 0.01 Hz) obtained after just a few measurement samples (not even one cycle of the fundamental comĆ ponent),

V

virtually no slew rate limitations,

V

the ability to produce an order value for every measurement sample point,

V

no phase distortion.

In order to use the Kalman filter the following conditions must apply -

194

d

The structure of the signal (sine wave) to be tracked must be accurately known.

d

The signals must be acquired at a constant sampling rate.

d

An accurate estimate of the instantaneous Rpm value is required when you are dealing with signals that vary with rotational speed.

The Lms Theory and Background Book

Harmonic tracking

13.2

Theoretical background The application of the Kalman filters to track harmonic components involves two stages. 1 Accurate determination of the Rpm If you want to track an order, then you must provide the corresponding Rpm/time trace. Your Rpm may have been determined using a Tacho signal which results in a pulse train or a swept sine function in which case you will need to convert it to a Rpm/time function. 2 The tracking of the specified waveform Section 13.2.2 describes the mathematical background to the operation of the tracking function. Some practical considerations are discussed in section 13.3.

13.2.1

Determination of the Rpm Since the Kalman filter is highly selective and accurate in tracking a target sigĆ nal buried in noise, it is crucial that the instantaneous RPM of the system is preĆ cisely modelled, otherwise the wrong component will be tracked. The rpm inĆ formation can be derived from the tachometer channel, which is sampled at the same rate as the measurement channels to obtain a small statistical variability in the period estimation. Clearly the tachometer events will occur at a lower rate and so to reduce the error on the period estimate, resampling is performed on the original tachometer signal. The first part of the process therefore is to convert the original tacho signal from a pulse train to an rpm/time function. The second step involves obtaining an equidistant function. Since all mechaniĆ cal systems have some inertia, it is reasonable to expect the speed to be a conĆ tinuous function, so a cubic spline with the appropriate boundary conditions can be used to obtain the required `sample-by-sample RPM' estimate of speed function.

13.2.2

Waveform tracking The Kalman filtering method involves setting up and solving a pair of equaĆ tions known as the Structural and the Data equations.

Part III Time data processing

195

Chapter 13 Harmonic tracking

The Structural equation This equation defines the shape or structure of the waveform you wish to track. A sine wave for example, x(t) of frequency  sampled at time t satisfies the following second order difference equation

x(nāt)Ă Ă 2 cos(2
āt)x((n  1)āt)  x((n  2)āt)
0

Eqn. 13-1

by dropping the time increment t this can be written more simply as

xā(n)Ă Ă c(n)xā(n  1)  xā(n  2)
0

Eqn. 13-2

where c(n) = cos (2
 t) When the instantaneous frequency  is known, equation 13-2 is a linear freĆ quency dependent constraint equation on the sine wave which is known as the structural equation. When tracking a sine wave which is changing in frequency, and which is conĆ taminated by noise and other sinusoids, a non homogeneity term (n) is introduced. This allows the sine wave to vary in frequency, amplitude and phase and Equation 13-2 then becomes

xā(n)Ă Ă c(n)xā(n  1)  xā(n  2)
(n)

Eqn. 13-3

(n) is a deterministic but unknown term which allows for deviations from the true stationary wave. It is also useful to define S (n) as the standard deviation of the non homogeneĆ ity of the structural equation.

The Data equation x(n) is the time history defined by the structural equation, but the measured

signal y(n) contains both the signal that matches the structural equation as well as noise and other periodic components.

196

The Lms Theory and Background Book

Harmonic tracking

yā(n)Ă
xā(n)  ā(n)

Eqn. 13-4

where (n) contains noise and periodic components at frequencies other than the target signal. Once again S (n) is defined as the standard deviation of the nuisance element of the data equation.

The Least squares formulation For any point in time (n), equations 13-3 and 13-4 provide linear equations for {x(n) x(n-1) x(n-2)}. Rearranging these equations gives an unweighted form of equation where the structural equation is on the top row and the data equaĆ tion on the bottom.









1Ą  c(n)Ą1 x(n  2) (n) Ąx(n  1)
y(n)  (n) ĄĄ1 x(n)



Eqn. 13-5

The error in equation 13-5 is made isotropic by applying a weighting factor r(n) which is defined as the ratio of the standard deviations of the errors in the structural and data equations.

rā(n)Ă


sā(n) s ā(n)

Eqn. 13-6

Equation 13-5 then becomes -









1Ą  c(n)Ą1 x(n  2) (n) Ąx(n  1)
r(n)ā(y(n)  (n)) ĄĄr(n) x(n)



Eqn. 13-7

The weighting function r(n) expresses the degree of confidence between the structural equation and data equation, or, the certainty of the presence of orders in the data. This function shapes the nature of the Kalman filter and influences its tracking characteristics. A small value for r(n) leads to a filter that is highly discriminating in frequency, but which takes time to converge. Conversely, fast convergence with low frequency resolution is achieved by choosing a large r(n).

Part III Time data processing

197

Chapter 13 Harmonic tracking

When applied to all observed time points Equation 13-7 provides a system of overdetermined equations which may be solved using standard least squares techniques.

198

The Lms Theory and Background Book

Harmonic tracking

13.3

Practical considerations This section considers some practical characteristics of the Kalman filter and the parameters that influence them.

Frequency resolution In principle the Kalman filter is capable of tracking sinusoidal components of any frequency up to half the sample frequency. In practice however, it has been found that the ability to distinguish between two closely spaced sine waves is inversely proportional to the total observation time. As a consequence, the obĆ servation time should be equal to the inverse of minimum frequency spacing required between components.

Filter characteristics It was mentioned above that the weighting r(n) used in Equation 13-7 can be used to influence the nature of the tracking filter used. This weighting can be adjusted through the specification of a harmonic confidence factor which is deĆ fined as the inverse of the weighting factor. s ā(n) HC
1
rā(n) s ā(n)

Eqn. 13-8

Applying a high value implies confidence in the harmonic (structural data) and assumes that the error in your measured data is high. In this case the filter will be narrow so that it is highly discriminating in frequency. This is obtained at the cost of time to converge in amplitude. Applying a low value implies that the error in the measured data is low and consequently a wider filter can be used which while less discriminating in frequency has the advantage that the amplitude converges more quickly. The three Kalman filters shown below are characterized by different harmonic confidence factors which influence the width of the filter.

Part III Time data processing

199

Chapter 13 Harmonic tracking

HC= 50 HC= 100 HC= 200

Figure 13-1

Effect of the Harmonic Confidence Factor

Bandwidth characteristics Equation 13-7 shows that the weighting function, r(n), which is the inverse of the harmonic confidence factor, can be different for every time point. This means that the bandwidth of the filter can vary as a function of the frequency or order being tracked. Using a frequency defined band width means that at low Rpm values, a numĆ ber of orders will be encompassed by the filter range. amp

Rpm amp orders

frequency

Rpm

orders

frequency

Figure 13-2 Defining the filter bandwidth in terms of frequency and amplitude.

Allowable slew rates The formulation of the Kalman filter assumes that the frequency of the signal to be tracked remains constant over three consecutive measurement points. When the frequency is varying, but the variation over these three points is less than the bandwidth of the filter then no problem arises. The minimum value of the bandwidth is equal to the inverse of the observation time T. If the sample rate is Fs then the slew rate must be less than Fsā/ā2T.

200

The Lms Theory and Background Book

Harmonic tracking

Tracking closely spaced order signals with a high slew rate requires sampling at a high frequency over a long period which imposes a heavy computational effort. However if you consider the significant slew rate encountered during the deceleration of gas turbines of 75Hz/sec over 5 seconds, from the above this implies a sample rate of 750Hz. It can be seen therefore that such an extreme slew rate does not impose any realistic limitation on the sample rate.

Part III Time data processing

201

Chapter 14

Counting and histogramming

This chapter provides an introduction to various counting methods and provides a reading list for further information at the end Counting of single events and occurrences Two–dimensional counting methods

203

Chapter 14 Counting and histogramming

14.1

Introduction In fatigue analysis, real life measurements of mechanical or thermal loads are used to assess and predict the damage inflicted by such loads over the life time of a product. Figure 14-1 shows such measurements made on a vehicle part over a period of around 5 minutes (330 seconds).

acceleration

0.4

(g)

time (s)

-0.4

Figure 14-1

Typical load/time data

In terms of fatigue analysis it is the occurrence of specific events that are of more significance than the frequency content of the loads. The approach used is to scan such time histories looking for typical fatigue-generating events and then to register how often they occur. These typical events can be demonĆ strated with a zoomed-in section of a load time history, shown in Figure 14-2.

Figure 14-2

Typical events in a data trace

The interesting events are:V

204

The occurrence of peaks at specific levels These are represented by the circles and are determined using ``Peak counting'' methods described in section 14.2.1.

The Lms Theory and Background Book

Counting and histogramming

V

The exceedence or crossing of specific levels. These are represented by the squares and are determined using ``Level cross'' counting methods described in section 14.2.2.

V

The occurrence of signal changes of a certain size. These are represented by the arrows and are determined using ``Range count methods' described in section 14.2.3

The determination of the signal characteristics based on the events mentioned above is a two stage process d

Stage 1, counting The data is scanned for the occurrence of one of the events listed above. This in effect reduces the full time history to a set of mechanical or therĆ mal load events.

d

Stage 2 histogramming This involves dividing the counted occurrences into classes where for each event, its number of occurrences is specified.

Part III Time data processing

205

Chapter 14 Counting and histogramming

14.2

One dimensional counting methods The procedures described above deal with the counting of `single events' or ocĆ currences which are further explored in this section. Section 14.3 describes a number of methods used to examine the occurrence of additional event circumstances. These methods are termed `Two dimensional counting methods'.

14.2.1

Peak count methods The turning points in a data trace are termed ``peaks"(maximums ) and ``valĆ leys" (minimums ). The number of times that peaks and valleys occur at speĆ cific levels is counted as shown below. You can choose to count both the peaks and the valleys (extrema) or just the peaks (maxima), or just the valleys (miniĆ ma). 2 1 0 -1 -2

Figure 14-3

Counting of peaks and valleys

A histogram is then created by calculating the distribution of the number of ocĆ currences as a function of the level at which the occurrence appeared. The FigĆ ure 14-4 shows the results of processing the above peak-valley reduction acĆ cording to the three types of counting methods.

206

The Lms Theory and Background Book

4

3

3

2 1 0

-2 -1

ÇÇ ÇÇ ÇÇ ÇÇ

0 level

1

Minima

Figure 14-4

14.2.2

2

2 1 0

Ç Ç Ç Ç

-2 -1

0 level

ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ

1

2

Maxima

ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉ ÉÉÉÉ ÉÉ ÉÉ ÉÉ ÉÉÉÉ ÉÉ ÉÉ

4 Nr of occurrences

Ç Ç Ç Ç Ç Ç Ç Ç

4

Nr of occurrences

Nr of occurrences

Counting and histogramming

3

2 1 0

-2 -1

0

1

2

level

Extrema

Histograms of peaks (maxima), valleys (minima) and both (extrema)

Level cross counting methods This procedure counts the number of times that the signal crosses various levĆ els. Distinctions can be made between an upward (positive ) and a downĆ ward (negative ) crossing as illustrated below. You can choose to count both the positive (up) crossings, the negative (down) crossings or both types. 2 1 0 -1 -2

Figure 14-5

Counting of level crossings

Peak counts and level cross counts are closely related. The number of positive crossings of a certain level is equal of the number of peaks above that level miĆ nus the number of valleys above it. This implies that a level cross count can be derived from a peak-valley count. A level crossing count is typically initiated by specifying a grid on top of the signal to determine the levels. The grid can be specified in ordinate units or as a percentage of the ordinate range. The resulting histograms for the above sigĆ nal when up, down and both types of crossings are counted are shown below.

Part III Time data processing

207

Chapter 14 Counting and histogramming

6

ÇÇ Ç ÇÇ Ç ÇÇ ÇÇ ÇÇ Ç ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇÇÇÇ 2

0 level

1

8

ÇÇ Ç ÇÇ ÇÇÇÇÇÇÇ Ç ÇÇÇÇÇÇÇ

4

-2 -1

6 4

0

2

14.2.3

-2 -1

0 level

1

8 6 4 2

2

up (+) crossings

Figure 14-6

10

Nr of occurrences

8

0

ÉÉ É ÉÉ É ÉÉ É ÉÉ ÉÉ É É É ÉÉ É ÉÉ ÉÉ ÉÉÉ ÉÉ ÉÉ ÉÉ ÉÉÉÉÉ

10 Nr of occurrences

Nr of occurrences

10

0

2

-2 -1

0 level

1

2

up (+) & down (-) crossings

down (-) crossings

Histograms of level crossing counts

Range counting methods A range count method will determine the number of times that a specific range change is observed between successive peak-valley sequences.

Counting of single ranges The range between successive peak-valley pairs is counted. Ranges are considĆ ered positive when the slope is rising and negative when the slope is falling. 4 1

+ 1

1 + 1

+ 1 + 1

1

1

+ 4

Figure 14-7

Counting of single peak-valley ranges

A histogram of the number of occurrences, as a function of the range, is generĆ ated.

208

The Lms Theory and Background Book

Counting and histogramming

Nr of occurrences

4 3

2 1 0

ÇÇ ÇÇ -4

-3

-2

ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ -1

0

ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ ÇÇ 1

2

3

ÇÇ ÇÇ 4

Range

Figure 14-8

Histogram of single peak-valley ranges

Counting of range–pairs The counting of single ranges (usually indicated as a range-count), is both simĆ ple and straightforward but sensitive to small variations of the signal. Thus in the analysis of the left hand signal illustrated in Figure 14-9, single range counting would result in a large number of relatively small ranges.

low pass filter

Figure 14-9

Sensitivity of single range counting to signal variation

If this signal were passed through a filter, suppressing the small load variaĆ tions, the resulting signal would reveal a count of only one very large range. As a consequence the two analysis results are completely different and the method is very sensitive to small signal variations. The range-pair counting method overcomes this sensitivity. Rather then splitĆ ting up the signal into consecutive ranges, it is interpreted in terms of a ``main" signal variation (or range) with a smaller cycle (range pair) superimposed on it.

=

+

R

Figure 14-10 Range pair counting

Part III Time data processing

209

Chapter 14 Counting and histogramming

If a pair of extremities are separated by a range that is less than the defined range of interest (R), then they are `filtered out' of the range count.

210

The Lms Theory and Background Book

Counting and histogramming

14.3

Two–dimensional counting methods The counting methods described so far, consider the occurrence of single events in isolation from any other circumstances which may affect these events. HowĆ ever, it is also meaningful to count events differently, depending on other cirĆ cumstances using `two-dimensional' methods. Such methods are discussed in this section.

14.3.1

From–to–counting Such a ``combined" event can be the occurrence of a peak at level j followed by a valley at level i. As an example, consider the combination of a valley at level A followed by a peak at level C as illustrated in Figure 14-11. 4

D 2

C B A

12

3

11

1

Figure 14-11 From-to counting

In this example, the Fromto sequence (12) is counted separately from the sequences (34) and (1112), although the ranges involved are identical (C-A=D-B). The result of such ``fromto'' counting can be presented in a so called MarkovMatrix A[i,j]. The element aij gives the number of peaks at level j followed by a valley at level i. The matrix of results of counting the events in Figure 14-11 are shown below.

Part III Time data processing

211

Chapter 14 Counting and histogramming

A

From j B C

D

A

X

0

1

0

1

0

B

0

X

1

2

3

2

C

1

1

X

2

2

4

D

1

2

1

X

0

To i

X

peaks

valleys

The lower left triangle of the Markov matrix contains the positive fromto events, the upper right triangle summarizes the negative transitions. The addiĆ tional separate columns contain the counting results for peaks and valleys at a particular level. These results are easily obtained for the triangles of the MarĆ kov matrix.

14.3.2

Range–mean counting Another example of a two-dimensional counting method results in the socalled Range-mean matrix. The variation or range (i-j) is associated with its corresponding mean value (i+j)/2. 4

D 2

C

C

C

B

B A

12

3 1

D-B

11

D-B

C-A

Figure 14-12 Range mean counting

Instead of considering the actual values of A and C, the Range-mean method will consider the values CA (the range) and B (= A+C / 2 the mean). Ranges, means and the number of occurrences can be displayed in a 3D format.

212

The Lms Theory and Background Book

Counting and histogramming

Number of events

Mean

Range

Figure 14-13 Display of range-mean counting

14.3.3

‘‘Range pair–range” or ‘‘Rainflow’’ method A two-dimensional counting method of special interest, especially for fatigue damage calculations, is the ``range pair-range" method. Such a method was also developed, simultaneously and independently in Japan, known as the ``Rainflow method". Both methods yield exactly the same results, i.e. they exĆ tract the same range-pairs and ranges from the signal, by combining the rangepair counting principle and the single range counting principle into one methĆ od. For further details see the references listed on page 217. Essentially the signal is split into separate cycles, having a specific amplitude (or range) and a mean. The result can be put directly into cumulative fatigue damage calculations according to Miner's rule and into simple crack growth calculations. Three steps are involved in the complete procedure. 1

Conversion of the load history into a peak-valley sequence. As the counting procedure considers only the values of successive peaks and valleys, the complete signal may first be reduced to a peak-valley seĆ quence. In doing this it is usual to apply a specific ``range-filter" or gate. For a range filter of size R, a peak (or valley) at a certain level is only recĆ ognized as such if the signal has dropped (or risen) to a level which is R lower (or higher) then the previous peak (or valley) level.

Part III Time data processing

213

Chapter 14 Counting and histogramming

e5 R

e3 R

e1 e0

e5

e4

e1

e6 R

R

e6

e0

e2

e2

Figure 14-14 Conversion of a load history to a peak valley sequence

In the above example e1 is counted as a peak because the signal drops by more then the range filter size R after it. After counting the first peak, the next valid valley is looked for, which in this case is e2. This point is validated as a valley as the signal rises by more then R to go to e3. The algorithm then searches for the next valid peak. The first peak encountered is e3, but this is not counted as a valid peak as the signal does not drop sufficiently before reaching the next exĆ tremum in the signal (e4). So the algorithm checks whether the following peak is a valid one. Peak e5 is regarded as valid since the drop in signal level following it, is greater than R. In this example the range filter eliminated the small signal variation (e3,e4) from the peak-valley sequence. Note that increasing the range filter eliminates only those transitions from the histogram for which the range is smaller than the new value of R. This is important for fatigue purposes since it proves that the filtering is not that sensitive to the range filter size. 2

Scanning of the entire signal for range-pairs. This phase of the counting procedure consists of taking a set of four conĆ secutive points, and check whether a range-pair is contained in it. If not, the search through the peak-valley sequence continues by shifting one data point ahead. Once a range-pair is detected, the pair is counted and removed from the sequence. After this, the next new set of four points is formed by adding the closest two previously scanned points, to the two remaining after removal of the range pair. The fact that earlier scanned points are re-considered, clearly distinguishes Range-pair range counting from single range counting.

3

214

Counting the ``Residue"

The Lms Theory and Background Book

Counting and histogramming

At the end of the second phase, a ``residue" of peaks and valleys is left which is analyzed according to the single range principle. It can be shown that this residue has a specific shape, namely a diverging part folĆ lowed by a converging part.

Example The following example shows how the range-pair range method operates. Consider the time signal shown beĆ low.

A peak-valley reduction with a range filter of size R, results in the peak-valley sequence shown below. S6 S2

R

S4

S8 S5

S3 S1

S7

The second phase (scanning of the range-pair occurrences) starts by looking at the 4 first extremes. In this group (S1,S2,S3,S4), a pair is counted if the two inĆ ner extremes (S2,S3,) fall within the range covered by the two outer extremes (S1, and S4),. If this is not (as in this example), then the algorithm moves one step forward and considers the extremes S2,S3,S4, and S5. These do not satisfy the condition either, so the extremes S3,S4,S5, and S6 are considered and this time a range pair is counted. S6 S2

S6 S2

S4

S3

S8

S8

S5

S3

S1

S1 S7

S4 S5

S7

Counting a range-pair implies deleting the counted extremes from the signal. ``Stepping backwards", the extremes S1,S2,S3, and S6 are now considered and another pair (S2,S3) is found.

Part III Time data processing

215

Chapter 14 Counting and histogramming

S6

S6

S2 S8

S8

S3 S1

S1

S2

S7

S7 S3

From the remaining four extremes, no ``pairs'' can be subtracted. This forms the residue which is further counted as single ``from-to-ranges''.

Further considerations The result of the range pair-range counting depends on the length of the data record being analyzed at one time because the largest range counted will be beĆ tween the lowest valley and the highest peak. This largest variation is often reĆ ferred to as the `half load cycle'. If the lowest valley occurs near the beginning of a very long load cycle, and the highest peak near the end, you should conĆ sider whether it makes physical sense to combine such occurrences, so remote in time into one cycle. The counting method is insensitive to the size of the range filter applied. The only effect of increasing the range filter size from R to 3R, for example, is that all elements in a From-to counting for which |from-to|<3*R, become zero. In other words, the choice of the range filter size is not critical.

216

The Lms Theory and Background Book

Counting and histogramming

14.4

References [1]

Fatigue load monitoring of tactical aircraft, de Jonghe J.B., 29th Meeting of the AGARD SMP, Istanbul, September 1969.

[2]

The monitoring of fatigue loads, de Jonghe J.B., IACS-Congress, Rome, September 1970 .

[3]

Statistical load data processing, van Dijk C.M, 6th ICAF Symposium MiĆ ami, Florida USA, May 1971 .

[4]

Fatigue of Metals subjected to varying stress, Matsuiski M. & Endo T., Kyushu district meeting, Japan Society of Mechanical Engineers, March 1968 .

[5]

Cycle counting and fatigue damage, Watson P., SEE Symposium of 12th February 1975, Journal of Society of Environmental Engineers, September 1976.

Part III Time data processing

217

Theory and Background

Part IV Analysis and design Chapter 15 Estimation of modal parameters . . . . . . . . . .

219

Chapter 16 Operational modal analysis . . . . . . . . . . . . . .

267

Chapter 17 Running modes analysis . . . . . . . . . . . . . . . .

281

Chapter 18 Modal validation . . . . . . . . . . . . . . . . . . . . . . . .

293

Chapter 19 Rigid body modes . . . . . . . . . . . . . . . . . . . . . .

309

Chapter 20 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

Chapter 21 Geometry concepts . . . . . . . . . . . . . . . . . . . . .

218

357

Chapter 15

Estimation of modal parameters

This chapter describes the basic principles involved in estimating modal parameters. The topic covered are : The definition and derivation of modal parameters Factors to consider in the estimation Descriptions of different parameter estimation techniques Calculation of static compensation modes

219

Chapter 15 Estimation of modal parameters

15.1

Estimation of modal parameters A modal analysis provides a set of modal parameters that characterize the dyĆ namic behavior of a structure. These modal parameters form the modal model and Figure 15-1 illustrates the process of arriving at the modal parameters.

Ą (jrĂ  )  (jrĂ  )

N

h ijĂ(j)Ă


k
1

ijk

ijk

*

* k

k

curve fit to estimate modal parameters

measure the frequency response function

input

FREQUENCY DAMPING MODE SHAPES

input

Figure 15-1 Derivation of modal parameters

If a structure exists on which measurements can be made, then it can be asĆ sumed that a parametric model can be defined that describes that data. The starting point is usually a set of measured data - most commonly frequency reĆ sponse functions (FRFs), or the time domain equivalent, impulse responses (IRs). For IRs the relation between modal parameters and the measurements is expressed in Equation 15-1.

N

h ijĂ(t)Ă
 Ą rĂ ijkeĂ  t  rĂ ijk *eĂ  t

k

* k

Eqn 15-1

k
1

The corresponding relation for FRFs is given in Equation 15-2.

220

The Lms Theory and Background Book

Estimation of modal parameters

 rĂ ijk  rĂ ijk ! h ijĂ(j)Ă
 Ą (j   ) (j   *) N

k
1

*

k

k



Eqn 15-2

where hij (t)

= IR between the response (or output) degree of freedom i and the refĆ erence (or input) DOF j

hij (j ) = FRF between the response DOF i and reference DOF j N

= number of modes of vibration that contribute to the structure's dyĆ namic response within the frequency range under consideration

r ijk

= residue value for mode k

k

= pole value for mode k.

* designates complex conjugate. The pole value can be expressed as shown in Equations 15-3 and 15-4.  k
 k  j dk

Eqn 15-3

where dk

= the damped natural frequency of mode k

k

= the damping factor of mode k

or  k
  k nk  j nk 1   2k

Eqn 15-4

where nk

= the undamped natural frequency of mode k

k

= damping ratio of mode

Equation 15-5 shows that the residue can be proven to be the product of three terms r ijk
a kĂv ikĂĂv jk

Part IV

Modal Analysis and Design

Eqn 15-5

221

Chapter 15 Estimation of modal parameters

where vik

= the mode shape coefficient at response DOF i of mode k

vjk

= the mode shape coefficient at reference DOF j of mode k

ak

= a complex scaling constant, whose value is determined by the scaling of the mode shapes

Note that the mode shape coefficients can be either real (normal mode shapes) or complex. If the mode shapes are real, the scaling constant can be expressed as, ak


1 2jm k dk

Eqn 15-6

where mk

= the modal mass of mode k

The poles, natural frequencies (damped and undamped), damping factors or ratios, mode shapes, and residues are commonly referred to as modal parameters (parameters of the modes of the structure). The fundamental problem of parameter estimation consists of adjusting (estiĆ mating) the parameters in the model, so that the data predicted by the model approximate (or curve-fit) the measured data as closely as possible. Modal paĆ rameters can be estimated using a number of techniques. These techniques are discussed in the following sections.

A note about units The frequency and damping values have a dimension of 1/time, and are thereĆ fore stored in Hz. The residues, as appearing in Equation 15-1 of 15-2, have the same dimension as the measurement data. As an aside, it is important to note that residues have a dimension. Residues are composed of a product of mode shape coefficients and a scaling constant, (Equation 15-5). The mode shape coefficients by themĆ selves do not have any dimension, nor absolute (or scaled) magnitude. DimenĆ sion, and therefore units will be viewed as attributes of the scaling constant. Finally, for multiple input analysis, the residues are written in factored form as the product of mode shapes with modal participation factors. Again, the prodĆ uct of the factors has a dimension and absolute magnitude. Formally, the mode shape coefficients will again be considered as without dimension and therefore units will be viewed as attributes of the residues.

222

The Lms Theory and Background Book

Estimation of modal parameters

15.2

Types of analysis The section discusses some general principles to be considered when performĆ ing a modal analysis. These topics include V

Using single or multiple degree of freedom methods in section 15.2.1

V

Making local or global estimates in section 15.2.2

V

Using multiple input analysis in section 15.2.3

V

Using time or frequency domain analysis in section 15.2.4

V

Special conditions which apply when performing vibro-acoustic analyĆ sis in section 15.2.5

The specific parameters estimation techniques are described in section 15.3

15.2.1

Single or multiple degree of freedom method If, in a given frequency band, only one mode is assumed to be important, then the parameters of this mode can be determined separately. This assumption is sometimes called the single degree of freedom (sDOF) assumption. d/f

min Figure 15-2

max

frequency

The single degree of freedom assumption

Under this assumption, the FRF equation 15-2 can be simplified to equation 15-7. This is assuming the data to have the dimension of displacement over force. h ij


Part IV

rĂ ijk (j   k)

Modal Analysis and Design



rĂ *ijk (j   *k)

Eqn 15-7

223

Chapter 15 Estimation of modal parameters

 min     max It is possible to compensate for the modes in the neighborhood of this band, by introducing so called upper and lower residual terms into the equation. h ij


rĂ ijk (j   k)



rĂ ijk * (j   *k)

 ur ij 

lr ij

Eqn 15-8

2

where urij = upper residual term (residual stiffness) used to approximate modes at frequencies above max. lrij = lower residual term (residual mass) used to approximate modes at frequencies below min Upper and lower residuals are illustrated in Figure 15-3. d/f

mass line upper residual urij lower  lr ij residual 2

Figure 15-3

stiffness line

frequency

Upper and lower residuals

Equation 15-7 can be further simplified by neglecting the complex conjugate term, and so becomes h ij


rĂ ijk (j   k)

Eqn 15-9

Single degree of freedom methods The single DOF assumption forms the basis for parameter estimation technĆ iques such as Peak picking, Mode picking and Circle fitting.

224

The Lms Theory and Background Book

Estimation of modal parameters

Multiple degree of freedom methods The sDOF assumption is valid only if the modes of the system are well deĆ coupled. In general this may not be the case. It then becomes necessary to approximate the data with a model that includes terms for several modes. The parameters of several modes are then estimated simultaneously with so-called multiple degree of freedom methods.

15.2.2

Local or global estimates If you recall the time domain relationship between modal parameters and meaĆ surement functions, h ijā(t)


N

 rijkeā t  r*ijkāeā t

k

* k

Eqn 15-10

k
1

you will see that the pole values k are independent of both the response and the reference DOFs. In other words the pole value k is a characteristic of the system and should be found in any function that is measured on the structure. When applying parameter estimation techniques, one of two strategies can be employed; making local or global estimates.

Part IV

Local estimates

Global estimates

Each data record hij is analyzed indiĆ vidually, and a potentially different estimate of the pole value k is found each time.

All the data records are analyzed siĆ multaneously in order to estimate the structure's characteristics.

Analyzing data in this manner proĆ duces as many estimates of each pole as there are data records. It is then left to the user to decide which estiĆ mate is the best or to somehow calcuĆ late the best average of all the estiĆ mates.

With this approach, a unique estimate of the pole values k ăis obtained. Such estimates are therefore called global estimates.

Peak picking and Circle fitting are techniques that calculate local estiĆ mates of pole values.

The Least Squares Complex ExponenĆ tial, Complex Mode Indicator FuncĆ tion and Direct Parameter IdentificaĆ tion methods allow you to obtain global estimates of structure characĆ teristics.

Modal Analysis and Design

225

Chapter 15 Estimation of modal parameters

15.2.3

Multiple input analysis Assume that data is available between Ni input DOFs and No output DOFs. The expression for each of the individual data records ( equation 15-10) can then be rewritten in matrix form for all the data records. [H]


N

 [Rk]eĂ t  [R*k]eĂ t

k

* k

Eqn 15-11

k
1

where [H] =(No ,Ni ) matrix with hij as elements [Rk ] =

(No ,Ni ) matrix with rijk as elements

Equation 15-5 can be used to express the residue matrix in factored form, [R k]
a kĂ{V} kĂV r k

Eqn 15-12

where {V}k = No vector (column) with mode shape coefficients at the output DOFs Vr k = Ni vector (row) with mode shape coefficients at the input DOFs If DOFs i and j are both output and input DOFs then the above equation imĆ plies Maxwell Betti reciprocity, r ijk
r jik

Eqn 15-13

This assumption is not essential however since the residue matrix can be exĆ pressed in a more general form, [R k]
Ă {V} kĂL k

Eqn 15-14

where L k is a vector (row) with Ni coefficients that express the parĆ ticipation of the mode k in response data relative to different input DOFs. These coefficients are called modal participation factors therefore. Note that if recĆ iprocity is assumed then the modal participation factors are proportional to the mode shape coefficient at the input DOFs.

226

The Lms Theory and Background Book

Estimation of modal parameters

Using the factored form of the residue matrix, equation 15-11 can be written as, N

 H 
 {V} kLkeĂ  t  {V *} kL* keĂ  t

* k

k

Eqn 15-15

k
1

If just the data between any output DOF and all input DOFs are considered then H i


N

 vikLkeĂ t  v*ikL*keĂ t

* k

k

Eqn 15-16

k
1

where H i = Ni vector of data between output DOF i and all input DOFs. It is essential in the model of equation 15-16 that both the poles and the modal participation factors are independent of the output DOF. In other words in this formulation the characteristics become L kĂe kt

Eqn 15-17

A multiple input modal parameter estimation technique is one that analyses data relative to several inputs simultaneously to estimate the characteristics exĆ pressed by equation 15-17 (i.e. both the pole values and the modal participaĆ tion factors). The basis for these techniques is a model expressed by equation 15-16. The identification of modal participation factors is essential for decoupling highly coupled or even repeated roots. To illustrate this consider a structure that has two modes with pole values 1 and 2 very close to each other. NeĆ glecting the other modes and the complex conjugate terms, the response data relative to the input DOF j can be expressed as {H} i
{V} 1l 1jeĂ 1t  {V} 2l 2jeĂ  2t...

Eqn 15-18

or since 1 " 2






{H} i
Ă {V} 1Ăl 1j  {V} 2Ăl 2j ĂeĂ t...

Part IV

Modal Analysis and Design

Eqn 15-19

227

Chapter 15 Estimation of modal parameters

The latter equation shows that in the response data relative to an input DOF j, a combination of the coupled modes is observed and not the individual modes. The combination coefficients for the modes are the modal participation factors l1j and l2j . The response data relative to another input DOF l, is expressed by an equation similar to equation 15-19. {H} i
Ă {V}1Ăl 1l  {V} 2Ăl 2lĂ Ăe t...

Eqn 15-20

The only difference between these last two equations is the modal participation factors l1l and l2l . If they are linearly independent of the modal participation factors for input i, then the modes will appear in a different combination in the response data relative to input l. As a multiple input parameter estimation technique analyses data relative to several inputs simultaneously, and the modĆ al participation factors are identified, then it is possible to detect highly coupled or repeated modes.

15.2.4

Time vs frequency domain implementation Using digital signal processing methods, only samples of a continuous function are available. For modal parameter estimation the sampled data consist most frequently of FRF measurements. Normally these are taken at equally spaced frequency lines. Testing techniques such as stepped sine excitation allow you to measure data at unequally spaced frequency lines. For modal parameter estimation applications with the data measured in the freĆ quency domain, introducing the sampled nature of the data transforms the equation for the model to -

 ijk h ij,nĂ(j)Ă
 Ą  (j n   ) N

k
1

rĂ

k

!  (j n  *k) rĂ ijk*

Eqn 15-21

where hij,n = samples of data in measured range.

228

The Lms Theory and Background Book

Estimation of modal parameters

n = sampled value of frequency in measured range. A frequency domain parameter estimation method uses data directly in the freĆ quency domain to estimate modal parameters. It is therefore irrelevant whethĆ er the frequency lines are equally spaced or not. They are based directly on the model expressed by equation 15-21. If the data are sampled at equally spaced frequency lines, then the FRF can be transformed back to the time domain to obtain a corresponding Impulse ReĆ sponse (IR). A Fast Fourier Transform (FFT) algorithm is used for this transĆ formation but the restriction on the number of frequency lines being equal to a power of 2 (e.g. 32, 64, 128...) is no longer valid. After transformation, a series of equally spaced samples of corresponding impulse response functions is obĆ tained. A time domain parameter estimation technique allows you to analyze such equally spaced time samples to estimate modal parameters. In practice, a variety of conditions mean that the frequency band over which data is analyzed is smaller than the full measurement band. This is illustrated in Figure 15-4.

hij

min

max analysis frequency band

frequency

measurement band

Figure 15-4

Analysis frequency band vs. measurement band

The analysis frequency band includes only three modes whereas the measureĆ ment band includes five. If the data is transformed from frequency to time doĆ main, then the time increment between samples will be determined by the analĆ ysis frequency band and not the measurement band. If the frequency band of analysis is bounded by max and min then t is determined from t


2
2( max   min )

Eqn 15-22

By substituting sampled time for continuous time

Part IV

Modal Analysis and Design

229

Chapter 15 Estimation of modal parameters

h ij,nĂ(t)Ă


N

 rĂijkeĂ nt  rĂ*ijkeĂ nt

k

* k

Eqn 15-23

k
1

or N

h ij,n
 rĂ ijkĂz nk  rĂ *ijkĂz *n k

Eqn 15-24

k
1

where z k
e  kāt

Eqn 15-25

Time domain parameter estimation methods are based on the model defined by equation 15-24. They analyze hij,n to estimate zk . k is then calculated from equation 15-25. Note however that this calculation is not unique since z k
e ( kjm2
t)t
e kt

Eqn 15-26

This implies that no poles outside the frequency band ă2 /
t can be identiĆ fied. In other words, with a time domain parameter estimation method, all esĆ timated poles are to be found in the frequency band of analysis ( min ,  max ). This may cause problems in estimating modal parameters if the data in the freĆ quency band of analysis is strongly influenced by modes outside this band (reĆ sidual effects). Since with frequency domain methods k is estimated directly, no such limitation arises. A frequency domain technique may therefore someĆ times be preferred over a time domain technique for analyzing data over a narĆ row frequency band, where residual effects are important.

15.2.5

Vibro–acoustic modal analysis Coupling between the structural dynamic behavior of a system and its interior acoustical characteristics can have an important impact in many applications. Based on combined vibrational and acoustical measurements with respect to acoustical or structural excitation, a mixed vibro-acoustical analysis can be perĆ formed. The finite element equation of motion is used to derive the equations describĆ ing the vibro-acoustical behavior:

230

The Lms Theory and Background Book

Estimation of modal parameters

  2M SĂ  iC SĂ  KS{x}Ă
{f} Ă {l p}

Eqn 15-27

with M S, C S, K S

the structural mass, damping and stiffness matrices

{f}

the externally applied forces

{lp }

the acoustical pressure loading vectors

In the fluidum the indirect acoustical formulation states:

 2M fĂ  iCfĂ  Kf{p}
{q. }  2{lf}

Eqn 15-28

with M f, C f, K f matrices describing the pressure-volume acceleration ω 2{lf }

the acoustical pressure loading vectors

Combining these equations with {l p}
Ă

 pĂdS

Eqn 15-29

Sb

{l f}
Ă

 Ăx ĂdS

Eqn 15-30

N

Sb

and rewriting the formulations results in the description of the vibro-acoustical coupled system:



px  iC0 C0 px   MM

KS  Kc 0 Kf

S

f

2

S

   

0 C Mf

x p


f .

q

Eqn 15-31

This represents a second order model formulation of the vibro-acoustical beĆ havior which is clearly non-symmetrical. The above equation also reflects the vibro-acoustical reciprocity principle which can be expressed as: ..

xj pi | q.
0
 . | fi
0 qi fj i

Eqn 15-32

Most of the multiple input - multiple output modal parameter algorithms do not require symmetry. So the non-symmetry of the basic set of equations and hence the modal description does not pose a problem in obtaining valid modal frequencies, damping factors and mode shapes.

Part IV

Modal Analysis and Design

231

Chapter 15 Estimation of modal parameters

Structural excitation can be substituted for acoustical excitation. The modal models derived from both are compatible but differ in a scaling factor per mode due to the special non-symmetry of the set of equations. To go from the structural formulation to the acoustical formulation a scaling factor which is the squared eigenvalue of the corresponding mode is required. This is fully explained in the paper `Vibro-acoustical Modal Analysis : ReciĆ procity, Model Symmetry and Model Validity' by K. Wyckaert and F. Augusztinovicz.

232

The Lms Theory and Background Book

Estimation of modal parameters

15.3

Parameter estimation methods A summary of different methods and their applications is given in Table 15.1. Method

Application

DOF

Domain Estimates

Inputs

Peak picking

frequency, damping

single

freq

local

single

Mode picking

mode shapes

single

freq

local

single

Circle fitting

frequency damping mode shapes

single

freq

local

single

Complex Mode Indicator Function

frequency damping mode shapes

multi

freq

global

single or multiple

Least Squares Complex Exponential

frequency damping modal participation factors

multi

time

global

single or multiple

Least Squares Frequency Domain

mode shapes

multi

freq

global

single or multiple

Frequency domain Direct Parameter identification

frequency damping modal participation factors

multi

freq

global

single or multiple

Table 15.1

Parameter estimation methods and application

Selection of a method A guide on which parameter estimation techniques method to adopt is outlined below. Details on all the methods are given in the following sections. SDOF Single degree of freedom curve fitters are rough and ready and will give you a quick impression of the most dominant modes (frequency damping and mode shapes) influencing a structure under test. As such they are useful in checking the measurement setup and can help assess: V

Part IV

whether all the transducers are working and correctly calibrated;

Modal Analysis and Design

233

Chapter 15 Estimation of modal parameters

V

whether the accelerometers are correctly labelled with their node and direction;

V

whether all the nodes are instrumented.

For this purpose it is recommended to identify real modes since these are the easiest to interpret when displayed. The circle fitter gives the most accurate estimates of the SDOF techniques, but may create large errors on nodal points of the mode shapes. Complex MIF This method can be used in the same way as the SDOF techniques to give you an idea of the most dominant modes and check the test setup. It has the advantage that multiple input FRFs can be used and the mode shape estimates are of a higher quality. Furthermore, it can extract a modal model that includes the most dominant modes in a particular frequency band. Time domain MDOF This is the most general purpose parameter estimation technique that is probĆ ably the standard tool used in modal analysis. It provides a complete and accuĆ rate modal model from MIMO FRFs. Its major weakness seems to be when analyzing heavily damped systems where the damping is greater than 5% such as in the case of a fully equipped car. Frequency domain MDOF The Frequency Domain Direct Parameter technique provides similar results to the Time domain technique described above, in terms of accuracy but is generĆ ally slower. It is weak when dealing with lightly damped systems (damping less than 0.3%) but fortunately performs better on heavily damped ones, thus complementing the other MDOF technique. Since it operates in the frequency domain it is able to analyze FRFs with an unequally spaced frequency axis.

15.3.1

Peak picking Peak picking is a single DOF method to make local estimates of frequency and damping. The method is based on the observation that the system reĆ sponse goes through an extremum in the neighborhood of the natural frequenĆ cies. For example, on a frequency response function (FRF) the real part will be zero around the natural frequency (minimum coincident part), the imaginary part will be maximal (peak quadrature) and the amplitude will also be maximal (peak amplitude). The frequency value where this extremum is observed is called the resonant frequency r and is a good estimate of the natural frequency of the mode nk for lightly damped systems.

234

The Lms Theory and Background Book

Estimation of modal parameters

A corresponding estimate of the damping can be found with the 3dB rule. The frequency values 1 and 2,on both sides of the peak of the FRF at which the amplitude is half the peak amplitude (3dB down) are introduced in the formula in equation 3.1 to yield the critical damping ratio. The method is also illusĆ trated in Figure 15-5 below. 1 and 2 are also called half power points. 


2  1 2 r

Eqn 15-33

hij dB ampl

3 dB

1 r 2 Figure 15-5

frequency

Half power (3 dB) method for damping estimates

Since the curve fitter locates the resonance frequency on a spectral line, signifiĆ cant errors can be introduced if the FRF has a low frequency resolution and the peaks of modes fall between two spectral lines. This can be compensated for by extrapolating the slopes on either side of the picked line to determine the amĆ plitude of the FRF more precisely. It may be necessary to deal with the situation when one of the half power points is not found. This may arise when the frequency of one mode is close to that of another mode, or it is near to the ends of the measured frequency range.

Note!

Peak picking is a single DOF method: it is therefore only suitable for data with well separated modes.

As this method yields local estimates, it requires only one data record to obtain frequency and damping values for all modes. However, if several data records are available, it may be that different records identify different modes.

Part IV

Modal Analysis and Design

235

Chapter 15 Estimation of modal parameters

15.3.2

Mode picking If you assume that the modes are uncoupled and lightly damped, the modal amplitude can be computed from the peak quadrature or peak amplitude of the FRF. With this assumption, the data in the neighborhood of the resonant freĆ quency can be approximated by h ij,n "

r ijk (j n   k)

Eqn 15-34

(see also equation 15-7) The amplitude is maximum at the resonant frequency. However for lightly damped modes, the resonant frequency, natural frequency and damped natural frequency are all approximately the same. Therefore, the amplitude at resoĆ nance or the modal amplitude is found at n which is equal to dk . By substituting dk for n in equation 15-34 the modal amplitude is given by 

r ijk k

Eqn 15-35

Note that from the modal amplitude a residue or mode shape estimate is obĆ tained by multiplying by the modal damping. To use the Mode picking method you must have an estimate of dk . This estiĆ mate can be obtained with the Peak picking method (see section 15.3.1) or other techniques. The Mode Picking method is obviously quite sensitive to frequency shifts in the data. If for example the resonant frequency of a mode in a data record is shifted a few spectral lines with respect to the frequency that is used as resoĆ nant frequency for that mode, then the modal amplitude would be erroneously picked. To accommodate situations where frequency shifts occur, you need to specify an allowed frequency shift around the resonant frequencies dk that are used to calculate the modal amplitudes. Rather than picking the modal ampliĆ tude at the resonant frequencies the method now scans a band around each modal frequency for each data record. The maximum amplitude in this band is used to determine the modal amplitude and thus the mode shape coefficient. Mode picking allows you to make a very quick determination of a modal modĆ el. The accuracy of this model however depends on how well the assumptions of the methods were applicable to the data.

236

The Lms Theory and Background Book

Estimation of modal parameters

15.3.3

Circle fitting The Circle fitting method is based on estimating a circle in the complex plane through data points in a band around a selected mode. The method was origiĆ nally developed by Kennedy and Pancu for lightly damped systems under the single DOF assumption. In the band around a mode, the data can be approxiĆ mately described by h ij,n


r ijk j n   k



r *ijk

Eqn 15-36

j n   *k

Making an abstraction of the indices i, j and k, introducing complex notation for the residue, and approximating the complex conjugate term by a complex constant, equation 15-36 transforms to hn


U  jV  R  jI    jĂ( n   d)

Eqn 15-37

It can be demonstrated that the modal parameters in this expression can be derived from the coefficients of a circle that is fitted to the data in the complex plane, as shown in Figure 15-6.






arctan V U

f

Re(h)

(R,I)

d


U 2  V 2 



(R  U2Ă, I  V2Ă)

  

Im(h)

f Figure 15-6



d

Relation between circle fitting parameters and modal parameters

The natural frequency d is determined by the maximum angular spacing method where the natural frequency is assumed to occur at the point of maxiĆ mum rate of change of angle between data points in the complex plane.

Part IV

Modal Analysis and Design

237

Chapter 15 Estimation of modal parameters

Having determined the natural frequency and assuming a lightly damped sysĆ tem, the damping is given by equation 15-38. 




2

 1 d

tan( 2) 2 tan( 2)

1

2

Eqn 15-38

The complex residue U + jV is determined from the diameter of the circle d, and the phase  as illustrated in Figure 15-6. U 2  V 2  V 
arctan U

Eqn 15-39

d




Eqn 15-40

Circle fitting is a basic sDOF parameter estimation method. It can be used to obtain frequency, damping and mode shape estimates. The method is fast, but should really be used interactively to obtain the best possible results.

15.3.4

Complex mode indicator function The Complex Mode Indicator Function method allows you to identify a modal model for a mechanical system where multiple reference FRFs were measured. The method provides a quick and easy way of determining the number of modes in a system and of detecting the presence of repeated roots. This inĆ formation can then be used as a basis for more sophisticated multiple input techniques such as LSCE or FDPI. However in cases where modes are well exĆ cited and obvious it can yield sufficiently accurate estimates of modal parameĆ ters. The FRF matrix of a system with No (output) and Ni (input) degrees of freedom can be expressed as follows 2N

Qr [ĂH()Ă]
 {ĂĂ} rĂ {ĂLĂ} Tr   r

Eqn 15-41

r
1

Or in matrix form as [ĂH()Ă]
[ĂĂ]Ă

238

 Q  Ă[ĂLĂ] r

r

T

Eqn 15-42

The Lms Theory and Background Book

Estimation of modal parameters

where [ĂHĂ()Ă]= the FRF matrix of size Ni by No [ĂĂ] = the mode shape matrix of size No by 2N Qr =

the scaling factor for the rth mode

 r = the system pole value for the rth mode [ĂLĂ]Ă T = the transposed modal participation factor matrix of size Ni by 2N Taking the singular value decomposition of the FRF matrix at each spectral line results in [ĂHĂ]
[ĂUĂ]Ă[ĂSĂ]Ă[ĂVĂ] H

Eqn 15-43

where [ĂĂUĂ]= the left singular matrix corresponding to the matrix of mode shape vectors [ĂĂSĂ]= the diagonal singular value matrix [ĂĂVĂ]= the right singular matrix corresponding to the matrix of modal parĆ ticipation vectors In comparing equations 15-42 and 15-43, the mode shape and modal participaĆ tion vectors in equation 15-42 are, through the singular value decomposition, scaled to be unitary vectors and the mass matrix in equation 15-43 is assumed to be an identity matrix, so that the orthogonality of modal vectors is still satisĆ fied. For any one mode, the natural frequency is the one where the maximum singuĆ lar value occurs. The Complex Mode Indicator Function is defined as the eigenvalues solved from the normal matrix, which is formed from the FRF matrix (ā[ĂHĂ]ā Hā[ĂHĂ]) at each spectral line. [ĂHĂ]Ă H[ĂHĂ]
[ĂVĂ]Ă[ĂSĂ]Ă 2[ĂVĂ] H

Eqn 15-44

CMIF k()
 k()
s k() 2ĄĄĄĄĄk
1, 2, .N i

Eqn 15-45

where k ()= the kth eigenvalue of the normal FRF matrix at frequency 

Part IV

Modal Analysis and Design

239

Chapter 15 Estimation of modal parameters

sk ()= the kth singular value of the FRF matrix at frequency  Ni =

the number of inputs

In practice the [ĂHĂ] Hā[ĂHĂ] ā matrix is calculated at each spectral line and the eiĆ genvalues are obtained. The CMIF is a plot of these values on a log scale as a function of frequency. The same number of CMIFs as there are references can be obtained. Distinct peaks indicate modes and their corresponding frequency, the damped natural frequency of the mode. This is illustrated in Figure 15-7. Peaks in the CMIF function can be searched for automatically whilst taking into account criteria that are used to eliminate spurious peaks due to noise or meaĆ surement errors. 1

CMIF log

.1

.01

.001

frequency

Figure 15-7

Example of a CMIF showing selected frequencies

When the frequencies have been selected, equations 15-43 and 15-44 can be used to yield the complex conjugate of the modal participation factors [ĂVĂ], and the as yet unscaled mode shape vectors [ĂUĂ]. The unscaled mode shape vectors and the modal participation factors are used to generate an enhanced FRF for each mode (r), defined by H HE r ()Ă
{ĂUĂ} r Ă[ĂH()Ă]Ă{ĂVĂ} r

Eqn 15-46

Since the mode shape vectors and modal participation factors are normalized to unitary vectors by the singular value decomposition, the enhanced FRF is acĆ tually the decoupled single mode response function HE r ()Ă


Qr   r

Eqn 15-47

A single degree of freedom method (such as the circle fitter technique) can now be applied to improve the accuracy of the natural frequency estimate and then to extract damping values and the scaling factor for the mode shape.

240

The Lms Theory and Background Book

Estimation of modal parameters

CMIF

1

.1

log .01

.001

frequency

amp

frequency

Figure 15-8

Example of a CMIF and the corresponding enhanced FRF

One CMIF can be calculated for each reference DOF. They can be sorted in terms of the magnitude of the eigenvalues. They can all be plotted as a funcĆ tion of frequency as shown in the example in Figure 15-9. CMIF 1

.1 CMIF_1

log .01

CMIF_2

.001 frequency

Figure 15-9

Part IV

Example of first and second order CMIFs

Modal Analysis and Design

241

Chapter 15 Estimation of modal parameters

Cross checking and tracking At any one frequency these functions will indicate how many significant indeĆ pendent phenomena are taking place as well as their relative importance. At a resonance, at least one CMIF will peak implying that at least one mode is active. At a different frequency however it may be that a different mode has increased its influence and is the major contributor to the response. Between resonances, a cross over point can occur where the contribution of two modes are equal. This can result in a higher order CMIF exhibiting peaks if they are sorted as shown in Figure 15-9 and in the effect of one CMIF exhibiting a dip at the same time as a lower order function is exhibiting a peak. A check on peaks in the second order CMIF functions can be made to deterĆ mine whether or not they are due to the cross over effect or a genuine pole of second order. This is done by calculating the MAC matrix using data on either side of the frequency of interest. MAC (1a,1b)

MAC (2a,1b)

MAC (1a,2b)

MAC (2a,2b)

Where a and b represent the frequencies and 1 and 2 the CMIF functions. CMIF 1 contains the larger values and CMIF 2 the smaller ones

CMIF_1

b

a

CMIF_2

When this MAC matrix approximates a unity matrix then the peak in CMIF_2 represents a resonance peak. The mode is not changing between freĆ quencies a and b. "1 "0

"0 "1

CMIF_1 CMIF_2

a

b

When this MAC matrix is anti diagoĆ nal then the peak in CMIF_2 repreĆ sents a cross over point. The mode is switching between frequencies a and b. "0 "1

"1 "0

Peak picking can be facilitated by using tracked CMIFs. This alters the display of the CMIFs for when the mode shapes represented by the two CMIFs are switched, the CMIFs are also switched. This is determined by the cross over check described above.

242

The Lms Theory and Background Book

Estimation of modal parameters

An example of the tracked versions of the CMIFs illustrated in Figure 15-9 is shown below. CMIF

1

.1

log .01

.001 frequency Figure 15-10 Example of first and second order tracked CMIFs

15.3.5

Least squares complex exponential The Least Squares Complex Exponential method allows you to estimate values of modal frequency and damping for several modes simultaneously. Since all the data is analyzed simultaneously, global estimates are obtained. To understand how the method works, recall the expression for an impulse reĆ sponse (IR) given below N

h ijĂ(t)Ă
 rĂ ijkeĂ  t  rĂ *ijkeĂ  t

k

* k

Eqn 15-48

k
1

It can be seen from this expression that the pole values k are not a function of a particular response (output) or reference (input) DOF. In other words the pole values are global (rather than local) characteristics of the structure. They are the same for any measured FRF on the structure. It should therefore be posĆ sible to use all the available data measured on the system to identify global estiĆ mates simultaneously. This method can be used with single and multiple inputs.

Part IV

Modal Analysis and Design

243

Chapter 15 Estimation of modal parameters

Model for continuous data A particular problem when trying to work with equation 15-48 to achieve the above objective is that it contains residues rijk which do depend on the response and reference DOFs. It is therefore essential to define another parametric modĆ el for the data hij , in which the coefficients are independent of response and refĆ erence DOFs and can be used to identify estimates for k. . It can be proved that such a model takes the form of a linear differential equation of order 2N with constant real coefficients (ddt) 2Nh ij  a 1(ddt) 2N1h ij   a 2Nh ij
0

Eqn 15-49

Indeed, equation 15-48 expresses the data as a linear superposition of a set of 2N damped complex exponentials occurring in complex conjugate pairs. Such complex exponentials can be viewed as the characteristic solutions of a linear differential equation with constant real coefficients (ddt) 2Nf (t)  a 1(ddt) 2N1f (t)   a 2Nf (t)
0

Eqn 15-50

The impulse response, being a linear superposition of characteristic solutions, is by itself also a characteristic solution. Therefore equation 15-49 is valid if the coefficients are such that  2N  a 1 2N1   a 2N
0 
 k, 


 *k, k


1 N

Eqn 15-51

Turning the reasoning around therefore, one could first try to estimate the coefĆ ficients in equation 15-49 using all available data. Estimates of the complex exĆ ponential coefficients k can then be found by solving equation 15-51.

Model for sampled data Measured data is however sampled, not continuous. So rather than working from equation 15-48 it is necessary to work with h ij,nĂ
Ă

N

 rijkĂznkĂ Ă r*ijkĂz*nk

k
1

Eqn 15-52

z k
e  kt Instead of damped complex exponentials, the characteristics are now power seĆ ries with base numbers zk .

244

The Lms Theory and Background Book

Estimation of modal parameters

Following a similar reasoning to that explained above for continuous data it can be proved that the sampled data is the solution of a linear finite difference equation with constant real coefficients of order 2N (instead of a differential equation as for continuous data). h ij,nĂ Ă a 1h ij,n1Ă Ă Ă Ă a 2Nh ij,n2NĂ
Ă 0

Eqn 15-53

The characteristics zk and therefore the poles k can be found by solving, 2N1   a 2N
0 z 2N k  a 1z k

Eqn 15-54

Practical implementation of the method The Least Squares Complex Exponential is a method that estimates the coeffiĆ cients in equation 15-53 using data measured on the system. In principle any data record hij,n ăcan be used. Applying the method to just a single data record at a time will result in local estimates of the poles. To estimate the coefficients in equation 15-53 in a least squares sense the equaĆ tions for all possible time points and all possible response and reference DOFs are to be solved simultaneously as indicated in equation 15-55. This equation system will be greatly overdetermined. To find the least squares solution the normal equations technique can be applied so that the final solution is calcuĆ lated from a compact equation with a square coefficient matrix, equation 15-56. The coefficient matrix in this equation is called a covariance matrix.

h11,2N1  )  h11,N 1  )   hij,n1   )  t

h N0N i,Nt1

h 11,0 ! s  h11,2N! ) )   )  h 11,N t2N  a 1 !  h 11,Nt     a  2 ) ) ) $ % $ ) % h ij,n2N     h ij,n a

 ) )  h N 0Ni,N t2N 

    hN N N 

  

2N

Eqn 15-55

)

0

i

t

where Nt= last available time sample N0 = number of response DOFs Ni = number of input DOFs We can write this in a simpler manner

Part IV

Modal Analysis and Design

245

Chapter 15 Estimation of modal parameters

r 1,1 .  ). .

sr 1,2 r 1,2N ! a1 !   r 1,0 ! r2,2 r 2,2N  a 2    r 2,0   ) %Ă=$ ) % ) ) ) $     r 2N,0 . . r 2N,2N a2N

Eqn 15-56



The coefficients in the covariance matrix are defined as r k,l


N0

Ni

Nt

   (hij,nkĂhij,nl)

Eqn 15-57

i
1 j
1 n
1

Building this covariance matrix is the first stage in applying the Least Squares Complex Exponential method. This phase is usually the most time consuming since all the available data is used to build the inner products expressed by equation 15-57. Note that after solving equation 15-56 all that is required to calculate the estiĆ mates of modal frequency and damping is to substitute the estimated coeffiĆ cients in equation 15-54 and to solve for zk .

Determining the optimum number of modes The solution of equation 15-56 results in least squares estimates of the coeffiĆ cients in the model expressed by equation 15-53. It is also possible therefore to calculate the corresponding least squares error. This error is of importance in determining the minimum number of modes in the data. In the preceding discussion it has been assumed that N modes are present in the data. However, the number of modes contained in the data is in fact unĆ known. It is preferable that this should be determined by the method itself. Using the Least Squares Complex Exponential method, this can be achieved by observing the evolution of the least squares error on the solutions of equation 15-56 as a function of the number of assumed modes. To do this, an equation like equation 15-56 is initially created, assuming a numĆ ber of modes N that is sufficiently large. A subset of such an equation is then taken to solve for the coefficients of a model that describes just one mode



r 1,1 r 1,2 . r 2,2

     r 1,0 a1 a 2 Ă=Ă  r 2,0

The corresponding least squares error is represented by 1. When 2 modes are assumed in the data then the sub set to be solved is

246

The Lms Theory and Background Book

Estimation of modal parameters

r 1,1 .  .  .

r 1,2 r 1,3 r 2,2 r 2,3 . r 3,3 . .

r 1,4 a ! 1!  r 1,0! r 2,4 a 2  r 2,0 $a %Ă=Ă$ r % r 3,4 3,0  3 r 4,4 a 4  r 4,0 



With corresponding least squares error 2, and so on. Now if a model is asĆ sumed with a number of modes equal to the number of modes that is present in the data then the corresponding least squares error should be significantly smaller than the error for models with fewer modes. A diagram that plots the least squares error for increasing number of modes is called the least squares error chart. Figure 15-11 shows a typical diagram if data is analyzed for a system with 4 modes (and 4 modes are observable from the data!). Noise on the data may cause the error diagram to show a significant drop at a certain number of modes, followed by a continued decrease of the error as the number of modes is increased. The problem now is to determine how many extra modes, or so called computational modes, are to be considered to comĆ pensate for the noise on the data so that the best estimates of modal frequency and damping can be obtained. This problem is also illustrated in Figure 15-11. Least Squares Error 

No noise on data Noise on data

1

2

3

4

5

6

7

Nr of modes

Figure 15-11 Least squares error diagram, system with 4 modes

To determine the optimal number of modes you could try to compare frequenĆ cy and damping estimates that are calculated from models with various numĆ ber of modes. Physical intuition would lead you to expect that estimates of freĆ quency and damping corresponding to true structural modes, should recur (in approximately the same place) as the number of modes is increased. ComputaĆ tional modes will not reappear with identical frequency and damping. A diaĆ gram that shows the evolution of frequency and damping as the number of modes is increased is called a Stabilization diagram. The optimal number of modes that can be calculated for use can then be seen, as those modes for which the frequency and damping values of the physical modes do not change signifiĆ cantly. In other words, those which have stabilized.

Part IV

Modal Analysis and Design

247

v f d d f f v f d v f f

s s s s s s s s

s s s s s s s s d v f f

s s s s s s s s s s v f o

number of modes

amplitude

Chapter 15 Estimation of modal parameters

frequency Figure 15-12 A stabilization diagram

Example Let two data records be measured on a system, both shown in Figure 15-13. h11 1 0

1

2

3

4

1

2

3

4

-1 h21 1 0

-1


t

Figure 15-13 Example least squares complex exponential

Let four data samples be measured of which the values are listed in the Table below.

248

The Lms Theory and Background Book

Estimation of modal parameters

n 0 1 2 3

h11 1 0 -1 0

h21 0 1 0 -1

Consider a model for 1 mode (N=1). Equations 15-55 and 15-56 become reĆ spectively

 0 1! a 1! 1 0 1 0     Ă=Ă $ a  1 0  2 0%  1 0 s1

20 02 aa Ă=Ă02 1 2

The solution is therefore a1 =0, a2 =1. Now equation 15-54 is used to calculate zk and so k, z2  1
0 z
* j The frequency and damping values follow from zĂ
Ă e t z
 j,Ă 
0  j


2t

z
 j,Ă 
0  j


2t

The solution indicates a mode with a period 4
t and zero damping. This is compatible with the trend of the cursor as shown in Figure 15-13.

15.3.5.1 Multiple input least squares complex exponential The Least Squares Complex Exponential method, described above, uses all data measured on a structure to estimate global estimates of modal frequency and damping. In principle, data relative to several reference DOFs can be used. However the model used by the previous method does not take specific advanĆ tage of this.

Part IV

Modal Analysis and Design

249

Chapter 15 Estimation of modal parameters

The multiple input Least Squares Complex Exponential, (or polyreference), is an extension of the Least Squares Complex Exponential that does allow consisĆ tent simultaneous analysis of data relative to several reference DOFs. The method computes global estimates of frequency and damping and also of modal participation factors. Modal participation factors are terms which exĆ press the participation of modes in the system response as a function of the refĆ erence (or input) DOF (see section 15.2.3). The simultaneous estimation of freĆ quency, damping and modal participation factors means that highly coupled, even repeated modes can be identified. The basis for the Multiple Input Least Squares Complex Exponential method is the model of the data introduced in section 15.2.3 equation 15-16. H i


N

 vikLkeĂ t  v*ikL*keĂ

k

* k

Eqn 15-58

k
1

where H i = Ni vector (row) of IRs between output DOF i and all input DOFs L k = vector of modal participation factor for mode k. If Ni reference DOFs are assumed then L k is of dimension Ni vik

= is the mode shape coefficient at response DOF i for mode k

Note that in this model, frequency, damping and modal participation factors are independent of the particular response DOF. It should therefore be possible to estimate these coefficients using all the available data simultaneously.

Model for sampled data The model expressed by equation 15-58 is not directly suitable for global esĆ timation of frequency, damping and modal participation factors as it still conĆ tains the mode shape coefficients that are dependant on the response DOF. Therefore a more suitable model must be derived. Introducing firstly the sampled nature of the data, equation 15-58 is rewritten as, H n i


N

 vikLkznk  v*ikL*kz*nk

k
1

Eqn 15-59

z k
e  kt It can be proved that if the data can be described by equation 15-59, it can also be described by the following model

250

The Lms Theory and Background Book

Estimation of modal parameters

H n iĂ Ă H n1A 1 Ă Ă HnpA pĂ
0

Eqn 15-60

if the following conditions are fulfilled A 1Ă  A p]
0 Lk[z pk  z p1 k

Eqn 15-61

pN i  2N

Eqn 15-62

(The proof of this follows from basic calculus along the same lines as for Least Squares Complex Exponential in section 15.3.5). Equation 15-60 represents, in matrix notation, a coupled set of Ni finite differĆ ence equations with constant coefficients. The coefficients A1 . . . Ap are thereĆ fore matrices of dimension (Ni Ni ). The condition expressed by equation 15-61 states that the terms [ Lk ] and zk nă are characteristic solutions of this system of finite difference equations. As equation 15-59 is a superposition of 2N of such terms, it is essential that the number of characteristic solutions of this system of equations pNi at least equals 2N as expressed by equation 15-62. Note finally, that if data for each reference DOF is treated individually, i.e. Ni = 1, then equation 15-60 and 15-61 simplify to equations 15-53 and 15-54. Thus the least squares complex exponential method is a special case of the multiple input least squares complex exponential method.

Practical implementation of the method To estimate the coefficients in equation 15-60 in a least squares sense the equaĆ tions for all possible time points and all possible response DOFs are to be solved simultaneously, as indicated by equation 15-63. A least squares solution is found, for example using the normal equations method, from equation 15-64. The coefficient matrix in this equation is again in the form of a covariance maĆ trix,

 Hp1 1 .   HN 1  1  .   Hn1i  .  H  N 1 N t

t

0

. . .

!   Hp1 ! A  .  1 !    H N    H N p 1   1 A  .  t. Ă=Ă  ) 2      H n i    H npi   A p  .  .      H   N N H N p N  H 0 1 . t

t

t

t

0

Eqn 15-63

0

where

Part IV

Modal Analysis and Design

251

Chapter 15 Estimation of modal parameters

Nt = the last available time sample N0 = the number of response DOFs R k,lĂ
Ă

R 1,1 .  )  .

N0

Nt

Ă  Ă(Ă[Hnk]ĂtiĂ[Hnl]iĂ)

Eqn 15-64

i
1 n
p

R 1,2 R 1,p A 1    R 1,0 R 2,2 R 2,pA 2  R 2,0 Ă 
Ă    ) ) )  )   )   R p,0 . . R p,p A p





Eqn 15-65



The order (p) of the finite difference equation is related to the number of modes in the data by equation 15-62. It is preferable that this be determined by the method itself. As the coefficients of the finite difference equation are solved for in a least squares sense, this can be done by observing the least squares error as a function of the assumed order. As an order is reached such that the model can describe as many modes as are present in the data, the error should drop considerably. Due to the condition expressed by equation 15-62 there is no linear relation beĆ tween the number of modes that can be described by the model and the order of the model. The relation between the number of modes, the order of the model and the number of reference DOFs is listed in Table 15.2. It can be seen that a model of order 8 can describe 11 or 12 modes if data for 3 inputs are anaĆ lyzed simultaneously. In the error diagrams therefore the same least squares error is shown for 11 and 12 modes. As for the Least Squares Complex Exponential method, a stabilization diagram can again be created to determine the optimal number of modes. As well as comparing frequency and damping values calculated from models of consecuĆ tive order it is now also possible to compare the stabilization of modal particiĆ pation factors. In section 15.2.3, the modal participation factors were shown to be proportional to the mode shape coefficients at the reference DOFs. They also represent a physical characteristic of the structure like the frequency and dampĆ ing. Therefore, the values corresponding to structural modes should also stabiĆ lize as the order of the model is increased. This additional criterion adds much to the readability of the stabilization diagram and to the ability to distinguish computational modes from physical modes Additionally, the modal participation factors can be used by themselves to identify physical modes. If they are normalized with respect to the largest, the values should all be approximately real, in phase or in anti-phase, for structurĆ al modes.

252

The Lms Theory and Background Book

Estimation of modal parameters

N

Ni=1

Ni=2

Ni=3

Ni=4

Ni=5

Ni=6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 51 52 54 56 58 60 62 64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

1 2 2 3 4 4 5 6 6 7 8 8 9 10 10 11 12 12 13 14 14 15 16 16 17 18 18 19 20 20 21 22

1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16

1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 9 9 10 10 10 11 11 12 12 12 13 13

1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11

Table 15.2

Relation between modal order (tabulated), number of modes (N) and number of reference DOFs (Ni )

Example To clarify the method, consider again the example discussed on page 248. Let the example system satisfy reciprocity so that h12 is also equal to h21 . The vecĆ tor [h12 h21 ] then represents the data between response DOF 1 and reference DOFs 1 and 2. Considering a model for 1 mode (so p= 1, as Ni = 2) equations 15-55 and 15-56 become respectively

Part IV

Modal Analysis and Design

253

Chapter 15 Estimation of modal parameters

 

1 0! a a 0  1! 11 12 0 1 a a Ă=Ă1 0  12 22  1 0 0  1



20 02 aa

11 12





a 12 0 2 a 22 Ă= 1 0



The resulting matrix polynomial is therefore

l 1Ăl 2

z 1 1z =Ă 00

and the solutions of this eigenvalue problem are z
* j, 
0 * j


2t

 L 
[* j, 1] Notice that the solution for the frequency and damping is the same as found with the Least Squares Complex Exponential (see page 249). In addition you also find an estimate of the modal participation factors. For this example they indicate that there should be a phase difference of 90_ in the system response between excitation from reference DOFs 1 and 2 as h11 is a cosine, and h12 a sine. This estimate seems to be correct.

15.3.6

Least squares frequency domain The Least Squares Frequency Domain method is a multiple DOF technique to estimate residues, or mode shape coefficients. The method requires that freĆ quency and damping values have already been estimated. It can be used with single or multiple inputs. Consider the model expressed by equation 15-66 N

h ijĂ(t)Ă
 rĂ ijkeĂ  t  rĂ *ijkeĂ  t

k

* k

Eqn 15-66

k
1

If estimates of the modal frequency and damping are available, then the resiĆ dues appear linearly as unknowns in this model.

254

The Lms Theory and Background Book

Estimation of modal parameters

To estimate the residues, equation 15-66 is transformed back to the frequency domain. Assuming sampled data therefore h ij,p


j N

k
1

r ijk p

 k



lr ij !  ur   ij 2 j p   *k p r *ijk

Eqn 15-67

where urij = an upper residual term used to approximate modes at frequencies above max lrij = an lower residual term used to approximate modes at frequencies below min These are illustrated in Figure 15-3. Note that the residues as well as lower and upper residuals are local characteristics; in other words, they depend on the particular response and reference DOF. The Least Squares Frequency Domain method is based on the model expressed by equation 15-67. Least squares estimates of residues, lower and upper residĆ uals are calculated by analyzing all data values in a selected frequency range.

15.3.6.1 Multiple input least squares frequency domain The multiple input Least Squares Frequency Domain method is a multiple DOF technique to estimate mode shapes. The method analyses data relative to sevĆ eral reference DOFs simultaneously to estimate mode shape coefficients that are independent of reference DOFs. Consider the model expressed by equation 15-58, H i


N

 Ă vikLkeĂ t  v*ikL*keĂ t

k

* k

Eqn 15-68

k
1

If estimates of frequency, damping and modal participation factors are availĆ able, then the mode shape coefficients appear linearly as the only unknowns in this model. Furthermore, they are only dependent on the response DOF (and not on the reference DOF) so that data relative to several reference DOFs can be analyzed simultaneously. To estimate the residues, equation 15-68 is transformed to the frequency doĆ main. Adding residual terms and assuming sampled data results in

Part IV

Modal Analysis and Design

255

Chapter 15 Estimation of modal parameters

H pi


 N

k
1

v ikL k v *ikL * k  j p   k j p   *k





LR i  2p

 UR i

Eqn 15-69

where [UR]Ăi = upper residuals between response DOF i and all reference DOFs, vector of dimension Ni [LR]ĂĂi = lower residuals between response DOF i and all reference DOFs, vecĆ tor of dimension Ni The multiple input LSFD method is based on equation 15-69.

15.3.7

Frequency domain direct parameter identification The Frequency domain Direct Parameter Identification (FDPI) technique allows you to estimate the natural frequencies, damping values and mode shapes of several modes simultaneously. If data relative to several references are availĆ able, a multiple input analysis will also extract values for the modal participaĆ tion factors. In this case, the FDPI technique offers the same capabilities as the LSCE time domain method.

Theoretical background The basis of the FDPI method is the second order differential equation for meĆ chanical structures ..

.

Eqn 15-70

My(t)  Cy(t)  Ky(t)
f (t)

When transformed into the frequency domain, this equation can be reformuĆ lated in terms of measured FRFs

[  2I  jA 1  A 0]Ă[H()]
jB 1  B 0

Eqn 15-71

where = frequency variable

256

The Lms Theory and Background Book

Estimation of modal parameters

A1 = M -1 C, mass modified damping matrix ( No by No ) A0 = M -1 K, mass modified stiffness matrix ( No by No ) H( ) = matrix of FRF's (No by Ni ) B0, B1 are the force distribution matrices (No by Ni ) Note that for the single input case, the H( ) ămatrix becomes a column vector of frequency dependent FRF's. Equation 15-71 is valid for every discrete frequency value  When these equaĆ tions are assembled for all available FRFs, including multiple input - multiple output test cases, the unknown matrix coefficients A0, A1 , B0, and B1 can be estiĆ mated from the measurement data H( ). Equation 15-71 thus means that the measurement dataăH( ) can be described by a second order linear model with constant matrix coefficients. From the identified matrices, the system's poles and mode shapes can be estimated via an eigenvalue and eigenvector decomĆ position of the system matrix.

IA 0A  1

0

Eqn 15-72

This will yield the diagonal matrix [] of poles and a matrix  of eigenvectors. It will become clear from the following section, that the matrix ăthus obtained is not equal to the matrix of mode shapes, although it is related to it. In a final step, the modal participation factors are estimated from another least squares problem, using the obtained [] and  matrices.

Data reduction Prior to estimating the system matrix, all available data are condensed via a projection on their principal components. For all response stations, a maximum of Nm ăprincipal components are first calculated and then analyzed. The obĆ tained matrix ă represents the modal matrix for this set of fictitious response stations. The data reduction procedure offers the following advantages

Part IV

V

the calculation time is drastically decreased for the estimation of model parameters. This is especially important for the calculation of least squares error charts and stabilization diagrams.

V

the number of contributing modes is more easily determined from the singular value analysis.

Modal Analysis and Design

257

Chapter 15 Estimation of modal parameters

Residual correction terms The FDPI technique operates directly on frequency domain data. It is therefore capable of taking into account the effects of modes outside the frequency band of analysis. This feature significantly improves the analysis results when modes below or above the selected band influence the data set. In the case where both upper and lower residual terms are included in the model, equaĆ tion 15-71 becomes [  2I  jA 1  A 0]ĂH()


Eqn 15-73

 2C 2   1C 1  C0  C 1   2C 2 The presence of these residual terms will influence the estimates for frequency, damping and mode shapes (as well as the modal participation factors for multiĆ ple input analysis). Determining the optimum number of modes As with the Least Squares Complex Exponential (LSCE) method, a least squares error chart can be built to determine the optimal number of modes in the seĆ lected frequency band. Because of the principal component projection, this chart may look somewhat different. For small models, only the first (most imĆ portant) principal data are used, and the global error will decrease drastically. As more and more principal components are included by estimating more modes, their information becomes less important, which may distort the least squares error chart. A more reliable tool for estimating the optimal number of nodes for the FDPI technique is the singular values diagram. As an alternative to the error diaĆ gram, and to some extent to the stabilization diagram too, the rank of the calcuĆ lated covariance matrix can be determined. The rank of the matrix is also a good indication of the optimal number of modes to be used in the analysis. The rank of the matrix can be determined using a singular value decomposiĆ tion. A diagram showing the normalized singular values in ascending order is called a singular values diagram: the rank of the matrix is determined at the point where the singular values become significantly smaller compared to the previous values. When building a stabilization diagram, (see LSCE method page 247), the same data are described by models of increasing order. An updating procedure is implemented to save calculation time. Pseudo–DOFs for small measurement sets Due to the type of identification algorithm, the FDPI technique can only estiĆ mate as many modes in the model as there are measurement Degrees of FreeĆ dom. This means that normally

258

The Lms Theory and Background Book

Estimation of modal parameters

Nm  N0 However, using a similar approach as for the time domain LSCE method, it is possible to create so-called pseudo-" Degrees of Freedom from the measureĆ ments that are available, thus generating enough new" measurements to allow a full identification on as few as one measurement.

Mode shape estimation Using the reduced mode shapes  for the principal responses, and the transĆ formation matrix between the principal and physical responses, the FDPI algoĆ rithm allows you to identify the complete mode shapes of the system by exĆ panding the reduced  matrix. This mode shape expansion offers several advantages :V

it is very fast (no least squares solution required as for the LSFD methĆ od)

V

it identifies a mode shape vector as a global direction in the modal space, rather than estimating its elements one by one via mutually inĆ dependent least squares problems.

If the mode shape expansion method is not employed then the LSFD technique is used to estimate mode shapes.

Normal modes From the meaning of the matrices [A0 ] and [A1 ] and the eigenvalue problem (15-72), it is possible to estimate damped (generally complex) mode shapes , or undamped real normal modes. Normal modes can be identified via the FDPI technique by solving an eigenvaĆ lue problem for the reduced mass and stiffness matrices only M 1ĂK n
 nĂ

2

Eqn 15-74

This eigenvalue problem is very much related to the one that is solved by FEM software packages that ignore the damping contribution in a system. This is an entirely different approach to the one that is used to estimate real modes via the LSFD technique. The latter technique estimates the real-valued mode shape coefficients that curve-fit the data set in a best least squares sense (proportional damping assumed), while the FDPI method uses an FEM-like approach. Damping values are computed by applying a circle-fitter to enhanced FRFs for each mode. The enhanced FRFs are calculated by projecting the principal FRFs on the reduced mode shapes.

Part IV

Modal Analysis and Design

259

Chapter 15 Estimation of modal parameters

15.4

Maximum likelihood method A multi-variable frequency-domain maximum likelihood (ML) estimator is proposed to identify the modal parameters together with their confidence interĆ vals. The solver is robust to errors in the non-parametric noise model and can handle measurements with a large dynamical range. Although the LSCE-LSFD approach has proven to be useful in solving many vibration problems, the method has some drawbacks:

15.4.1

V

the polyreference LSCE estimator does not always work well when the number of references (inputs) is larger than 3 for example

V

the frequencies should be uniformly distributed

V

the method is not able to handle noisy measurements properly, which can result in unclear stabilization plots and

V

the method does not deliver confidence intervals on the estimated moĆ dal parameters.

Theoretical aspects A scalar matrix-fraction description ć better known as a common-denominator model ć will be used. The Frequency Response Function (FRF) between output o and input i is modeled as ^

H oiĂ( f)


N oiĂ( f)

Eqn 15-75

DĂ( f)

for i = 1, . . . . . , Ni and o = 1, . . . . . , No with n

N oiĂ( f)
 ! jā( f)āA j j
0

the numerator polynomial between output o and input i and

260

The Lms Theory and Background Book

Estimation of modal parameters

n

Dā( f)
 !ā( f)āB oij j
0

the common-denominator polynomial. The polynomial basis functions j ( f) are given by ! j( f)
e iāā fāT sā.j for a discrete-time model (with Ts the sampling period). The complex-valued coeffiĆ cients Aj and Boij are the parameters to be estimated. The approach used to opĆ timize the computation speed and memory requirements will first be explained for the Least Squares Solver and then these results will be extrapolated to the ML estimator. The Least–Squares Solver ^

Replacing the model H oiĂ( f) in equation 15-75 by the measured FRF H oiĂ( f) gives, after multiplication with the denominator polynomial, n

n

j
0

j
0

 !jĂ(f)ĂBoij   !jĂ(f)ĂHoj(f)ĂAj # 0

Eqn 15-76

for i = 1, . . . . . , Ni 0 = 1, . . . . . , No and f= 1, . . . . . , Nf Note that equation 15-76 can be multiplied with a weighting function Woi (f ). The quality of the estimate can often be improved by using an adequate weighting function. As the elements in equation 15-76 are linear in the parameters, they can be reĆ formulated as

X1 0  0) X) 2  0 0

0 0 )

X N 0N i

 B1  Y1  B  Y2    )2  B # 0 )  X N0N i  N0N i  A 

with

A0 B oi0 B  A1 .. ,ą A
 B k
 .  oi1  .   ..  B oin An

Part IV

Modal Analysis and Design

261

Chapter 15 Estimation of modal parameters

X kā( f)
W oiĂ( f)ā[!( f)Ă, ! 1( f)Ă,ā .ā.ā.ā.Ă, ! n(f)] Y k( f)
 X kā( f)ā.āH oiā( f) and k
(o  1)Ni  i
1,ā .ā.ā.ā, N oāN i The (complex) Jacobian matrix J of this least-squares problem

X1 0 .ā.ā. 0 0  0 X2 .. J
 .. .. . . .  XN N 0 0  o

i

Y1  Y2  ..  .  Y NoN i

Eqn 15-77



has Nf No Ni rows and (n+1)(No Ni +1) columns (with Nf >> n, where n is the order of the polynomials). Because every element in equation 15-76 has been weighted with Woi (f ), the Xk 's in equation 15-77 can all be different.

The Maximum–Likelihood Solver Using referenced measurements (e.g., FRF data) makes it easier to get global estimates from measurements that were obtained by roving the sensors over the structure under test (which is a common practice in experimental modal analyĆ sis). Because of this, the FRFs will be used here as primary data instead of the input/output spectra (i.e. non-referenced data). However, one should take care that the FRFs are not contaminated by systematic errors.

The ML equations Assuming the different FRFs to be uncorrelated, the (negative) log-likelihood function reduces to No

l ML()


Ni

Nf

 Ă Ă  Ă

^

|ĂH oiā(,ā  f)  H oiā( f)Ă| 2

o
1 i
1 f
1

var{Hoi( f)}

Eqn 15-78

, .ā.ā.ā.ā,Ă B TN oN ,Ă A TĂ] T is given by minimizing equaĆ The ML estimate of 
[ĂB T 1 i tion 15-78. This can be done by means of a Gauss-Newton optimization algoĆ rithm, which takes advantage of the quadratic form of the cost function (15-78). The Gauss-Newton iterations are given by H (a ) solve āJ H for m mĂJ mĂ m
ā j mĂr m

262

The Lms Theory and Background Book

Estimation of modal parameters

(b ) set  m1
 m   m with r m
r( m), J m
+r()+| m and

 H^11(, 1)  H11(1) !   var{H 11(1)}     ..   .  ^   H11(, N )  H11(N ā)     var{H ( )}  11 N     .. . r()
$ % ^ H (,  )  H ( )   12 1  12 1    var{H 12(1)}   ..   .   . ^  HN N (, N )  HN N (N ā)    var{H ( )}  NN N  f

f

f

o

i

o

f

o

i

i

f

f

Deriving confidence interval The covariance matrix of the ML estimate ^ is usually close to the correĆ ML sponding Cramér-Rav Lower Bound (CRLB) : cov{^ }  CRLB ML

A good approximation of this Cramér-Rav Lower Bound is given by

1 CRLB " [J H mĂJ m]

Eqn 15-79

with Jm the Jacobian matrix evaluated in the last iteration step of the GaussNewton algorithm. As one is mainly interested in the uncertainty on the resoĆ nance frequencies and damping ratios, only the covariance matrix of the deĆ nominator coefficients is in fact required. Hence, it is not necessary to invert the full matrix to obtain the uncertainty on the poles (or on the resonance frequencies and the damping ratios).

Part IV

Modal Analysis and Design

263

Chapter 15 Estimation of modal parameters

15.5

Calculation of static compensation modes Modal synthesis can be used to couple substructures together in low frequency ranges. For this, modal models for each of the substructures are required as separate disconnected items. However the results of this coupling may be less than optimal due to truncation errors. Truncation errors arise because only a limited number of modes are taken into account. To improve the results, both static and dynamic compensation terms can be used. Eqn 15-80

[H Cā()] #ă [H Rā()]ă ă [ĂH 0Ă]   2[ĂH 1Ă]

exact FRF supposition

modal FRF

static compensation term

dynamic compensation term

Truncation errors can be approximated by a quadratic function using a taylor expansion. It has been shown that there is a good correspondence between the real truncation error and the quadratic estimation. Static compensation terms can be derived from direct (driving point) FRFs. These static compensation terms are calculated using the upper residual terms which were obtained while fitting the FRF matrix of the coupling points (drivĆ ing points and cross terms). The upper residual terms are converted by means of a singular value compensation into regular mode shapes and participation factors. These mode shapes and participation factors can be used afterwards for modal substructuring, in addition to the regular modes of the two substrucĆ tures. Frequency of the static compensation modes The frequency of the static compensation modes ( 0) must be significantly higher than the frequency band of the modes, which are taken into account during the substructuring calculations. The upper limit of the frequency band used in modal substructuring is defined by the frequency of the upper residual (upper residual )  0
10.0Ă upperĂresidual The Singular Value decomposition (SVD) In order to calculate the static compensation terms, a singular value decomposiĆ tion has to be applied on the upper residual term matrix. This is obtained by putting all upper residual terms together in one big matrix.

264

The Lms Theory and Background Book

Estimation of modal parameters

R upperĂresidual
U

 VĂT

The mode shape values of the static compensation mode () are related to the left singular vector, the singular value, and the frequency value ( 0)  j
U j  j  0 The participation factor values (L) can be derived from the mode shape values () Lj


j  2Ăm rĂ 0 j

m r
 V    U  Lj


Part IV

Modal Analysis and Design

j    ĂV 2Ă 0 j 0 j

265

Chapter 16

Operational modal analysis

This chapter describes the theoretical and technical background reĆ lating to operational modal analysis. Reasons for performing operational modal analysis Theoretical aspects

267

Chapter 16 Operational modal analysis

16.1

Why operational modal analysis? Traditional modal model identification methods and procedures are based on forced excitation laboratory tests during which Frequency Response Functions (FRFs) are measured. However, the real loading conditions to which a strucĆ ture is subjected often differs considerably from those used in laboratory testĆ ing. Since all real-world systems are to a certain extent non-linear, the models obtained under real loading will be linearized for much more representative working points. Additionally, environmental influences on system behavior (such as pre-stress of suspensions, load-induced stiffening and aero-elastic inĆ teraction) will be taken into account. In many cases, such as small excitation of off-shore platforms or traffic/wind excitation of civil constructions, forced excitation tests are very difficult, if not impossible, to conduct, at least with standard testing equipment. In such situaĆ tions operational data are often the only ones available. It is also the case that large in-operation data sets are measured anyway, for level verification, operating field shape analysis and other purposes. Hence, extending classical operating data analysis procedures with modal parameter identification capabilities will allow a better exploitation of these data. Finally, the availability of in-operation established models opens the way for in situ model-based diagnosis and damage detection. Hence, a considerable inĆ terest exists in extracting valid models directly from operating data.

Traditional processing of operational data An accepted way of dealing with operational analysis in industry is based on a peak-picking technique applied to the auto-and crosspowers of the operational responses. Such processing results in the so-called Running Mode Analysis". By selecting the peaks in the spectra, approximate estimates for the resonance frequencies and operational deflection shapes can be obtained. These shapes can then be compared to or even decomposed into the laboratory modal results. Correlation of the operating data set with the modal database measured in the lab allows an assessment the modes which are dominant for a particular operĆ ating condition. In case of partially correlated inputs (e.g. road analysis), princiĆ pal component techniques are employed to decompose the multi-reference probĆ lem into subsets of single reference problems, which can be analyzed in parallel. These decomposed sets of data can be fed to an animation program, to interpret the operational deflection shapes for each principal component as a function of frequency.

268

The Lms Theory and Background Book

Operational modal analysis

The auto-and crosspower peak-picking method requires considerable engiĆ neering skill to select the peaks which correspond to system resonances. In addition, no information about the damping of the modes is obtained and the operational deflections shapes may differ significantly from the real mode shapes in case of closely spaced modes. Pre-knowledge of a modal model deĆ rived from FRF measurements in the lab is often indispensable to successfully perform a conventional operational (running modes) analysis. Curve-fitting techniques therefore, which allow modal parameters to be exĆ tracted directly from the operational data would be of a great use for the engiĆ neer. Such techniques would identify the dominant modes excited under drivĆ ing conditions and this information might even be used to improve some traditional FRF tests in the laboratory.

Using Operational modal analysis The purpose of this procedure is to extract modal frequencies, damping and mode shapes from data taken under operating conditions. This means that unĆ der the influence of its natural excitation such as airflow around the structure (e.g. wind turbines, aeroplanes, helicopters), road input, liquid flow (in pipes), road traffic (e.g. bridges), internal excitation (rotating machinery). Theoretically, one could consider the case where the input forces are measured in such conditions which means that conventional FRF processing and modal analysis techniques could be used. However the Operational modal analysis software is aimed specifically at applications where the inputs can not be meaĆ sured, and works when only responses such as accelerations signals are availĆ able. The ideal situation is when the input has a flat spectrum. Three methods are discussed, all of which use time domain correlation funcĆ tions. These auto- and cross-correlation functions can be calculated directly from raw time data or be derived from measured auto- and cross powers by an inverse FFT processing.

Part IV

Modal Analysis and Design

269

Chapter 16 Operational modal analysis

16.2

Theoretical aspects This section describes the mathematical background to the methods used to identify modal parameters from operational data. Over recent years, several modal parameter estimation techniques have been proposed and studied for modal parameter extraction from output-only data. These include Auto-Regressive Moving Averaging models (ARMA), Natural Excitation Technique (NExT) Stochastic subspace methods. The Natural Excitation Technique (NExT) The underlying principle of the NExT technique is that correlation functions between the responses can be expressed as a sum of decaying sinusoids. Each decaying sinusoid has a damped natural frequency and damping ratio that is identical to the one of the corresponding structural mode. Consequently, conĆ ventional modal parameter techniques such as polyreference Least-Squares Complex Exponential (LSCE) can be used for output-only system identificaĆ tion. Stochastic subspace methods With the subspace approach, first a reduced set of system states is derived, and then a state space model is identified. From the state space model, the modal parameters are derived. The terminology subspace" comes mainly from the control theory ć it is a family name" which groups methods that use Singular Values Decomposition in the identification process. Two subspace techniques, referred to as the Balanced Realization (BR) and the Canonical Variate Analysis (CVA) are provided.

16.2.1

Stochastic substate identification methods The following stochastic discrete time state space model is considered {x k1}
[A]Ă{x k}  {w k}

Eqn 16-1

{y k}
[C]Ă{x k}  {v k} where

{xk } represents the state vector of dimension n, {yk } is the output vector of dimension Nresp {wk }, {vk }, are zero-mean, white vector sequences, which represent the process noise and measurement noise respectively.

270

The Lms Theory and Background Book

Operational modal analysis

For p and q large enough, the matrices [A] and [C] are respectively the state space matrix and the output matrix. Along with this model, the observability matrix [Op ] of order p and the controllability matrix [Cq ] of order q are defined :

 [C]  [C][A]  q1 [O p]
  .. p1ā; [Cqā]
[Ă[G]ă[A][G]...ă[A] [G]Ă] [C][A] 

Eqn 16-2

T where [G]
E[Ă{x k1}{y k} Ă] and E[.] denotes the expectation operator. The matrices [Op ] and [Cq ] are assumed to be of rank 2Nm , where Nm is the number of system modes.

The dynamics of the system are completely characterized by the eigenvalues and the observed parts of the eigenvectors of the [A] matrix. The eigenvalue decomposition of [A] is given by [A]
[āā][ā ā][āā] 1

Eqn 16-3

Complex eigenvectors and eigenvalues in equation 16-3 always appear in comĆ plex conjugate pairs. The discrete eigenvalues λr on the diagonal of [Λ] can be transformed into continuous eigenvalues or system poles µr by using the folĆ lowing equation  r
eĂ rĂt &  r
 r  i r
1 ln( r) t

Eqn 16-4

where r is the damping factor and r the damped natural frequency of the r-th mode. The damping ratio ξr of the r-th mode is given by r


r 2r  2r

Eqn 16-5

The mode shape {}r of the r-th mode at the sensor locations are the observed parts of the system eigenvectors {}r of [], given by the following equation {"} r
[C]Ă{}

Eqn 16-6

The extracted mode shapes can not be mass-normalized as this requires the measurement of the input force.

Part IV

Modal Analysis and Design

271

Chapter 16 Operational modal analysis

The stochastic realization problem The problem considered here is the estimation of the matrices [A] and [C] in equation 16-2, up to a similarity transformation, using only the output meaĆ surements { yk }. This problem is known as the stochastic realization problem and has been addressed by many researchers from the control departments as well as the statistics community [ 4, 5 and 6]. Two correlation-driven subspace algorithms are briefly discussed below, known as the Balanced Realization (BR) and the Canonical Variate Analysis (CVA). Given a sequence of correlations [R k]
EĂ {y km}Ă{y m} TrefĂ

Eqn 16-7

where {yk }ref is a vector containing Nref outputs serving as references. For p ≥ q, let [Hp,q ] be the following block-Hankel matrix :

[R 1] [R 2] . [R q]  [R 2] [R 3] .. [R q1]   .. [H p,q]
 .. ă .. ă ..ă  . . . . .   [R p] [R p1] [R pq1]  

Eqn 16-8

Direct computations of the [Rk ] from the model equations lead to the following factorization property Eqn 16-9

[H p,q]
[O p][C q]

Let [W1 ] and [W2 ] be two user-defined invertible weighting matrices of size pNresp and qNresp , respectively. Pre-and post multiplying the Hankel matrix with [W1 ] and [W2 ] and performing a SVD decomposition on the weighted Hankel matrix gives the following





[S 1] [0] [W 1]Ă[H p,q][W 2]
[Ă[U 1]ă[U 2]Ă] [0] [0] T

[V 1]T
[U ]Ă[S ]Ă[V ]T [V ]T 1 1 1  2 

Eqn 16-10

where [S1 ] contains n non-zero singular values in decreasing order, the n colĆ umns of [U1 ] are the corresponding left singular vectors and the n columns of [V1 ] are the corresponding right singular vectors. On the other hand, the factorization property of the weighted Hankel matrix results in

272

The Lms Theory and Background Book

Operational modal analysis

[W 1]Ă[H p,q]Ă[W 2] T
[W 1]Ă[O p]Ă[C q]Ă[W 2] T

Eqn 16-11

From equations 16-10 and 16-11, it can be easily seen that the observability maĆ trix can be recovered, up to a similarity transformation, as [O p]
[W 1] 1[U 1][S 1] 12

Eqn 16-12

The system matrices are then estimated, up to similarity transformation, using the shift structure of [Op ]. So, [C]
{firstĂblockĂofĂrowO p}

Eqn 16-13

and [A] is computed as the solution of

O,p1
[Op1]Ă[A]

Eqn 16-14

where [Op-1 ] is the matrix obtained by deleting the last block row of [Op ] and [Op-1 ↑] is the upper shifted matrix by one block row. Different choices of weighting will lead to different stochastic subspace identifiĆ cation methods. Two particular choices for the weighting matrices give rise to the Balance Realization and the Canonical Variate Analysis methods.

Balanced Realization (BR) [W 1]
[I] and[W 2]
[I]

Eqn 16-15

So no weighting is involved.

Canonical Variate Analysis (CVA) CVA requires that all responses are serving as references, so {yk }={yk }ref . ConseĆ quently, the correlation matrix [Rk ] given by equation 16-7 is square. Define then the following Toeplitz matrices [R p1] [R 1]  [R 0] [R 1]T [R p1]T  [R 0] .. . [R 0] .. [R p2] [R 1] [R 1]T [R 0] . [R p2] T       .. . . . . [ ]
 . ă . ă .ă ă ă ..ă ..  .. ă;ă [ ]
 .. . . . . . .   .  [R ] [R ] T T [R ] [R ] [R 0] T  p2 [R 0]   p1  p1 p2 Eqn 16-16

Part IV

Modal Analysis and Design

273

Chapter 16 Operational modal analysis

Let the full-rank factorization of [ℜ+] and [ℜ-] be [ ]
[L ]Ă[L ] TĄ; [ ]
[L ]Ă[L ] T

Eqn 16-17

In case of CVA, the weighting is as follows [W 1]
[L ] 1ă andă[W 2]
[L ] 1

Eqn 16-18

With this weighting, the singular values in equation 16-10 correspond to the so-called canonical angles. A physical interpretation of the CVA weighting is that the system modes are balanced in terms of energy. Modes which are less well excited in operational conditions might be better identified.

Practical implementation of correlation–driven stochastic subspace methods Equation 16-10 only holds for `true' block-Hankel matrices and for a finite orĆ der system. In practice, the system has a larger, possibly infinite order and the Hankel and Toeplitz matrices in equations 16-8 and 16-16 will be filled up with `empirical' correlations, which are computed as follows : [R k]
1 M ^

M

 {ymk}Ă{ym}Tref

Eqn 16-19

m
0

where M is the number of data samples. Although equation 16-19 is a preferred estimator for the correlation functions as no leakage errors are made and as it can also be used for non-stationary data, the evaluation of equation (19) in the time domain is not really efficient in computational effort. A faster estimator for the correlation functions can be imĆ plemented by taking the inverse FFT of auto-and crosspower spectra which are calculated on the basis of the FFT and segment averaging. This however asĆ sumes stationary signals and time windowing (e.g. Hanning) is needed to avoid leakage. The SVD decomposition of the weighted empirical Hankel matrix will then reĆ sult in the following ^ T ^  ^ ^ ^ ^ ^ ^ [S 1] [0] [V 1]  [W 1]ā[H p,q]ā[W 2]
[Ă[U 1]Ą[U 2]Ă] ă ^ Ă ^ Ă
[U1]ā[S 1]ā[V 1] T  [U 2]ā[S 2]ā[V 2] T T  [0] [S2][V 2]  ^

^

^

Eqn 16-20 with

274

The Lms Theory and Background Book

Operational modal analysis

^

[S 1]
diagĂ( 1---  n)Ă,ą  1   2---  n  0 ^

[S 2]
diagĂ( n1---  pRrespĂ)Ă,ą  n1   n2---  pR resp  0

Eqn 16-21

Identification of a model with order n is done by truncating the singular values, so by keeping [S1 ]. The observability matrix is then approximated by ^

^

^

[O p]
[W 1] 1Ă[U 1]Ă[S 1] 12

Eqn 16-22

As the model order is typically unknown, inspection of the singular values might help the engineer to select n such that n  n+1 In practice, this criteria is not often of great use as no significant drop in the singular values can be obĆ served. Other techniques such as stabilization diagrams are then needed in orĆ der to find the correct model order. The remaining steps of the algorithm are similar to those described in equations 16-11 to 16-18, where theoretical quantities are replaced with empirical ones.

16.2.2

Natural Excitation Techniques Subtitled : The Polyreference Least Squares Complex Exponential (LSCE) method apĆ plied to auto-and crosscorrelation functions Polyreference LSCE applied to impulse response functions is a well-known technique in conventional modal analysis, yielding global estimates of poles and the modal participation factors [7]. It has been shown that, under the asĆ sumption that the system is excited by stationary white noise, correlation funcĆ tions between the response signals can also be expressed as a sum of decaying sinusoids [ 8 ]. Each decaying sinusoid has a damped natural frequency and damping ratio that is identical to that of a corresponding structural mode. Consequently, the classical modal parameter techniques using impulse repines functions as input, like Polyreference LSCE, Eigensystem Realization Algorithm (ERA) and Ibrahim Time Domain are also appropriate to extract the modal parameters from response-only data measured under operational conditions. This technique is also referred to as NExT, standing for Natural Excitation TechĆ nique. An interesting remark is that the ERA method applied to correlation functions instead of impulse response functions is basically the same as the BalĆ anced Realization method.

Part IV

Modal Analysis and Design

275

Chapter 16 Operational modal analysis

Mathematically, the Polyreference LSCE will decompose the correlation funcĆ tions as a sum of decaying sinusoids. So, [R k]


[R k]


Nm

 {"}rĂeĂ kt{L}Tr  {"}*r ĂeĂ kt{L}T*r ăăor r

* r

r
1

Eqn 16-23

Nm

 {"}rĂkr{L}Tr  {"}*r Ăk*r {L}T*r

r
1

where  r
e rĂt and {L}r is a column vector of Nref constant multipliers which are constant for all response stations for the r-th mode. {Note that in conventional modal analysis, these constant multipliers are the modal participation factors.} The combinations of complex exponential and constant multipliers,  r{L} TrĂorĂ *r {L} T* r are a solution of the following matrix finite difference equaĆ tion of order t T T  krā{L} Trā[I]   k1 ā{L} Trā[F 1]    kt r r ā{L} r [F t]
{0}

Eqn 16-24

where [F1 ]...[Ft ] are coefficient matrices with dimension Nref x Nref . In case the system has Nm physical modes, the order t in equation 16-24 should be theoretically equal to 2Nm/Nref in order to find the 2Nm characteristic poles. In practice, over specification of the model order will be needed. Since the correlation functions are a linear combination of the characteristic T * T* solutions of equation 16-24,  r{L} r ĂorĂ r {L} r , they are also a solution of that equation. Hence, [R k]Ă[I]  [R k1]Ă[F 1] ---  [R kt]Ă[F t]
0

Eqn 16-25

Equation 16-25 which uses all response stations simultaneously enables a globĆ al least squares estimate of the coefficient matrices [F1 ]... [Ft ]... The overdeterĆ mination is also achieved by considering all available or selected time intervals. Once the coefficient matrices are known, equation 16-24 can be reformulated into a generalized eigenvalue problem resulting in Nref t eigenvalues lr, yielding estimates for the system poles µr and the corresponding left eigenvectors {L}r T . The selection of outputs which function as references have to be chosen in such a way that they contain all of the relevant modal information. In fact, the selecĆ tion of output-reference channels is similar to choosing the input-reference locations in a traditional modal test.

276

The Lms Theory and Background Book

Operational modal analysis

Extraction of mode shapes in a second step and model validation Contrary to the stochastic subspace methods, the Polyreference LSCE does not yield the mode shapes. So, a second step is needed to extract the mode shapes using the identified modal frequencies and modal damping ratios. For outputonly data, it has been shown [ 9 ] that this can be done by fitting the auto-and crosspower spectra between the responses and the responses serving as referĆ ences : X mn(j)


A mn* B mn B mn* r r r r  jAmn   

 r j   *  j  r  j   * Nm

r
1

r

r

Eqn 16-26

where Xmn (jω) is the crosspower between m-th response station and the n-th response station serving as a reference. In case of autopowers (m=n), Ar mn equals Br mn. The residue Ar mn is proportionĆ al to the m-th component of the mode shape {}r and the residue Br mn is proĆ portional to the n-th component of the mode shape {}r. Consequently, by fitĆ ting the crosspowers between all response stations and one reference station, the complete mode shape can be derived. The power spectra fitting step offers the advantage that not all responses should be included in the time-domain parameter extraction scheme and that consequently, mode shapes of a large number of response stations can be easily processed by consecutively fitting the spectra. Additionally, it provides a graphical quality check by overlaying the actual test data with the synthesized data. In comparison with modal FRF synthesis, it can be observed in equation 16-26 that two additional terms as function of -jw need to be included for a corĆ rect synthesis of the auto-and crosspowers which are assumed to be estimated on the basis of the FFT and segment averaging. If Xmn (jw) would not be calcuĆ lated with the FFT segment averaging approach, but as the FFT of the correlaĆ tion function between response m and response n estimated using equation 16-19, the last 2 terms in equation 16-26 can be neglected.

16.2.3

Selection of the modal parameter identification method This section discusses the criteria for selecting a particular method LSCE - LSFD This classical Least Squares Complex Exponential method is adapted to work on Auto-correlation and Cross-correlation instead of FRFs or ImĆ pulse Response functions.

Part IV

Modal Analysis and Design

277

Chapter 16 Operational modal analysis

A subset of the response functions can be selected as references in the computation of the cross power functions. The responses chosen as referĆ ences should contain all of the relevant modal information, as is required for the input-reference locations in a traditional modal test. Mode shapes are identified in a secondary process using the Least Squares Frequency Domain procedure. For the theoretical background on this method see section 16.2.2 BR (Balanced Reduction) This is one of the subspace" techniques which identifies frequency, damping and mode shapes. A subset of the response functions can be selected as references. These are used in the computation of the cross power functions from the original time domain data. This method is useful in identifying the most dominant modes occurring under operational conditions. CVA (Canonical Variate Analysis) This is the second of the subspace" techniques which identifies frequency damping and mode shapes. In this case all the response functions must be selected as references which are used in the computation of the cross power functions from the original time domain data. Thus this method requires more computational effort but this algorithm will give equal importance" to all modes and can identify modes which are not well excited under operational conditions. For the theoretical background on subspace methods see section 16.2.1.

278

The Lms Theory and Background Book

Operational modal analysis

16.3

References [1]

LMS CADA-X Running modes manual, 1997.

[2] Otte D., Development and Evaluation of Singular Value Analysis MethodĆ ologies for Studying Multivariate Noise and Vibration Problems, PhD K.U.LeuĆ ven, 1994. [3] Otte D., Van de Ponseele P., Leuridan J., Operational Deflection Shapes in Multisource Environments, Proc. 8th International Modal Analysis Conference, p. 413-421, Florida, 1990. [4] Abdelghani M., Basseville M., Benveniste A., In-operation Damage MonĆ itoring and Diagnostics of Vibrating Structures, with Applications to Offshore Structures and Rotating Machinery", Proc. of IMAC XV, Orlando, 1997. [5] Desai U.B., Debajyoti P., Kirkpatrick R.D., A realization approach to stoĆ chastic model reduction", Int. J. Control, Vol. 42, No. 4, pp. 821-838, 1985. [6] Kung S., A new identification and model reduction algorithm via singuĆ lar value decomposition", Proc. 12th Asilomar Conf. Circuits, Systems and Computers, pp. 705-714, Pacific Groves, 1978. [7] Brown D., Allemang R., Zimmerman R., and Mergeay, M., Parameter EsĆ timation Techniques for Modal Analysis", SAE Paper 790221, pp. 19, 1979. [8] James G.H. III, Carne T.G., and Laufer J.P., The Natural Excitation TechĆ nique (NExT) for Modal Parameter Extraction from Operating Structures, the international Journal of Analytical and Experimental Modal Analysis", Vol. 10, no 4, pp. 260-277, 1995. [9] Hermans L., Van der Auweraer H., On the Use of Auto-and Cross-corĆ relation functions to extract modal parameters from output-only data, Proc. of the 6th International conference on Recent Advances in Structural Dynamics, Work in progress Paper, 1997. [10] Van der Auweraer H., Wyckaert K., Hendricx W., From Sound Quality to the Engineering of Solutions for NVH Problems: Case Studies", Acustica/Acta Acustica, Vol. 83, N° 5, pp. 796-804, 1997. [11] Wyckaert K., Van der Auweraer H., Hendricx W., Correlation of AcoustiĆ cal Modal Analysis with Operating Data for Road Noise Problems", Proc. 3rd International Congress on Air- and Structure-Borne Sound and Vibration, Montreal (CND), June 13-15, 1994, pp. 931-940, 1994. [12] Wyckaert K., Hendricx W., Transmission Path Analysis in View of Active Cancellation of Road Induced Noise in Automotive Vehicles", 3rd International Congress on Air- and Structure-Borne Sound and Vibration, Montreal (CND), June 13-15, 1994, pp. 1437-1445, 1994.

Part IV

Modal Analysis and Design

279

Chapter 16 Operational modal analysis

[13] Van der Auweraer H., Ishaque K., Leuridan J., Signal Processing and System Identification Techniques for Flutter Test Data Analysis", Proc. 15th Int. Seminar of Modal Analysis, K.U.Leuven, pp. 517-538, Leuven, 1990. [14] Van der Auweraer H, Guillaume P., A Maximum Likelihood Parameter Estimation Technique to Analyse Multiple Input/Multiple Output Flutter Test Data", AGARD Structures and Materials Panel Specialists' Meeting on AdĆ vanced Aeroservoelastic Testing and Data Analysis, Paper no 12, May, 1995. [15] Van der Auweraer H., Leuridan J., Pintelon R., Schoukens J., A FrequenĆ cy Domain Maximum Likelihood Identification Scheme with application to Flight Flutter Data Analysis", Proc. 8-th IMAC, pp. 1252-1261, Kissimmee, 1990.

280

The Lms Theory and Background Book

Chapter 17

Running modes analysis

This chapter describes the basic principles involved in running mode analysis. It includes the following topics: The definition of running modes analysis The type of measurement data required for running mode analysis The identification and scaling of running modes The interpretation and validation of running modes

281

Chapter 17 Running modes analysis

17.1

Running mode analysis The aim of modal analysis is to identify a modal model that describes the dyĆ namic behavior of a (mechanical) system. This behavior is identified by means of the transfer function measured between any two degrees of freedom of the system. The outcome of a modal analysis therefore is the estimated modal parameters of the system, which are the natural frequencies (n ), damping ratios () and scaled mode shapes (Vik ). One of the most common ways of estimating the modal parameters is based upon the measurement of FRFs between one or more input (reference DOFs) and all response DOFs of interest. These measurements are made under well defined and controlled conditions, where all input and output signals are meaĆ sured and no unknown forces (external or internal) are acting on the system. The modal model is (ideally) valid under any circumstances; that is to say, whatever the frequency contents, level or nature of the acting forces. This makes modal analysis a very powerful tool, and the modal model (once identiĆ fied) can be used in a number of ways, such as trouble shooting, forced reĆ sponse prediction, sensitivity analysis or modification prediction. For many reasons, a complete modal analysis can be impracticable. It may be that the cost of the test setup is too high, the measurement object (e.g. a protoĆ type) cannot be made available for the period of time required to perform a modal analysis, or it is found to be simply impossible to isolate the object from all the forces acting on the system and excite it artificially. In this case, it is possible to take measurements of the system while it is operatĆ ing. A number of output signals can be measured (one at each response DOF), while the system is operating under stationary conditions. This provides a set of measurements (Xi ()) as a function of frequency. The measured quantity Xi ( ) at DOF i can be any number of things: displaceĆ ment, acceleration, voltage, angular position or acceleration, for example. It is however measured for one particular operating condition, with an unknown level or nature of the acting forces or inputs. If you are interested in a particular phenomenon at a well defined frequency, it is very often most helpful to see what the output levels are at that frequency for each measurement DOF. So you might, for example, want to know what the harmonic motion of measurement point 13 is at 85.6 Hz, or perhaps its level of acceleration. These values can then be assembled in a vector {X}, having one element for each of the measurement DOFs.

282

The Lms Theory and Background Book

Running modes analysis

Animating the system's wire frame model can lead to a better understanding of these phenomena. This makes it possible to show each motion (or acceleration) level at the corresponding DOF, in a cyclic manner. Because of the external reĆ semblance of the animated representation of the vector quantity {X} with the mode shape vector {V}, the vector {X} is called a running mode, or an operational deflection shape. These running modes must be interpreted entirely differently from modal modes. Running modes only reflect the cyclic motion of each DOF under specifĆ ic operational conditions, and at a specific frequency. Using a modal model based on displacement/force frequency response functions {H}, the displacement runĆ ning mode {X} can be described as follows. {X i( p)}
{Hi1( p)}F 1( p)  {H i2( p)}F 2( p)   {H im( p)}F m( p)

 2N


 Ą k
1

V ikV 1k !  2N Ą V ikV mk !ĂF ( ) ĂF ( )   p  jp    m p j p   k k  1



k
1

Eqn 17-1

Eqn 17-2

where, i = the DOF counter p = the particular angular frequency Fj () = the force input spectrum at DOF j m = the number of acting forces The above equation clearly shows that running modes:

Part IV

V

can be identified at any of the measured frequencies p , whereas a moĆ dal mode has a fixed natural frequency determined by the structural characteristics of the system (mass, size, Young's modulus, etc.).

V

depend on the level and nature of the acting force(s).

V

depend on the structural characteristics of the system, through its FRF behavior.

V

depend on the frequency contents of each of the acting forces : if F3 (p ) happens to be zero at p , it will not contribute to the running mode {x(p )}.

V

will be dominant at structural resonances (p " k ), but also at peaks in the acting force spectra.

Modal Analysis and Design

283

Chapter 17 Running modes analysis

17.2

Measuring running modes Ideally, all response spectra for a running mode analysis would be acquired: V

simultaneously

V

in a short period of time in which the operating conditions of the test object remain constant

V

with signals having a high signal to noise ratio, so that no averaging is required.

In practice, the number of acquisition channels on the measurement system limĆ its the number of response signals which can be measured simultaneously, and so different sets of responses have to be measured at different periods of time. Additionally, if a relatively high level of noise is present on the signals, an averĆ aging procedure may be necessary during the acquisition of the response sigĆ nals. Because of varying operation conditions, it is usual to choose a specific reĆ sponse DOF as a reference station and then measure the responses relative to this reference. If the operating conditions then change slightly from one meaĆ surement to the next, this will hopefully affect all response signals in the same way and the change will be cancelled out because of the relative nature of the measurements. This procedure also guarantees a fixed phase relationship beĆ tween the different response signals, using the phase of the reference signal as a reference. The two measured functions available for running mode analysis are: transmisĆ sibility functions and crosspower spectra.

17.2.1

Transmissibility functions When the response signals are related to the reference by simply dividing each response signal frequency spectrum by the reference frequency spectrum, the result is the transmissibility function (T) T ijĂ()Ă
Ă

X iĂ()

X jĂ()

Eqn 17-3

where j is the reference station.

284

The Lms Theory and Background Book

Running modes analysis

When averaging is involved, transmissibilities can be calculated from measured crosspower and autopower spectra.

T ij()Ă
Ă

G ijĂ() G jjĂ()

Eqn 17-4

The transmissibility function represents the complex ratio (amplitude and phase) between two spectra. A peak in this function may thus be caused either by a peak in the numerator crosspower (i.e. a structural resonance or peak in the excitation spectrum), or a zero (anti-resonance) in the denominator autoĆ power spectrum. As resonance peaks will occur at the same frequencies for cross and autopower spectra, while antiresonances do not, the denominator zeĆ ros will cause more peaks in Tij. Resonance peaks tend to cancel each other out. In the case of Frequency Response Functions (acceleration over force), different estimators (H1 , H2 , HV ) can be used to estimate the transmissibility functions. In practice, the difference between these different methods of estimating Tij () is small when the coherence function is high (near 100 %). When estimating the transmissibility functions from Equation 17-4 above, the coherence function () can also be calculated using the following equation.

 2ijĂ()Ă
Ă

ĂGijĂ()2 G iiĂ()Ă.ĂG jjĂ()

Eqn 17-5

The coherence function expresses the linear relationship between both response signals of the measured system. This coherence function is expected to be high, since both responses are caused by the same acting forces. In practice, however, it can be low for the same reasons as those affecting the measurement of FRFs, that is to say due to low signal to noise ratio for one or both of the signals, bad signal conditioning, etc. Another interesting reason why the coherence between two measured signals may be low, can be derived from equation 17-1, when it is substituted in equaĆ tion 17-3. The linear relationship (and hence the coherence) will vary as a funcĆ tion of the weighting factors Fj ( ), this can be because of changing operating conditions during the averaging process for example. High coherence function values in the frequency regions of interest therefore indicate both a high quality of the measurement signals and stationary operating conditions.

Part IV

Modal Analysis and Design

285

Chapter 17 Running modes analysis

Absolutely scaled running mode coefficients for each DOF i can be obtained by multiplying the transmissibility spectra by the RMS value of the reference autoĆ power spectrum.

 X iĂ()  Ă
Ă T ijĂ()Ă.Ă G jjĂ()

Eqn 17-6

When the measured autopower spectrum has units of displacement squared, the scaled running mode will be expressed in units of displacement (for examĆ ple, meters, or inches), if the transmissibility functions themselves are dimenĆ sionless. Displacement running modes can be converted to velocity or acceleraĆ tions by simply multiplying by j or (j)2. For a certain value of  (say o), the following relationships apply.

 X iĂ( 0)  Ă
Ă T ijĂ( 0)Ă.Ă G jjĂ(0)

[m]

Eqn 17-7

XiĂ(0)  Ă
Ă  XiĂ(0) Ă.ĂjĂ(0)

[m/s]

Eqn 17-8

XĂ(0)  Ă
Ă X iĂ( 0) Ă.ĂjĂ(0)

[m/s 2]

Eqn 17-9

.

..

17.2.2

.

Crosspower spectra When it can be assumed that the operating conditions are not going to change while measuring all response signals, then it is possible to measure just crossĆ power spectra between each response DOF i and a certain reference DOF j G ijĂ()Ă
Ă X iĂ()ĂX *j Ă()

Eqn 17-10

where * denotes the complex conjugate. Compared to transmissibility functions, crosspower functions have the advanĆ tage that peaks clearly indicate high response levels (which may still be caused by a structural resonance or a peak in the acting force spectrum). This techĆ nique is especially useful when all the response signals are measured simultaĆ neously by a multi-channel measurement system. In this case, the operating conditions are indeed the same for all response DOFs.

286

The Lms Theory and Background Book

Running modes analysis

Absolutely scaled running modes can, in this case, be obtained again by means of the autopower spectrum of the reference station j {X iĂ()}Ă
Ă

G ijĂ()

GjjĂ()

Eqn 17-11

When displacements were measured, the running mode coefficient will have units of displacement. Equations 17-8 and 17-9 can be used to derive velocity or acceleration values.

Part IV

Modal Analysis and Design

287

Chapter 17 Running modes analysis

17.3

Identification and scaling of running modes Unlike modal modes, a running mode can be identified at any arbitrary freĆ quency of the measured spectra. Simple peak picking and mode picking methods can be used to extract the sampled values, corresponding to a certain spectral line from the measured spectra. They can then be scaled, and assembled into a vector which can be listed, or animated using a 3D wire frame model of the measured object. For a measurement blocksize of 1024 (512 spectral lines), it is thus possible to identify 512 running modes - or even more when interpolating between the spectral lines.

Note!

17.3.1

There is no such quantity as damping defined for a running mode. Similarly other modal parameter concepts such as residues or modal participation factors have no meaning for running mode analysis.

Scaling of running modes It is possible to scale the identified running modes to values with absolute meaning. The scaling of running modes coefficients that have been determined using peak picking methods, depends upon the nature of the measurement data (e.g. transmissibilities, or autopowers). Several ways of scaling running modes can be considered

288

V

If transmissibility spectra were measured, then scaling can be perĆ formed using the reference autopower spectrum, as described in equaĆ tion 17-6.

V

If crosspowers were measured, then equation 17-11 can be applied to scale the running modes, again using the reference autopower specĆ trum.

V

It is possible to convert between displacement, velocity and acceleraĆ tion coefficients using equations 17-7, 17-8 and 17-9 where it is posĆ sible to integrate or differentiate once or twice.

The Lms Theory and Background Book

Running modes analysis

V

A number of running modes can be scaled manually, by entering a complex scale factor. Each individual mode shape coefficient will be multiplied with this scaling factor.

V

Finally, a very general scaling mechanism can be used to scale a numĆ ber of running modes using a spectrum. Individual running mode coĆ efficients will be multiplied by the (possibly complex) value of the spectrum block, belonging to the spectral line that corresponds to the frequency of that particular mode.

Each one of the above scaling methods may change and influence the units of the scaled running mode. The scaling factor's units will be incorporated into the mode shape coefficient units, which were initially obtained from the meaĆ surement data.

Part IV

Modal Analysis and Design

289

Chapter 17 Running modes analysis

17.4

Interpretation of results A set of functions exists, that are designed to assess the validity of modes. These include the functions of Modal Scale Factor, Modal Assurance Criterion and Modal decomposition.

Modal Scale Factors and Modal Assurance Criterion Both the Modal Scale Factor and Modal Assurance Criterion are mathematical tools used to compare two vectors of equal length. They can be used to compare running and modal, mode shape information. The Modal Scale Factor between columns l and j of mode shape k or MSFjlo is the ratio between two vectors. Although this ratio should be independent of the row index i (the response station), a least squares estimate has to be comĆ puted for it when more than one output station coefficient is available.

MSF jlk


{V jk} t*Ă{V lk}

Eqn 17-12

{V jk} t Ă{V jk} *

where {V jk } is the jth column of [Vk ]. The corresponding Modal Assurance Criterion expresses the degree of confiĆ dence in this calculation, which is obtained using equation 17-13.

MAC jlk


({V jk} t*Ă{V lk}) 2 ({V jk} t Ă{V jk})Ą({V lk} t {V lk}) *

*

Eqn 17-13

If a linear relationship exists between the two complex vectors {V jk } and {V lk }, then the MSF is the corresponding proportionality constant between them, and the MAC value will be near to one. If they are linearly independent, the MAC value will be small (near zero), and the MSF not very meaningful. Modal Scale Factors and Modal Assurance Criterion values can be used to compare an obtained modal model with the accepted running modes. The MAC values for corresponding modeshapes should be near 100 % and the MSF between corresponding vectors should be close to unity. When multiple inputs are used, the MSF can be calculated for each input, while the corresponding MAC will be the same for all of them.

290

The Lms Theory and Background Book

Running modes analysis

Modal decomposition When a modal model for the same DOFs is available for a measured object, it is possible to compare modal and running modes and to track down resonance phenomena causing a particular running mode to become predominant. This is termed Modal decomposition. By using a decomposition of each running mode in a linear combination of the modal modes, it becomes clear whether or not a running mode originates primarily from a resonance phenomenon. The modal modes form what is termed the `basis' group of modes. The runĆ ning modes are in a separate group that is to be decomposed. The following formula applies. {X i( 0)}
a 1{V 1}  a 2{V 2}   a n{V n}  Rest

Eqn 17-14

Where Xi is the i th mode of the group to be decomposed (running modes) Vi is the i th mode of the basis group (modal modes) ai are the scaling coefficients needed to satisfy the above equation. The scaling coefficients are rescaled relative to the maximum value. {X i( 0)}






Eqn 17-15

 {Xi( 0)}  [a 1{V 1}   a n{V n}]   {X i( 0)} 

Eqn 17-16

a a max a 1 .100%{V 1}  a n .100%{V n}   Rest a max max 100%

The Rest" is expressed as a relative error Rest
100%

Note!

Part IV

Take care when interpreting these values since resemblance of the modal and the running mode may purely be coincidental. A running mode at 56 Hz will have no connection with a modal mode at 200 Hz even if they look alike.

Modal Analysis and Design

291

Chapter 18

Modal validation

This document describes tools used to verify the validity of a modal model. Modal Scale Factors and Modal Assurance Criterion Mode participation Reciprocity Scaling Modal Phase Collinearity and Mean Phase Deviation Comparison of modal models Mode Indicator Functions Summation of FRFs Synthesis of FRFs

293

Chapter 18 Modal validation

18.1

Introduction A number of means are available to validate the accuracy of modal models of frequencies, damping values, mode shapes and modal participation factors. These tools are V

Modal Scale Factors between modes and corresponding correlation factors (Modal Assurance Criterion, MAC) described in section 18.2.

V

Mode participation described in section 18.3.

V

Reciprocity between inputs and outputs, described in section 18.4.

V

Generalized modal parameters, described in section 18.5. (Scaling)

V

Mode complexity, described in section 18.6.

V

Modal Phase Collinearity and Mean Phase Deviation indices, deĆ scribed in section 18.7.

V

Comparison of modal models described in section 18.8.

V

Mode Indicator Functions, described in section 18.9.

V

Summation of FRF data in the Index table, described in section 18.10.

V

Synthesis of FRFs described in section 18.11

Some validation procedures allow you to convert the complex mode shape vecĆ tors to normalized ones. Normalized mode shapes are obtained from the amĆ plitudes of the complex mode shape coefficients after a rotation over their weighted mean phase angle in the complex plane.

294

The Lms Theory and Background Book

Modal validation

18.2

MSF and MAC Modal Scale Factors and Modal Assurance Criterion The FRF between input j and output i on a structure can be written in partial fraction expansion form as

 rĂ ijk h ij,n
 j   N

k
1

n

k



rĂ *ijk

!  

j n   *k

Eqn 18-1

The matrix of FRFs is then expressed as

[H]


 j[RK] N

k
1

n

k



[R K] * j n   *k



Eqn 18-2

where [RK ] represents the matrix of residues. When Maxwell's reciprocity prinĆ ciple holds for the tested structure this residue matrix is symmetric and can be rewritten as R k
a k{V k}Ă{V k} t

Eqn 18-3

The ratio between two residue elements on the same row i but on two different columns j and l can be computed as r iāj,āk v jāk r iāl,āk
v lāk
MSF jālāk

Eqn 18-4

This ratio MSFjlk is called the Modal Scale Factor between columns l and j of mode k. Although this ratio should be independent of the row index i (the reĆ sponse station), a least squares estimate has to be computed for it when more than one output station residue coefficient is available

MSF jlk


{R jk} t*Ă{R lk} {R jk} t Ă{R jk} *

Eqn 18-5

where {R jk } is the jth column of [Rk ].

Part IV

Modal Analysis and Design

295

Chapter 18 Modal validation

The corresponding Modal Assurance Criterion expresses a degree of confidence for this calculation :

MAC jlk


({R jk} t*Ă{R lk}) 2

Eqn 18-6

({R jk} t Ă{R jk})Ą({R lk} t {R lk}) *

*

If a linear relationship exists between the two complex vectors {R jk} and {R lk} the MSF is the corresponding proportionality constant between them and the MAC value will be near to one. If they are linearly independent, the MAC valĆ ue will be small (near zero), and the MSF not very meaningful. In a more general way, the MAC concept can be applied on two arbitrary comĆ plex vectors. This is useful in comparing two arbitrary scaled mode shape vecĆ tors since similar mode shapes have a high MAC value. Modal Scale Factors and Modal Assurance Criterion values can be used to compare two modal models obtained from two different modal parameter esĆ timation processes on the same test data for example. When comparing mode shapes, the MAC values for corresponding modes should be near 100 % and the MSF between corresponding residue vectors (mode shapes, scaled by the modal participation factors) should be close to unity. When multiple inputs were used, this MSF can be calculated for each input while the corresponding MAC will be the same for all of them. A second application for the MAC value is derived from the orthogonality of mode shape vectors when weighted by the mass matrix: {V k} tĂ[M]Ă{V}
m kĄwhenĄk
1

Eqn 18-7


0Ăotherwise where mk represents the modal mass for mode k.

Even when no exact mass matrix is available, it can usually be assumed to be almost diagonal with more or less equal elements. In this case, the calculation of the MAC value between two different modes is approximately equivalent to checking their orthogonality. For more specific information on using MSF and MAC for interpreting results in a running mode analysis see section 17.4.

296

The Lms Theory and Background Book

Modal validation

18.3

Mode participation The relative importance of different modes in a certain frequency band can be investigated using the concept of modal participation. For each mode, the sum of all residue values for a specific reference expresses that mode's contribution to the response. At the same time these sums can be added over all references, to evaluate the importance of each mode.

Note!

These evaluations are only meaningful when the same response and reference stations are included for all modes. When a comparison is made of the residue sums for one mode at all the referĆ ences, it evaluates the reference point selection for that mode. The reference with the highest residue sum is the best one to excite that mode. When these sums are added together for all references, the importance of the modes themselves is evaluated. The mode with the highest result is the most important one. Finally the sums of residues can be added for all modes. Comparison of these results between different inputs allows you to evaluate the selection of referĆ ence stations in a global sense for all modes.

Part IV

Modal Analysis and Design

297

Chapter 18 Modal validation

18.4

Reciprocity between inputs and outputs Reciprocity is one of the fundamental assumptions of modal analysis theory. This section discusses the reciprocity of FRFs and the reciprocity of the modal model.

Reciprocity of FRFs Reciprocity of FRFs means that measuring the response at DOF i while exciting at DOF j is the same as measuring the response at DOF j while exciting at DOF i This is expressed mathematically as h iĂjĂ()
h jĂiĂ()

Eqn 18-8

This means that the FRF matrix is symmetric. Note that this property is inĆ herently assumed when performing hammer impact testing to measure FRFs or impulse responses.

Reciprocity in the modal model Using the modal model for the FRF matrix *  {V} kL k {V} *kL k!   H Ă
Ă   j   k j   *k  k
1 N

Eqn 18-9

it becomes clear that, when this matrix is symmetric, the role of mode shape vectors and modal participation vectors can be switched. Making an abstracĆ tion of the absolute scaling of residues, this property can be expressed as folĆ lows. For a reciprocal test structure, the modal participation factors should be proporĆ tional to the mode shape coefficients at the input stations. Using this proportionality between mode shapes and modal participation facĆ tors, reciprocity can be checked for each mode when data for more than one inĆ put station has been used for the modal parameter estimation.

298

The Lms Theory and Background Book

Modal validation

If reciprocity exists then it is possible to correctly synthesize the transfer funcĆ tion between any pair of response and reference DOFs. This is done by comĆ puting a scaling factor between the driving point mode shape and the modal participation factor. This same scaling factor is then used as a reference to deĆ rive the necessary participation factor from the available mode shape coeffiĆ cient. If reciprocity is not satisfied then really only the transfer functions between the measured response and reference DOFs can be correctly synthesized. If reciprocĆ ity is required then it can be imposed on the model, and a number of options are available to calculate the proportionality factor needed to do this. 1 Select one driving point for each mode. The best choice in this case is the one with the largest driving point residue, since it is the one that best excites and is observed from that input DOF. 2 Select one specific driving point for all modes. Other participation facĆ tors are disregarded for scaling. 3 Compute a reciprocal scale factor (RSF) using a least squares average of all the driving point data as defined by the following formula for n driving points.



n

 v*i Ăli

RSF


i
1 n

 v*i Ăvi

i
1

where

vi = the mode shape coefficient li = the modal participation factor

Part IV

Modal Analysis and Design

299

Chapter 18 Modal validation

18.5

Generalized modal parameters This section deals with mode shape scaling and generalized parameters (modal mass). The residueărij,k between locations i and j for mode k can be written as the product of a scaling factor ak (which is independent of the location) and the moĆ dal vector components in both locations. If the structure is proportionally damped, the modal vectors of the structure are real whereas the residues are purely imaginary. As a consequence, the scaling factor ak , is also purely imagiĆ nary. r iāj,k
a kĂv ikĂv jāk ak


Eqn 18-10

1 2j dkm k

Equation 18-1 can then be rewritten as 1  vikv jk H ij(j)
 2j m j   N

k
1

dk



k

n

k



  j n   *k v *ikv *jk

Eqn 18-11

where mk = modal mass of mode k dk = damped natural frequency of mode k
nk 1   2k

k = the critical damping ratio of mode k nk = the undamped natural frequency of mode k At this point, it should be pointed out that equation 18-11 contains N more paĆ rameters than equation 18-1, i.e. one more parameter per mode. This is due to the fact that residues are scaled quantities whereas the modal vectors are deterĆ mined within a scaling factor only. In equation 18-11 the modal mass values play the role of the scaling constants. It is clear that the value of the modal mass depends on the scaling scheme that was used to obtain the numerical valĆ ues of the modal vector amplitudes. When the residues of a proportionally damped structure are known, equations 18-10 and 18-11 can therefore be used to compute the modal mass and the moĆ dal vector amplitudes once a scaling method is proposed. Indeed residues, moĆ dal vectors and modal mass are related by following equation

300

The Lms Theory and Background Book

Modal validation

r ijk


v ikv jk 2j dkm k

Eqn 18-12

To compute the amplitudes of one modal vector and the corresponding modal mass from a set of residues with respect to a given input location j you need one additional equation since the set of equations that can be written for all outĆ put locationsăi in the form of equation 18-12 is undetermined. Therefore N equations in N +1 unknowns are obtained. This last equation will actually deĆ termine the scaling of the modal vector. Note that an eigenvector determines only a direction in the state space and has no absolutely scaled amplitude, while a residue has a magnitude with physical meaning. The scaling of the eigenvectors will determine the modal mass. MoĆ dal stiffness is determined as the modal mass multiplied by the natural freĆ quency squared. Modal damping is twice the modal mass multiplied by the natural frequency and the damping ratio. V

Unity mass In this case the mode shapes and participation factors are scaled such that the modal mass (mk ) in equation 18-12 is equal to 1.

V

Unity stiffness In this case the mode shapes and participation factors are scaled such that the modal stiffness (kk = mk k 2 ) is scaled to 1.

V

Unity modal A In this case the mode shapes and participation factors are scaled such that the scaling factor (ak ) is scaled to 1. This scaling factor is indepenĆ dent of the DOFs.

V

Unity length In this case the mode shapes and participation factors are scaled such that the squared norm of the vector vik is scaled to unity. N0

Ă v2ikĂ
1

i
1

Part IV

V

Unity maximum In this case the mode shapes and participation factors are scaled such that the vector vik is scaled to 1 where i is the DOF with the largest mode shape amplitude.

V

Unity component In this case the mode shapes and participation factors are scaled such that the vector vik is scaled to 1 where i is any DOF selected by the user.

Modal Analysis and Design

301

Chapter 18 Modal validation

18.6

Mode complexity When a mass is added to a mechanical structure at a certain measurement point then the damped natural frequencies for all modes will shift downwards. This theoretical characteristic forms the basis of a criterion for the evaluation of estiĆ mated mode shape vectors. For each response station, the sensitivity of each natural frequency to a mass increase at that station can be calculated and should be negative. A quantity called the Mode Overcomplexity Value" (MOV) is defined as the (weighted) percentage of the response stations for which a mass addition indeed decreases the natural frequency for a specific mode, N0

 wiĂaik

MOV k


i
0

Ă xĂ100%

N0

w i
0

Eqn 18-13

i

where wi

is the weighting factor = 1 for unweighted calculations = |vik |2 for weighted calculations

aik

= 1 if the k th frequency sensitivity to a mass addition in point i is negative = 0 otherwise

This MOV index should be high (near 100 %) for high quality modes. If this index is low the considered mode shape vector is either computational or wrongly estimated. It is called overcomplex", which means that the phase angle of some modal coefficients exceeds a reasonable limit. However if this MOV is low for all modes for a specific input station (say, beĆ low 10%), this might indicate that the excitation force direction was wrongly entered while measuring the FRFs for that input station. This error may be corrected by changing the signs of the modal participation factors for all modes for that particular input.

302

The Lms Theory and Background Book

Modal validation

18.7

Modal phase collinearity For lightly or proportionally damped structures, the estimated mode shapes should be purely normal. This means that the phase angle between two differĆ ent complex mode shape coefficients of the same mode (i.e. for two different response stations) should be either 0_, 180_ or -180_. An indicator called the ``Modal Phase Collinearity" (MPC) index expresses the linear functional relaĆ tionship between the real and the imaginary parts of the unscaled mode shape vector. This index should be high (near 100%) for real normal modes. A low MPC inĆ dex indicates a rather complex mode, either because of local damping elements in the tested structure or because of an erroneous measurement or analysis proĆ cedure.

Mean phase deviation Another indicator for the complexity of unscaled mode shape vectors is the Mean Phase Deviation (MPD). This index is the statistical variance of the phase angles for each mode shape coefficient from their mean value, and indicates the phase scatter of a mode shape. This MPD value should be low (near 0_) for real normal modes.

Part IV

Modal Analysis and Design

303

Chapter 18 Modal validation

18.8

Comparison of models When you have two groups of modes representing the same modal space then you can compare the two groups. The comparison concerns the damped freĆ quencies, the damping values, the modal phase collinearities and the MAC valĆ ues of the two groups. This is a useful way of comparing sets of modes generĆ ated from the same data but using different estimation techniques for example.

304

The Lms Theory and Background Book

Modal validation

18.9

Mode indicator functions Mode Indicator Functions (MIFs) are frequency domain functions that exhibit local minima at the natural frequencies of real normal modes. The number of MIFs that can be computed for a given data set equals the number of input locations that are available. The so-called primary MIF will exhibit a local minimum at each of the structure's natural frequencies. The secondary MIF will have local minima only in the case of repeated roots. Depending on the number of input locations for which data is available, higher order MIFs can be computed to determine the multiplicity of the repeated root. So a root with a multiplicity of four will cause a minimum in the first, second, third and fourth MIF for example. An example of a MIF is shown below.

Given a structure's FRF matrix [H], describing its input-output characteristics and a force vector, {F}, the output or response {X} can be computed from the folĆ lowing equation {X}
H{F}

Eqn 18-14

Removing the brackets from the notation, equation 18-14 can be split into real and imaginary parts X r  jX i
(H r  jH i)Ă(F r  jF i)

Eqn 18-15

For real normal modes, the structural response must lag the excitation forces by 90_. Therefore, when the structure is excited at the correct frequency according to one of these modes (modal tuning) the contribution of the real part of the reĆ sponse vector X to its total length must become minimal. Mathematically this can be formulated in the following minimisation problem

Part IV

Modal Analysis and Design

305

Chapter 18 Modal validation



X trX r min |FF| ( 1Ă X t X  X tX r r i i



Eqn 18-16

Substituting the expression for the real and imaginary parts of the response 18-15 in this expression yields





FH trH rF min Ă |FF| ( 1 F t(H t H  H tH )F r r i i

Eqn 18-17

The solution of equation 18-17 reduces to finding minima of the frequency functions that are built from eigenvalues. The following eigenvalue problem is formulated at each spectral line under investigation H trĂH rĂF
Ă(H trĂH r  H tiĂH iĂ)ĂF

Eqn 18-18

The square matrices Hr t Hr and Hi t Hi have as many rows and columns as the number of input or reference locations that were used to create them (i.e. the number of columns of the FRF matrix that were measured). The primary Mode Indicator Function is now constructed from the smallest eigenvalue of expresĆ sion 18-18 at each spectral line. It exhibits noticeable local minima at the freĆ quencies where real normal modes exist. A second MIF can be constructed usĆ ing the second smallest eigenvalue of 18-18 for each spectral line. It will contain noticeable local minima if the structure has repeated modes. This can be repeated for all other eigenvalues of equation 18-18. The number of funcĆ tions that can be constructed is equal to the number of eigenvalues, which is the same as the number of input stations. From these functions, you can then deĆ duce the multiplicity of each of the normal modes.

306

The Lms Theory and Background Book

Modal validation

18.10

Summation of FRFs An important indication of the accuracy of the natural frequency estimates is their coincidence with resonance peaks in the FRF measurements. These resoĆ nance peaks can be enhanced by a summation of all available data, either by real or imaginary parts. Graphically comparing this summation of FRFs with values of the natural freĆ quencies of modes in a display module can be useful. Problems like missing modes, erroneous frequency estimates or shifting resonances because of mass loading by the transducers can easily be detected this way.

Part IV

Modal Analysis and Design

307

Chapter 18 Modal validation

18.11

Synthesis of FRFs The FRFs that you have obtained from a modal model can be synthesized in a number of ways. Scaled mode shapes (i.e. mode shapes and modal participaĆ tion factors) have to be available for at least one input station for which a mode shape coefficient is also available. Using the Maxwell-Betti reciprocity princiĆ ple between inputs and outputs (section 18.4) it is however possible to calculate the FRF between any two measurement stations.

Correlation and errors It is also possible to assess correlation and error values relating to the measured and synthesized FRFs. The correlation is the normalized complex product of the synthesized and meaĆ sured values. |

Ă SiĂxĂM*i |2 i

correlationĂ
Ă

 S ĂxĂS* ! M ĂxĂM* !  i i  i  i i  i

Eqn 18-19

with Si = the complex value of the synthesized FRF at spectral line i Mi = the complex value of the measured FRF at spectral line i The LS error is the least square difference normalized to the synthesized values.

Ă SiĂ Ă Mi ĂxĂ SiĂ Ă Mi * LSĂerrorĂ
Ă i  SiĂxĂS*i

Eqn 18-20

i

A listing of FRFs where the correlation is lower than a specified percentage and which exhibit an error higher than a specified percentage provides useful inforĆ mation on the quality if the synthesized FRF.

308

The Lms Theory and Background Book

Chapter 19

Rigid body modes

In this chapter the behavior of a structure as a rigid body is disĆ cussed. The following topics are covered The calculation of rigid body properties of a structure from FRF measurements Rigid body analysis to determine rigid body modes

309

Chapter 19 Rigid body modes

19.1

Calculation of rigid body properties This section discusses the theory used in the calculation of rigid body properĆ ties. Experimental frequency response functions (FRFs) can be used to derive structural modes of a structure and the inertia properties of a system. These properties are: the moments of inertia, the products of inertia and the principal moments of inertia. In general two types of method are applied. 1 A first type determines the inertia characteristics using the rigid body mode shapes obtained from test data. This is the Modal Model Method described in reference [1]. 2 The second type starts from the mass line, i.e. the FRF inertia restraint of the softly suspended structure. This mass line is used in a set of kinematic and dynamic equations, from which the rigid body characteristics (mass, center of gravity, principal directions and moments of inertia) can be determined (reference [2]). Some of these methods also look for the suspension stiffĆ nesses while others consider the mass of the system as known (reference [3]). This type of method is described in more detail below. acc/force first deformation mode frequency band Rigid body mode mass line frequency

Figure 19-1

Rigid body modes

Derivation of rigid body properties from measured FRFs Input data FRFs are required in order to determine the rigid body properties. The input format is required to be acceleration/force, and if this is not the case then a transformation can be applied. Rotational or scalar (acoustic) measurements are not used in the rigid body calculations.

310

The Lms Theory and Background Book

Rigid body modes

In theory 2 excitations and 6 responses are needed for the calculations. PractiĆ cal tests show that best results are obtained with at least 6 excitations (e.g. 2 nodes in 3 directions) and 12 responses need to be measured. Reference axis system All the rigid body properties are calculated relative to a reference axis. The refĆ erence axis system is defined by the three coordinate values of its origin and three euler angles representing its rotation. Specification of the frequency band Rigid body properties are calculated in a global (least squares) sense over a speĆ cified frequency band between the last rigid body mode and the first deformaĆ tion mode (see Figure 19-1). Mass line value The mass line" value which is needed for the calculations, can be derived from the measured FRFs in three ways: 1)

When the rigid body modes and deformation modes are sufficiently spaced, the amplitude values (with the sign of the real part) of the origiĆ nal, unchanged measured FRFs can be used. In this case there is no need to have the deformation modes available for the rigid body modes analyĆ sis.

2)

When the spacing between rigid body modes and deformation modes is not sufficient, the FRFs have to be corrected. In this case the influence of the first deformation modes, if significant, can be subtracted from the original FRFs. The amplitude values (with the sign of the real part) of synthesized FRFs are used.

3)

If accurate measured FRFs are not available in the frequency range directĆ ly above the rigid body modes, then lower residual terms which lie in a frequency band which contains the first deformation modes can be used. Residual terms can be determined from a modal analysis. Lower residuĆ als represent the influence of the modes below the deformation modes, and are therefore representative of the rigid body modes.

Calculation of the rigid body properties 1

Calculation of the reference acceleration matrix

1.1

Coordinate transformation If nodes, corresponding to the response DOFs used do not have global diĆ rections or when a reference (not coincident with the global origin) is speĆ cified, then a rotation of the measured accelerations according to the globĆ al/reference axis system is needed.

Part IV

Modal Analysis and Design

311

Chapter 19 Rigid body modes

All three directions (+X, +Y, +Z) are required. For the three measured (loĆ cal) accelerations of output node o": ..

..

{X} g
[āTā] 1 o Ă{X} l

Eqn 19-1

where .. {X} g is the global acceleration vector ..

{X} l is the local acceleration vector [T] 1 o is the rotation matrix (global to local) of node o" When a reference is specified, which does not coincide with the global oriĆ gin, the three measured accelerations of output node o" are also rotated according to the axis of the reference system. ..

..

{X} r
([āTā] 1 o Ă[āTā] r)Ă{X} l

Eqn 19-2

where [T] r is the rotation matrix (global to local) of node r". 1.2

System of equations For all spectral lines of the selected band, for all response nodes P, Q,... and for all inputs 1, 2, ... under consideration ..

X.. 1P X1P  .. X1P  .. X1Q  .. X  .. 1Q  X1Q  )

..

x

X 2Px ---

y

X 2Py

z

X 2Pz

x

X 2Qx

y

X 2Qy

z

.. ..

.. .. ..

X 2Qz )

..

1 0 0 0 Z P  Y P X.. 1g  --- 0 1 0  ZP 0 X P  X 1g    .. 0 0 1 YP  XP 0  ---  X 1g     ..    1 0 0 0 Z  Y
Q Q  ---    .. 1g  0 1 0  ZQ 0 XQ  ---   .. 1g  0 0 1 Y Q  XQ  0   1g --- ) ) )    ) ---

..

x y z

x y z

X 2g x --- X 2g y --- ..



X 2g z ---  ..  2g x ---  ..  2g y --- Eqn 19-3 ..



 2g z --- ) --- ..

acceleration of input 1 towards global axis system

where XP, YP, and ZP are the global coordinates of node P (or towards the reference axis system). This over-determined system of equations (number of output DOFs is higher than or equals 6) is solved for each spectral line in a least square sense. In this way at each spectral line, the reference acceleration matrix is found. Further, a general solution of the reference acceleration matrix over the total frequency band is calculated by solving in a least squares sense the global set of equations containing all outputs and all spectral lines.

312

The Lms Theory and Background Book

Rigid body modes

2

Calculation of the reference force matrix

2.1

Coordinate transformation For input force 1 in the local X-direction of node i":

1.0 ! {F 1}
[T] 1 i $0.0% 0.0

Eqn 19-4

[T] 1 i is the rotation matrix (global to local) of node i" When the reference r" is not coincident with the global origin: {F 1}


1.0 ! $0.0% 0.0

([T] r[T] 1 i )

Eqn 19-5

[T] 1 r is the rotation matrix (global to local) of reference node r" Similar equations are used when the input has Y-direction or Z-direction. 2.2

System of equations For all inputs 1, 2 . . .

F1g ! 1 0 0 F1g   0 1 0  F1g   0 0 1   $M1g %
 0  Z1 Y 1  Ą{F1}  M1g   Z1 0  X1     Y 1 X1 0  M 1g  x y z

Eqn 19-6

x y z

reference force matrix towards global axis system for input 1

{F1} is the applied force at input 1 X1, Y1 and Z1 are the global coordinates of node corresponding with input 1. 3

Calculation of the co-ordinates of the center of gravity and moments and products of inertia For (i) each input and for each spectral line (ii) each input over the total band:

Part IV

Modal Analysis and Design

313

Chapter 19 Rigid body modes

Fg  m.ag !  0  mz F g  m.a g m z 0  Fg  m.ag     my mx $ Mg %
 Fg  0   0  Mg     Fg Mg   Fg  mFg x

x

y

y

z

z

z

x

z

y

y

z

x

Xcog! m y 0 0 0 0 0 0  Y gog   0 0  Z cog  m x 0 0 0 0 0 0 0 0 0 0 0   I xx  I  F g  x 0 0   y 0   z Ą$ Iyy % zz  Fg 0 y 0  x  z 0   I xy   0 0 0  z 0   y   x    Iyz  I xz  y

x

Eqn 19-7 Xcog, Ycog and Zcog are the global coordinates of the center of gravity Ixx, Iyy Izz are the moments of inertia towards the global axis system Ixy, Iyz Ixz are the products of inertia towards the global axis system. This set of equations can be solved in two steps. First, the coordinates of the center of gravity can be solved from the first three equations (per refĆ erence). Afterwards, these values can be filled in the last equations to solve the inertia moments and products. Step 1 for each input and for each spectral line and for each input over the total band: Fg  m.a g !  0  m z m y  x cog  ycog!   F  m.a m 0  m g g
ă z x $   $z % Fg  m.ag %   m m 0 y x  cog   x

x

y

y

z

z

Eqn 19-8

Step 2 for each input and for each spectral line and for each input over the total band:

Ixx! I yy  Mg  ycogĂFg  zcogĂFg ! x 0 0  y 0  z   I zz   $Mg  x cogĂFg  zcogĂFg %
 0 y 0  x  z 0 ă$Ixy% Mg  x cogĂFg  ycogĂFg    0 0 z 0  y  x  Iyz  I xz x

z

y

y

z

x

z

y

x

Eqn 19-9

At each spectral line, these over-determined sets of equations (number of inputs larger than or equal to 2) are solved in a Least Square sense. Also a global solution for these rigid body properties over the total band can be found out of the global acceleration matrix over the total frequency band (see equation 19-3).

314

The Lms Theory and Background Book

Rigid body modes

If wanted, only the second set of equations is solved. In this case the coorĆ dinates of the center of gravity are presumed to be known and specified by the user. 4

Calculation of the co-ordinates of the center of gravity and moments and products of inertia In general: {L g}
[A]{ g}

Lx!  Ixx  Ixy  Ixzx! $Ly%
 Iyx Iyy  IyzĂ$y% L z  I zx  I zy I zz  z

Eqn 19-10

{Lg } is the vector of total impulse towards the global (reference) axis sysĆ tem [A] is the matrix of inertia (symmetrical) {g } is the vector of velocity This is an eigenvalue problem, where the Eigenvalues : I1, I2, I3: are 3 principal moments of inertia the Eigenvectors : {e1}, {e2}, {e3} : are directions of the 3 principal axes of inertia.

Part IV

Modal Analysis and Design

315

Chapter 19 Rigid body modes

19.2

Rigid body mode analysis A rigid body is a (part of a) structure that does not deform of itself, but that moves periodically as a whole at a certain frequency. The modal parameters for such a rigid body mode are determined not by the dynamics of the structure itself, but by the dynamic properties of the boundary conditions of that structure. This includes the way it is attached to its surĆ roundings (or the rest of the structure), the stiffness and damping characterisĆ tics of suspending elements, its global mass, etc... A rigid body can be compared to a simple system with a mass attached to a fixed point by a spring and a damper element. It has 6 modes of vibration i.e. translation along the X, Y, and Z axes, and rotaĆ tion about these axes. Every mode which is measured for such a system will be a linear combination of these 6 modes. Section 19.1 describes how it is possible to calculate the inertia properties of a structure based on measured FRFs. This enables you to calculate the center of gravity, moments of inertia and the principle axes as well as synthesized rigid body modes. This section discusses how rigid body modes are used and describes two methĆ ods by which the modes can be determined; namely V

decomposition of measured modes into rigid body modes

V

synthesis of rigid body modes based on geometrical data

Use of rigid body analysis In modal analysis applications, the fact that (part of) a structure acts as a rigid body up to a certain frequency can be used in different ways. 1 Debugging the measurement set–up Rigid body modes can be used to verify the measurement set-up when the frequency range of measured FRFs covers a rigid body of the entire structure in its suspension (elastic cords or air bags for example). In this case, a simple peak picking procedure and an animation of the resulting mode will indicate which measurement points are not moving in line" with the rest of the structure. Deviations from this rigid body motion can be caused by

316

V

non-measured nodes (not moving at all)

V

wrong response point identification (moving out of line)

The Lms Theory and Background Book

Rigid body modes

V

wrong response direction (moving in opposite direction)

V

bad transducers or wrong calibration values (wrong amplitude)

V

other measurement errors

Obvious errors, as in the first 4 cases, can be easily detected by curve-fitting a rigid body mode of the structure. 2 Completion of non–measured DOFs Mode shape coefficients for non-measured points and/or directions can be calculated based on the assumptions that the resulting deformed mode shape should still be a rigid body. This can be achieved by first calculating the weighting coefficients for each of the 6 rigid body motions of the strucĆ ture from the measured data and then applying the same weighting to obtain the motion of the non-measured DOFs. This takes into account the geometry constraints and thus preserves the rigid body motion of the structure. This feature is useful to complete sparsely measured rigid parts of a wire frame model for animation. 3 Correction of measurement errors Using the same approach as described under 2 it is also possible to re-calcuĆ late mode shape coefficients for measured DOFs and compare them to the actually measured ones to evaluate measurement errors (as under 1) or meaĆ surement noise. It is even possible to replace the measured data by the calĆ culated data and smooth the mode shapes to obtain good rigid body motion for (parts of) the structure. 4 Synthesis of modes based on the geometry of the structure Rigid body modes can be calculated for a structure based on the structure's mass, moments of inertia, the boundary conditions and values for frequency and damping specified by the user. This is useful when coupling two subĆ structures for example for which the modal parameters have been obtained separately.

19.2.1

Decomposition of measured modes into rigid body modes The decomposition into rigid body mode is quite simple and involves the folĆ lowing steps. 1 Use the geometry data to construct the 6 rigid body motions of the structure.

Part IV

Modal Analysis and Design

317

Chapter 19 Rigid body modes

2 Decompose a given mode shape in these 6 modes. This involves solving a system of linear equation and can only be accomplished if enough equations can be built. This means that at least 6 measured DOFs must be available and that the equations must be linearly independent. This means for examĆ ple that it is not possible to calculate the contribution of a rotation about the Z axis from data for 2 points on that axis, even if they both have all 3 DOFs measured. 3 Calculate the mode shape coefficients for the requested DOFs based upon the geometry and the 6 weighting coefficients.

Limitations Calculating the rigid body motion for a part of the structure (for example one single component) can sometimes prove a little awkward. The component will indeed move as a rigid body but is not constrained to still be connected to the rest of the structure. When applied to the tail wing of an airplane for example this wing may rotate about a horizontal axis through the middle of the wing but may no longer be connected to the fuselage at its base. The same may hapĆ pen to an engine block of a car which may be disconnected from the supports when a rigid body motion is applied to it.

19.2.2

Synthesis of rigid body modes based on geometrical data The synthesis of rigid body modes for a `free-free' structure is based on the translation along and the rotation about the three principal axes of inertia. The position of these three axes as well as the principal moments of inertia about them and the mass are required for the calculation of the rigid modes. The damping and frequency are specified by the user. The residues are calculated as follows R trans
1 2m R rot


r fĂr x 2I

where

318

The Lms Theory and Background Book

Rigid body modes

m = the total mass  = the user defined damped natural frequency rf = the perpendicular distance from the reference DOF to the respective axis of inertia rx = the perpendicular distance from the response DOF to the respective axis of inertia I = the moment of inertia about the respective axis of inertia. Rigid body modes are useful in completing the modal model of a structure that is being used for structural modification purposes.

Part IV

Modal Analysis and Design

319

Chapter 19 Rigid body modes

19.3

320

References [1]

Toivola, J. and Nuutila, O. Comparison of three Methods for Determining Rigid Body Inertia Properties from Frequency Response Functions Tampere University of Technology, P.O. Box 589, SF-33101 Tampere, Finland,

[2]

Okuzumi, H. Identification of the Rigid Body Characteristics of a Powerplant by Using Experimental Obtained Transfer Functions Central Engineering Laboratories, Nissan Motor Co., Ltd., Jun 1991

[3]

Lemaire, G. and Gielen, L. Het bepalen van de inertie-parameters van een star lichaam door middel van transfertfuncties Eindwerk katholieke hogeschool Brugge-Oostende dep. industriele wetenschappen en technologie, 1995-1996

[4]

LMS International LMS CADA-X Modal Analysis Manual Revision 3.4 LMS International, Leuven, Belgium, pp 2.6-2.7, pp 3.24-3.32, 1996

[5]

LMS International How to Add Rigid Body Modes to an Existing Modal Model in CADA-X LMS International Consulting reports, Ref. DVDB/sh/911295, Leuven, Belgium, 22 pp, 1991

The Lms Theory and Background Book

Chapter 20

Design

This chapter discusses the three types of analysis that can be perĆ formed to determine the effect of design changes on the modal beĆ havior of a structure. These are Sensitivity Modification prediction Forced response

321

Chapter 20 Design

20.1

Using the modal model for modal design Correctly scaled mode shapes are an absolute pre-requisite of the correct apĆ plication of the design procedures described here. The dynamic behavior of a structure can be fully described and modelled thereĆ fore if the poles #k ), and the residues rijk for each mode k and each pair of reĆ sponse and reference DOFs i and j are known. In practise however the modal model is often defined by the poles (frequency and damping values) and the residues for only one (or a few) reference staĆ tion(s) j. The question now arises as to how this limited modal model can be used for the prediction of responses when forces are acting on a degree of freeĆ dom for which residues are not readily available. The residues required beĆ tween any two degrees of freedom can be derived as follows. For a linear structure which obeys the Maxwell-Betti reciprocity principle beĆ tween inputs and outputs, the FRF between two DOFs i and j can be obtained by exciting the structure at DOF j and measuring the response at DOF i, or by exciting at DOF i and measuring the response at j: H ijĂ()Ă
H jiĂ(Ă)

Eqn 20-1

In other words, the FRF matrix for a reciprocal structure is symmetric. Under these circumstances, the residue for each mode k between two response DOFs m and n can be obtained from the ones between each of them and the available set of residues for reference j: r mjk
a kĂ$mkĂ$ jk

Eqn 20-2

r njk
a kĂ$nkĂ$ jk

Eqn 20-3

where

322

rmjk

is the known residue between DOFs m and j

rnjk

is the known residue between DOFs n and j

$mk

is the unknown mode shape coefficient at response DOF m

$nk

is the unknown mode shape coefficient at response DOF n

The Lms Theory and Background Book

Design

$jk

is the unknown mode shape coefficient at response DOF j

The required residue is then r mnkĂ
Ă a kĂ$ mkĂ$nkĂ
Ă a kĂ

$ mkĂ$ jk $ nkĂ$jk r mjkĂrnjk $jk Ă a kĂ a kĂ$jk Ă
Ă r jjk

Eqn 20-4

where rjjk is the known driving point residue. The starting point for modal synthesis applications is the available modal modĆ el for the structure to be modified or for each of the substructures to be asĆ sembled. It is important however that some conditions are met. V

In order to be able to scale the included mode shapes correctly, they must include driving point coefficients.

V

Mode shape coefficients need only be available for the Degrees Of FreeĆ dom which are affected by the structural changes.

The information used to obtain this scaling are: poles, (unscaled) mode shapes and modal participation factors for a number of reference stations. The reĆ quired scaled mode shape coefficients can be obtained from this information as follows For Ni points for which output data are also available (i.e. driving points), a vector of complex modal participation factors Lkj for each mode k can be built: L kĂ
Ă L 1ĂL 2...ĂL N i

k

Eqn 20-5

The corresponding unscaled mode shape coefficients Wik are assembled in a column vector {W}k

 W1 !  W2  { W } kĂ
Ă $ .. % W.  N k

Eqn 20-6

i

The residues Rk are defined as the product of mode shapes and modal particiĆ pation factors : [ R ] kĂ
Ă { W } kĂL k

Part IV

Modal Analysis and Design

Eqn 20-7

323

Chapter 20 Design

The scaled mode shapes {V}k , used in the theoretical derivation of the previous chapter are related to the unscaled mode shapes {W}k via a complex scaling facĆ tor k for each mode : { V } kĂ
Ă  kĂ{ W } k

Eqn20-8

From the definition of residues these mode shapes are scaled such that R kĂ
Ă {W} kĂL kĂ
Ă {V} kĂ{V} tk

Eqn 20-9

or from equation 20-8 t

 2kĂ{ W } kĂ{W } kĂ
{ W } kĂĂL k

 kĂ




* Ă  ...Ă Ă W N kĂL N ik i W *1kW1kĂ Ă W *2kW 2kĂ Ă ...Ă Ă W *NiĂkW N ik

W *1kĂLikĂ

W *2kĂL 2kĂ

Eqn 20-10

In the special case where only one input is considered, i.e. only one set of resiĆ dues is available, the scaling factor becomes -

 kĂ
Ă



L 1k W 1k

Eqn 20-11

The scaling of equation 20-8 actually converts the generally valid modal model of mode shape vectors W and modal participation factors L to a model of scaled mode shape vectors V, in which the modal participation factors are absorbed via equation 20-10. Obviously some information is lost by removing the scalĆ ing factors L from the model, and as a consequence, the resulting model is only valid for reciprocal structures with a symmetric FRF matrix. The calculation of the scaling factor k according to equation 20-10 is in fact the best compromise in a least squares sense to approximate a non-reciprocal modal model by a reĆ duced reciprocal one.

324

The Lms Theory and Background Book

Design

20.2

Sensitivity An experimental modal analysis of a structure results in a dynamic model in terms of modal parameters. The qualitative information contained in this modĆ el can be used to identify dynamic problems for example by animation of the mode shapes. Through physical insight and expertise structural modifications can be proposed to overcome specific dynamic problems. For structures with complex dynamic behavior, predictions about the effect of physical changes on modal parameters are usually very difficult - if not imposĆ sible - to make. When unsatisfactory dynamic behavior is detected or susĆ pected the designer can use trial and error procedures to try out a number of modifications, but there is no guarantee that any of these attempts will yield satisfying results. On the other hand numerical techniques can be employed which use the quantitative results of a modal test to evaluate the effects of structural changes. These structural changes can be imposed by modifying the physical characterĆ istics of the structure in terms of its inertia, stiffness and damping. A SensitivĆ ity analysis allows you to see how changes in these physical characteristics afĆ fect particular modes at various points on the structure. It computes only the sensitivity of the modal model to structural alterations, and does not involve actually applying any changes. A Sensitivity analysis provides you with the means of determining the points where such modifications will have most efĆ fect.

20.2.1

Mathematical background to sensitivity analysis Determining the sensitivity of a DOF to various parameters involves (in a mathematical sense) evaluating the partial derivatives of the eigen properties of a matrix with respect to its individual elements. Modal parameters are related to the Frequency Response Function as follows.

H ij()Ă


 Ą jrĂijk

2N

k
1

k

Eqn 20-12

The partial derivative of this equation to a physical parameter P, can be comĆ puted as follows

Part IV

Modal Analysis and Design

325

Chapter 20 Design

āH ij āP

Ă


2N



k
1

ār ijk Ă 1  j   k āP

2N

r ijk ā k  (j  2  ) āP

k
1

Eqn 20-13

k

P can be a mass at one DOF or damping or stiffness between a pair of DOFs. The dynamic stiffness matrix Q is given by Q
  2M cc  jC ccK cc where M C K c that

Eqn 20-14

is the mass matrix is the damping matrix is the stiffness matrix is a subscript denoting that only elements in the matrices are affected by P will be considered.

Using this equation and the theory of adjoined matrices, equation 20-13 can be rewritten in the form āH ij āP

Ă
{H ic} t

āQ {H cj} āP

Eqn 20-15

Using equation 20-12 equation 20-15 becomes

āH ij āP

t

2N r cjk !  2N r ick ! Ă āQĂ  Ă
$ % $ j  k āP (j   k)%  k
1 k
1

Eqn 20-16

Splitting up equation 20-16 into partial fractions, and identifying the correĆ sponding terms of equation 20-13, gives the sensitivities for frequency (20-17) and mode shape (20-18).

 

ā k 1 Ă{r } tĂ āQ Ă
 ick r ijk āP āP

326

j
jdk

{r cjk}

Eqn 20-17

The Lms Theory and Background Book

Design

 

ār ijk

āQ Ă
 {r ick} tĂ  j āP āP 

 2N

m
1

j
jdk

 

{r cjk}Ă 

 {rick}tāāQP  2N

m
1

j
jdk



r cjm m  k



t

r ick āQ Ă m  k āP

j
j dk

{r cjm}

Eqn 20-18

So from equations 20-17 and 20-18, the residues rick and rcjk for each DOF c that is influenced by the structural change are required in order to calculate the sensitivity to that change. Even if not all the residues are available, the MaxĆ well-Betti reciprocity principle can be used to calculate the required values. The residue rick to be derived for any reference DOF c when the residues for DOFs i and c are available for an arbitrary reference j on condition that the drivĆ ing point residue rjjk is also available. The driving point residue is also required if the mode shapes are to be correctly scaled. From the general formula of equation 20-18, it is now possible to calculate the sensitivity value of a mode shape coefficient for DOF i when a structural change is considered for the parameter P, which will affect DOFs a and b. The corresponding scaled mode shape coefficients for each mode in the modal modĆ el are required. From the definition of the dynamic stiffness matrix Q, the three specific cases of P being a mass, a linear spring (stiffness) or a viscous damper can be considered.

Mass This is the case where P is a mass at a specific DOF a. Equations 20-17 and 20-18 are then simplified to ā k Ă
 2kĂ$ 2ak ām a ā$ ik Ă
  kā$ 2akā$ ik  $ak ām a

Eqn 20-19 2N



m
1

 2k $ amā$ im k  m

Eqn 20-20

Stiffness This is the case where P is a linear spring between DOFs a and b. Equations 20-17 and 20-18 are then simplified to

Part IV

Modal Analysis and Design

327

Chapter 20 Design

ā k Ă
 ($ ak  $ bkā) 2 āk ab ā$ ik Ă
 ($ ak  $ bk)ā āk ab

Eqn 20-21 2N

)$ im  ($am $bm 

m
1

k

Eqn 20-22

m

Note that if DOF b is a fixed point (ground") then $bm = $bk = 0

Damping This is the case where P is a viscous damper between DOFs a and b. Equations 20-17 and 20-18 then become ā k Ă
  k($ ak  $ bkā)2 āc ab ā$ ik ($  $ bk) 2 Ă
 ak $ik  ($ ak  $ bk)ā 2 āc ab

Eqn 20-23 2N



k

 m m
1 k

($ am  $ bm)$ im

Eqn 20-24

The imaginary parts of equation 20-19, 20-21 and 20-23 are used to compute the sensitivities of the damped natural frequencies. The corresponding real parts express the sensitivities of damping factors or exponential decay rates.

328

The Lms Theory and Background Book

Design

20.3

Modification prediction This section describes the use of a dynamics modification theory to predict the effect of structural modifications on a mechanical structure's modal parameters. These modifications can take the form of local mass, stiffness and/or damping, FEM-like rod, truss, beam or plate reinforcements. In addition to local modifiĆ cations, a substructure assembly theory allows you to predict the modal model for a structure that consists of an assembly of substructures. Modification prediction allows you to evaluate: V

the effect of structural modifications

V

the effect of any number and type of connections between any number of substructures (only if installed)

V

the dynamics of small scale models, built up from lumped massspring-dash pot elements

Such an analysis avoids time consuming experimental trial and error proceĆ dures of modifying prototypes or scale models of mechanical structures, meaĆ suring and analyzing the dynamic behavior and evaluating the effects of these modifications.

20.3.1

Mathematical background The starting point for the structural modification and substructure theory is the modal model described in section 15.1. The first section of this theoretical background deals with the coupling and modification of substructures using flexible coupling and general viscous damping. It continues with the cases of rigid coupling and flexible coupling with proportional damping.

Modal models for the assembly of substructures with flexible coupling and viscous damping Modal models of substructures Consider two structures, 1 and 2. They obey the following equations of motion in the Laplace domain :

Part IV

Modal Analysis and Design

329

Chapter 20 Design

s 2ĂM 1ĂĂx 1Ă  sC 1ĂĂx 1  K 1ĂĂx 1
Ă Ăf 1Ă

Eqn 20-25

s 2ĂM 2ĂĂx 2Ă  sC 2ĂĂx 2  K 2Ăx 2
Ă Ăf 2Ă

Eqn 20-26

The matrices Mi , Ci and Ki are the mass, damping and stiffness matrices of the structure 1 or 2 corresponding to the subscript i. General viscous damping is allowed. The system matrices are symmetric. The displacement vectors are {x1 } and {x2 }, and the force vectors {f1 } and {f2 } respectively. The modal parameters for substructure 1 will first be derived in a general way. For substructure 2 the same method can be used but will not be entirely reĆ peated. The transformation to decouple the equations of motion can be found by adĆ ding a set of dummy equations (Duncan's method) : sĂM 1Ăx 1 ĂĂ sĂM 1ĂĂx 1 Ă
Ă 0Ă

Eqn 20-27

The system equations for substructure 1 become : sA 1Ăy 1Ă Ă B 1 ĂĂy 1Ă
Ă p 1

Eqn 20-28

where



0 M1 A 1Ă
Ă M C 1 1



sx 

y 1Ă
Ă x 1 1 Ă

B 1Ă
Ă



 M1 0 0 K1



 0

p 1Ă
Ă f 1

The matrices A1 and B1 are diagonalized by the transformation matrix V1 , the matrix of eigenvectors of substructure 1. The corresponding eigenvalues are stored in the diagonal matrix 1 . Due to the addition of equation 20-27 there are twice as many eigenvalues as there are degrees of freedom. They appear in complex conjugate pairs. The matrices A1 and B1 are diagonalized by post- and pre-multiplication by the eigenvector matrix V1 and its transpose : V t1ĂA 1ĂV 1Ă
Ă a 1

330

Eqn 20-29

The Lms Theory and Background Book

Design

V t1ĂB 1ĂV 1Ă
Ă b 1

Eqn 20-30

The matrix of eigenvectors V1 defines a coordinate transformation from physiĆ cal co-ordinates {y1 } to modal coordinates {q1 } :

y 1Ă
Ă V 1Ăq1

Eqn 20-31

Using expressions 20-29 and 20-30 in the equation of motion 20-28 after premultiplication with the transpose of V1 and substitution with expression 20-31 one obtains the equations of motion in modal coordinates for substructure 1 : sĂa 1Ăq 1Ă Ă b 1Ăq 1Ă
Ă V t1Ăp1

Eqn 20-32

It can be seen that the equations of motion in modal space are uncoupled. The same procedure can be repeated for substructure 2, yielding a diagonal eiĆ genvalue matrix  2 and an eigenvector matrix V2 . The eigenvector matrix V2 defines a transformation to modal coordinates {q2 }. The equations of motion for substructure 2 in modal space are : sĂa 2Ăq 2Ă Ă b 2Ăq 2Ă
Ă V t2Ăp2

Eqn 20-33

Substructure assembly The system matrices of both substructures can be merged to give a structure composed of two dynamically independent substructures. For this assembled structure one can easily derive the modal parameters since they are the same as those of the two substructures but gathered in one eigenvalue matrix and one eigenvector matrix. More explicitly this substructuring yields the following system matrices : AĂ
Ă



A1 0 0 A2



BĂ
Ă

  B1 0 0 B2

and Ă

Eqn 20-34

y 

 y Ă
Ă y 1 2

p 

 p Ă
Ă p 1 2

which yields as equation :

Part IV

Modal Analysis and Design

331

Chapter 20 Design

sAĂ{ y }Ă  BĂ{ y }Ă
Ă p 

Eqn 20-35

It can be verified that the matrices of equation 20-35 are diagonalized by the eigenvector matrix V composed as follows :



V1 0 VĂ
Ă 0 V 2



Eqn 20-36

and that the eigenvalue diagonal matrix is :



1

 Ă
Ă 0

0 2



Eqn 20-37

This yields a transformation to modal coordinates : { y } Ă
Ă VĂq ĂĂ

Eqn 20-38

where



q  q Ă
Ă q 1 2

An expression of the type of equation 20-33 using the eigenvector and eigenvaĆ lue matrices, yields : sa Ă q Ă Ă bĂq Ă
Ă V tĂ p 

Eqn 20-39

A close look at the matrix of eigenvectors V shows that the two substructures 1 and 2 are still dynamically independent. Indeed, any force at any point of one substructure will not induce any motion at any point of the other substructure. The two substructures can now be connected with flexible connections modĆ elled as springs and dampers. With the connection matrices Kc and Cc equation 20-35 becomes: s(AĂ Ă A cĂ)Ă{ y }Ă  (BĂ  B cĂ)Ă{ y }Ă
p 

332

Eqn 20-40

The Lms Theory and Background Book

Design

where

0 0 A cĂ
Ă  0 0

0 Cc 0  Cc

0 0  0  Cc  0 0  0 Cc 

0 0 B cĂ
Ă 0 0

0 Kc 0  Kc

0 0  0  Kc  0 0  0 Kc 

The system matrices of the connected substructures will no longer be diagonalĆ ized by the transformation matrix V as the unconnected substructures were. This is due to the introduction of the connection stiffness and/or damping valĆ ues. Modification of structures Before decoupling the equations of motion of the connected substructures a number of modifications to each substructure can be added. Let the structural modifications be gathered in the modification matrices M 1,Ă C 1,Ă K 1ĂandĂM 2,Ă C 2,Ă K 2

Eqn 20-41

These changes can be brought together in system matrices for the modificaĆ tions: 0 M A
 0 0

1

   

0 M 1 0 0 C 1 0 0 0 M2 0 M 2 C2

M 0 B
 0 0

1

   

0 0 0 K 1 0 0 0  M 2 0 0 0 K 2

Eqn 20-42

It is clear from the matrices of previous expression that the modifications are not coupling the substructures, they are only modifying each substructure sepaĆ rately. When the modifications of expression 20-42 are added to the system equation of the connected structure (Eqn. 20-40), one obtains the final equation in physiĆ cal coordinates sĂ(A  A c  A)Ă{ y }Ă Ă (B  B c  B)Ă{y }Ă
Ă  p 

Eqn 20-43

Uncoupling the equations of motion Using the coordinate transformation of the original unconnected substructures (expression 20-36) and premultiplying with Vt , one derives a new set of equaĆ tions of motion in modal coordinates :

Part IV

Modal Analysis and Design

333

Chapter 20 Design

sA m q Ă Ă B m q Ă
Ă V t p 

Eqn 20-44

where A mĂ
Ă aĂ Ă V tA cVĂ Ă V tAV B mĂ
Ă bĂ Ă V tB cVĂ Ă V tBV The matrices Am and Bm for the modified structure can again be diagonalized by a general eigenvalue decomposition. When the new eigenvalues and eigenĆ vectors are represented by ' and W, one has : W tA mWĂ
Ă a W tB mWĂ
Ă b Consider then the transformation :  q Ă
Ă WĂ q ĂĂ

Eqn 20-45

Substituting equation 20-44 and premultiplying with W t yields : saĂ q Ă Ă bĂ q Ă
Ă W tV tĂ p 

Eqn 20-46

The transformation matrices V and W can be combined in one matrix Vi as Eqn 20-47

VĂ
Ă VW which then gives the following transformation equation : { y }Ă
Ă VĂ q 

Eqn 20-48

Equation 20-48 is the transformation between physical coordinates and modal coordinates of the connected and modified substructures. With this coordinate transformation the uncoupled equations of motion are : saĂ q Ă Ă bĂ q Ă
Ă V tĂ p 

Eqn 20-49

The natural frequencies and the damping factors can be found as the imaginary resp. the real part of the eigenvalues in  . The mode shapes are the columns of the matrix Vi.

334

The Lms Theory and Background Book

Design

Flexible coupling with proportional damping The theory discussed above relates to flexible coupling with general viscous damping. In this section we consider the case of zero and proportional dampĆ ing. Recall the general equation of motion for viscous damping (s 2[M]  s[C]  [K])Ă{X}
{F}

Eqn 20-50

Zero damping In case of no damping : [C] = [0], next eigenvalue problem is to be solved with eigenvalues:  r2 and with eigenvectors : {}r. (s 2[M]  [K]){X}
{0}

Eqn 20-51

This system has purely imaginary poles, occurring in complex conjugate pairs.  1
j 1, ...,  N
j N *

*

 1
 j 1Ă, ...,Ă  N
 j N

Eqn 20-52 Eqn 20-53

The modal vectors are real, called normal modes (phase: +/ - 180_). The equation of motion can be diagonalized, based on the orthogonality of the modal vectors. Transformation to modal coordinates leads to an equation of motion, with diagonal system matrices, being the modal mass and modal stifĆ fness maĆ trices: Eqn 20-54 ["]
[{" 1}ā...ā{" N}] ["] tĂ[M]Ă["]
m ąąą["] tĂ[K]Ă["]
k

Eqn 20-55

{X}
["]{q}

Eqn 20-56

(  2m   k){q}
{0}

Eqn 20-57

whereĂ :Ă k
m ā r 2

Eqn 20-58

Propotional damping In case of proportional damping, the damping system matrix is a linear comĆ bination of the mass system matrix and the stiffness system matrix:

Part IV

Modal Analysis and Design

335

Chapter 20 Design

Eqn 20-59

[C]
[M]  [K] This leads to the next equation of motion: 2 (ā(s  s)[M]  [K]ā)ā{X}
{0} s  1

Eqn 20-60

The eigenvalues are related to the complex poles 2

 r   r
  r2  r  1

Eqn 20-61

The complex poles are solved from the real eigenvalues (-ωn) and the damping factors (α, β). When more than two original modes are taken into account (in practical cases, this is always the case), the damping factors can solved in a least squares way from the modal masses, modal stiffnesses and modal dampĆ ing factors. Modal synthesis Only mass and stiffness coupling modifications, ∆M, ∆K and not damping coupling modifications can be applied. The equation of motion of the coupled system are (s 2([M]  [M])  [K]  [K]){X}
{0}

Eqn 20-62

(  2(m   [m])  k  [k]){q}
{0}

Eqn 20-63

["] m
["][q r] m

Eqn 20-64

[m]
[V] tā[M][V]Ą andĄ[k]
[V] tā[K][V]

Eqn 20-65

In modal space:

Where :

The eigenvalues and eigenvectors of this equation, back-transformed from moĆ dal to physical space, are the modal parameters of the coupled system. In case of proportional damping, the complex poles can be solved from the eiĆ genvalues and the proportional damping factors: α and β. The option to use proportional damping is provided when modes are preĆ dicted. It reduces the computation time when dealing with large structures with numerous modifications and mode shapes containing a lot of DOFs. At least two original modes must be used in order to determine  and .

336

The Lms Theory and Background Book

Design

Rigid coupling The above theory relates to flexible coupling, but it is also possible to place constraints on DOFs connecting substructures to create rigid coupling between them, or to constrain a single DOF, thus fixing it rigidly to `ground'. In this case the restrained DOFs will have zero displacement. Constraints on the physical degrees of freedom are [R]{Y}
{0}

Eqn 20-66

Performing a modal transformation: Eqn 20-67

{Y}
["]{q} yields constraints in modal space: [R]["]{q}
[T]{q}
{0}

Eqn 20-68

The modal coordinates are split up into dependent modal coordinates qd an inĆ dependent modal coordinates qi. The constraint matrix [T] is also split up

 

qd [[T d][T i]] q
{0} i

  

Eqn 20-69



{q d}  [T d] 1ă[T i]
{q i}
[T]{q i} I  {q i}

Eqn 20-70

The choice of the dependent modal coordinates has to be made to lead to a non-singular [Td]. This leads to the new eigenvalue problem: (s[T] ta [T]  [T] tb[T]){q i}
{0}

Eqn 20-71

When the eigenvalues and the eigenvectors with the independent modal coordinates qi are solved, the dependent modal coordinates qd of the eigenvecĆ tors can be calculated. In a last step, the mode shapes in physical coordinates are found by the inverse modal transformation. Constraints can be defined in the same way as other structural modifications.

Part IV

Modal Analysis and Design

337

Chapter 20 Design

20.3.2

Implementation of Modification prediction This section discusses some of the more practical aspects of performing modifiĆ cation prediction. This process allows you to compute the natural frequencies, damping values and scaled mode shapes for a modified mechanical structure which is possibly build up from a number of substructures.

20.3.2.1 Retrieval of the modal model The starting point for modal synthesis applications is the available modal modĆ el for the structure to be modified or for each of the substructures to be asĆ sembled. All modal parameters (natural frequencies, damping values, and scaled mode shapes) have to be available for the calculation procedure. It is important howĆ ever that some conditions are met 1 Driving point coefficients In order to be able to scale the included mode shapes correctly, they must inĆ clude driving point coefficients. This means that for at least one record of the modal participation factor table, the force input (reference) identifier should match with a record of the mode shape table for the same mode and neither one of them should be equal to zero. Note that this driving point Degree Of Freedom can be different for each of the included modes. 2 Matching DOFs for modes and modifications Mode shape coefficients need only be available for the Degrees Of Freedom which are affected by the structural changes. This means those for which mass, stiffness or damping modifications are to be considered or to which structural elements are to be attached. Moreover, it is perfectly possible to use incomplete mode shape vectors missing some coefficients for irrelevant DeĆ grees Of Freedom. To obtain correct results, the modal model should include all structural modes to accurately describe the dynamic response for the frequency band of interest. This aspect is especially important when an experimental modal model was obĆ tained from a set of FRFs relative to only one reference station, which happened not to excite some structural modes. This may arise if the reference station was located on or near a nodal point for these modes. In this case the modal model may be well suited to describe the measured FRFs but not the dynamic behavĆ ior of the structure as such.

338

The Lms Theory and Background Book

Design

A similar problem occurs for out-of-band effects caused by the presence of modes above or below the frequency band of experimental modal parameter estimation. Some of the frequency domain techniques for estimating mode shape coefficients allow correction terms (residual masses and flexibilities) to compensate for these residual effects. Using these corrections it is often posĆ sible to curve-fit the measurement data fairly accurately. Unfortunately, these residual terms cannot be scaled correctly for other reference stations as is done for the mode shape coefficients in the previous sections. They cannot therefore be included in the calculations. For this reason, it is advisable to use a suffiĆ ciently large modal model i.e. one with at least one mode below and one mode above the frequency band of interest. When using a modal model for a limited frequency band it is possible that imĆ portant structural modifications would generate modes with a natural frequenĆ cy outside the range of this frequency band. Since the original modal models are not valid at these frequencies, the predicted results will not be very reliable. It is therefore advisable to either include all modes for the frequency band of the resulting modal model or to keep the structural modifications small enough to avoid these problems. In any case you should not attach too much confiĆ dence to modes with natural frequencies outside the frequency band of the original modal model. Included mode shapes are correctly scaled. To obtain correctly scaled mode shapes, the original mode shapes should be scaled in a consistent unit set which respects the consistency of physical quantities: poles, response engineering units per Volt, etc.... A correct calibration of measurement signal transducers and acquisition equipment is required to attach any absolute scaling values to the obtained results.

20.3.3

Definition of modifications to the model At each of the available Degrees Of Freedom of the modal model you can deĆ fine one or more local modifications to influence the dynamic behavior of the mechanical structure. The structure can also be modified by the addition of complete substructures for which modal models exist and by the use of constraints providing rigid coupling.

20.3.3.1 Mass modifications A point mass can be added to a node on the structure. To add a mass modificaĆ tion you simply have to specify the node and the mass.

Part IV

Modal Analysis and Design

339

Chapter 20 Design

20.3.3.2 Stiffness modifications A stiffness connection (spring) can be added between any two Degrees Of FreeĆ dom of the structure. To add a stiffness modification you have to specify, the DOFs between which the stiffness is to be applied and the stiffness value. Note that stiffness (with mass) can also be added to a structure through the addition of a truss or a rod.

20.3.3.3 Damping modifications A damping element (dashpot) can be added between any two Degrees Of FreeĆ dom of the structure. To add a stiffness modification you have to specify the DOFs between which the damping is to be applied, and the damping value. Note that damping can also be added to the structure through the addition of a tuned absorber.

20.3.3.4 Truss elements A truss element can be defined as a doubly hinged rod between two points. Forces located at the ends of the truss element (nodal forces) are directed along the axis of the rod. Since trusses are modelled with hinges at the end, they canĆ not withstand transversal forces. Bending and torsion moments cannot be transmitted from one element to the next. It provides a means of adding stiffness and mass between two points by the addition of a connection for which you know the physical characteristics. To add a truss element you have to specify; The nodes between which the truss is to be fixed and the physical characteristics of the truss. A truss element is characterized by its - cross sectional area A - material's Young's modulus of elasticity E - mass density d These must all expressed in the active unit system.

340

The Lms Theory and Background Book

Design

A truss element between two nodes is translated into elementary mass and stiffness modifications. The longitudinal stiffness is related to a 6 by 6 stiffness matrix for 6 Degrees Of Freedom (3 for each node). This matrix is obtained by projecting the longitudinal stiffness along each of the 3 coordinate axes.

20.3.3.5 Rod elements A rod element can be added between any two separate nodes on the structure. Rods are modelled with hinges at their ends so (modal) forces acting on the ends are directed along the axis of the rod. Bending and torsion moments canĆ not be transmitted from one element to the next. In effect it provides a means of adding stiffness and mass between two points by the addition of a connection for which you know the mass and the stiffness. To add a rod element you have to specify the nodes between which the rod is to be fixed and the physical characteristics of the rod. A rod element is characterĆ ized by its - longitudinal stiffness Kij - its mass M. The longitudinal stiffness is related to a 6 by 6 stiffness matrix for 6 Degrees Of Freedom (3 for each node). This matrix is obtained by projecting the longitudiĆ nal stiffness along each of the 3 co-ordinate axes. The mass M is divided into two equal parts at both ends of the rod.

20.3.3.6 Beam elements A beam element is an element that can transfer translational forces and moĆ ments of bending and torsion. To add a beam element you have to specify the following parameters which are illustrated below :

Part IV

V

the two end nodes (n1, n2)

V

the area of its cross section (A)

V

the material's Young's modulus (E)

Modal Analysis and Design

341

Chapter 20 Design

V

the material's mass density (m)

V

the material's shear modulus (G)

V

the moment of inertia for bending in two planes (Ip, Ib)

V

the moment of inertia for torsion (It)

V

a reference node to define the orientation of the moments of inertia for bending (r) material : E,G,m

cross sectional area A Ip

ÉÉÉ ÉÉÉ ÉÉÉ Ib

It

Reference Node (r)

The reference node together with the two end nodes defines the so-called referĆ ence plane. The moments of inertia for bending are defined in two directions : Ib for bending in the reference plane Ip for bending in a plane perpendicular to the reference plane The 2 end nodes have six Degrees Of Freedom each: 3 translational and 3 rotaĆ tions. A beam element can therefore transmit six forces to another beam eleĆ ment: 3 translational forces and 3 moments. For end nodes that are not conĆ nected to another beam only the translational forces can be transmitted as for example in the case for a stand-alone beam. In the same way, beams that are positioned on a straight line (colinear beams) will not be subjected to torsion.

20.3.3.7 Plate membrane elements A plate membrane element is a two dimensional quadrilateral element capable of transferring both bending forces (perpendicular to the plane of the plate) and membrane forces (in the same plane as the plate). To add a plate element you have to specify the following parameters which are illustrated below :

342

The Lms Theory and Background Book

Design

V

The name of the plate

V

The four corner nodes c1, c2, c3, and c4

V

The plate thickness (t) expressed in the appropriate user unit

V

The number of divisions along the first side, between c1 and c2 (a)

V

The number of divisions along the second side, between c2 and c3 (b)

V

The connection nodes n1, n2 and n3

V

Material properties of the plate i.e. Young's Modulus (E), Poisson's raĆ tio (), mass density (m). These must all be expressed in the appropriate unit. c1

n 3

c4

n

a

1

n

c2

2

b

c3

thickness t

When a plate is defined with a and b divisions along its two sides, a mesh of (a x b) rectangles is created as shown in the diagram. As the corner nodes already exist this means that ((a+1).(b+1) - 4) new nodes are generated. If there are connection nodes defined then the mesh point situated closest to a connection node is replaced by that node. The plate so defined should comply with the following conditions (1 ) d

the mesh elements should not deviate too much from a rectangular form, i.e. each corner angle should be #900

(1) The calculation of the mass and stiffness matrices of a plate membrane described here is based on the plate theory of Mindlin.

Part IV

Modal Analysis and Design

343

Chapter 20 Design

d

the mesh elements should be approximately square, i.e. the ratio of length/width should be # 1

d

the plate should not be too thick, i.e. the ratio of length/thickness should be 

5

Each of the corner nodes of the mesh elements has 6 Degrees Of Freedom - 3 translations and 3 rotations - and so can transmit six forces to another mesh eleĆ ment. This is also the case between elements of different plate membranes, as long as they are connected either at a corner or at a common connection node.

20.3.3.8 Tuned absorbers A tuned absorber is a single Degree Of Freedom system consisting of a rigid mass which is connected by a spring and a dashpot to a more complex strucĆ ture. m

ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ ÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ a

The parameters m, k and c of this SDOF system are designed such that the moĆ tion of the coupling point in the direction of this absorber is decreased (damped) as much as possible for a certain frequency, typically at resonance. xa e jwt k

c

xr e jwt

If the motion of the coupling point in the direction of the absorber is designated by xa and the frequency to be damped by f (= /2
) then the following formuĆ lae apply for the equations of motion of m (xr is the relative displacement beĆ tween the absorber's mass and the attachment point).

344

The Lms Theory and Background Book

Design

(kx r  cjx r)Ăe jtĂ
Ă mĂ(x a  x r)Ă 2Ăe jt

Eqn 20-72

When this equation is solved for xr : x rĂ
Ă

m 2Ăx 2  m 2Ă  jcĂ Ă k

Eqn 20-73

The force acting on the attachment point is Fe jtĂ
Ă (k  jc)Ăx rĂe jt

Eqn 20-74

From equations 20-73 and 20-74 FĂ
Ă

(k  jc)m Ă  2Ăx a 2  m  jc  k

Eqn 20-75

This force can be imagined as being generated by the inertia of an equivalent mass meq , which is rigidly attached to the attachment point : FĂ
Ă m eqĂ 2Ăxa

m eqĂ
Ă

(kĂ  jc)Ăm  mĂ 2Ă Ă jcĂ Ă k

Eqn 20-76

Eqn 20-77

It can be shown that if no damping is used (c=0) the mass and stiffness of the absorber can be designed such that the vibration of the attachment point is eliminated entirely (xa = 0). This happens if the natural frequency of the abĆ sorber equals the forcing frequency . The most practical application of a tuned absorber is the reduction of vibration levels at a resonance frequency n . In this case, the absorber's own natural freĆ quency for optimal tuning is  naĂ
Ă

Part IV

Modal Analysis and Design

mĂk Ă
1Ă ĂĂn 

Eqn 20-78

345

Chapter 20 Design

where ! is the ratio between the absorber's mass and the equivaĆ lent" mass of the system at resonance : Ă
Ă mm

Eqn 20-79

eq

An optimal damping ratio for the absorber is then obtained from : % optĂ
Ă

c Ă
Ă  2 km



3 8(1  ) 3

Eqn 20-80

From equations 20-78, 20-79 and 20-80 the physical parameters m, c and k of the attached absorber can be computed if the following values are known. meq the equivalent mass (see further) n

the target frequency of tuning, natural frequency of mode to be tuned

m

the absorber's mass to be specified by user

The equivalent mass of the system for a certain mode can be obtained as folĆ lows: m eqĂ
Ă

1Ă .V 2i Ă.Ă2Ăjd.

Eqn 20-81

where Vi is the scaled mode shape coefficient of the mode to be tuned at the attachment point d

is the damped natural frequency of the mode to be tuned.

20.3.3.9 Constraints Physical constraints can be defined between separate DOFs or between one DOF and itself. Defining a constraint between two separate DOFs, applies a rigid coupling beĆ tween them. Defining a constraint between a DOF and itself effectively fixes it to `ground'.

346

The Lms Theory and Background Book

Design

20.3.4

Modification prediction calculation Once the required modifications have been defined the modification prediction calculation process can be started. For the simplified case of two substructures which are possibly modified (symĆ bol
) and connected to each other (subscript c), the following procedure is folĆ lowed to predict the modal model of the resulting structure : 1. Retrieve the modal models for each substructure. Build the diagonal maĆ trices  1 and  2 of poles and the (possibly complex) modal matrices V1 and V2 of scaled mode shapes. 2. Join both modal models into the global matrices   (equation 20-37) and V (20-36). 3. Define the connecting elements (springs and dash pots) between both subĆ structures. This yields matrices Ac and Bc (equation 20-40). 4. Define the necessary modifications and join them into matrices A and B (equation 20-42). 5. Use the modal matrix V to transform the connection and modification maĆ trices to the modal space. 6. Add the diagonalized matrices in modal space (equation 20-44) to yield the system matrix of the resulting structure. 7. Calculate the modal model via an eigenvalue and eigenvector decomposition of the resulting system matrix. This yields the complex poles (natural freĆ quencies and damping factors) and the mode shapes.

Numerical problems The eigenvalue problem mentioned above that is to be solved for the modified system, can be subject to numerical problems. These can arise from two sources.

Part IV

V

The presence of unbalanced structural modifications, such as those introducing large amounts of stiffness to simulate a fixation or local heavy dampers.

V

A wide range of original natural frequencies. This can occur especially when rigid body modes of free-free systems (virtually at 0Hz) are imĆ ported from an FE code and mixed with flexible modes at high freĆ quencies. More specifically in this case it is the ratio of the highest to lowest natural frequency that is the relevant factor.

Modal Analysis and Design

347

Chapter 20 Design

In practice these numerical problems are manifested in the modified modal model by unrealistic modal parameters or missing modes. While it is impossiĆ ble to eliminate such problems, they can be reported during the modification prediction calculation. The criterion used in this respect is the condition number of the system matrix. The system matrix is the one whose eigenvalues and eigenvectors yield the moĆ dal parameters. If this condition number exceeds a certain (critical) value this is reported to the user. The critical value used has been established by empiriĆ cal tests and is by default set to 1e+8.

20.3.5

Units of scaling In order to obtain correct modification prediction results, it is absolutely necesĆ sary to maintain a correct scaling of the original modal model using a consisĆ tent unit set. The scaled mode shapes of the original structure have a physical dimension reĆ lated to the measurement data from which they were extracted by modal paĆ rameter estimation techniques. Since this modal model is a valid description for the relation between input forces and response displacements, the applied modifications should be defined in a unit set which is consistent for these quanĆ tities. The same rule applies to the interpretation of the resulting modal model. Erroneous results are bound to occur when the original mode shape vectors are not scaled correctly. This might arise because of the incorrect definition of the reference point for the data (wrong driving point residue), not using the correct transducer sensitivity or calibration factors for the experimental FRFs (force as well as response transducers), or the use of an inconsistent unit set during the modal test or analysis phase. These errors may cause an entirely wrong transĆ formation of the applied physical modifications to the modal space and a small mass modification for example may grow out of proportion because of this bad scaling.

348

The Lms Theory and Background Book

Design

Example of the application of a beam element The following example will illustrate the procedure. Suppose the dynamic beĆ havior of an isotropic plate is to be influenced by a rib fixed to the plate as shown below. 5 4

main plate

3

2 1

I cross section beam

elem 1

elem 2

2

elem 3

elem 4

5

nodes

4 3

1

The procedure becomes : 1 Discretization of the rib into 4 beam elements, interconnected at nodes correĆ sponding to measurement points of the experimental analysis. 2 Definition or calculation of the following physical parameters. A

= cross section of the beam

It

= moment of inertia for torsion

Ib

= moment of inertia for bending in the reference plane, defined by the nodes n1, n2 and r

E

= Young's modulus of elasticity

G

= shear modulus

L

= length of the beam

1, 2, 3 = orientation of the local beam reference system in the global system. This information is derived from the position of the three nodes n1, n2 and r as shown in Figure 20-1.

Part IV

Modal Analysis and Design

349

Chapter 20 Design

m

= material's mass density

From the geometrical properties of the beam the user can calculate the cross sectional area and the different moments of inertia. Tables listing characterĆ istics of various types can be found. z

2

r

1 3

n2

n1

y x

Figure 20-1 Stiffening rib orientation and local co-ordinate system (Axes 1 2 and 3) 3 Construction of the element matrices for each beam element. An element stiffness (full) and mass (diagonal) matrix can be built from the relations between the 6 forces and 6 Degrees Of Freedom at each end node (U1 , V1 , W1 , 1 , 1 , and "1 for node 1, u2 , v2 , w2 ,  & 2 and "2 for node 2) U1! U 2!      V1%
T ĂĂ$V 2
T 2ĂĂĂĂT & translation $ % 1 W1   W 2 

T t1

R t1 T t2 R t2 T1 R1

 1! 2!      1%
R 1ĂĂ$ 2
R 2ĂĂĂĂĂT & rotation $ % "1   " 2 

T2 R1

Figure 20-2 Element matrices for nodes 1 and 2 4 Assembly of the element matrices as shown below

350

The Lms Theory and Background Book

Design

t t t t t t t t t t T1 R1 T2 R2 T3 R3 T4 R4 T5 R5 T1 R1 T1 R1 T1 R1 T1 R1 T1 R1 Figure 20-3 Assembly of element matrices 5 Perform a static condensation (see below) of the rotational DOFs. 6 Add the condensed matrices to the system matrices and continue the calculaĆ tion procedure as for other (lumped) modifications. Remarks : V

The element matrices of a beam model must be assembled before conĆ densation and addition to the system matrices to allow moments to be transmitted between different elements.

V

It is important to keep in mind that the basic assumption in beambending analysis is that a plane section originally normal to the neutral axis remains plane during deformation. This assumption is true proĆ vided that the ratio of beam length to beam height is greater than 2. Furthermore, shear effects do not contribute to the elements of the stiffĆ ness matrix.

V

Care should be taken with the input of moments of inertia. In the exĆ ample stated above the distance between the axis of the plate and the axis of the beam must be taken into account.

Static condensation Static condensation in a dynamic analysis is based upon the assumption that the mass at some Degrees Of Freedom can be neglected without a significant loss of accuracy on the dynamic model in the frequency range of interest. More explicitly, for the beam elements in the application of interest consider the rotaĆ tional Degrees Of Freedom to be without mass. The assembled mass and stiffĆ ness matrices of the entire beam can then be partitioned as follows,

Part IV

Modal Analysis and Design

351

Chapter 20 Design

KTT K   RT

K TR M  [0] TT  K RRĂ;Ă [ 0   ] [0]

Eqn 20-82

where T

refers to the translational DOFs

R

refers to the rotational DOFs.

The modal parameters describing the dynamic behavior of this structure are then obtained by solving following eigenvalue problem,

KTT K   RT

 

 

K TR V T MTT [0]Ă V T ĂĂ V Ă
Ă  2ĂĂ   K RR  [0] [0] V R  R

Eqn 20-83

From the bottom half of equation 20-83 a relation between the translational and the rotational DOFs is derived.

K RTĂV TĂ Ă K RRĂV RĂ
Ă {0}

Eqn 20-84

which can be solved to express the rotational DOFs in terms of the translational ones, 1

V RĂ
Ă Ă K RR ĂK RTĂV T

Eqn 20-85

Introduction of equation 20-85 into equation 20-83 yields :

K TĂĂV TĂ
Ă 2ĂM TTĂĂV TĂ

Eqn 20-86

with

K TĂĂ
K TTĂ Ă K TRĂĂK RRĂ 1ĂKRT

352

Eqn 20-87

The Lms Theory and Background Book

Design

The matrices [KT] and [MTT] of equation 20-86 are used to dynamically model the beam structure. The model will only be valid in the frequency range where the mass effects of the rotational DOFs are negligible. Mass effects only conĆ tribute significantly to the dynamic behavior around and above those resoĆ nances where they are capable of storing a considerable amount of kinetic enerĆ gy. Note that [KT] as expressed in equation 20-87 can only be computed if [KRR] is non-singular. The stiffness matrix is singular if rigid body motion is possible. The rigid body mode of a beam along its longitudinal axis is not naturally elimĆ inated by constraining its three translational DOFs so causing in general a first order singularity. With such configurations it will not be possible to store torĆ sional deformation energy in the beam therefore the corresponding off-diagoĆ nal elements of the assembled stiffness matrix can be neglected and the diagoĆ nal elements made relatively small. In this way the matrix becomes invertible and the predicted dynamic behavior will reflect the inability to store torsional deformation energy in the beam. This operation will, however, not be necesĆ sary when the beam is two or three dimensional, as in such cases, rigid body motion through rotation around one of the axes is no longer possible.

Part IV

Modal Analysis and Design

353

Chapter 20 Design

20.4

Forced response Experimental modal analysis results in a dynamic model described by the moĆ dal parameters, damped natural frequency, exponential decay rate and scaled mode shapes (residues). These modal parameters provide valuable insight into the dynamic behavior of a structure. Problem areas can be identified by aniĆ mating the mode shapes and the relative importance of the mode shapes can be assessed by comparing their amplitudes. In most cases however the designer is less interested in dynamic characteristics themselves than in knowing how the structure is going to behave under normal operating conditions. The important points to determine are V

what will happen under dynamic loading conditions ?

V

which of the natural frequencies will dominate the response ?

V

which points will exhibit large deformations ?

V

how will the structure will deform at particular frequencies ?

The natural frequency of the modes of vibration which seem to be the most imĆ portant parameters in the modal model may well not dominate the response if conditions are such that they are not excited. The Forced response functions enable you to answer these questions by deterĆ mining the response of the modal model to known force spectra.

20.4.1

Mathematical background for forced response The structure's modal model forms the input for the computation of its dynamĆ ic response and is the starting point for the forced response analysis. The equations of motion of a linear, time invariant mechanical structure are exĆ pressed in the frequency domain as follows:

 X() Ă
Ă  H() Ă F() 

Eqn 20-88

where {X()} is the response spectra vector (N0 by 1) [H()] is the Frequency Response Function matrix (N0 by N0 ) {F()} is the applied force spectra vector (N0 by 1).

354

The Lms Theory and Background Book

Design

These quantities are complex-valued functions of the frequency variable  and are valid for every value of  for which these functions are known. When the response at one specific degree of freedom (DOF), say i, is needed the above equations become: X i()Ă
Ă

N0

Ă HijĂ()ĂFj()

Eqn 20-89

j
1

This means that the response at DOF i can be written as a linear combination of the applied forces, each weighted by the corresponding FRF between input DOF j and output DOF i. These frequency dependent weighting factors deĆ scribe the dynamic flexibility between two degrees of freedom i and j of a meĆ chanical structure. When the modal model for that structure is available, e.g. from modal test data or finite element calculations, the FRF can be modelled as given by H ij()Ă
Ă

 j rijk 

2N

Eqn 20-90

k

k
1

Using equation 20-89, it is now possible to predict the dynamic response at DOF i when the structure is subjected to a number of simultaneous loads at DOFs j for which scaled mode shape coefficients (residues) are also available in the modal model. X i()Ă
Ă

Ă jvikvjk !ĂFj() N0

2N

j
1 k
1



k

Eqn 20-91

Even if not all the residues are available, the Maxwell-Betti reciprocity princiĆ ple can be used to calculate the required values. Equation 20-4 allows the resiĆ due rick to be derived for any reference DOF c when the residues for DOFs i and c are available for an arbitrary reference j on condition that the driving point resiĆ due rjjk is also available. The driving point residue is also required if the mode shapes are to be correctly scaled. Equation 20-91 represents the response at all DOFs to all forces with a contribuĆ tion from all modes. The contribution of each mode is given by -

Part IV

Modal Analysis and Design

355

Chapter 20 Design

mode k ; 0 to N

; N to 2N

1 f k()
j   k f k()


N0

N0

j
1

j
1

N0

N0

 vjkFj()
pk()  vjkFj() 



Eqn 20-92

1 v *jkF j()
p k() v *jkF j() j   *k j
1 j
1 (complex conjugate modes)

The response for each DOF then taking into account the contribution of each mode is then given by X i()Ă
Ă

356

N

2N

k
1

k
N

 Ă vikĂfk() Ă  Ă v*ikĂfk()

Eqn 20-93

The Lms Theory and Background Book

Chapter 21

Geometry concepts

This chapter describes the basic concepts involved in the definition of the geometry of a structure. the geometry of a test structure the definition of nodes

357

Chapter 21 Geometry concepts

21.1

The geometry of a test structure A geometrical representation of a test structure is necessary for the display and animation of mode shapes, and for the implementation of design modifications. This chapter discusses the basics regarding the geometry definition of a model for a test structure. The most important part of the model is the nodes. These define the points where measurements will be taken on the structure, and the points where the mode shape deformation are calculated. It is common practice to defined conĆ nections or edges between specific nodes to form a wire frame model of the structure. In addition surfaces can be defined, that aid in the visual representaĆ tion of the structure. z y node

surface

Figure 21-1

x

connection

A wire frame model of a structure

Note that the definition of nodes and meshes for acoustic measurements are deĆ scribed in the Acoustic" documentation.

358

The Lms Theory and Background Book

Geometry concepts

21.2

Nodes A node is defined by its location and its orientation.

Location The location of a node in the 3D space is defined by a set of 3 real numbers known as the coordinates. Coordinates are always defined relative to a referĆ ence coordinate system. The reference coordinates are normally shown along with the model in the the display window. The origin of the global coordinate system is the origin of the 3D space that contains the test structure and the global symmetry of the strucĆ ture should be considered when defining this. The reference coordinate system can be either Cartesian, cylindrical or spheriĆ cal. Z

Z

Z z y

y

y x

x

x

"

Right handed Cartesian

y

r

r

x

Figure 21-2



z

Cylindrical

" Spherical

Coordinate systems

So as an example, the same node defined in each of the coordinate systems would appear as follows. Cartesian

X

Y

Z

1

1

1

Cylindrical

r

"

Z

2

45°

1

Spherical

r

"



3

45°

55°

Orientation Nodal orientation is defined using a Cartesian coordinate system. In many apĆ plications the orientation of the node defines the measurement directions.

Part IV

Modal Analysis and Design

359

Chapter 21 Geometry concepts

z x Figure 21-3

y

Nodal coordinate system

The origin of the nodal coordinate system coincides with the node's location. If the principal axes of the nodal coordinate system are not coincident with the measurement directions, in either a positive or a negative sense, then the differĆ ence must be defined with Euler angles.

Euler angles Three Euler angles are used to define the orientation of a one coordinate sysĆ tem, relative to a reference coordinate system with the same origin. "xy The first angle, ă"xy (Euler XY) is a rotation about the Zr axis of the reference system. (PosiĆ tive from Xr axis to Yr axis). This generates a first intermediate system indicated by a single quote ' on the axis labels.

Zr z'

y'

Xr "xy

"xz The second angle "xz (Euler XZ) is a rotation about the y' axis of the first intermediate system. (Positive from the x' axis to the z' axis). This genĆ erates a second intermediate system, indicated by two quotes " on the axis labels.

360

+

z'

z"

y'y''

x" + "xz

Yr

x'

x'

The Lms Theory and Background Book

Geometry concepts

"yz

Z

Finally the third angle, "yz (Euler YZ) is a rotaĆ tion about the x" axis of the second intermediate system, positive from the y" axis to the z" axis. This last orientation generates the desired new coordinate system orientation.

x''X

z''

Y + "yz y''

Degrees Of Freedom (DOFs) The Degrees Of Freedom of a node represent the directions in which a node is free to move. Each node therefore has a maximum of 7 Degrees Of Freedom; 3 translational, 3 rotational and a scalar DOF(Sc). Z RZ scalar RX X Figure 21-4

Part IV

RY Y

Degrees of freedom

Modal Analysis and Design

361