Structures Lab 1 - Cantilever Flexure Beam

Structures Lab 1 Cantilever Beam Vibration Test

David Clark Group 1 MAE 449 – Aerospace Laboratory

Abstract The following exercise observes the lateral modes of vibration of a thin steel cantilever specimen. Using accurate geometric and material properties, the natural frequency response for the cantilever beam can be calculated. For the steel beam used in the experiment, the first natural response was approximately 8.5 Hz. The experimental error experienced at low frequencies was as much as 50%, however at high frequencies, the error was only 5%.

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Contents Abstract .................................................................................................................................................. 2 Introduction and Background................................................................................................................. 4 Introduction........................................................................................................................................ 4 Equipment and Procedure ..................................................................................................................... 6 Equipment .......................................................................................................................................... 6 Experiment Setup ............................................................................................................................... 6 Basic Procedure .................................................................................................................................. 6 Data, Calculations, and Analysis ............................................................................................................. 6 Experimental Results and Error .............................................................................................................. 7 Discussion and Conclusions .................................................................................................................... 8 References ............................................................................................................................................ 10 MathCAD Work..................................................................................................................................... 10

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Introduction and Background Introduction The following laboratory procedure outlines a method for observing and measuring several lateral modes of vibration of a thin steel cantilever specimen. To study the vibration modes, an imaginary cut can be made through the cross section of a cantilever beam. At this cutting plane, the reactions can be expressed as a shear force and a bending moment under a simple point-load configuration. Using simple identities from intermediate mechanics of materials, the displacement for any lateral section of the beam can be determined. Using calculus to find the conditions at which these displacements are maximized, the natural frequencies for a specific geometry with certain properties can be determined. More specifically, the natural frequency is ultimately a function of the following parameters. •

E : The modulus of elasticity

•

I : The moment of inertia perpendicular to the bending axis.

•

ρ : A derived parameter representing the mass per unit length of the beam

•

l : The length of the beam

For a simply supported beam, the following relation may be derived. ݀ସ ‫ݕ‬ − ߚସ‫ = ݕ‬0 ݀‫ ݔ‬ସ Equation 1

where the parameter ϐ represents ߚସ = ߩ

߱ଶ ‫ܫܧ‬

Equation 2

Using boundary conditions and knowledge of solving differential equations, the solution can be expressed using trigonometric expressions.

coshሺߚ݈ሻ cosሺߚ݈ሻ + 1 = 0 Equation 3

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This solution can now easily be solved numerically. The graph and table below visualizes the output of equation 3, as well as solves for the values of ϐl.

Equation 3 Graphical Results 400 300 200 Result

100 0 -100 0

5

10

15

20

-200 -300 -400

ϐl

Figure 1

1.875103894221328 ‫ ۍ‬4.694091132974175 ‫ې‬ ‫ێ‬ ‫ۑ‬ 7.854757438237613 ‫ۑ‬ ߚ݈ = ‫ێ‬ ‫݀ܽݎ‬ ‫ێ‬10.995540734875467‫ۑ‬ ‫ ێ‬14.13716839104647 ‫ۑ‬ ‫ۏ‬17.278759532088237‫ے‬ Equation 4

Mode

angle (βl)

angle (βl)2

cosh(βl)

cos(βl)

1 2 3 4 5 6

1.8751038942E+00 4.6940911330E+00 7.8547574382E+00 1.0995540735E+01 1.4137168391E+01 1.7278759532E+01

3.5160146141E+00 2.2034491565E+01 6.1697214414E+01 1.2090191605E+02 1.9985953012E+02 2.9855553097E+02

3.3374178330E+00 5.4654287011E+01 1.2889850544E+03 2.9803870738E+04 6.8970635290E+05 1.5960258579E+07

-2.9963262857E-01 -1.8296826374E-02 -7.7580418530E-04 -3.3552688870E-05 -1.4498923301E-06 -6.2655664442E-08

Mode 1 2 3 4 5 6

cosh(βl) x cos(βl) -9.9999927793E-01 -1.0000000000E+00 -1.0000000000E+00 -1.0000000020E+00 -9.9999995109E-01 -1.0000006059E+00

cosh(βl) x cos(βl) + 1 7.2206555923E-07 -2.6223467842E-13 3.3504310437E-12 -2.0044048643E-09 4.8913099682E-08 -6.0591884488E-07

ωn/ω1 1.0000000000E+00 6.2668941921E+00 1.7547485203E+01 3.4386067557E+01 5.6842633507E+01 8.4913051774E+01

fn/f1 1.0000000000E+00 6.2668941921E+00 1.7547485203E+01 3.4386067557E+01 5.6842633507E+01 8.4913051774E+01

Table 1

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Equipment and Procedure Equipment The following experiment used the following equipment: •

Thin steel beam approximately 9” x 1” x 0.016”

•

Piezoelectric material

•

Variable electronic function generator

•

Calipers / Ruler

•

Cantilever fixture

Experiment Setup The steel beam is secured by the cantilever fixture. The piezoelectric material is mounted on the beam such that an electrical pulse may cause the beam to flex upon receiving a pulse from the wave generator.

Basic Procedure The electronic function generator is adjusted to output varying pulse outputs. The beam is monitored as the range is adjusted. Upon outputting a natural frequency of the beam, the steel will flex a noticeable amount at low frequencies. Adjusting the output between small ranges allows for the discovery of a natural harmonic response. At higher frequencies, the arrival at a natural response will cause the beam to emit audible noise. The frequencies at which both phenomena occur are recorded as the experimental natural frequencies.

Data, Calculations, and Analysis The dimensions of the beam were measured as follows: •

Length, L = 9.125 inches, or 2.318x10-1 m

•

Width, W = 1.206 inches, or 3.063x10-2 m

•

Thickness, T = 0.41mm, or 4.1x10-4 m

The modulus of elasticity and density of steel are as follows: •

E = 2.034x1011 Pa 6|Page

•

ρsteel = 7.85x103 kg/m3

The moment of inertia for the cross-sectional area perpendicular to the axis of bending, I, is ‫=ܫ‬

1 1 ሺ3.063 × 10ିଶ ݉ሻሺ4.1 × 10ିସ ݉ሻଷ = 1.759 × 10ିଵଷ ݉ସ ܹ ܶଷ = 12 12 Equation 5

The volume of the beam, which is used later to find the mass of the beam, can both be expressed as

ܸ = ‫ = ܶ × ܹ × ܮ‬ሺ2.318 × 10ିଵ ݉ሻሺ3.063 × 10ିଵ ݉ሻሺ4.1 × 10ିସ ݉ሻ = 2.911 × 10ି଺ ݉ଷ Equation 6

݉ܽ‫ߩ = ݏݏ‬௦௧௘௘௟ ܸ = ൬7.85 × 10ଷ

݇݃ ൰ ሺ2.911 × 10ି଺ ݉ଷ ሻ = 2.285 × 10ିଶ ݇݃ ݉ଷ Equation 7

The derived parameter, ρ, is therefore

݉ܽ‫ ݏݏ‬2.285 × 10ିଶ ݇݃ ݇݃ ߩ= = = 0.099 2.318 × 10ିଵ ݉ ‫ܮ‬ ݉ଷ Equation 8

The natural frequency, finally, can be found using the following equation. ‫ܫܧ‬ ߱௡ = ሺߚ௡ ‫ܮ‬ሻଶ ඨ ସ ߩ‫ܮ‬ Equation 9

Substituting the previous values into equation 8, the natural frequencies of the beam can be expressed as ߱௡೟೓೐೚ೝ೐೟೔೎ೌ೗

6.276 ‫ۍ‬39.33‫ې‬ ‫ێ‬ ‫ۑ‬ 110.1‫ۑ‬ =‫ێ‬ ‫ݖܪ‬ ‫ێ‬215.8‫ۑ‬ ‫ێ‬356.7‫ۑ‬ ‫ۏ‬532.9‫ے‬

Equation 10

Experimental Results and Error 7|Page

The following frequencies were recorded as the natural frequencies using the experimental procedure explained previously. ߱௡ೌ೎೟ೠೌ೗

8.5 ‫ ۍ‬23 ‫ې‬ ‫ێ‬ ‫ۑ‬ 130‫ۑ‬ =‫ێ‬ ‫ݖܪ‬ ‫ێ‬230‫ۑ‬ ‫ێ‬370‫ۑ‬ ‫ۏ‬560‫ے‬

Equation 11

35.4 ‫ۍ‬41.5‫ې‬ ห߱௡೟೓೐೚ೝ೐೟೔೎ೌ೗ − ߱௡ೌ೎೟ೠೌ೗ ห ‫ێ‬18.0‫ۑ‬ ‫ۑ‬% ‫= ݎ݋ݎݎܧ‬ =‫ێ‬ ߱௡೟೓೐೚ೝ೐೟೔೎ೌ೗ ‫ێ‬6.58‫ۑ‬ ‫ێ‬3.72‫ۑ‬ ‫ۏ‬5.09‫ے‬

Discussion and Conclusions The table below catalogs the results. Mode 1 2 3 4 5 6

ftheory 6.276 39.33 110.1 215.8 356.7 532.9

fexperimental 8.5 23 130 230 370 560

fexpected 8.5 53.3 149.2 292.3 483.2 721.8

ωtheory ωexperimental 39.4 53.4 247.1 144.5 691.8 816.8 1355.9 1445.1 2241.2 2324.8 3348.3 3518.6

ωexpected 53.4 334.7 937.2 1836.5 3035.8 4535.0

Table 2

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Natural Frequency Modes 800 700

Frequency (Hz)

600 500 400

ftheory

300

fexperimental fexpected

200 100 0 1

2

3

4

5

6

Mode

Figure 2

Natural Frequency (in rad/s) 5000 4500

Response (rad/s)

4000 3500 3000 2500

ωtheory

2000

ωexperimental

1500

ωexpected

1000 500 0 0

1

2

3

4

5

6

7

Mode

Figure 3

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References

“Structures Lab 1 – Cantilever Beam Vibration Test.” Handout

MathCAD Work

The length of the beam −1

L := 9.125in = 2.31775 × 10

−2

W := 1.206in = 3.0632 × 10

−4

T := 0.00041m = 4.1 × 10

m

m

m

Modulus of Elasticity for the beam 6

11

E := 29.5⋅ 10 psi = 2.034 × 10

Pa

Moment of inertia I :=

1 12

3

− 13

⋅ W⋅ T = 1.759 × 10

4

⋅m

The density of steel ρSteel := 7.85

gm 3

3 kg

= 7.85 × 10

cm

3

m

The volume of the beam −6

V := L⋅ W⋅ T = 2.911 × 10

3

⋅m

The mass of the beam −2

mass := ρSteel⋅ V = 2.285 × 10

kg

The parameter, ρ, which is mass per length ρlength :=

mass L

= 0.099

kg m

10 | P a g e

γ, which I will use for the right hand side of the natural frequency equation E⋅ I

gam :=

4

ρlength⋅ L

= 11.215

1 s

The product of βl,

 1.875103894221328   4.694091132974175    7.854757438237613   βl := ⋅ rad  10.995540734875467   14.13716839104647     17.278759532088237  The natural frequencies are, therefore...

 39.432     247.115   691.929  rad 2 3 ⋅ ωnthrad := ( βl) ⋅ ( gam) =  1.356 10 ×   s 3

2.241 × 10

3  3.348 × 10 

 6.276  39.33 110.124 1  ωnth := ωnthrad⋅  = ⋅ Hz 215.799  2⋅ π rad  356.731

 532.894 

11 | P a g e

 8.5  23 130 ωnexp := Hz 230 370

 560   35.442  −41.52 Error :=

ωnexp − ωnth ωnth

18.049 = 6.581

⋅%

3.72

 5.087 

12 | P a g e