Lab5b-guideline For Vibration

MAE 244

Introduction to Vibration

Lab 5-B .

Guidelines for Vibration Experiment Introduction This lab gives the procedure for identifying the system parameters applicable to a simple mass-spring vibration system. Experimental vibration (modal analysis) represents an interdisciplinary field that brings together the signal conditioning, the theory of mechanics, vibrations, acoustics and control theory from mechanical engineering, and the parameter estimation approach of applied mathematics. An experimental approach is provided for determining the modal parameters (frequencies, damping factors, modal vectors and modal scaling) of a linear, time invariant system. Objectives The objective of this lab is to examine the response of a single degree of freedom vibratory system under certain initial conditions: initial displacement and velocity. You will have the opportunity to calculate the system parameters (mass, spring constant, damping) and the dynamic response parameters (natural frequency, amplitude of vibration, period). Then you will verify the accuracy of the theoretical equations by comparing the analytically predicted response to the experimental response for the specific initial condition of the system under study. Equipment The experimental dynamic system comprises the three subsystems, electromechanical apparatus, real time controller and software system (Figure 1).

i.e.,

The electromechanical apparatus consists of the rectilinear mechanism, actuator and

1

MAE 244

Introduction to Vibration

Lab 5-B .

sensor. The design features a brushless DC servomotor, precision rack and pinion drive, highresolution encoders, and reconfigurable system type. The real time controller unit contains the digital signal processor (DSP) based real time processor, servo/actuator interfaces, servo amplifiers, and auxiliary power supplies. DSP is capable of controlling the system high sampling rates allowing the driving function implementation to be modeled as being continuous (rather than discrete) time. The controller also supports such functions as data acquisition, driving function shape generator, and system diagnosis. A logic gate array performs motor commutation and encoder pulse decoding. Two auxiliary digital-to-analog converters (DAC) provide real-time analog signal measurement. The software program runs on a WindowsTM PC. It supports driving function specification, input shape definition, data acquisition, plotting, system execution commands and more. Experimental Procedure 1. Turn on the power of ecp real time controller (Press the Black switch on the front panel). This may prevent from the sudden movement of the vibration system. 2. Turn on the PC computer. 3. Run ECP32 software. 4. You can follow the following steps to make your configuration. Or you can load the configuration directly by clicking menu: File→ Load settings..., select “Fall2005.cfg”, then go to Step (10). It is better to go through the configuration once and know what is going on. 5. Click menu Setup →User Unit... to change the displacement unit to centimeter. 6. Click menu: Command →Trajectory..., On pop-up dialog window, make sure the “step” is selected in the selection group. Then click “Setup” button. In the new pop-up window, input: Step size (cm) :0 Dwell Time (msec) : 4000 Number of reps :1 7. Click “OK” button, then click “OK” again to exit Trajectory configuration. This set up the controller to the mode for acquiring 4 sec of data on command hut without actually introducing driving force (0 magnitude of force) (via the drive motor). This procedure may be repeated and the duration adjusted to vary the data acquisition period. 8. Click menu: Data→ Setup Data Acquisition..., Acquire data only from Encoder 2. 9. Click menu: Plotting→ Setup Plot.... On the pop-up window, make sure “Left Axis” has only “Encoder 2 position”. You can add or delete items. Click “Add to left axis” button to add interested items to “Left Axis” list. You can practice to see different choice. In this test, we only use Encoder 2. 10. Click menu: Utility → Reset Controller. The reading of the controller is set to zero. 11. You can save your current configuration through menu: File→ Save Setting... Real-time plot 12. Setup a real-time plot of the vibration system: Menu: Plotting→ Real time plot ..., in the pop-up dialog, Make sure the Encode 2 Position is displayed in the Left Axis. Click “Plot Data” button. If you have not seen “Enc 2 Position” from the new created window, you need to do step 12 again till you see the right setup. 13. Click menu: Plotting → Axis Scaling..., in the new window, you need to change the

2

Introduction to Vibration

MAE 244

Lab 5-B .

displace scale of y axis to: From: -3, To: 3. (cm). This will change the scale of the ordinate coordinate system so we can see the vibration response. Move the middle cartridge of the rectilinear system and see the real time vibration graph. Vibration system set-up 14. Secure two big blocks (500±5g/each, 1.kg total) on the middle mass carriage. 15. Use either one of the spring (either 800N/m or 400N/m nominally). 16. Move the middle carriage a little bit; check the response on the real-time plot to see the vibration phenomenon. Move the carriage to different position 2.5cm, 2.0cm. 1.5cm, 1cm and 0.5cm. Please do NOT exceed the limit switch. The smaller amplitude of the later cycles becomes dominated by nonlinear (Coulomb) friction effects. Data recording 17. Close real-time plot window. Conduct the test and record data in a file. 18. One student should move and hold the middle carriage to a position approximately at 2.5cm. Another student Click menu: Command→ Execute, on the pop-up dialog, click “Run” button to start. After about 0.5 to 1 second, another student releases the mass. The initial position will be recorded in your original data. 19. The mass will oscillate and attenuate while encoder data is collected to record this response. After a while, when “upload data successfully” window show up, Click “OK” button. 20. Click menu: Plotting→ Setup Plot.... and then click “OK”. Subsequently click “Plot Data” button located lower right corner of the pop-up dialog, you will see the time response plot. 21. Click menu: Data → Export Raw Data ... to save your data in ASCII format. So you can reproduce the plot in Excel or other software. Name your data file properly, such as group11.txt, group1-2.txt. 22. Repeat your tests according to the Experimental data sheet, Change the spring as necessary and connect or disconnect the dashpot as indicated. You need to use the highest stiffness (nominally 800N/m), and least stiffness (nominally 175N/m). Use the hex wrench to connect and disconnect the dashpot. It is assumed that with or without the dashpot, the mass does NOT change. Data processing Attention: The third column “Encoder 2 pos” in the raw data file has to be converted the “cm” unit (The number should be divided by 2266) 23. Measure the initial cycle amplitude Xo and the cycle amplitude Xn after n cycles measured. Use only amplitudes ≥0.25cm in your damping ratio calculation. You should count the steady state error (See Fig.2). Use the logarithmic following decrement equation to calculate ζ :

δ=

δ =

1 x0 ln n xn 2πς 1−ς

2

⇒

ζ =

δ 4π 2 + δ 2

3

Introduction to Vibration

MAE 244

Lab 5-B .

Figure 2 typical time response for the damped vibration system 24. Obtain the period from your vibration response graph. It is better to count more than one cycle and get the time interval. 25. Convert period t into frequency f in Hz and ω in radians/sec. 2 26. Now you can calculate natural frequency from this equation: ω d = ω 1 − ζ

Thus natural frequency is,

ω=

ωd 1− ζ 2

27. For the first two tests with same spring but different mass, you need to calculate the carriage mass mc and the spring stiffness constant after getting the natural frequency. Assume the mass of the block is accurate.

k = ω1 mc + m1 k = ω2 mc + m 2 Solve these two equations to get k and mc. Compare the one with the given value. The carriage mass calculated here will include all inertias from all connected elements. 4

Introduction to Vibration

MAE 244

Lab 5-B .

e.g. motor pinion and armature. Then the critical damping coefficient can be calculate using, c c = 2 mk = 2mω that is, c c1 = 2(mc + m1 )ω1 c c 2 = 2(mc + m2 )ω 2 The damping coefficient is calculated using the damping factor (ratio): c = ζ ⋅ cc 28. Theoretical natural frequency can be calculated using this equation:

ω=

k m

Report (Discussion items indicated in italics) 1. Based on the recorded curves, please calculate the following parameters from each curve: δ = logarithmic decrement ζ = damping factor τ = period cc = critical damping constant c = damping constant ω = natural frequency ωd = damped circular frequency 2. List eight groups of data (calculated in Step 1) in a table to compare the effects of the experimental condition (mass, m, spring constant, k, and the use of dash port) on the characteristic parameters of vibration (δ cc, τ ω and ωd) 3. Discuss the effects of the experimental condition on the characteristic parameters (δ cc τ ω and ωd). 4. Compare the theoretical frequency (ω) with the experimental frequency (ωd).

5