Lab 10: Vibration Analysis of a Weighted Cantilever Beam

1. ME 322: InstrumentationLecture 31 April 9, 2014 Professor Miles Greiner

2. Announcements/Reminders • This week: Lab 9.1 Open-ended Extra-Credit • LabVIEW Hands-On Seminar • Extra-Credit • Friday, April 18, 2014, 2-4 PM, Place TBA • Sign-up on WebCampus • If enough interest then we may offer a second session • Noon-2 • HW 10 due Friday • I revised the Lab 10 Instructions, so please let me know about mistakes or needed clarifications. • Did you know? • HW solutions are posted on WebCampus • Exam solution posted outside PE 213 (my office)

3. Lab 10 Vibration of a Weighted Cantilever Beam LE LB • Accelerometer Calibration Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm • C = 616.7 mV/g • Use calibration constant for the issued accelerometer • Inverted Transfer function: a = V/C • Measure: E, W, T, LB, LE, LT, MT, MW • Estimate uncertainties of each Clamp MW W T E (Lab 5) Accelerometer LT MT

4. Table 1 Measured and Calculated Beam Properties • The value and uncertainty in E were determined in Lab 5 • W and T were measured using micrometers whose uncertainty were determined in Lab 4 • LT, LE, and LB were measured using a tape measure (readability = 1/16 in) • MT and MW were measured using an analytical balance (readability = 0.1 g)

5. Figure 2 VI Block Diagram

6. Figure 1 VI Front Panel

7. Disturb Beam and Measure a(t) • Use a sufficiently high sampling rate to capture the peaks • fS> 2fM • Looks like • Expect , • Measure f from spectral analysis ( fM) • Find b from exponential fit to acceleration peaks

8. Time and Frequency Dependent Data • http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Lab%20Index.htm • Plot aversus t • Time increment Dt = 1/fS • Plot aRMSversus f • Frequency increment Df= 1/T1 • Measured Damped (natural) Frequency, fM • Frequency with peak aRMS • Uncertainty • Exponential Decay Constant b (Is it constant?) • Show how to find acceleration peaks versus time • Use AND statements to find accelerations that are larger than the ones before and after it • Use If statements to select those accelerations and times • Sort the results by time • Plot and create new data sets before and after 2.46 sec • Fit data to y = Aebx to find b

9. Figure 4 Acceleration Oscillatory Amplitude Versus Frequency • The sampling period and frequency were T1= 10 sec and fS = 200 Hz. • As a result the system is capable of detecting frequencies between 0.1 and 100 Hz, with a resolution of 0.1 Hz. • The frequency with the peak oscillatory amplitude is fM = 8.70 ± 0.05 Hz. This frequency is easily detected from this plot.

10. Fig. 5 Peak Acceleration versus Time • The exponential decay changed at t = 2.46 sec • During the first and second periods the decay rates are • b1 = -0.292 1/s • b2 = -0.196 1/s

11. Predictdamped natural frequency f from mass, dimension and elastic modulus measurements • How to find equivalent (or effective) mass MEQ, damping coefficient lEQ, and spring constant kEQ for the weighted and damped cantilever beam?

12. Equivalent Endpoint Mass LE LB Clamp • Beam is not massless, so its mass affects its motion and natural frequency • mass of weight, accelerometer, pin, nut • Weight them together on analytical balance (uncertainty = 0.1 g) LT MT ME Beam Mass MB

13. Intermediate Mass, • How to find uncertainty in MEQ? • Power Product or Linear Sum? • Power product or linear sum? • Power product or linear sum? • Power product or linear sum?

14. Uncertainty

15. Beam Equivalent Spring Constant, KEQ F LB • From Solid Mechanics: • E = Elastic modulus measured in Lab 5 • Power product? d

16. Predicted Frequencies • Undamped • Power Product? • Damped • Power product? • If , then , and

17. Table 2 Calculated Values and Uncertainties • The equivalent mass is not strongly affected by the intermediate mass • The predicted undamped and damped frequencies, fOP and fP, are essentially the same (frequency is unaffected by damping). • The confidence interval for the predicted damped frequency fP = 9.0 ± 0.2 Hz does not include the measure value fM= 8.70 ± 0.05 Hz.

19. Measurements and Uncertainties • Lengths • W, T, wW, wT: Lab 4 • LT, LE, LB: Ruler w = 1/16 inch • Masses • MT Total beam mass • MW End components measured together • Uncertainty 0.1 g

21. Lab 10 Vibration of a Weighted Cantilever Beam http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%2010%20Vibrating%20Beam/Accelerometer.pdf

22. Measure a(t) • Dominant frequency ~ 10 Hz • Sampling rate: fS ~1000 Hz (to reliable observe peaks) • Sampling time: T1 ~10 sec (to resolve dominant frequency to with 0.1 • Find damping coefficient and damped natural frequency and compare to predictions • How to predict? t (s) Fit to data: find b and f

23. Predicted behavior Prediction: a(t) = 0 What are the effective values of m, k,  ? ME Equivalent Point end Mass Uniform MB MACC = mass of accelerometer, pin, nut

24. Lab 10 F Beam Spring ConstKeq d Beam cross-section moment of Inertia Determined E and its uncertainty in Lab 4 (you will be given the same beam) Lengths W, T, WW, WT LT, LE, LB - ruler W0 = ± Lab 4 • inch Measure with a ruler in this lab, ± 1/16 inch Masses MT ≡ Beam total mass MW ≡ End components – Mass, nut, bolt, accelerometer

25. Lab 10 Modulus from Lab 4 E, WE Power Product

26. Uncertainty

27. Predicted Undamped Frequency

28. Predicted Damped Frequency 𝜆 = ? = f(Frictional Heating, Fluid Mechanics, Acoustics) • Hard to predict, but we can measure it.

29. Predicted Damped Frequency If then

30. VI • Statistics, time (frequency) of Maximum