9. VIBRATIONS EXPERIMENT OBJECTIVES: 1. Solve a second order non-homogenous differential equation describing the displacement of a specimen experiencing sinusoidal excitation 2. Determine the frequency response of the specimen 3. Find viscous dampening coefficient

10. VIBRATIONS EXPERIMENT OBJECTIVES: 1. Solve a second order non-homogenous differential equation describing the displacement of a specimen experiencing sinusoidal excitation 2. Determine the frequency response of the specimen 3. Find viscous dampening coefficient

11. VIBRATIONS EXPERIMENT Testing equipment used: Accelerometer Force Transducer Signal Conditioner Oscilloscope Computer Function Generator/Shaker Signal Amplifier Data Acquisition: Data Sampling: Other Testing Equipment: Data Acquisition: Data Sampling: Other Testing Equipment: Strobe Light Setup:

12. VIBRATIONS EXPERIMENT Setup: Testing equipment used: Data Acquisition: Accelerometer Force Transducer Data Sampling: Signal Conditioner Oscilloscope Computer Other Testing Equipment: Function Generator Signal Amplifier Accelerometer Force Transducer Signal Conditioner Oscilloscope Computer Signal Amplifier Data Acquisition: Data Sampling: Other Testing Equipment: Strobe Light Function Generator/Shaker

13. VIBRATIONS EXPERIMENT Setup: Preparing to test: Specimen Force Transducer Accelerometer Shaker Wiring: Signal conditioner Function generator Signal amplifier oscilloscope Computer

14. VIBRATIONS EXPERIMENT Objective 1: Solving the Differential Equation: Differential Equation Solver

15. VIBRATIONS EXPERIMENT X Data: Having solved the differential equation, we then plotted displacement per unit force As a function of frequency. Frequency Response:

16. VIBRATIONS EXPERIMENT Data: Displacement, Force, and Frequency

17. VIBRATIONS EXPERIMENT X Data: Having solved the differential equation, we then plotted displacement per unit force As a function of frequency. Frequency Response: Phase angle and frequency

18. VIBRATIONS EXPERIMENT Phase Angle and Frequency

19. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

20. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

21. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

22. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

23. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

24. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

25. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

26. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

27. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

28. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

29. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

30. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

31. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

32. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

33. Mode 1: 64.7 Hz 2 Nodes Mode 2: 204 Hz 3 Nodes Mode 3: 380 Hz 4 Nodes Mode shapes Data: Frequency Response:

34. Mode 1: 64.7 Hz 2 Nodes Mode 2: 204 Hz 2 Nodes Mode 3: 380 Hz 4 Nodes VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Mode shapes

35. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

36. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

37. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

38. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

39. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

40. VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response:

41. Determining dampening coefficient Determining dampening coefficient: Half power method VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: (Note f 2 and f 1 are the Frequencies at 70.7% Of maximum magnitude)

42. Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Log decrement method Solving for zeta where x 0 and x n are amplitudes 10 cycles (n) apart:

43. Log decrement method Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Log decrement method Best guess

44. Log decrement method Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Log decrement method Best guess

45. Log decrement method Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Log decrement method Best guess

46. VIBRATIONS EXPERIMENT

48. Log decrement method Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Log decrement method Best guess   

49. Log decrement method Determining dampening coefficient Determining dampening coefficient: VIBRATIONS EXPERIMENT Mode shapes Data: Frequency Response: Half power method Best guess Data Validity Most data was sound The specimen began to bounce around at frequencies below 55 Hz. We therefore ruled that data as invalid. Below 55 Hz

50. VIBRATIONS EXPERIMENT Frequency Response Domain Change Error Analysis O-Scope Traces

52. Domain change

53. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains:

54. VIBRATIONS EXPERIMENT Complete Frequency Response Trying to express the frequency response on one graph: Change domains: Vector with length X/F At angle  to Y Axis At any given frequency , we have…..

56. Error Analysis Finding the error in the dampening coefficient:

57. Error Analysis DAMPENING CALCULATION ERROR: HALF POWER METHOD We calculated the quantitative errors for the half power method dampening coefficient below:

58. Error Analysis DAMPENING CALCULATION ERROR: HALF POWER METHOD We calculated the quantitative errors for the half power method dampening coefficient below: Since the dampening equation is a quotient of two smaller pieces, we found the relative error in the dampening.

60. 64.7Hz

61. 130 Hz

62. 630 Hz

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