Chapter 10 Vibration Measurement and Applications

1. 10 Chapter 10 Vibration Measurement and Applications

2. Chapter Outline 10.1 Introduction 10.2 Transducers 10.3 Vibration Pickups 10.4 Frequency-Measuring Instruments 10.5 Vibration Exciters 10.6 Signal Analysis 10.7 Dynamic Testing of Machines and Structure 10.8 Experimental Modal Analysis 10.9 Machine-Condition Monitoring and Diagnosis

3. 10.1 10.1Introduction

4. 10.1 Introduction Why we need to measure vibrations: To detect shifts in ωn which indicates possible failure To select operational speeds to avoid resonance Measured values may be different from theoretical values To design active vibration isolation systems To identify mass, stiffness and damping of a system To verify the approximated model

5. 10.1 Introduction Type of vibration measuring instrument used will depend on: Expected range of frequencies and amplitudes Size of machine/structure involved Conditions of operation of the machine/structure Type of data processing used

6. 10.2 10.2Transducers

7. 10.2 Transducers A device that transforms values of physical variables into electrical signals Following slides show some common transducers for measuring vibration

8. 10.2 Transducers Variable Resistance Transducers Mechanical motion changes electrical resistance, which cause a change in voltage or current Strain gage is a fine wire bonded to surface where strain is to be measured.

9. 10.2 Transducers Variable Resistance Transducers Surface and wire both undergo same strain. Resulting change in wire resistance: where K = Gage factor of the wire R = Initial resistance ΔR = Change in resistance L = Initial length of wire ΔL = Change in length of wire v = Poisson’s ratio of the wire r = Resistivity of the wire Δr = Change in resistivity of the wire ≈ 0 for Advance

10. 10.2 Transducers Variable Resistance Transducers Strain: The following figure shows a vibration pickup:

11. 10.2 Transducers Variable Resistance Transducers ΔR can be measured using a Wheatstone bridge as shown:

12. 10.2 Transducers Variable Resistance Transducers Initially, resistances are adjusted so that E=0 R1R3 = R2R4 When Ri change by ΔRi,

13. 10.2 Transducers Variable Resistance Transducers If the leads are connected between pts a and b, R1=Rg, ΔR1,= ΔRg, ΔR2= ΔR3= ΔR4=0 where Rg is the initial resistance of the gauge. Hence E can be calibrated to read ε directly.

14. 10.2 Transducers Piezoelectric Transducers Certain materials generate electrical charge when subjected to deformation or stress. Charge generated due to force: where k =piezoelectric constant A =area on which Fx acts px =pressure due to Fx.

15. 10.2 Transducers Piezoelectric Transducers E=vtpx v = voltage sensitivity t = thickness of crystal A piezoelectric accelerometer is shown. Output voltage proportional to acceleration

16. 10.2 Transducers Example 10.1 Output Voltage of a Piezoelectric Transducer A quartz crystal having a thickness of 2.5mm is subjected to a pressure of 50psi. Find the output voltage if the voltage sensitivity is 0.055 V-m/N.

17. 10.2 Transducers Example 10.1 Output Voltage of a Piezoelectric Transducer Solution E = vtpx =(0.055)(0.00254)(344738) = 47.4015V

18. 10.2 Transducers Electrodynamic Transducers Voltage E is generated when the coil moves in a magnetic field as shown. E = Dlv where D = magnetic flux density l = length of conductor v = velocity of conductor relative to magnetic field

19. 10.2 Transducers Linear Variable Differential Transformer Transducer Output voltage depends on the axial displacement of the core. Insensitive to temp and high output.

20. 10.3Vibration Pickups 10.3

21. 10.3 Vibration Pickups Most common pickups are seismic instruments as shown Bottom ends of spring and dashpot have same motion as the cage Vibration will excite the suspended mass Displacement of mass relative to cage: z = x – y

22. 10.3 Vibration Pickups Y(t) = Ysinωt Equation of motion of mass m: Steady-state solution:

23. 10.3 Vibration Pickups

24. 10.3 Vibration Pickups Vibrometer Measures displacement of a vibrating body Z/Y ≈ 1 when ω/ωn ≥ 3 (range II) In practice Z may not be equal to Y as r may not be large, to prevent the equipment from getting bulky

25. 10.3 Vibration Pickups Example 10.2 Amplitude by Vibrometer A vibrometer having a natural frequency of 4 rad/s and ζ = 0.2 is attached to a structure that performs a harmonic motion. If the difference between the mximum and the minimum recorded values is 8 mm, find the amplitude of motion of the vibrating structure when its frequency is 40 rad/s.

26. 10.3 Vibration Pickups Example 10.2 Amplitude by Vibrometer Solution Amplitude of recorded motion: Amplitude of vibration of structure: Y = Z/1.0093 = 3.9631 mm

27. 10.3 Vibration Pickups Vibrometer Measures acceleration of a vibrating body.

28. 10.3 Vibration Pickups Vibrometer If 0.65< ζ < 0.7, Accelerometers are preferred due their small size.

29. 10.3 Vibration Pickups Example 10.3 Design of an Accelerometer An accelerometer has a suspended mass of 0.01 kg with a damped natural frequency of vibration of 150 Hz. When mounted on an engine undergoing an acceleration of 1 g at an operating speed of 6000 rpm, the acceleration is recorded as 9.5 m/s2 by the instrument. Find the damping constant and the spring stiffness of the accelerometer.

30. 10.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution

31. 10.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution Substitute (E.2) into (E.1): 1.5801ζ4 – 2.2714ζ2 + 0.7576 = 0 Solution gives ζ2 = 0.7253, 0.9547 Choosing ζ= 0.7253 arbitrarily,

32. 10.3 Vibration Pickups Example 10.3 Design of an Accelerometer Solution Measures velocity of vibrating body: Velocity:

33. 10.3 Vibration Pickups Example 10.4 Design of a Velometer Design a velometer if the maximum error is to be limited to 1% of the true velocity. The natural frequency of the velometer is to be 80Hz and the suspended mass is to be 0.05 kg.

34. 10.3 Vibration Pickups Example 10.4 Design of a Velometer Solution We have Maximum

35. 10.3 Vibration Pickups Example 10.4 Design of a Velometer Solution Substitute (E.2) into (E.1),

36. 10.3 Vibration Pickups Example 10.4 Design of a Velometer Solution R = 1.01 or 0.99 for 1% error ζ4 – ζ2 + 0.245075 = 0 and ζ4 – ζ2+ 0.255075=0 ζ2 = 0.570178, 0.429821 or ζ = 0.755101, 0.655607

37. 10.3 Vibration Pickups Example 10.4 Design of a Velometer Solution Choosing ζ = 0.755101 arbitrarily,

38. 10.3 Vibration Pickups Phase Distortion All vibrating-measuring instruments have phase lag. If the vibration consists of 2 or more harmonic components, the recorded graph will not give an accurate picture – phase distortion Consider vibration signal of the form as shown:

39. 10.3 Vibration Pickups Phase Distortion Let phase shift = 90° for first harmonic Let phase shift = 180° for third harmonic Corresponding time lags: t1= 90°/ω, t2 = 180°/ω Output signal is as shown:

40. 10.3 Vibration Pickups Phase Distortion In general, let the complex wave be y(t) = a1sinωt + a2sin2ωt + … Output of vibrometer becomes: z(t) = a1sin(ωt – Φ1) + a2sin(2ωt – Φ2) + … where

41. 10.3 Vibration Pickups Phase Distortion Φj ≈ π since ω/ωn is large. z(t) ≈ – [a1sinωt + a2sin2ωt + …] ≈ -y(t) Thus the output record can be easily corrected. Similarly we can show that output of velometer isAccelerometer: Let the acceleration curve beOutput of accelerometer:

42. 10.3 Vibration Pickups Phase Distortion Since Φ varies almost linearly from 0° to 90° for ζ = 0.7, Φ≈αr = α(ω/ωn) = βω where α and β are constants. Time lag is independent of frequency. Thus output of accelerometer represents the true acceleration being measured.

43. 10.4Frequency-Measuring Instruments 10.4

44. 10.4 Frequency-Measuring Instruments Single-reed instrument or Fullarton Tachometer Clamped end pressed against vibrating body Adjust l until free end shows largest amplitude of vibration Read off frequency

45. 10.4 Frequency-Measuring Instruments Multi-reed Instrument or Frahm Tachometer Clamped end pressed against vibrating body Frequency read directly off strip whose free end shows largest amplitude of vibration

46. 10.4 Frequency-Measuring Instruments Stroboscope Produces light pulses A vibrating object viewed with it will appear stationary when frequency of pulse is equal to vibration frequency

47. 10.5Vibration Exciters 10.5

48. 10.5 Vibration Exciters Used to determine dynamic characteristics of machines and structures and fatigue testing of materials Can be mechanical, electromagnetic, electrodynamic or hydraulic type

49. 10.5 Vibration Exciters Mechanical Exciters Force can be applied as an inertia force Force can be applied as an elastic spring force for frequency <30 Hz and loads <700N

50. 10.5 Vibration Exciters Mechanical Exciters The unbalance created by two masses rotating at the same speed in opposite directions can be used as a mechanical exciter.