1. 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 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

4. 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

5. 10.2 Transducers
A device that transforms values of physical variables into electrical signals
Following slides show some common transducers for measuring vibration

6. 10.2.1 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.

7. 10.2.1 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

8. 10.2.1 Variable Resistance Transducers
Strain
The following figure shows a vibration pickup:

9. 10.2.1 Variable Resistance Transducers
ΔR can be measured using a Wheatstone bridge as shown:

10. 10.2.1 Variable Resistance Transducers
Initially, resistances are adjusted so that E=0
R1R3 = R2R4
When Ri change by ΔRi,
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.

11. 10.2.2 Piezoelectric Transducers
Certain materials generate electrical charge when subjected to deformation or stress.
Charge generated due to force:
where k is the piezoelectric constant
A is the area on which Fx acts,
px is the pressure due to Fx.

12. 10.2.2 Piezoelectric Transducers
E=vtpx
v is voltage sensitivity
t is thickness of crystal
A piezoelectric accelerometer is shown.
Output voltage proportional to acceleration

13. Example 10.1
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 V-m/N
Solution
E = vtpx =(0.055)( )(344738) = V

14. 10.2.3 Electrodynamic Transducers
Voltage E is generated when the coil moves in a magnetic field as shown.
E = Dlv
D is magnetic flux density
l is length of conductor
v is velocity of conductor relative to magnetic field

15. 10.2.4 Linear variable differential transformer transducer
Output voltage depends on the axial displacement of the core
Insensitive to temp and high output

16. 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

17. 10.3 Vibration Pickups Y(t) = Ysinωt Equation of motion of mass m:
Steady-state solution:

18. 10.3 Vibration Pickups

19. 10.3 Vibration Pickups

20. 10.3.1 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

21. Example 10.2
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.

22. Solution Amplitude of recorded motion:
Amplitude of vibration of structure:
Y = Z/ = mm

23. Accelerometer
Measures acceleration of a vibrating body.

24. 10.3.2 Accelerometer If 0.65< ζ < 0.7,
Accelerometers are preferred due their small size.

25. Example 10.3
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.

26. Solution

27. Solution Substitute (E.2) into (E.1): 1.5801ζ4 – 2.2714ζ2 + 0.7576 = 0
Solution gives ζ2 = ,
Choosing ζ= arbitrarily,

28. Velometer
Measures velocity of vibrating body
Velocity

29. Example 10.4
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.
Solution:

30. Solution
Maximum
Substitute (E.2) into (E.1)

31. Solution R = 1.01 or 0.99 for 1% error
ζ4 – ζ = 0 and ζ4 – ζ =0
ζ2 = , or
ζ = ,
Choosing ζ = arbitrarily,

32. 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:

33. 10.3.4 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:

34. 10.3.4 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

35. 10.3.4 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
Accelerometer: Let the acceleration curve be
Output of accelerometer:

36. 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.

37. 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.

38. 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

39. 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

40. 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.

41. 10.5.1 Mechanical Exciters Force applied as an inertia force:
Force applied as an elastic spring force:
Used for frequency <30 Hz and loads <700N

42. Mechanical Exciters
Makes use of unbalance created by 2 masses rotating at same speed in opposite directions
F(t) = 2mRω2cosωt

43. 10.5.2 Electrodynamic Shaker
2 resonant frequencies as shown:

44. 10.6 Signal Analysis
Acceleration-time history of a frame subjected to excessive vibration:
Transformed to frequency domain:

45. Spectrum Analyzers
Separates energy of signal into various frequency bands
Real-time analyzers useful for machine health monitoring
2 types of real-time analysis procedures: digital filtering method and Fast Fourier Transform method
Basic component of spectrum analyzer: Bandpass filter

46. Bandpass Filter
Permits passage of frequencies over a band and rejects all other frequency components
Response of a filter:

47. Bandpass Filter
fu and fl are upper and lower cutoff frequencies respectively
fc is centre (tuned) frequency
Ripples within band is minimum for a good bandpass filter
2 types of bandpass filters: constant percent bandwidth filters and constant bandwidth filters
Constant percent: (fu – fl)/fc is a constant
E.g. octave, one-half-octave filters
Constant bandwidth: fu – fl is independent of fc

48. 10.6.3 Constant Percent Bandwidth and Constant Bandwidth Analyzers
Spectrum analyzer with a set of octave and 1/3-octave band filters can be use for signal analysis
Lower cutoff freq of a filter = upper cutoff freq of previous filter. Filter characteristics as shown:

49. 10.6.3 Constant Percent Bandwidth and Constant Bandwidth Analyzers
Constant bandwidth analyzer used to obtain more detailed analysis than constant percent bandwidth analyzer
Wave or heterodyne analyzer is a constant bandwidth analyzer with a continuously varying centre frequency

50. 10.7 Dynamic Testing of Machines and Structures
Involves finding the deformation of machines/structures at a critical frequency
Approaches:
Operational Deflection Shape measurements
Modal Testing

51. 10.7.1 Using Operational Deflection Shape measurements
Forced dynamic deflection shape measured under steady-state frequency of system
Valid only for forces/frequency associated with operating conditions
If a particular location has excessive deflection, we can stiffen that location

52. Modal Testing
Any dynamic response of a machine/structure can be obtained as a combination of its modes
Knowledge of the mode shapes, modal frequencies and modal damping ratio will describe completely the machine dynamics.

53. 10.8 Experimental Modal Analysis
When a system is excited, its response exhibits a sharp peak at resonance
Phase of response changes by 180° as forcing frequency crosses the natural frequency
Equipment needed:
Exciter to apply known input force
Transducer to convert physical motion into electrical signal
Signal conditioning amplifier
Analyzer with suitable software

54. 10.8.2 Necessary Equipment Exciter
Can be an electromagnetic shaker or impact hammer
Shaker is attached to the structure through a stringer, to control the direction of the force
Impact hammer is a hammer with built-in force transducer in its head
Portable, inexpensive and much faster to use than a shaker
But often cannot impart sufficient energy and difficult to control direction of applied force

55. 10.8.2 Necessary Equipment Transducer
Piezoelectric transducers most popular
Strain gauges can also be used
Signal conditioner
Outgoing impedance of tranducers not suitable for direct input into analyzers.
Charge or voltage amplifiers are used to match and amplify the signals before analysis
Analyzer
FFT analyzer commonly used
Analyzed signals used to find natural frequencies, damping ratios and mode shapes

56. Necessary Equipment
General arrangement for experimental modal analysis:

57. 10.8.3 Digital Signal Processing
x(t) represents analog signal, xi = x(ti) represents corresponding digital record

58. 10.8.3 Digital Signal Processing
N is fixed for a given analyzer.
Equations can be expressed as

59. 10.8.4 Analysis of Random Signals
Input and output data usually contain random noise.
If x(t) is random signal, its average is

60. 10.8.4 Analysis of Random Signals
Define a new variable x(t) as
Mean square value

61. 10.8.4 Analysis of Random Signals
Autocorrelation function
If x(t) is purely random, R(t)  0 as T  ∞
If x(t) is periodic, R(t) will also be periodic.

62. 10.8.4 Analysis of Random Signals
Power spectral density (PSD)
Cross-correlation function
Cross-PSD

63. 10.8.4 Analysis of Random Signals
If f(τ+t) is replaced by x(τ+t), we get Rxx(t) which leads to Sxx(ω)
Frequency response function H(iω) is related to PSD as

64. 10.8.4 Analysis of Random Signals
Coherence function
β = 0 if x and f are pure noises.
β = 1 if x and f are not contaminated at all.
Typical coherence function:

65. 10.8.5 Determination of Modal Data from observed peaks
Let the graph of H(iω) be as shown:
4 peaks suggesting a 4-DOF system

66. 10.8.5 Determination of Modal Data from observed peaks
Partition into several frequency ranges
Each range is consider a 1-DOF system
Damping ratio corresponding to peak j:
When damping is small, ωj ≈ ωn

67. Example 10.5
The graphs showing the variations of the magnitude of the response and its phase angle with the frequency of a single DOF system provides the frequency response of the system.

68. Example 10.5
Instead of dealing with the magnitude curves directly, if the logarithms of the magnitude ratios (in decibels) are used, the resulting plots are called Bode diagrams. Find the natural frequency and damping ratio of a system whose Bode diagram is as shown.

69. Solution ωn = 10Hz, ω1 = 9.6 Hz, ω2 = 10.5 Hz, Peak response = -35 dB
Damping ratio

70. 10.8.6 Determination of Modal data from Nyquist Plot
Real parts of frequency-response function of 1-DOF system plotted along horizontal axis
Imaginary parts of frequency-response function of 1-DOF system plotted along vertical axis
Frequency-response function

71. 10.8.6 Determination of Modal data from Nyquist Plot

72. 10.8.6 Determination of Modal data from Nyquist Plot
Properties of Nyquist Circle
u and v are large when r=1.
1-r2 = (1+r)(1-r) ≈ 2(1-r) and 2ζr ≈ 2ζ

73. 10.8.6 Determination of Modal data from Nyquist Plot
Once H(iω) is measured, use least square approach to fit a circle. Intersection of circle with –ve Im axis corresponds to H(iωn)
Bandwidth is the difference of the frequencies at the 2 horizontal diametral points
Damping ratio

74. 10.8.7 Measurement of Mode Shapes
Undamped multi-DOF system:
Free harmonic vibration:
Orthogonal relations for mode shapes:

75. 10.8.7 Measurement of Mode Shapes
When forcing functions are harmonic,

76. 10.8.7 Measurement of Mode Shapes
Further normalized [Y] as

77. 10.8.7 Measurement of Mode Shapes
Damped multi-DOF system:
Assume proportional damping:
Undamped mode shapes of the system will diagonalize the damping matrix:

78. 10.8.7 Measurement of Mode Shapes
Frequency-response function when is harmonic:
When mass-normalized mode shapes are used:

79. 10.8.7 Measurement of Mode Shapes
Substituting ω=ωi into the equation, we get:

80. 10.9 Machine Condition Monitoring and Diagnosis
Machine operations will cause misalignments, cracks, unbalances etc in machine parts
Vibration level will increase until machine failure occurs

81. 10.9.1 Vibration Severity Criteria
Vibration severity charts can be used as a guide to determine machine condition
RMS value of vibratory velocity is compared against the criteria set by the standards
However the overall velocity signal used for comparison may not give sufficient warning of the imminent damage.

82. 10.9.2 Machine Maintenance Techniques
Life of machine follows the bathtub curve:

83. 10.9.2 Machine Maintenance Techniques
Breakdown maintenance:
Allow the machine to fail and then replace it with a new machine
This strategy is used when machine is inexpensive and no other damage is caused by the breakdown.

84. 10.9.2 Machine Maintenance Techniques
Preventive maintenance:
Maintenance performed at fixed intervals
Intervals determined statistically from past experience
This method is uneconomical
Condition-based/Predictive maintenance:
Replace fixed-interval overhaul with fixed-interval measurements
Can extrapolate measured vibration levels to predict when they will reach unacceptable values

85. 10.9.2 Machine Maintenance Techniques
Maintenance costs are greatly reduced

86. 10.9.3 Machine Condition Monitoring Technique
Following methods are used to monitor machine conditions:
Aural and visual – a skilled technician will listen and see the vibrations produced by the machine
Operational variables monitoring – performance is monitored wrt intended duty. Deviation denotes a malfunction.

87. 10.9.3 Machine Condition Monitoring Technique
Temperature monitoring – rapid increase in temperature is an indication of malfunction
Wear debris found in lubricating oils can be used to assess extent of damage by observing concentration, size, shape and colour of the particles
Available vibration monitoring techniques

88. 10.9.4 Vibration Monitoring Techniques
Time domain analysis
E.g. following is an acceleration waveform of a gearbox. Pinion is coupled to 2685 rpm motor.
Period of waveform same as period of pinion.
This implies a broken gear tooth on the pinion.

89. 10.9.4 Vibration Monitoring Techniques
Peak level, RMS level and crest factor may be used as indices to identify damage
Changes in Lissajous figures can be used to identify faults.

90. 10.9.4 Vibration Monitoring Techniques
Statistical Methods
Waveform corresponding to good components will have bell-shaped probability density curve
Any deviations can be due to component failure
First 4 moments of the curve are called the mean, standard deviation, skewness and kurtosis.
Kurtosis is defined as
Increase in value of kurtosis can be due to machine component failure

91. 10.9.4 Vibration Monitoring Techniques
Frequency Domain Analysis
Vibration spectrum is unique to that particular machine. Its shape changes as faults starts developing
Nature and location of the fault can be detected by comparing the frequency spectrum of the damaged machine with that of the machine in good condition.

92. 10.9.4 Vibration Monitoring Techniques
Each rotating element generates identifiable frequency. Thus changes in the spectrum at a given freq can be attributed to the corresponding element.

93. 10.9.4 Vibration Monitoring Techniques
Quefrency Domain Analysis
Quefrency is the x-axis for cepstrum
Cepstrum c(τ) is the inverse fourier transform of the log of the power spectrum SX(ω)
Cepstrum can detect any periodicity in the spectrum caused by component failure

94. 10.9.4 Vibration Monitoring Techniques
2nd gear was at fault although 1st gear was engaged.

95. 10.9.5 Instrumentation Systems
3 types – basic system, portable system, computer-based system
Basic system consists of vibration meter, stroboscope and headset
Portable system consists of portable FFT vibration analyzer based on battery power
Computer-based system consists of FFT vibration analyzer coupled with computer for maintaining centralized database and provide diagnostic capabilities

96. 10.9.6 Choice of Monitoring Parameter
Piezoelectric accelerometers are commonly used
Can choose between acceleration, velocity and displacement to monitor
Velocity is commonly used as the parameter for monitoring the machine conditions because the velocity spectrum is the flattest
Any change in the amplitude can be observed easily in a flatter spectrum