vibration measurement, analysis and control - … (vibration measurement and... · vibration...
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Condition-based maintenance : Vibration measurement, analysis and control for the
PMM course
Classnotes
Prepared for
CE@UP
DIRECTORS:
M Heyns Pr.Eng., Ph.D.,
(Managing)
CJ Botha B.Eng(Hons):
Industrial
Document No:
Revision:
Date:
IM-TR000
0.0
February 2018
mailto:[email protected]
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Table of Contents 1. INTRODUCTION ............................................................................................................................ 6
1.1. Course objective .................................................................................................................... 6 1.2. Goal ........................................................................................................................................ 6 1.3. Main topics of the course ....................................................................................................... 6 1.4. Mandatory reading ................................................................................................................. 7
2. SINGLE DEGREE OF FREEDOM SYSTEMS ............................................................................... 7 2.1. Objective ................................................................................................................................ 7 2.2. After completion you will be able to ....................................................................................... 7 2.3. Why study vibration as part of this course ............................................................................. 7 2.4. Causes of vibration ................................................................................................................ 7 2.5. Elements in a single degree of freedom system .................................................................... 7 2.6. Base and force excitation ....................................................................................................... 9 2.7. Natural frequency ................................................................................................................. 10 2.8. Damping factor ..................................................................................................................... 10 2.9. Frequency ratio .................................................................................................................... 10 2.10. Frequency response function for base excitation ................................................................ 11 2.11. Frequency response function for force excitation ................................................................ 14 2.12. Determination of damping from the response of single degree of freedom systems .......... 16 2.13. Damping factor from the logarithmic decrement .................................................................. 18 2.14. Conclusion............................................................................................................................ 18
3. EFFECT OF VIBRATION ON MAN .............................................................................................. 19 3.1. Natural frequencies of the human body ............................................................................... 19 3.2. Quantification of effects on health ........................................................................................ 20 3.3. Hand-arm vibration syndrome and white finger disease ...................................................... 20 3.4. Whiplash............................................................................................................................... 21 3.5. Conclusion............................................................................................................................ 21 3.6. References ........................................................................................................................... 22
4. VIBRATION CONTROL ................................................................................................................ 22 4.1. Consequence of vibration .................................................................................................... 22 4.2. Control strategies for vibration ............................................................................................. 22 4.3. Critical speeds of rotors ....................................................................................................... 22 4.4. Vibration isolation of a machine on a rigid base .................................................................. 23 4.5. Problems .............................................................................................................................. 24 4.6. Other vibration isolation techniques for information only ..................................................... 25 4.7. Conclusion............................................................................................................................ 25
5. VIBRATION ABSORBERS ........................................................................................................... 25 5.1. Mathematical model for an undamped dynamic vibration absorber .................................... 25 5.2. Conclusion............................................................................................................................ 27
6. CONTINUOUS SYSTEMS ........................................................................................................... 28 6.1. Objective .............................................................................................................................. 28 6.2. After this section you will be able to: .................................................................................... 28 6.3. Natural mode shapes and natural frequencies .................................................................... 28 6.4. Effect of axial force on lateral vibration of a beam ............................................................... 29 6.5. Effect of tension on the natural frequency of a taut string ................................................... 29 6.6. Modal analysis ..................................................................................................................... 29 6.7. Gyroscopic effects ................................................................................................................ 29 6.8. Conclusion............................................................................................................................ 29
7. TRANSDUCERS ........................................................................................................................... 30 7.1. Objective .............................................................................................................................. 30 7.2. After completion you will be able to: .................................................................................... 30 7.3. Measurement system layout ................................................................................................ 30 7.4. Typical transducers .............................................................................................................. 30 7.5. What to measure between displacement, velocity & acceleration ....................................... 30 7.6. Typical applications of vibration transducers ....................................................................... 31 7.7. Vibration transducer construction types ............................................................................... 31 7.8. Transducer sensitivity .......................................................................................................... 32 7.9. Transducer calibration .......................................................................................................... 33 7.10. Transducer mounting considerations ................................................................................... 33 7.11. Selecting transducers ........................................................................................................... 34 7.12. Class problem ...................................................................................................................... 35
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7.13. Conclusion............................................................................................................................ 35 8. ANALOGUE TO DIGITAL CONVERSION.................................................................................... 35 9. TIME DOMAIN ANALYSIS ........................................................................................................... 36
9.1. Introduction........................................................................................................................... 36 9.2. Objective .............................................................................................................................. 36 9.3. After completion ................................................................................................................... 36 9.4. Classification of signal types ................................................................................................ 36
9.4.1. Stationary .................................................................................................................... 36 9.4.2. Ergodic ........................................................................................................................ 36
9.5. Measured or calculated time signal ..................................................................................... 36 9.6. Signal statistics .................................................................................................................... 37
9.6.1. Peak value ................................................................................................................... 37 9.6.2. Peak-to-peak value...................................................................................................... 37 9.6.3. Mean ............................................................................................................................ 37 9.6.4. Root-mean-square....................................................................................................... 37 9.6.5. Crest factor .................................................................................................................. 38 9.6.6. Variance and standard deviation ................................................................................. 38 9.6.7. Kurtosis ........................................................................................................................ 38
9.7. Class problem ...................................................................................................................... 38 9.8. Auto-correlation and cross-correlation ................................................................................. 39 9.9. Averaging in the time domain .............................................................................................. 40 9.10. Orbital analysis ..................................................................................................................... 40 9.11. Conclusion............................................................................................................................ 40
10. FREQUENCY DOMAIN ANALYSIS ......................................................................................... 41 10.1. Objective .............................................................................................................................. 41 10.2. After completion you will be able to: .................................................................................... 41 10.3. Harmonic functions and harmonic analysis ......................................................................... 41 10.4. Order tracking ...................................................................................................................... 44 10.5. Time windows ...................................................................................................................... 44 10.6. Frequency domain data is used: .......................................................................................... 44 10.7. Cepstrum analysis ................................................................................................................ 45 10.8. Spectrum averaging ............................................................................................................. 45 10.9. Number of averages and measurement time....................................................................... 46 10.10. Overlapping ...................................................................................................................... 46 10.11. Class problem .................................................................................................................. 46 10.12. Problem for PMM student on Frequency Domain Analysis ............................................. 46 10.13. Conclusion ....................................................................................................................... 46
11. MAKING GOOD MEASUREMENTS ........................................................................................ 47 12. VIBRATION MONITORING ...................................................................................................... 47
12.1. Objective .............................................................................................................................. 47 12.2. Key concepts in vibration monitoring ................................................................................... 47 12.3. Vibration severity chart ......................................................................................................... 48 12.4. Typical vibration alarm levels ............................................................................................... 49 12.5. Trending analysis ................................................................................................................. 49 12.6. Trending spectra .................................................................................................................. 50 12.7. Quefrency domain analysis .................................................................................................. 52 12.8. Fault diagnosis ..................................................................................................................... 52 12.9. Conclusion............................................................................................................................ 52
13. BALANCING OF RIGID ROTORS ........................................................................................... 52 14. ALIGNMENT OF MACHINES .................................................................................................. 52
14.1. Alignment and soft foot ........................................................................................................ 52 14.2. Thermal growth .................................................................................................................... 52
15. FAULT DIAGNOSIS ................................................................................................................. 52 15.1. Objective .............................................................................................................................. 53
15.1.1. Force unbalance .......................................................................................................... 53 15.1.2. Couple unbalance........................................................................................................ 53 15.1.3. Dynamic unbalance ..................................................................................................... 54 15.1.4. Overhung rotor unbalance ........................................................................................... 54 15.1.5. Eccentric rotor ............................................................................................................. 55 15.1.6. Bent shaft .................................................................................................................... 55 15.1.7. Angular misalignment .................................................................................................. 56
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15.1.8. Parallel misalignment .................................................................................................. 56 15.1.9. Misaligned bearing cocked on shaft ............................................................................ 57 15.1.10. Mechanical Looseness ................................................................................................ 57 15.1.11. Mechanical Looseness Type C ................................................................................... 59 15.1.12. Rolling element bearing frequencies ........................................................................... 60 15.1.13. Resonance .................................................................................................................. 61 15.1.14. Rotor rub ...................................................................................................................... 61 15.1.15. Wear and clearance problems .................................................................................... 62 15.1.16. Oil whirl ........................................................................................................................ 63 15.1.17. Oil whip ........................................................................................................................ 63 15.1.18. Oil whirl and whip ........................................................................................................ 63 15.1.19. Blade and vane pass ................................................................................................... 64
15.2. Conclusions .......................................................................................................................... 64 16. PREDICTIVE MAINTENANCE ................................................................................................. 65
16.1. Objective .............................................................................................................................. 65 16.2. Maintenance management types ......................................................................................... 65 16.3. Benefits of predictive maintenance ...................................................................................... 65 16.4. Predictive maintenance techniques ..................................................................................... 65 16.5. Selecting a predictive maintenance system ......................................................................... 66 16.6. Establishing a predictive maintenance programme ............................................................. 66 16.7. Objectives, goals and benefits from predictive maintenance ............................................... 66 16.8. Management support ........................................................................................................... 67 16.9. Personnel ............................................................................................................................. 67 16.10. Data collection ................................................................................................................. 67 16.11. Data base ......................................................................................................................... 67 16.12. On which equipment must I start? ................................................................................... 67
17. CASE STUDIES FROM CLASS FOR DISCUSSION .............................................................. 67 18. FUTURE TRENDS ................................................................................................................... 67
List of Tables Table 1: Considerations for good measurements ................................................................................ 47
List of Figures Figure 1: Mathematical model of based excited single degree of freedom system ............................... 8 Figure 2: Force-deflection of a linear spring ........................................................................................... 8 Figure 3: Force-velocity curve of a viscous damper ............................................................................... 9 Figure 4: Force excitation of a single degree of freedom system .......................................................... 9 Figure 5: Force and base excitation of a single degree of freedom system ........................................ 10 Figure 6: Mathematical model of a base excited single degree of freedom system ............................ 11 Figure 7: Transmissibility amplitude on a log-log scale with damping factors 0.05, 0.2 and 1.0 ........ 12 Figure 8: Transmissibility amplitude on a linear scale for damping factors 0.05, 0.2 and 1.0 ............. 12 Figure 9: Transmissibility phase for damping factors 0, 0.05, 0.20 and 1.00 ....................................... 13 Figure 10: Force transmitted to the base of a single degree of freedom system under base excitation
........................................................................................................................................................ 14 Figure 11: Mathematical model and frequency response function for force excitation ........................ 15 Figure 12: Magnification factor amplitude for force excited single degree of freedom system ............ 15 Figure 13: Magnification factor phase angle for force excited single degree of freedom system ........ 16 Figure 14: Damping factor estimation from the magnification factor .................................................... 17 Figure 15: Damping from the free vibration of single degree of freedom systems .............................. 18 Figure 16: Natural frequencies in the human body .............................................................................. 19 Figure 17: Whiplash on human beings (Harris's Shock & Vibration Handbook) .................................. 21 Figure 18: Transmissibility function and the amplification and vibration isolation regions ................... 24 Figure 19: Mathematical model of an undamped dynamic absorber ................................................... 26 Figure 20: Steady-state response of a tuned undamped dynamic vibration absorber ........................ 27 Figure 21: Schematic layout of a measurement system ...................................................................... 30 Figure 22: Displacement, velocity and acceleration response as function of frequency ..................... 31 Figure 23: Transducer mounting options with natural frequencies ...................................................... 34 Figure 24: Classification of signal types ............................................................................................... 36
https://investmech-my.sharepoint.com/personal/mheyns_investmech_com/Documents/COURSES/Vibration%20measurement%20&%20Analysis/Handout%20notes/Investmech%20(Vibration%20measurement%20and%20analysis%20for%20PMM)%20TN%20R0.0.docx#_Toc507535969https://investmech-my.sharepoint.com/personal/mheyns_investmech_com/Documents/COURSES/Vibration%20measurement%20&%20Analysis/Handout%20notes/Investmech%20(Vibration%20measurement%20and%20analysis%20for%20PMM)%20TN%20R0.0.docx#_Toc507535986
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Figure 25: Example signal .................................................................................................................... 37 Figure 26: Cross-correlation from two pipe acceleration signals ......................................................... 39 Figure 27: 1s section of the 10s signal ................................................................................................. 42 Figure 28: 0.1s section of the 10s signal .............................................................................................. 42 Figure 29: Root-mean-square spectrum of an arbitrary signal............................................................. 43 Figure 30: A few time windows ............................................................................................................. 44 Figure 31: Cepstrum analysis ............................................................................................................... 45 Figure 32: Overlapping explained ........................................................................................................ 46 Figure 33: Layout of a digital measurement system ............................................................................ 47 Figure 34: Root-mean-square value trended over time ....................................................................... 50 Figure 35: Stages of rolling element bearing failure ............................................................................. 60
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1. INTRODUCTION
This document summarises the notes used for the vibration measurement and analysis section in condition-based maintenance module. Extractions from slides used in class as well as other additional information were used in the construction of these notes.
Slides file: Investmech - Vibration (00 Course detail) R0.0
1.1. Course objective
To give the necessary understanding of vibration, how it is measured and analysed for design, control, fault diagnosis, and vibration monitoring purposes
The course will focus on:
o The basics of the vibration of single- and multiple degree of freedom systems
o Vibration measurement and analysis in the time and frequency domains
o Fault diagnosis from vibration results
1.2. Goal
After completion of this course, you will be able to:
Understand the behaviour and characteristics of single degree of freedom systems
o It is not expected from condition-based maintenance students to carry out all the calculations. Only focus on resonance & damping factor equations to understand effects of mass, stiffness and damping.
Describe the different transducers used
Design vibration isolation and absorbers
o Condition-based maintenance students only need to understand the concepts.
Describe the characteristics of vibration
List causes of vibration
Design a vibration measurement system
Analyse vibration signals:
o For fault diagnosis
o For condition monitoring
o For design
Describe the dynamic behaviour of continuous systems and calculate mode shapes and natural frequencies
o Condition-based maintenance students only focus on natural frequencies and natural modes.
Explain the effect of vibration on man
Discuss maintenance management techniques
1.3. Main topics of the course
The main topics of this section of the course, in the order to be presented, are as follows:
1. Single degree of freedom systems
2. Effect of vibration on man
3. Vibration control
4. Vibration absorbers (only limited focus for condition-based maintenance students)
5. Continuous systems
6. Transducers and vibration equipment
7. Analogue to digital conversion (only limited focus for CBM students)
8. Time domain analysis
9. Frequency domain analysis (only limited focus for CBD students)
10. Making good measurements
11. Vibration monitoring
12. Balancing of rigid rotors (not for for CBD students)
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13. Fault diagnosis
14. Predictive maintenance
15. Case studies
16. Future trends
1.4. Mandatory reading
Study these documents: http://www.trolex.com/silo/files/Vibration%20Application%20Data%281%29.pdf
http://www.southernwater.co.uk/pdf/environment/inYourArea/crossSolent/Chapter_14_Noise.pdf
2. SINGLE DEGREE OF FREEDOM SYSTEMS
Slide files: Investmech - Vibration (01 Single degree of freedom systems) R0.0
2.1. Objective
To give an understanding of the free and forced vibration response of single degree of freedom systems and how to characterise the system elements
2.2. After completion you will be able to
Describe what vibration is
Sketch a mathematical model for a single degree of freedom system
Set up the equations of motion of a single degree of freedom system
Describe: Linear stiffness coefficient, mass, viscous damping coefficient, damping factor, natural frequency, excitation frequency, frequency ratio, frequency response, phase angle, base excitation, force excitation, free response
List methods which may be used to estimate the viscous damping in a single degree of freedom system
Remember: vibration is motion. You can measure acceleration, velocity or displacement and calculate the dependent values.
2.3. Why study vibration as part of this course
Many transducers may be modelled as a single degree of freedom system
Mechanical systems may be modelled as a combination of single degree of freedom systems
Resonance can cause failure in mechanical systems
Vibration control
It is a powerful fault diagnosis technique
Most efficient in condition-based maintenance
Quantifies vibration effect on humans
2.4. Causes of vibration
There are many causes of vibration of which the following is just a small list: Unbalance; Misalignment; Loose elements; Resonance; Bearing defects; Cracks; Cavitation; Gear meshing; Road excitation; Lubrication effects; Wind loads; Earth quakes; Fan blades passing openings, etc.; Driving vehicle over road; etc.
Any excitation that causes motion (and/or deflection) causes vibration.
2.5. Elements in a single degree of freedom system
A single degree of freedom system consists of: 1. Mass, m, in kg. 2. Linear stiffness coefficient, k, in N/m. 3. Viscous damping coefficient, c, in Ns/m.
http://www.trolex.com/silo/files/Vibration%20Application%20Data(1).pdfhttp://www.trolex.com/silo/files/Vibration%20Application%20Data(1).pdfhttp://www.trolex.com/silo/files/Vibration%20Application%20Data(1).pdfhttp://www.southernwater.co.uk/pdf/environment/inYourArea/crossSolent/Chapter_14_Noise.pdfhttp://www.southernwater.co.uk/pdf/environment/inYourArea/crossSolent/Chapter_14_Noise.pdf
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Figure 1: Mathematical model of based excited single degree of freedom system
Figure 2: Force-deflection of a linear spring
()
y()
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Figure 3: Force-velocity curve of a viscous damper
2.6. Base and force excitation
In most instances we have a combination of force and base excitation.
Figure 4: Force excitation of a single degree of freedom system
()
()
Base fixed
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Figure 5: Force and base excitation of a single degree of freedom system
2.7. Natural frequency
The natural frequency of a single degree of freedom system is given by:
fn =n2
=1
2
k
m
2.8. Damping factor
The equivalent viscous damping factor for a single degree of freedom system is given by:
=c
2km
2.9. Frequency ratio
The frequency ratio is defined as the ration between excitation and natural frequency:
r =Excitation frequency
Natural frequency
=f
fn
The frequency ratio characteristics are calculated from the equation of motion of a single degree of freedom system.
()
()
()
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2.10. Frequency response function for base excitation
In this case the base displacement is given as y(t) as shown in Figure 6 in which case F(t) = 0.
The transmissibility amplitude is shown in Figures 7 and 8 on log-log and linear scales respectively, from which the following may be concluded (add from class discussion):
1. For = 2 the amplitude of the transmissibility function is the same for all damping factors. 2. At resonance ( = 1), damping is the best way to control vibration, and an increase in damping
results in a reduction in transmissibility.
3. For > 2, an increase in damping results in a reduction in transmissibility. This is the phenomenon that will result in a smoother ride in a vehicle with low damping factor compared to same vehicle with higher damping factor.
()
()
() The frequency response function is given by the transmissibility function:
=
=
+
( 2) +
=1 + 2
(1 2) + 2
The transmissibility amplitude is:
=
= 1 + (2)2
(1 2)2 + (2)2
Figure 6: Mathematical model of a base excited single degree of freedom system
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Figure 7: Transmissibility amplitude on a log-log scale with damping factors 0.05, 0.2 and 1.0
Figure 8: Transmissibility amplitude on a linear scale for damping factors 0.05, 0.2 and 1.0
10-2
10-1
100
101
10-2
10-1
100
101
Transmissibility Amplitude
Frequency ratio r
Tr
= 0.05
= 0.2
= 1.0
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
12Transmissibility Amplitude
Frequency ratio r
Tr
= 0.05
= 0.2
= 1.0
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Figure 9: Transmissibility phase for damping factors 0, 0.05, 0.20 and 1.00
The force transmitted through the suspension to the base and vice versa is given by the following equation giving the ration between the transmitted force amplitude FT and the static force kY:
FTkY
= r21 + 2ri
(1 r2) + 2ri
The amplitude of this transmitted force ration is shown in Figure 10 from which the following can be seen (use the space below to add inputs during class):
1. For = 2 the amplitude of the transmitted force is the same for all damping factors. 2. At resonance ( = 1), damping is the best way to control transmitted force, and an increase in
damping results in a reduction in transmitted force.
3. For > 2, an increase in damping results in an increase in transmitted force.
0 0.5 1 1.5 2 2.5 3 3.5 4-180
-160
-140
-120
-100
-80
-60
-40
-20
0Transmissibility Phase Angle
Frequency ratio r
Phase A
ngle
[D
egre
es]
= 0.00
= 0.05
= 0.20
= 1.00
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Figure 10: Force transmitted to the base of a single degree of freedom system under base excitation
2.11. Frequency response function for force excitation
The mathematical model and frequency response function for force excitation is shown in Figure 11 for which the magnification factor amplitude and phase angle is shown in Figures 12 and 13 respectively. The following conclusion can be made (add from presentation in class):
1. An increase in damping factor results in a reduction of the magnification factor at all frequencies.
2. At resonance the phase angle between the static displacement and excitation force is 90 .
3. Above resonance the phase angle approach 180 .
4. The magnification factor is sensitive for damping at resonance where an increase in damping factor result in a reduction.
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
12Transmitted Force Ratio
Frequency ratio r
abs(F
T/k
Y)
= 0.05
= 0.2
= 0.5
= 1
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Figure 11: Mathematical model and frequency response function for force excitation
Figure 12: Magnification factor amplitude for force excited single degree of freedom system
()
()
Base fixed
The frequency response function is:
=
1
1 2 + 2
The magnification factor is defined as the magnitude of the inverse of this equation:
=
=
=
1
1 2 + 2
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Figure 13: Magnification factor phase angle for force excited single degree of freedom system
2.12. Determination of damping from the response of single degree of freedom systems
The magnification factor for a single degree of freedom system is shown in Figure 14. The damping can be calculated from the peak value at resonance or from the bandwidth of the half power points for damping factors less than 0.05:
1
2
2 1
2
=1
2 1
For the magnification factor for the blue line at damping factor = 0.05 we have:
= 1
2 0.05= 10
1
2 1=
1
1.046 0.946= 10
For the green line: = 0.2 = 1
20.2= 2.5 and
1
2 1=
1
1.152 0.716= 2.39
0 0.5 1 1.5 2 2.5 3 3.5 40
20
40
60
80
100
120
140
160
180Magnification factor Phase Angle
Frequency ratio r
Phase A
ngle
[D
egre
es]
= 0.05
= 0.2
= 0.5
= 1
= 3
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Figure 14: Damping factor estimation from the magnification factor
Note, you can always use the FRF to find damping factor Nelder-Mead
Bandwidth for damping
estimation taken at
2
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2.13. Damping factor from the logarithmic decrement
Figure 15 shows the free response of an underdamped single degree of freedom system with initial displacement. The viscous damping factor for an under-damped system can be determined from the logarithmic decrement as follows:
=1
mln (
xixi+m
)
=2
1 2
Figure 15: Damping from the free vibration of single degree of freedom systems
2.14. Conclusion
For a single degree of freedom system:
o A natural frequency exists which depends on the stiffness and mass
o Resonance occurs when the system is excited close to resonance
o For base excitation, the transmissibility is low for frequency ratios exceeding 2
o Damping affects the peak amplitude at resonance
o The viscous damping factor can be found from the amplitude and bandwidth of the transfer function, or, the logarithmic decrement
To control the amplitude of vibration of a single degree of freedom system:
o Change natural frequency
Change mass
Change stiffness
o Change damping only efficient at damping
o Change excitation frequency to move away from resonance
=2
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3. EFFECT OF VIBRATION ON MAN
Slides file: Investmech - Vibration (02 Effect of vibration on man) R0.0
The effect of vibration on man background information:
A human beings response to vibration can be modelled as a number of single degree of freedom systems
Time and frequency domain characteristics of excitation vibration are used to quantify the perceived effects of vibration
Effects are:
o Motion sickness: 0.1 to 0.63 Hz
Ships, vehicles with very soft suspensions
o Whole-body in buildings, vehicles or on platforms:
1 to 80 Hz
o Shock
Short duration events
Use ISO2631 (Guide for the evaluation of human exposure to whole-body vibration) and/or BS6841 (British Standard guide to measurement and evaluation of human exposure to whole-body mechanical vibration and repeated shock) as standards.
3.1. Natural frequencies of the human body
Figure 16 shows some natural frequencies in the human body. The actual values differ from person to person (because mass, stiffness and even damping can be different).
Figure 16: Natural frequencies in the human body
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3.2. Quantification of effects on health
Effects: Health effects:
Vibration exposure = vibration + time Motion sickness
Suspected health effects (prolonged exposure) Tissue damage = vibration exposure + time Lumbar spinal disorders Haemorrhoids painful, swollen veins in lower portion of rectum or anus Hernias protrusion of an organ or fascia of an organ through the wall of the cavity that
normally contains it Digestive problems Urinary problems
Performance effects: Control errors Tracking errors increase up to 40% compared to non-vibration performance
Sinusoidal vibration in range 4 20 Hz with accelerations > 0.2g worse than random vibration (Hedge, 2010).
Visual performance disrupted most between 1025 Hz
3.3. Hand-arm vibration syndrome and white finger disease
Please see the slides on the website for figures as used in class.
White finger syndrome (Raynauds syndrome)
Most common condition
Symptoms:
Whitening (blanching) of one or more fingers when exposed to cold
Tingling & loss of sensation
Loss of light touch
Pain and cold sensation between periodic white finger attacks
Loss of grip strength
Bone cysts in fingers & wrists
Changes in tendons, muscles, bones & joints, and can affect nervous system
Stages of White Finger (Hedge, 2010):
Stage 0: No symptoms
Intermittent tingling
Intermittent numbness
Tingling & numbness
Stage 1: Blanching of 1 or more fingertips with(out) tingling & numbness
Stage 2: Blanching of 1 or more fingers with numbness
Usually during winter only
Slight interference with home & social activities
Restricted hobbies
Stage 3: Extensive blanching with frequent episodes during both winter & summer
Definite interference with work, home & social activities, restricted hobbies
Stage 4: Extensive blanching of most fingers
Frequent episodes during winter & summer
Finger ulceration
Gangrene
Occupation change required to avoid further vibration exposure
According to Hedge the white finger disease prevalence is (2010):
50% x 146 tree fellers in British Columbia had Raynaudss phenomenon
Affected 75% x workers with over 20 years of experience
45% x 58 rock drillers had attacks of white finger
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25% of workers with < 5 years of experience
80% of workers with > 16 years experience
Reducing injuries can be achieved by:
Anti-vibration gloves
Vibration isolated handles
Using remote controlled equipment
Reducing exposure times
Please read the following:
1. http://www.hse.gov.uk/vibration/hav/roadshow/bmb2.pdf
3.4. Whiplash
Source: http://papers.sae.org/2011-01-0270/
Provides means of directly measuring pocketing behaviour and relating it to the relevant low-speed rear impact test results. USD 20
Head injury biomechanics
Gadd severity index (GSI) is an extension of the WSTC and is a weighted approach taking the form:
SI = a(t)ndt
a is the acceleration response function
n is the weighting factor
n = 2.5 offered as approximation of the slope of a log-log plot of the WSTC
t is time
SI = 1000 is suggested as injury threshold
For example, for fore & aft accelerations the limits shown in Figure 17 applies. Please see the slide on the website for more information if required.
Figure 17: Whiplash on human beings (Harris's Shock & Vibration Handbook)
3.5. Conclusion
Vibration can have severe effects on humans and need to be measured, monitored and controlled
Whiplash can lead to serious injuries
Measure vibration continuously and integrate with operator warning system to prevent shock and vibration driven injuries
= ()
= 382.5(0.1 0)= 890
http://www.hse.gov.uk/vibration/hav/roadshow/bmb2.pdfhttp://papers.sae.org/2011-01-0270/
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3.6. References
HARRIS, C.M. & PIERSOL, A.G. 2002. Harris shock and vibration handbook. McGraw-Hill.
HEDGE, A. 2010. Human Vibration Vibration Issues. Cornell University, DEA 3250/6510
RAO, S.S. Mechanical Vibrations. McGraw Hill.
http://papers.sae.org/2011-01-0270/
ISO 2631-5
www.sae.org provides standards and publications on the issue
4. VIBRATION CONTROL
Slides file: Investmech - Vibration (03 Vibration control) R0.0
Objective
To discuss various procedures to control vibration
After completion you will be able to:
List consequences of vibration
Explain different ways to control vibration
Design vibration isolation for a single degree of freedom system
4.1. Consequence of vibration
Vibration can cause inter alia:
Structural damage
Fatigue
Crack propagation
Human pain and discomfort
High maintenance costs, etc.
Bearing failures
Component failures
Inaccuracies of instruments (like electron microscopes)
Think of the requirements at nuclear plants
You can add to this list
4.2. Control strategies for vibration
Control strategies for vibration (The focus of the CBM course is on the red items below):
Eliminate the source
Shift natural frequency to get away from resonance OR shift excitation frequency (e.g. change speed)
Introduce damping
Mount machine on vibration isolators
Use tuned vibration absorbers
Do active vibration control
DO DAMAGE TOLERANT DESIGN
Fatigue design according to responses
Follow fracture control programme
4.3. Critical speeds of rotors
The PMM student should only be aware of the critical speeds of rotating machinery which is the speeds at which the system has natural frequencies. Therefore, the rotating system should be taken quickly through its critical speeds. See the slides on the website for more information.
http://papers.sae.org/2011-01-0270/http://www.sae.org/
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4.4. Vibration isolation of a machine on a rigid base
The mathematical models for base and force excitation of a single degree of freedom system is repeated below.
For the force excitation we have an excitation force F(t) = Fosin (t) and a force FT(t) which is transmitted through the suspension to the structure. For the base excitation we have the response x(t) and the base excitation y(t). For both these cases, the transmissibility or transmission ration is:
Tr = FTFo
= X
Y =
1 + (2r)2
(1 r2)2 + (2r)2
This frequency dependent function was plotted for different equivalent viscous damping factor and frequency ration values as shown in Figure 18 from which the following regions can be identified:
1. The amplification region is for frequency ratios 0 r 2
2. The isolation region is for frequency ratio r > 2 Therefore, to have isolation, a system need to be operated above its natural frequency (or critical speed if its a rotating system).
()
()
Base fixed
Mathematical model for force and base excitation
()
()
()
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Figure 18: Transmissibility function and the amplification and vibration isolation regions
4.5. Problems
A 50kg exhaust fan operating at 1500 rpm is to be supported by four springs each having a linear stiffness coefficient of k. If only 20% of the unbalanced force of the fan may be transmitted to the base, what is the linear stiffness coefficient of the springs which may be used?
Use the graph and calculations.
Divide in groups of not more than 7 and quickly do this calculation. The lecturer will guide the class through the process.
Notes on the answer may be made here:
0 0.5 1 1.5 2 2.5 3 3.5 40
2
4
6
8
10
12Transmissibility Amplitude
Frequency ratio r
Tr
= 0.05
= 0.2
= 1.0
Isolation region Amplification region
Isolation region: > 2
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4.6. Other vibration isolation techniques for information only
The following will only be presented in class and is not part of the PMM syllabus.
1. Vibration isolation with flexible foundations
2. Vibration isolation with partially flexible foundations
3. Shock isolation
4. Shock spectrum
4.7. Conclusion
There are many causes of vibration which may be used to identify problems in single degree of freedom systems under certain circumstances.
Vibration can be controlled at the source by eliminating the problem, or machines can be isolated from the vibrating foundations.
Vibration can be controlled by operating away from natural frequencies that cause resonance (change speed, or change natural frequency).
To isolate vibration, low stiffness and damping coefficients must be used and the frequency ratio must
be > 2.
5. VIBRATION ABSORBERS
Slides in file: Investmech - Vibration (04 Vibration absorbers) R0.0
Objective
To introduce the concept of vibration absorbers
After completion you will be able to:
Explain how vibration absorbers work This is the only requirement for CBM students.
Calculate the response of a machine on which vibration absorbers are mounted
Apply the equations to solve the characteristics for optimally tuned damped vibration absorbers
5.1. Mathematical model for an undamped dynamic vibration absorber
A tuned undamped dynamic vibration absorber (mathematical model shown in Figure 19) has the following characteristics:
1 = k1m1
, 2 = k2m2
1 = 2 ; = m2m1
st =F0k1
a = 2; n = 1
f =an
; g =
n; cc = 2m2n; =
c2cc
The equations of motion for this two-degree of freedom system is:
m1x1 = Foeiwt k1x1 + k2(x2 x1)
m2x2 = k2(x1 x2)
That can be written as:
[k1 + k2 m1
2 k2k2 k2 m2
2] [X1X2
] = [Fo0
]
The natural frequencies 1 and 2 of the two-degree of freedom system is given by:
1, 2 = eig [k1 + k2 k2
k2 k2]
This equation can be solved in matlab for any input force Fo. The Matlab code is as follows:
close all, clear all,
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Fo=1; k1=1000; m1=10; m2=1; r1=0.1; r2=2; ds=Fo/k1; k2=k1/m1*m2;
w1=sqrt(k1/m1); w=linspace(r1*w1,r2*w1,1000);
for i=1:length(w)
A=[k1+k2-m1*w(i)^2 -k2
-k2 k2-m2*w(i)^2];
C=[1
0];
B=inv(A)*C;
X0(i)=Fo/(k1-m1*w(i)^2); X1(i)=B(1); X2(i)=B(2);
end
semilogy(w/w1,abs(X1/ds),'LineWidth',2);xlabel('Frequency ratio');ylabel('|X_1/d_s|');
hold on, semilogy(w/w1,abs(X0/ds), '--r','LineWidth',2)
grid, figure
plot(w/w1,abs(X1/ds),'LineWidth',2);xlabel('Frequency ratio');ylabel('|X_1/d_s|');
axis([r1 r2 0 5]), hold on, plot(w/w1,abs(X0/ds), '--r','LineWidth',2)
W1=min(eig([k1+k2 -k2; -k2 k2]))
W2=max(eig([k1+k2 -k2; -k2 k2]))
The response is typically as shown in Figure 20 from which the following can be seen:
1. The response X1 comes to zero at the excitation frequency of r = 1 of the original single degree of freedom system when the absorbers are mounted.
2. The original natural frequency moved to a lower frequency because of the addition of an extra mass and a new natural frequency is formed at higher frequency.
Figure 19: Mathematical model of an undamped dynamic absorber
12
12
2
1()
2()
sin
2
2
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Figure 20: Steady-state response of a tuned undamped dynamic vibration absorber
5.2. Conclusion
Tuned damped vibration absorbers:
Introduce another natural frequency for single degree of freedom systems it makes it a two degree of freedom system
Is only used to control vibration problems caused by resonance
In the undamped case, can theoretically produce an additional force 180 out of phase of the excitation force, of equal magnitude, forcing the vibrating mass to standstill
0 0.5 1 1.5 210
-3
10-2
10-1
100
101
102
103
104
Frequency ratio
|X1/d
s|
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Frequency ratio
|X1/d
s|
Graph on the right is a zoom on a linear scale
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6. CONTINUOUS SYSTEMS
Slides file: Investmech - Vibration (05 Continuous systems) R0.0.
6.1. Objective
To introduce the vibration of continuum systems
6.2. After this section you will be able to:
Calculate the natural frequencies and the position of the mode shapes for the transverse vibration of a beam
Find the mode shape of the vibration of a beam from frequency response function measurements (PMM students only need to explain the process in general)
Define the following: natural frequencies, natural mode shapes, harmonics, frequency response function, spectrum amplitude, phase angle
6.3. Natural mode shapes and natural frequencies
See the slides presented in class on animations of structures.
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6.4. Effect of axial force on lateral vibration of a beam
A compressive force (+P) reduces the natural frequency (think of guitar string)
A tensile force (-P) increases the natural frequency (think of guitar string)
o The equation is
= 2
2
(4 2
)
= 2
2
o Where :
o If = 0, = simply supported beam
o If = 0, that of a taut string (tense guitar string)
o If < 0, increases as tensile force stiffens the beam
o If = , approaches zero for n=1
6.5. Effect of tension on the natural frequency of a taut string
The natural frequency is: =
2
The characteristic speed is: =
/=
Where:
Tension of the string [N]
Mass of the string [kg]
String density [kg/m]
There will be a wave standing over the length L of the string
where =
2 and = 1,2,3,
For the first bending mode =1
2 because only half the
transverse wave stands between the constraints.
6.6. Modal analysis
Modal analysis is the procedure used to find the natural frequencies and the natural modes of continuous systems
The modal mass, stiffness and damping are calculated from the results
In this process, the structure is excited with a modal hammer, or an electrodynamic actuator
The input force and the resulting response are measured
The modal characteristics are calculated from the resulting transfer function
6.7. Gyroscopic effects
When a rotating mass is mounted on a rigid shaft, it may cause a reduction in the natural frequency because the inertial effect introduced by the rotating mass introduces additional inertia
On a flexible shaft the effect might be the other way around for some vibration modes of the shaft
This problem is solved by introducing boundary conditions and solving the equations of motion
This concept should be treated in depth which will not be done in this course
6.8. Conclusion
Continuous systems have an infinite number of natural frequencies and mode shapes
A vibration mode shape is found from the amplitude and phase of the frequency response function
A two channel FFT Analyser can be used to find the mode shape of a structure by moving the response accelerometer to different positions
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Modal analysis is the technique used by exciting structure and measuring excitation force & response to calculated dynamic characteristics
7. TRANSDUCERS
Slides file: Investmech - Vibration (06 Transducers) R0.0
7.1. Objective
To discuss vibration measurement systems and the important specifications for the selection of vibration sensors and equipment
7.2. After completion you will be able to:
Make a decision on when to measure displacement, velocity or acceleration
List the different types of transducers and their advantages and disadvantages
Select accurate mounting techniques
Calculate the sensitivity of a transducer through field calibration
7.3. Measurement system layout
Figure 21: Schematic layout of a measurement system
7.4. Typical transducers
Excitation transducers (see figures in the slides)
Load cells
Response transducers
Displacement
Velocity
Acceleration
Strain gauges
Fibre optic strain transducers
Other transducers
Thermocouples
Pressure transducers
Flow meters
In vibration analysis the typical quantities that are measured are displacement, velocity, acceleration and/or strain
7.5. What to measure between displacement, velocity & acceleration
For sinusoidal vibration, displacement is given as:
x = X sin t
Velocity is then:
x = X cos t
Response Transducer
Excitation Transducer
Signal conditioner
Analyzer
Signal conditioner
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and acceleration:
x = 2X sin t
Based on this, as shown in Figure 22:
Displacement transducers : is used for low frequencies (0 - 1 000Hz)
Accelerometers : is used for high frequencies (0 to >70 kHz)
Velocity meters : is used for frequencies in between (10 to 2 000 Hz)
Accelerometers are the most widely used vibration transducer.
Figure 22: Displacement, velocity and acceleration response as function of frequency
7.6. Typical applications of vibration transducers
Machinery vibration
Vibration control
Modal analysis and structural testing
Seismic vibration
Package testing
Shock
Off-shore structures
Vehicle vibrations
Structural vibration
Hand held tool vibration
etc.
7.7. Vibration transducer construction types
Variable Resistance
Piezoelectric
Capacitive
Electrodynamic
Linear Variable Differential Transformer (LVDT)
Eddy Current
Lazer
Strain gauge
Fibre optic
Stroboscopes
Tachometers
Response of different physical quantities
0
2
4
6
8
10
10 100 1000
Frequency [Hz]
Log
Displ.
Vel.
Acc.
= () = log()
= log()
2
2 = log(2)
Sketch for = 2
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7.8. Transducer sensitivity
The sensitivity is defined as:
Analogue signal = (Sensitivity) (Physical quantity)
Found by calibration
To calculate the physical quantity from the measured signal:
o Physical quantity - (Analogue signal) / (Sensitivity)
Units of the sensitivity is (Volts)/(Quantity measured by the transducer)
The sensitivity for an accelerometer is as follows as a function of the frequency
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7.9. Transducer calibration
Types of Calibration
Absolute
Single channel
Reliable source is required
Referential
Calibrated transducer for standard
Accurate matching required
Ratio
Transducer pair related by constitutive law
Force = mass acceleration
7.10. Transducer mounting considerations
Mounting Considerations
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Frequency Range
Accuracy
Stability
Repeatability
Setup time
Movement time
Environment
Grease, oil, grit, grease, etc.
Moisture
Temperature
Ability to position
Surface damage
Please read the following website for more information: http://www.pcb.com/techsupport/tech_accel.php
7.11. Selecting transducers
For good measurements, pay attention to the following:
http://www.dytran.com/img/tech/a8.pdf
http://www.imi-sensors.com/Mounting_Techniques.aspx
Figure 23: Transducer mounting options with natural frequencies
http://www.pcb.com/techsupport/tech_accel.php
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Mass, Sensitivity, Frequency Response, Contact area, Base strain, Thermal shock sensitivity, Case Isolation, Shielding, Ground isolation, Stability, Repeatability, Compatibility, Water proof, etc.
7.12. Class problem
Divide into groups and discuss the following:
Time allowed: 15 minutes
Feedback will be given by two lecturer indicated groups
1. Suggest a method to do a quick calibration on:
1.1. a capacitive type accelerometer they can measure gravity
1.2. a piezoelectric type accelerometer
2. How will you calibrate:
2.1 an eddy current type displacement transducer
2.2 a LVDT
3. An accelerometer measures an acceleration amplitude of 8g @ a frequency of 10Hz. What is the displacement amplitude of the vibration?
7.13. Conclusion
We have seen that:
displacement, velocity and/or acceleration may be used to quantify the vibration at a point
the selection of the physical quantity that we want to measure depend on the frequency range
accelerometers have a wide frequency range where they can be applied
the frequency range of a transducer influences the sensitivity at that frequency
the frequency range and mounting accuracy controls the mounting technique
there are accurate techniques to do field calibration on accelerometers
This space below is left open for the students in-class notes on Analogue-do-digital conversion (ADC). Only a quick 15 minutes will be spent on this to make the student aware of problems in this field.
8. ANALOGUE TO DIGITAL CONVERSION
Slides file: Investmech - Vibration (07 Analogue to digital conversion) R0.0
The student is only quickly introduced to this and this is not part of the PMM syllabus. For more information, please download the slides from the indicated website.
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9. TIME DOMAIN ANALYSIS
Slides file: Investmech - Vibration (08 Time domain analysis) R0.0
9.1. Introduction
The time signal contains information which can be used successfully in fault diagnosis and vibration monitoring programmes.
9.2. Objective
Give the necessary theoretical background of analysis in the time domain.
9.3. After completion
After completion you will be able to calculate and explain the significance of the following statistical parameters from a signal: peak, peak-to-peak, arithmetic mean, root-mean-square, crest factor, kurtosis, variance, standard deviation, auto-correlation, cross-correlation, averaging.
9.4. Classification of signal types
9.4.1. Stationary
For stationary signals, do time and frequency analysis. Take note of the effect of ergodic and non-ergodic effects.
9.4.2. Ergodic
A signal is ergodic if its statistical properties (such as mean and variance) can be deduced from a single, sufficiently long sample of the process
Figure 24: Classification of signal types
9.5. Measured or calculated time signal
Figure 25 shows the detail of an example signal.
What is the peak-value? 5 at the red dot, or -10 at the yellow dot? This depends entirely on the type of data shown. For acceleration of rotating machinery, it will be -10, given as the absolute value = 10.
The peak-to-peak value is 5 (-10) = 15 N in this case
The mean was calculated as -2.33 N and standard deviation 5.00 N
Signal
Deterministic[r.m.s., peak-to-peak, mean-square analyses]
Periodic
Sinusoidal
Complex
Quasi-periodic
Non-periodic
Transient
Shock
Random
Stationary
[Statistical parameters stays constant as function of time]
Ergodic
[Statistical parameters and auto-correlation do not vary for time and different sample functions]
Self-stationary
Strongly
Weakly
Non-ergodic
Non-stationary
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Time F
[s] [N]
0 5
0,1 0
0,2 5
0,3 -7
0,4 2
0,5 -1
0,6 -10
0,7 4
0,8 -3
0,9 2
1 2
1,1 -9
1,2 -3
1,3 -3
1,4 -9
1,5 2
1,6 -8
1,7 -1
1,8 -5
1,9 -2
2 -10
Mean -2,33 N
Std.Dev. 5,00 N
5
0
5
-7
2
-1
-10
4
-3
2 2
-9
-3 -3
-9
2
-8
-1
-5
-2
-10
-12
-10
-8
-6
-4
-2
0
2
4
6
0 0,5 1 1,5 2 2,5
Figure 25: Example signal
9.6. Signal statistics
9.6.1. Peak value
Peak value (Vp) : maximum absolute value of the signal
Maximum(abs(signal(t)))
9.6.2. Peak-to-peak value
Peak-to-peak value (Vp-p) : maximum minimum value
9.6.3. Mean
= limT
1
T f(t)dt
T
0
=1
N f(i)
N
i=1
For a random signal the mean is zero
9.6.4. Root-mean-square
RMS = limT
1
T f(t)2dt
T
0
= 1
N f(i)2
N
i=1
Gives the intensity of the data which is an indication of the energy
This is an Overall Value, that is, one value that describes the characteristic of all the values
With band-pass filter will give narrow-band intensity
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9.6.5. Crest factor
CF =Peak of the signal
RMS
For a pure sine wave cf = 2
A cf > 3 indicates on irregularities in the signal
The cf is not monotome
o Will not necessarily increase with an increase in RMS
Used to describe the peakiness of a function/signal
9.6.6. Variance and standard deviation
2 = limT
1
T(f(t) )2dt
T
0
=1
N(f(i) )2
N
i=1
Variance = (standard deviation)2 = 2
The standard deviation quantifies the distribution of data points around the mean.
9.6.7. Kurtosis
KU =1
4lim
T
1
T(f(t) )4dt
T
0
=1
4
1
N(f(i) )4
N
i=1
The kurtosis is not monotome
Describe the peakiness of a signal
For sine wave KU=2
For a random signal KU=1.5
9.7. Class problem
Space allowed for class problem on time domain analysis
Sample record given: -1; -0,5; 0; 0,5; 1; 0,75; 0,5; 0,2; -0,2
Peak-to-peak value
Mean
RMS
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Standard deviation
Crest factor
9.8. Auto-correlation and cross-correlation
This is not part of the PMM syllabus. The lecturer will however demonstrate the power of cross-correlation calculations to obtain the epicentre of cracks in pressure vessels from recorded acceleration signals.
The cross-correlation is defined as:
T
Tfx dttxtf
TR
0)()(
1lim)(
The cross-correlation define the general dependence of the values of one signal at a particular time with the values of another signal as a function of the time shift. The answer is a peak at the time shift between the two signals. The technique can be used to detect leaks in a pipe as shown in Figure 26. Because the signals travel to Sensor 1 and Sensor 2 at the same speed, there will be a time shift in the measured signals that can be used with the known length of the pipe to determine the epicentre of the leak.
Figure 26: Cross-correlation from two pipe acceleration signals
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1
0
1
2
3
4
5
Time shift [s]
Cro
ss-c
orr
ela
tio
n
Random signal with 1 s time lag
1 2
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9.9. Averaging in the time domain
1. If averaging is done on a random signal, the result will be zero
2. Averaging in the time domain is applied where the signal repeat itself after a period of time
3. To do averaging, it is important that the sampling is triggered at the same position on the signal for each sample record
4. Analysers can also hold information before the sampling is triggered and display that by setting proper delays
9.10. Orbital analysis
9.11. Conclusion
The time signal contains important information which can be quantified by statistical quantities.
It is good practise to look at the time signals before signal processing is done on a measured signal. The shape and statistical parameters may describe certain faults in the system under investigation.
Correlation functions can find deterministic signals in a dominant random signal.
Pay attention when you are averaging in the time domain.
For example, a broken gear tooth will result in impacts clearly visible on the acceleration time signal.
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10. FREQUENCY DOMAIN ANALYSIS
Slides file: Investmech - Vibration (09 Frequency domain analysis) R0.0
10.1. Objective
To give a thorough understanding of the transformation of time domain data to frequency domain data
10.2. After completion you will be able to:
Explain what the frequency domain is and what information of the time signal is given
Explain : frequency, bandwidth, amplitude, phase angle, spectrum, auto-spectrum, cross-spectrum, power spectral density, frequency response function, transfer function, cepstrum, windowing, order tracking, averaging, overlapping
Note, CBM students will only be introduced to the principle and not all detail.
10.3. Harmonic functions and harmonic analysis
The basis of frequency domain analysis is that any periodic function of time can be represented as an infinite sum of sine and cosine terms.
In Fourier transform analysis the amplitudes, frequencies and phase angles of a finite number of sine or cosine terms are calculated which will reproduce the original signal when they are summed in the time domain.
If digital data is used, the bandwidth of the result will be 1/2 the sampling frequency.
For example, say a signal consists of the following:
y = Aisin2Fi
5
i=1
+ random
Where:
A = [1 2 1 3 5]
and
F = [50 75 100 150 175]
A 10 second time signal was prepared at sampling frequency Fs = 500 Hz shown in Figures 27 and 28.
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Figure 27: 1s section of the 10s signal
Figure 28: 0.1s section of the 10s signal
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-15
-10
-5
0
5
10
15
Time [s]
Accele
ration [
g]
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-15
-10
-5
0
5
10
15
Time [s]
Accele
ration [
g]
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A NFFT = 1024 point spectrum was calculated that will yield a spectrum frequency resolution of Fs
NFFT=
500
1024= 0.4883 Hz. Overlapping was adjusted at
NFFT
2 points. A hamming window was used that yields
the root-mean-square spectrum shown in Figure 29 from which the peaks at the relevant frequencies are shown with the noise floor caused by the random values added to the signal.
The Matlab code that was used:
close all Fs=500; NFFT=1024; t=0:1/Fs:10; A=[1 2 1 3 5]; F=[50 75 100 150 175]; y=zeros(1,length(t)); [M,N]=size(A); for i=1:N y=y+A(i)*sin(2*pi*F(i)*t); end y=y+2*randn(1,length(t)); plot(t,y) axis([0 1 min(y) max(y)]); xlabel('Time [s]'),ylabel('Acceleration [g]') figure plot(t,y) axis([0 0.1 min(y) max(y)]); xlabel('Time [s]'),ylabel('Acceleration [g]') figure [Pxx,F] = pwelch(y,hamming(NFFT),NFFT/2,NFFT,Fs); Arms=sqrt(Pxx*Fs/NFFT); plot(F(1:length(F)),Arms(1:length(Arms)),'LineWidth',2); % Plot the results. xlabel('Frequency [Hz]'); ylabel('Acceleration [g-RMS]');
Figure 29: Root-mean-square spectrum of an arbitrary signal
0 50 100 150 200 2500
0.5
1
1.5
2
2.5
3
Frequency [Hz]
Accele
ration [
g-R
MS
]
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10.4. Order tracking
Order tracking is the process where the energy at a frequency which is an order of a reference frequency, is computed
o This can be done in the time domain and in the frequency domain
In the time domain the centre frequency of analogue narrow band filters are adjusted and the RMS value of the passed signal calculated
In the frequency domain the signal is transferred to the frequency domain, the frequency of the harmonic calculated, and the energy at/or a given region around that frequency found
10.5. Time windows
Time windows are applied to prevent leakage on the spectrum. In Fourier Transformation it is assumed that the signal is periodic and therefore, zero at the beginning and end of the sample record. Energy from resonance (of the algorithm) is smeared into adjacent spectral lines if the signal does not fit in the sample record. To reduce this problem, the sample record is forced to zero at the ends using time windows applicable for the type of analysis that will be done.
Figure 30: A few time windows
10.6. Frequency domain data is used:
1. To find the amplitude vs. frequency content of a signal
2. To find the phase angle vs. frequency content of a signal
3. To compute frequency response functions
a. Now, if we have the amplitude and phase angle of two channels at each frequency line, the transmissibility from the one channel to the next channel can be calculated. This is called the frequency response function.
50 100 150 200 250 300 350 400 450 5000
0.2
0.4
0.6
0.8
1
Hamming
Hann
Rectangular
Flat top
Exponential
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10.7. Cepstrum analysis
The Cepstum is the power spectrum of the logarithm of the power spectrum. It is used to detect periodicity in the spectrum. For example, Figure 31 shows the spectrum of a bad and good gearbox. Note the high number of peaks on the bad gearbox velocity spectrum. When the Cepstrum is calculated, Graph c shows that the peaks are harmonics of a peak with period 95.9 ms (which is 10.4 Hz). This is then the fault frequency that needs further investigation.
Figure 31: Cepstrum analysis
10.8. Spectrum averaging
When an instantaneous spectrum is calculated from a time signal, the spectrum will sometimes not be very smooth. To get clear spectra, a series of spectra are calculated and the average found.
Typical averaging types are:
RMS averaging
Linear averaging
Peak hold averaging
Exponential averaging
The number of averages, and consequently the time length of the signal depends on the:
Lowest frequency which must be found
Sensitivity of the transducers
Overlapping factor
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10.9. Number of averages and measurement time
The following guidelines may be used to find the number of averages:
If you want repeatability of data, remember to take the same number of averages each time you measure
The test time over which data must be taken for transfer functions is estimated by:
T 50 to 100
Br
where Br is the half power bandwidth of the first mode of the structure for no overlapping.
Experience has shown that 8 and more averages give sufficient results root-cause analysis and condition based decisions
10.10. Overlapping
In this method the signal is divided into overlapping segments where the length of the overlapping is specified as a percentage of the number of data points per sample used in the FFT. More PSDs can now be averaged for the same length of signal. Shorter testing times are therefore required to have the same statistic accuracy. An overlap factor of 50% is suggested.
10.11. Class problem
During a preliminary test a typical auto-spectrum shows a half-power bandwidth of approximately 0.3 Hz for the first natural frequency. What is the maximum recording time which must be used for no overlapping?
Figure 32: Overlapping explained
10.12. Problem for PMM student on Frequency Domain Analysis
During a preliminary test a typical auto-spectrum shows a half-power bandwidth of approximately 0.3 Hz for the first natural frequency.
What is the maximum recording time which must be used for no overlapping?
Answer:
10.13. Conclusion
This section showed that:
Any time signal can be represented as the finite sum of sine and cosine terms
A signal can be forced to be periodic with proper windows
The bandwidth that is computed is less than 1/2 the sampling frequency used
Different averaging techniques can be used to smooth spectral data
The frequency response function can be computed between any two points and the result will be the amplitude ratio and the phase angle at a particular frequency
The test time on a structure depend on the half-power-bandwidth of the lowest peak
50% overlapping 0 N
FFT done on these
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11. MAKING GOOD MEASUREMENTS
Slides file: Investmech - Vibration (10 Making good measurements) R0.0
Figure 33: Layout of a digital measurement system
Table 1 summarises the areas to consider for making good measurements.
Table 1: Considerations for good measurements
Transducers Sensitivity Frequency range Mounting Isolation and water resistance Noise
Signal processing Number of averages Bandwidth Signal type Windows
Input and ADC Input ranges AC/DC coupling Anti-aliasing Delays Triggers
Data interpretation Repeatability Linearity Measurement observation Coherence
12. VIBRATION MONITORING
Slides filename: Investmech - Vibration (11 Vibration monitoring) R0.0
12.1. Objective
To give an introduction to condition monitoring and how vibration monitoring can be used to quantify the condition of mechanical systems
After completion you will be able to:
List different condition monitoring techniques
Explain: overall vibration, narrow band vibration, demodulation, trending, enveloping, on-line systems
12.2. Key concepts in vibration monitoring
Overall value monitoring
Spectral analysis
Noise measurements
Time domain analysis
Frequency domain analysis
Trending analysis
o Compares relative changes over time
Alert and Alarm limits
Vibration severity
o Indication of the severity of machine vibration
o Standards exist against which peak/rms values may be compared
TransducerSignal
conditioningAnti-aliasing
filter
Analogue to digital converter
Digital signal processing
(time, stats, fft, etc.)
Display/storage
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o Vibration severity charts may be used to quantify severity of vibration
12.3. Vibration severity chart
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12.4. Typical vibration alarm levels
The following is just to give a guideline.
CLASS Description Peak [mm/s]
RMS [mm/s]
I Machines driven by electric motors 15 kW & < 75 kW on solid foundations
75 kW on rigid heavy foundations
>15 kW on flexible foundations (anti-vibration pads)
25 18
IV
Machines driven by electric motors:
>300 kW on foundations relatively soft in direction of measurement (e.g. turbines with outputs > 10 MW)
40 28
http://www.scribd.com/doc/3870904/Introduction-to-Vibration
12.5. Trending analysis
In trending:
1. Plot the overall value versus time.
2. Define alert and alarm levels to instigate action.
http://www.scribd.com/doc/3870904/Introduction-to-Vibration
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12.6. Trending spectra
There is a possibility that a growing peak may be missed.
Also do trending of spectra to ensure that growing peaks that do not influence the overall values significantly are detected and monitored.
Figure 34: Root-mean-square value trended over time
1 2 3 4 5 6 74.4
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12.7. Quefrency domain analysis
The Cepstrum is defined as the inverse Fourier transform of the logarithm of the power spectrum
Referred to as Cepstrum
Terms used in cepstrum:
o Quefrency - Frequency
o Rahmonics - Harmonics
o Gamnitude - Magnitude
o Saphe - Phase
12.8. Fault diagnosis
List of possible faults in rotating machinery supplied
Capture a signal and investigate the result in the time domain
Do spectrum analysis and relate peaks to fault frequencies in the system
Monitor the changes in the peaks of the spectrum
Quefrency analysis can be used as powerful tool where the spectrum has many peaks
12.9. Conclusion
Vibration based condition monitoring is a powerful tool in condition monitoring and fault diagnosis, and can be done as follows:
Overall values may be used to quantify the condition of a machine:
These must be trended, compared against OEM limits, and/or compare against severity charts.
Spectral analysis are used where the cause of high vibration levels must be found
Can also be used in trending peaks at specific frequencies.
Standards and severity charts may be used to quantify the condition of machines
Where none of these exist, a baseline value may be found from history, or, the first measurements taken used and the baseline updated as a function of time. A baseline value/spectrum forms the cornerstone of vibration-based condition monitoring.
13. BALANCING OF RIGID ROTORS
This section is not part of the PMM syllabus and will not be discussed in class.
14. ALIGNMENT OF MACHINES
14.1. Alignment and soft foot
Videos to watch:
1. Typical alignment steps: https://youtu.be/U_04dRQZUD4
2. Misalignment by dial gauges: https://youtu.be/8A2kpOad4VA
3. SKF Shaft Alignment Tool TKSA 51: https://youtu.be/cx10vvtlSVE
4. https://youtu.be/2FNGRGCo8sk
14.2. Thermal growth
Continuous monitoring provides a sound method to measure thermal growth accurately. Temperature based manual or finite element analysis based calculations can provide an acceptable range, but, then both transverse and angular dependence on temperature need to be modelled.
Look at the following videos:
1. https://youtu.be/96MeQrvH2-g
15. FAULT DIAGNOSIS
Slides filename: Investmech - Vibration (13 Fault diagnosis) R0.0
https://youtu.be/U_04dRQZUD4https://youtu.be/8A2kpOad4VAhttps://youtu.be/cx10vvtlSVEhttps://youtu.be/2FNGRGCo8skhttps://youtu.be/96MeQrvH2-g
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15.1. Objective
To quantify vibration responses caused by faults in systems
Outcome:
You will be able to identify the potential cause of a vibration response
15.1.1. Force unbalance
Cause radial vibration
In-phase and steady
Amplitude 2 below first critical
Peak at 1xRPM and dominate
Correct with one mass in one plane at CoG
0 phase difference between OB&IB
90 phase difference between horizontal and vertical at each bearing
15.1.2. Couple unbalance
Radial & axial vibration
180 out-of-phase motion on same shaft in same direction
Peak at 1x RPM
Amplitude 2 below first critical
Balance weights in at least two planes
180 phase difference between OB&IB verticals & horizontals
90 phase difference between vertical and horizontal at same bearing
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
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15.1.3. Dynamic unbalance
Dominant type found
Combination of force & couple unbalance
Causes radial vibration
Peak at 1xRPM and dominate spectrum
Radial phase difference between IB&OB horizontals from 0 to 180
IB&OB horizontal & vertical phase differences same
90 phase difference between horizontal & vertical at each bearing
15.1.4. Overhung rotor unbalance
Cause axial & radial vibration
In-phase response
Radial phase might be unsteady
Peaks at 1xRPM radial & axial
Horizontal phase difference match vertical phase difference
Force &