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]

Confidential 2

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

Confidential Page 3 of 68

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


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


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


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)


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


Figure 1: Mathematical model of based excited single degree of freedom system

Figure 2: Force-deflection of a linear spring

()

y()


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


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.

()

()

()


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


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


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


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


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


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


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


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


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


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


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/


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/


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

()

()

()


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


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


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,


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


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


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.


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


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


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


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


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


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


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.


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


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


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


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


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.


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.


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]


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

]


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


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


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


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


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


o Vibration severity charts may be used to quantify severity of vibration

12.3. Vibration severity chart


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


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

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Confidential 51

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Confidential 52

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


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

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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 &