10台灣師範大學機電科技學系

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台灣師範大學機電科技學系 -1-

10

10.1 Introduction 10.2 Transducers 10.3 Vibration Pickups

10.4 Frequency-Measuring Instruments 10.5 Vibration Exciters

10.6 Signal Analysis

10.7 Dynamic Testing of Machines and Structure 10.8 Experimental Modal Analysis

10.9 Machine-Condition Monitoring and Diagnosis

C. R. Yang, NTNU MT

10.1 Introduction

10.1 10.1 Introduction

• Why we need to measure vibrations:

–To detect shifts in ωnwhich indicates possible failure –To select operational speeds to avoid resonance

–Measured values may be different from theoretical values –To design active vibration isolation systems

–To identify mass, stiffness and damping of a system –To verify the approximated model

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台灣師範大學機電科技學系 C. R. Yang, NTNU MT

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10.1 Introduction

• Type of vibration measuring instrument used will depend on:

–Expected range of frequencies and amplitudes –Size of machine/structure involved

–Conditions of operation of the machine/structure –Type of data processing used

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10.2 Transducers

10.2

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10.2 Transducers

• A device that transforms values of physical variables into electrical signals

• Following slides show some common transducers for measuring vibration

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10.2 Transducers

• Variable Resistance Transducers

Mechanical motion changes electrical resistance, which cause a change in voltage or current

Strain gage is a fine wire bonded to surface where strain is to be measured.

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Surface and wire both undergo same strain.

Resulting change in wire resistance:

where K = Gage factor of the wire R = Initial resistance ΔR = Change in resistance L = Initial length of wire ΔL = Change in length of wire v = Poisson’s ratio of the wire r = Resistivity of the wire

Δr = Change in resistivity of the wire ≈ 0 for Advance

L v L r v r L L

R

K R 1 2 1 2

/

/  







 



Strain:

The following figure shows a vibration pickup:

RK R L

L





10.2 Transducers

ΔR can be measured using a Wheatstone bridge as shown:



R R



R R



^V

R R R

E R 



 







 

4 3 2 1

4 2 3 1

10.2 Transducers

Initially, resistances are adjusted so that E=0 R1R3= R2R4

When Richange by ΔRi,

  

3 4



²

4 3 2 2 1

2 1 0 4

4 3

3 2

2 1

1

0 where

R R

R R R R

R r R R

R R

R R Vr R

E  

 



 



   





(4)

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10.2 Transducers

If the leads are connected between pts a and b, R1=Rg, ΔR1,= ΔRg, ΔR2= ΔR3= ΔR4=0

where Rgis the initial resistance of the gauge.

Hence E can be calibrated to read ε directly.



 ₀

0

or

E KVr

Vr K E R

R

g

g    

 

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10.2 Transducers

• Piezoelectric Transducers

Certain materials generate electrical charge when subjected to deformation or stress.

Charge generated due to force:

where k =piezoelectric constant A =area on which Fxacts px=pressure due to Fx.

x x

x

kF kAp

Q  

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10.2 Transducers

• Piezoelectric Transducers E=vtpx

v = voltage sensitivity t = thickness of crystal

A piezoelectric accelerometer is shown.

Output voltage proportional to acceleration

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10.2 Transducers

Example 10.1

Output Voltage of a Piezoelectric Transducer

A quartz crystal having a thickness of 2.5mm is subjected to a pressure of 50psi. Find the output voltage if the voltage sensitivity is 0.055 V-m/N.

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Example 10.1

Output Voltage of a Piezoelectric Transducer Solution

E = vtpx =(0.055)(0.00254)(344738) = 47.4015V

• Electrodynamic Transducers

Voltage E is generated when the coil moves in a magnetic field as shown.

E = Dlv

where D = magnetic flux density l = length of conductor v = velocity of conductor

relative to magnetic field I F v DlE

10.2 Transducers

• Linear Variable Differential Transformer Transducer Output voltage depends on the axial displacement of the core.

Insensitive to temp and high output.

10.3 Vibration Pickups

10.3

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10.3 Vibration Pickups

• Most common pickups are seismic instruments as shown

• Bottom ends of spring and dashpot have same motion as the cage

• Vibration will excite the suspended mass

• Displacement of mass relative to cage: z = x – y

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10.3 Vibration Pickups

• Y(t) = Ysinωt

• Equation of motion of mass m:

• Steady-state solution:

   

t Y m kz z c z m

y m kz z c z m

y x k y x c x m



 sin

or 0



2

















 



  t  Z   t   

z sin

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10.3 Vibration Pickups

    ^{ }

n

m

r c

r r m

k c

r r

Y r c

m k Z Y

 







 





2 ,

1 tan 2 tan

2 1

2 1 2

1

2 2 2

2

2 2 2 2

2



 

 





 

 

 





 



 



 



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10.3 Vibration Pickups

• Vibrometer

Measures displacement of a vibrating body Z/Y ≈ 1 when ω/ωn≥ 3 (range II)

In practice Z may not be equal to Y as r may not be large, to prevent the equipment from getting bulky

   



¹



^r

^{ }

² ¹

if sin

2 2 2

2

 

 



r r

t Y t

z

  

(7)

Example 10.2

Amplitude by Vibrometer

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

Example 10.2

Amplitude by Vibrometer Solution

Amplitude of recorded motion:

Amplitude of vibration of structure:

Y = Z/1.0093 = 3.9631 mm



1 10

   

2

  

0.2 10



¹^.⁰⁰⁹³ ⁴^mm

10

2 2 2

2



 

  Y Y

Z

10.3 Vibration Pickups

• Vibrometer

Measures acceleration of a vibrating body.

   

       







 









 







 



t Y t z

r r

t t Y

z

n n

sin , 1 2 1

1 If

2 1

sin

2 2

2 2 2

2 2 2 2 2

10.3 Vibration Pickups

• Vibrometer If 0.65< ζ < 0.7,

Accelerometers are preferred due their small size.



1

  

2 ¹^.⁰⁴^for⁰ ⁰^.⁶ 96 1

. 0

2 2 2



 

  r

r



(8)

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10.3 Vibration Pickups

Example 10.3

Design of an Accelerometer

An accelerometer has a suspended mass of 0.01 kg with a damped natural frequency of vibration of 150 Hz. When mounted on an engine undergoing an acceleration of 1 g at an operating speed of 6000 rpm, the acceleration is recorded as 9.5 m/s²by the instrument. Find the damping constant and the spring stiffness of the accelerometer.

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10.3 Vibration Pickups

Example 10.3

Design of an Accelerometer Solution

   

     

 



¹



^(E.2)

4444 . 0 r or 1 6667 . 0

6667 . 48 0 . 942

32 . 628 1

1 Thus

rad/s 48 . 942 2 150 1

rad/s 32 . 60 628

2 6000 speed Operating

(E.1) 0663 . 1 9684 . 0 / 1 2 1

or

9684 . 81 0 . 9

5 . 9 value True

value M easured 2

1 1

2 2

2

2 d 2

2 d

2 2 2

2 2 2 2

















 















 

















r

r r

r r r

n n

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10.3 Vibration Pickups

Example 10.3

Substitute (E.2) into (E.1): 1.5801ζ⁴– 2.2714ζ²+ 0.7576 = 0 Solution gives ζ²= 0.7253, 0.9547

Choosing ζ= 0.7253 arbitrarily,

  

   

s/m - N 8571 . 19

7253 . 0 8889 . 1368 01 . 0 2 2

constant Damping

N/m 5628 . 18738 8889 . 1368 01 . 0

rad/s 8889 . 1368 7253 . 0 1

48 . 942 1

2 2

d n



 









 

n n

m c

m k

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10.3 Vibration Pickups

Example 10.3

Measures velocity of vibrating body:

Velocity:

  t Y t y    cos 

       

   

      





 







 



 

 

t Y t z

r r

r

t r

r Y t r

z

cos

then , 1 2 1

If

cos 2 1

2 2 2

2 2 2 2

2



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Example 10.4

Design of a Velometer

Design a velometer if the maximum error is to be limited to 1% of the true velocity. The natural frequency of the velometer is to be 80Hz and the suspended mass is to be 0.05 kg.

Example 10.4

Design of a Velometer Solution

We have

Maximum

       

   

^True^velocity ^(E.1)

velocity Recorded 2

1

cos 2 1

2 2 2

2 2 2 2

2

 

 

 

 

r r R r

t r r

Y t r

z





 

 

(E.2) 2 1

1



2

 

r^ r

10.3 Vibration Pickups

Example 10.4

Substitute (E.2) into (E.1),

R

 







 





 



 



 

 





 



 







4 2

2 2 2 2

2

4 4 1

2 1 4 1 2

1 1 1

2 1

1



 



10.3 Vibration Pickups

Example 10.4

R = 1.01 or 0.99 for 1% error

ζ⁴– ζ²+ 0.245075 = 0 and ζ⁴– ζ²+ 0.255075=0 ζ²= 0.570178, 0.429821 or

ζ = 0.755101, 0.655607

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10.3 Vibration Pickups

Example 10.4

Choosing ζ = 0.755101 arbitrarily,

 

  

   

s/m - N 9556 . 37

05 . 0 656 . 502 755101 . 0 2 2 c

N/m 1527 . 12633 656 . 502 05 . 0

rad/s 656 . 502 2 80

n 2 2



m m

k n

n









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10.3 Vibration Pickups

• Phase Distortion

All vibrating-measuring instruments have phase lag.

If the vibration consists of 2 or more harmonic components, the recorded graph will not give an accurate picture – phase distortion Consider vibration signal of the form as shown:

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10.3 Vibration Pickups

Let phase shift = 90° for first harmonic Let phase shift = 180° for third harmonic

Corresponding time lags: t1= 90° /ω, t2= 180° /ω Output signal is as shown:

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10.3 Vibration Pickups

In general, let the complex wave be y(t) = a1sinωt + a2sin2ωt + … Output of vibrometer becomes:

z(t) = a1sin(ωt – Φ1) + a2sin(2ωt – Φ2) + … where

,...

2 , 1 , 1

2

tan ₂ 



 







 





 j

j j

n n j







(11)

Φj≈ π since ω/ωnis large.

z(t) ≈ – [a1sinωt + a2sin2ωt + …] ≈ -y(t) Thus the output record can be easily corrected.

Similarly we can show that output of velometer is Accelerometer: Let the acceleration curve be

Output of accelerometer:

 

t y

 

t z 

   





t a ta t

y ₁²sin ₂ 2²sin2

       





zt a₁²sint₁ a₂2²sin2t₂ 

Since Φ varies almost linearly from 0° to 90° for ζ = 0.7, Φ ≈ αr

= α(ω/ωn) = βω where α and β are constants.

Time lag is independent of frequency.

Thus output of accelerometer represents the true acceleration being measured.

 





  

 t

       

 

   



























t a

a

t a

t a t z

where 2

sin 2 sin

2 2 sin 2 sin

2 2 2

1

2 2 2

1 2





10.4 Frequency-Measuring Instruments

10.4 10.4 Frequency-Measuring Instruments

• Single-reed instrument or Fullarton Tachometer

Clamped end pressed against vibrating body

Adjust l until free end shows largest amplitude of vibration

Read off frequency

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10.4 Frequency-Measuring Instruments

• Multi-reed Instrument or Frahm Tachometer

Clamped end pressed against vibrating body

Frequency read directly off strip whose free end shows largest amplitude of vibration

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10.4 Frequency-Measuring Instruments

• Stroboscope

Produces light pulses

A vibrating object viewed with it will appear stationary when frequency of pulse is equal to vibration frequency

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10.5 Vibration Exciters

10.5

台灣師範大學機電科技學系

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10.5 Vibration Exciters

• Used to determine dynamic characteristics of machines and structures and fatigue testing of materials

• Can be mechanical, electromagnetic, electrodynamic or hydraulic type

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• Mechanical Exciters

Force can be applied as an inertia force

Force can be applied as an elastic spring force for frequency <30 Hz and loads <700N

• Mechanical Exciters

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

10.5 Vibration Exciters

• Electrodynamic Shaker

The electrodynamic shaker can be considered as the reverse of an electrodynamic transducer.

2 resonant frequencies are shown below.

DIl F

10.6 Signal Analysis

10.6

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10.6 Signal Analysis

• Acceleration-time history of a frame subjected to excessive vibration:

• Transformed to frequency domain:

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10.6 Signal Analysis

• Spectrum Analyzers

Separates energy of signal into various frequency bands Real-time analyzers useful for machine health monitoring 2 types of real-time analysis procedures: digital filtering method and Fast Fourier Transform method

Basic component of spectrum analyzer: Bandpass filter

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10.6 Signal Analysis

• Bandpass Filter

Permits passage of frequencies over a band and rejects all other frequency components

Response of a filter:

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10.6 Signal Analysis

• Bandpass Filter

fuand flare upper and lower cutoff frequencies respectively fc is centre (tuned) frequency

Ripples within band is minimum for a good bandpass filter

2 types of bandpass filters: constant percent bandwidth filters and constant bandwidth filters

Constant percent: (fu– fl)/fcis a constant E.g. octave, one-half-octave filters

Constant bandwidth: fu– flis independent of fc

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• Constant Percent Bandwidth and Constant Bandwidth Analyzers

Spectrum analyzer with a set of octave and 1/3-octave band filters can be use for signal analysis

Lower cutoff freq of a filter = upper cutoff freq of previous filter.

Filter characteristics as shown

• Constant Percent Bandwidth and Constant Bandwidth Analyzers

Constant bandwidth analyzer used to obtain more detailed analysis than constant percent bandwidth analyzer

Wave or heterodyne analyzer is a constant bandwidth analyzer with a continuously varying centre frequency

10.7 Dynamic Testing of Machines and Structures

10.7 10.7 Dynamic Testing of Machines and Structures

• Involves finding the deformation of machines/structures at a critical frequency

• Approaches:

Operational Deflection Shape measurements

Modal Testing

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10.7 Dynamic Testing of Machines and Structures

• Using Operational Deflection Shape Measurements

Forced dynamic deflection shape measured under steady-state frequency of system.

Valid only for forces/frequency associated with operating conditions.

If a particular location has excessive deflection, we can stiffen that location.

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10.7 Dynamic Testing of Machines and Structures

• Modal Testing

Any dynamic response of a machine/structure can be obtained as a combination of its modes.

Knowledge of the mode shapes, modal frequencies and modal damping ratio will describe completely the machine dynamics.

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10.8 Experimental Modal Analysis

10.8

台灣師範大學機電科技學系

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10.8 Experimental Modal Analysis

• When a system is excited, its response exhibits a sharp peak at resonance

• Phase of response changes by 180° as forcing frequency crosses the natural frequency

• Equipment needed:

Exciter to apply known input force

Transducer to convert physical motion into electrical signal

Signal conditioning amplifier

Analyzer with suitable software

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• Necessary Equipment Exciter

Can be an electromagnetic shaker or impact hammer

Shaker is attached to the structure through a stringer, to control the direction of the force

Impact hammer is a hammer with built-in force transducer in its head Portable, inexpensive and much faster to use than a shaker

But often cannot impart sufficient energy and difficult to control direction of applied force

• Necessary Equipment Transducer

Piezoelectric transducers most popular Strain gauges can also be used Signal conditioner

Outgoing impedance of tranducers not suitable for direct input into analyzers.

Charge or voltage amplifiers are used to match and amplify the signals before analysis

10.8 Experimental Modal Analysis

• Necessary Equipment

Analyzer

FFT analyzer commonly used

Analyzed signals used to find natural frequencies, damping ratios and mode shapes

10.8 Experimental Modal Analysis

• Necessary Equipment

General arrangement for experimental modal analysis:

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10.8 Experimental Modal Analysis

• Digital Signal Processing

x(t) represents analog signal, xi= x(ti) represents corresponding digital record.

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10.8 Experimental Modal Analysis

• Digital Signal Processing We have

N is fixed for a given analyzer and equations can be expressed as

 





 



 



 





N

j

j j i N

j

j j i N

j j N

i

j i j i j

j

T x it b N T x it a N N x a

N T j

b it T a it t a

x x

1 1

1 0

2 /

1 0

sin2 1

2 , 1 cos 1 , where

, , 2 , 1 2 ; 2 sin

2 cos



 

   



N N



^T

T N

b b b a a a a d

x x x X X

A d

2 / 1 1 2 / 2 1 0

2 1

1 where ,



   





 ^

-71-

10.8 Experimental Modal Analysis

• Analysis of Random Signals

Input and output data usually contain random noise.

If x(t) is random signal, its average is

   

 

j N

N j T T

t N x x

dt t T x t x





 









1 0

signal, 1 digital For

1

lim lim

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10.8 Experimental Modal Analysis

• Analysis of Random Signals Define a new variable x(t) as

Mean square value

      t y t y t

x  

   

 

2 RMS

2

1 2

0 2 2

value square mean Root

signal, 1 digital For

1

lim lim

x x

t N x x

dt t T x t

x

j N

N j T T

















(19)

• Analysis of Random Signals Autocorrelation function

If x(t) is purely random, R(t)  0 as T  ∞ If x(t) is periodic, R(t) will also be periodic.

     

  





 





 







n N

j n j j T

T

x n x t N n R

d t x T x x t R

0 0

2

, 1 signal, digital For

lim1

  

• Analysis of Random Signals Power spectral density (PSD):

Cross-correlation function:

Cross-PSD:

   

t N S x

d e R

S ⁱ

 







^



2

: form Digital

2 1

 



 

 ^{ }

  

^

 









 

R e ^{ }d

S_xf _xf ⁱ

2 1

 

^ 

   

^



T

xf T x f td

t T

R 0

lim1   

10.8 Experimental Modal Analysis

• Analysis of Random Signals

If f(τ+t) is replaced by x(τ+t), we get Rxx(t) which leads to Sxx(ω).

Frequency response function H(iω) is related to PSD as

     

        



xf xx

ff fx

ff xx

S i H S

S H S



²

10.8 Experimental Modal Analysis

• Analysis of Random Signals Coherence function:

β = 0 if x and f are pure noises.

β = 1 if x and f are not contaminated at all.

Typical coherence function:

   

 



 



ff xx

xf xx

xf ff fx

S S

S ²





 

















(20)

-77-

10.8 Experimental Modal Analysis

• Determination of Modal Data from Observed Peaks

Let the graph of H(iω) be as shown below.

4 peaks suggesting a 4-DOF system.

-78-

10.8 Experimental Modal Analysis

• Determination of Modal Data from Observed Peaks

Partition into several frequency ranges.

Each range is consider a 1-DOF system Damping ratio corresponding to peak j:

When damping is small, ωj≈ ωn

   

 

^{ }

 

2

satisfy and

where 2

2 1

2 1 1

2

j j

j

j j j

i i H H i

H 



 



 



 

-79-

10.8 Experimental Modal Analysis

Example 10.5

Determination of Damping Ratio from Bode Diagram

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

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

-80-

10.8 Experimental Modal Analysis

Example 10.5

Determination of Damping Ratio from Bode Diagram

(21)

Example 10.5

Determination of Damping Ratio from Bode Diagram Solution

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



10.0



⁰^.⁰⁴⁵

2 6 . 9 5 . 10 2

1

2   





n



 

• Determination of Modal Data from Nyquist Plot

Real parts of frequency-response function of 1-DOF system plotted along horizontal axis

Imaginary parts of frequency-response function of 1-DOF system plotted along vertical axis

Frequency-response function:

 



²



² ² ²



²



² ² ²

2 2

4 1 2 , 4 1

1

where 2

1 1

r r v r r r

u r

r iv r u i i r

n





 







 





 





 

 

10.8 Experimental Modal Analysis

Properties of Nyquist Circle:

•u and v are large when r=1

•1-r²= (1+r)(1-r) ≈ 2(1-r) and 2ζr ≈ 2ζ

 

     

2 2 2

2 2 2 2 2

2

4 1 4

1

1 2 , 1

2 1



 







 



 





 





 



v u

r v r

u r

(22)

-85-

10.8 Experimental Modal Analysis

Once H(iω) is measured, use least square approach to fit a circle.

Intersection of circle with –ve Im axis corresponds to H(iωn) Bandwidth is the difference of the frequencies at the 2 horizontal diametral points

Damping ratio:

   

n



  2

1 2



-86-

10.8 Experimental Modal Analysis

• Measurement of Mode Shapes Undamped multi-DOF system:

Free harmonic vibration:

Orthogonal relations for mode shapes:

    m x    k x   f 

   



k 



i²m



y^i⁰^

      

i i i

T T

M K

K Y

k Y

M Y

m Y



2

diag diag



-87-

10.8 Experimental Modal Analysis

• Measurement of Mode Shapes

When forcing functions are harmonic,

         ^{ } 

          

     

   











 













N

i i i

q i p i

q j N j q F

p pq pq

T

t i t

i t

i

M K

y y

F X

Y M K Y

e F e

F m k e X t x

j

1 2

; , , 2 , 1

; 0 with 2 1

~ 1 ~

~ 2



















^ ^





 



-88-

10.8 Experimental Modal Analysis

Further normalized [Y] as

    

     







 







N

i i

i q i p pq

N Y M

1 2 2

2 / 1 2

] 1

[





 





 





(23)

• Measurement of Mode Shapes Damped multi-DOF system:

Assume proportional damping:

Undamped mode shapes of the system will diagonalize the damping matrix:

     

mxcxkxf

     

c ak bm

    

Y^Tc Y diag

 

k

Frequency-response function when is harmonic:

When mass-normalized mode shapes are used:

f

         



  



 ^N

i i i i

i q i p pq

pq K M i C

y y

1

2 ~



 









  _    



 



^N

i i i i

i q i p

pq ₁ ² ²

i ~

2   





 





10.8 Experimental Modal Analysis

Substituting ω=ωiinto the equation, we get:

       

   

i p i q i i pq

^{ }

i i i

i q i p i pq i pq

i i i

q i p i i

pq i pq

H H H i













 













 





2

2 2 2

2

2 or

or 2

2~





 





10.9

Machine-Condition Monitoring and Diagnosis

10.9

(24)

-93-

10.9 Machine-Condition Monitoring and Diagnosis

• Machine operations will cause misalignments, cracks, unbalances etc in machine parts

• Vibration level will increase until machine failure occurs

-94-

10.9 Machine-Condition Monitoring and Diagnosis

• Vibration Severity Criteria

Vibration severity charts can be used as a guide to determine machine condition.

RMS value of vibratory velocity is compared against the criteria set by the standards.

However the overall velocity signal used for comparison may not give sufficient warning of the imminent damage.

-95-

10.9 Machine-Condition Monitoring and Diagnosis

• Machine Maintenance Techniques Life of machine follows the bathtub curve:

-96-

10.9 Machine-Condition Monitoring and Diagnosis

• Machine Maintenance Techniques

Breakdown maintenance:

Allow the machine to fail and then replace it with a new machine.

This strategy is used when machine is inexpensive and no other damage is caused by the breakdown.

(25)

Preventive maintenance:

Maintenance performed at fixed intervals.

Intervals determined statistically from past experience.

This method is uneconomical.

Condition-based/Predictive maintenance:

Replace fixed-interval overhaul with fixed-interval measurements.

Can extrapolate measured vibration levels to predict when they will reach unacceptable values.

Maintenance costs are greatly reduced.

10.9 Machine-Condition Monitoring and Diagnosis

• Machine Condition Monitoring Technique

Following methods are used to monitor machine conditions:

Aural and visual – a skilled technician will listen and see the vibrations produced by the machine

Operational variables monitoring – performance is monitored wrt intended duty. Deviation denotes a malfunction.

(26)

-101-

10.9 Machine-Condition Monitoring and Diagnosis

• Machine Condition Monitoring Technique

Temperature monitoring – rapid increase in temperature is an indication of malfunction.

Wear debris found in lubricating oils can be used to assess extent of damage by observing concentration, size, shape and colour of the particles.

Available vibration monitoring techniques.

-102-

10.9 Machine-Condition Monitoring and Diagnosis

• Vibration Monitoring Techniques

Time domain analysis

E.g. following is an acceleration waveform of a gearbox. Pinion is coupled to 2685 rpm motor.

Period of waveform same as period of pinion.

This implies a broken gear tooth on the pinion.

-103-

10.9 Machine-Condition Monitoring and Diagnosis

Statistical Methods

Peak level, RMS level and crest factor may be used as indices to identify damage.

Changes in Lissajous figures can be used to identify faults.

-104-

10.9 Machine Condition Monitoring and Diagnosis

• Vibration Monitoring Techniques Statistical Methods

Waveform corresponding to good components will have bell-shaped probability density curve

Any deviations can be due to component failure

First 4 moments of the curve are called the mean, standard deviation, skewness and kurtosis.

Kurtosis is defined as

Increase in value of kurtosis can be due to machine component failure

   



^



 x x f x dx

k 1

₄ ⁴



(27)

Frequency Domain Analysis

Vibration spectrum is unique to that particular machine. Its shape changes as faults starts developing.

Nature and location of the fault can be detected by comparing the frequency spectrum of the damaged machine with that of the machine in good condition.

Frequency Domain Analysis

Each rotating element generates identifiable frequency.

Thus changes in the spectrum at a given freq can be attributed to the corresponding element.

10.9 Machine-Condition Monitoring and Diagnosis

Quefrency Domain Analysis

Quefrency is the x-axis for cepstrum.

Cepstrum c(τ) is the inverse fourier transform of the log of the power spectrum SX(ω).

Cepstrum can detect any periodicity in the spectrum caused by component failure.

           

 



  





 ^

X

T T

t i X

S F c

dt e t T x t x F t

x F S

log

1 where

1

2 2 2







10.9 Machine-Condition Monitoring and Diagnosis

• Vibration Monitoring Techniques Quefrency Domain Analysis

2^ndgear was at fault although 1^stgear was engaged.

(28)

-109-

10.9 Machine-Condition Monitoring and Diagnosis

• Instrumentation Systems

Quefrency Domain Analysis

3 types – basic system, portable system, computer-based system.

Basic system consists of vibration meter, stroboscope and headset.

Portable system consists of portable FFT vibration analyzer based on battery power.

Computer-based system consists of FFT vibration analyzer coupled with computer for maintaining centralized database and provide diagnostic capabilities.

-110-

10.9 Machine-Condition Monitoring and Diagnosis

• Instrumentation Systems

Piezoelectric accelerometers are commonly used.

Can choose between acceleration, velocity and displacement to monitor.

Velocity is commonly used as the parameter for monitoring the machine conditions because the velocity spectrum is the flattest.

Any change in the amplitude can be observed easily in a flatter spectrum.