1

Vibration Measurement

2

Introduction• Why we need to measure vibrations:

– To detect shifts in ωn which indicates possible failure

– To select operational speeds to avoid resonance

– Measured values may be different from theoretical values

– To design active vibration isolation systems– To identify mass, stiffness and damping of a

system– To verify the approximated model

3

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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Equipmental Modal Testing• General arrangement for experimental modal analysis:

f(t)

)(tx&&

v(t)

5

Vibration Measurement

• Dynamics testing is a valuable complement to dynamics analysis

• Modal testing– Natural frequencies– Damping – Mode shapes

• Operational testing– In-service loads (often random)

6

Transducers

• Translates motion or forces into electrical signals

• Size of transducer is important (micro)– Ideally does not influence the structure’s

dynamics through added mass or stiffness– Analytical models often include effects of

transducer mass

7

m

Cantilever beam

Strain gage

base)(const tx=ε

)(tx

Variable resistance transducer

Sec. 10.2

8

Variable resistance transducer

http://www.blh.de

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Qx=d F=d A p

Vout(t)=ν h p(t)

F=A p

Fh

Piezoelectric transducer

d = piezoelectric constant

v =voltage sensitivity coeff

h = thickness of transducer

http://www.bksv.com/

13

Electrodynamic transducer

B l =V/v=F/I

F=Bl I

Vout(t)=B l v(t)

VoutSensor: vel. →Volt

Exciter: I → F

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Response Transducers (acceleration)

• Accelerometers are very common– Based on piezoelectric elements

piezoresistive, capacitive, etc.– See Figures 10.12 in book

• Accelerometers themselves are m/c/k systems– Must watch for internal resonances

www.bksv.com

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Response Transducers (velocity)

• MHD (magneto-hydrodynamics) ATA Sensors Inc, USAhttp://www.aptec.com/Sensors2/ars01_01s_mhd_angular_rate_sensor.htm

• Lasers vibrometer– Doppler shifts– Can scan large structures– Line of sight– Noncontact

6

Response Transducers (displacement)

• LVDT– linear variable differential transformers– Magnets and coils– Noisy; best for low-freq, high-

displacement application• RVDT

– rotary variable differential transformers• Optical sensors (noncontact)

http://www.transtekinc.com/

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Linear variable differential transformer transducer

• Output voltage depends on the axial displacement of the core• Insensitive to temp and high output

http://www.daytronic.com/Products/trans/lvdt/default.htm

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Modal Testing

The use of our analytical methods to date to interpret vibration

measurementsA standard skill used in industry

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Measurement Hardware

Structure

Exciter

Powersupply

Signal generator

SC SC

Signal Analyzer

Modal Software

Display

AccelerometerLaser, etc

Load cell

SC = Singal Conditioning

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Exciters• Electromagnetic shakers which may apply a

force through a range of frequencies (harmonic or random inputs)

• Instrumented hammers which simulate an impact (recall the impulse response)

B&K 4808,

5~10 kHz, 112 Newtons

22

Mechanical Exciters• Force applied as an

inertia force:

• Force applied as an elastic spring force:

• Used for frequency <30 Hz and loads <700N Scotch yoke

mechanism

23

Mechanical Exciters

• Makes use of unbalance created by 2 masses rotating at same speed in opposite directions, load: 250N-25kN

• F(t) = 2mRω2cosωt

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Signal Conditioning• The direct output of a transducer not

usually well suited for input into an analyzer• Impedance miss matched, voltage or

current levels too low• SC is a charge or voltage amp designed to

take an accelerometer signal and match it to the input requirements of the analyzer

B&K 2635 charge amplifierhttp://www.bksv.com/3073.asp

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Analyzer

• Electronic boxes (really dedicated computers) which gather signals and manipulate them mathematically

• Like all other computer based technologies the analyzer “boxes” have evolved almost in to chip sized devices

• Essentially their main source of manipulation is digital Fourier Transforms for manipulating the vibration data in the frequency domain

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Fourier Series of F(t) of period T

F(t ) =a0

2+ (an cosnωT t +

n=1

∞

∑ bn sin nωT t)

where ωT = 2πT

a0 =2T

F(t)dt0

T

∫

an =2T

F(t)cosnωTtdt0

T

∫ n = 1,2,3....

bn =2T

F(t)sin nωT tdt0

T

∫ n =1,2,3....

27

Basic idea of the Analyzer

• Analog voltage in from force f(t) and one of x(t), v(t) or a(t) transducers

• Signals are filtered, “digitized” and transformed to the frequency domian

• Manipulated to produce digital frequency response functions from which vibration data is extracted

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Digital signal processing

• The analyzer takes signals form the transducer and puts the signal into a form that can be mathematically manipulated

• This of course is best performed with digital computers, hence we rely on some basic principles of DSP (Matlab)

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Forming a digital signal:Analog to Digital Conversion

• The analog signal x(t) is sampled at many equally spaced time intervals to produce the digital record

{x(t1), x(t2), ….x(tN)}where x(tk) is the discrete value of x(t) at time tk and N is the number of samples taken

• Constructed by A/D converter (eg gated)

30

Sampling rate

• Aliasing is caused by not sampling enough of the signal so the digital record does not catch the details of the analog signal

• Problem is solved by choosing a sample rate of 2.5 times the largest frequency of interest and by

• Anti-aliasing filters (cut off filter) allowing a sane choice of sampling rate

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Digital Fourier Transform

xk = x(tk ) = a0

2+ ai cos 2πitk

T+ bi sin 2πitk

T⎛ ⎝

⎞ ⎠ ,k =1,2...N

i =1

N / 2

∑where

a0 =1N

xkk =1

N

∑

ai =1N

xkk=1

N

∑ cos2πik

T

bi = 1N

xkk =1

N

∑ sin 2πikT

32

FFT/DFT Analyzer

• Above becomes the matrix equation x=Ca where

C contains the sin and cos termsx is the vector of samples anda is the vector of Fourier Coefficients

• The analyzer computes the coefficients in the DFT formula by a=C-1x

• N is fixed by hardware (a power of 2)

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Spectral Leakage

• If the signal is not periodic in N samples (signal cut off mid period) the DFT will produce extra frequencies, called leakage

• Fixed by windowing, multiply signal by a function which is zero at the end points, or tapers off (many kinds, depending on nature of signal)

• Downside, it adds damping to signal

Random Signal Analysis

Autocorrelation : Rxx (τ) = limT → ∞

1T

x(t)x( t + τ)dτ0

T

∫

Power Spectral Density (PSD) : Sxx (ω) =12π

Rxx (τ)e− jωτdτ−∞

∞

∫

Crosscorrelation : Rxf (τ) = limT → ∞

1T

x(t) f (t + τ)dτ0

T

∫

Cross Spectral Density : Sxf (ω) = 12π

Rxf (τ)e− jωτdτ−∞

∞

∫

Tells how fast x(t) is changing

Fourier transform of R

Tells how fast one signal changesrelative to another

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R and S are available from DFT and connect response to system frequency

response function

Sfx (ω) = H ( jω)Sff

Sxx (ω) = H( jω)Sxf

Sxx (ω) = H( jω) 2 Sff

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G(s) =1

ms2 + cs + k

G( jω) = H(ω) =1

k − mω2 + cjω

h(t) =1

mωd

e− ζωt sinωdt

LT[h(t)] = G(s) =1

ms2 + cs + kf (t) deterministic f (t) randomX(s) = G(s)F(s) Sxx(ω) = H(ω)2 Sff (ω)

x(t) = h(t − τ) f (τ)dτ0

t

∫ E[x 2 ] = H(ω) 2 Sff (ω)dω−∞

∞

∫

Transfer function

Frequency response function

Impulse response function

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The way it works

• PSD’s calculated in analyzer• Used to form H(w) for force in, response

out (velocity, position or acceleration)• H(w) used to extract modal data of

structure in a number of different ways, forming the topic of Modal Testing

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Coherence

γ 2 =Sxf (ω)

2

Sxx(ω)Sff (ω)

0 ≤ γ 2 ≤1should be 1, especially near resonance

• Compute H(w) a number of different ways

• Compare the various measured values of H(w)

• Indicates how good the measurement is

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Transfer function nomenclature

massapparent inertanceonAcceleratiimpedancemobilityVelocity

stiffness dynamiccompliancentDisplacemerecipricalstandardresponse

)()()(),(

)()(),(

)()( 2

sGsFsXsorsG

sFssXorsG

sFsX

===

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Measured FRF

Illustrates peak picking methodof determining modal parameters

Natural frequency taken as peak value

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Damping ratio from peak picking

H(ωa ) = H(ωb) =H (ωd )

2

ζ = ωb −ωa

2ωd

• Use the two frequencies determined from the half power (0.707) points

• Gives the modal damping ratio

42

Example 1 : Response with two modes:

First mode second mode

ζ1 =ωb − ωa

2ω1

= 10.16 − 9.7520

= 0.02

ζ2 = ωb − ωa

2ω2

=21.67 −17.10

40= 0.11

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Mode shape measurement

122

12

12

2

])2(diag[ )()(

)()(

,

−

−

−

+−=

+−=

+−=

=+−

==++

Tiii

tjtj

SjSCjMK

CjMKCjMK

eeKCM

ωζωω

ωωωα

ωω

ωω

ωω

fufu

uxfxxx &&&

44

S = M1 / 2P, P = [u1 : u2 :Lun ]

α(ω) = S−Tdiag1

ωi2 − ω2 + 2ζiωiωj

⎡

⎣ ⎢ ⎤

⎦ ⎥ S−1

α(ω) =uiu i

T

ωi2 − ω2 + 2ζiωiωji=1

n

∑in element form at resonance

uiu i

Tsr

= 2ζiω i2 Hsr (ω i), where Hsr = αsr

Measurement points

45

Need only one column to get mode shape

Suppose ui = [a1 a2 a3]T

uiuiT =

a12 a1a2 a1a3

a2a1 a22 a2 a3

a3a1 a3a2 a32

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

really only 3 unkowns a1, a2 and a3. So just threeelements of α(ω) need be measured per mode to get the mode shape

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Accel. #1

Accel. #2

Accel. #3

ForceTransd.

Shaker

Freq Response Fun.

Measure MDOF FRF

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Impulse Response Testing

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0

k/2

x

k/2c

m

y

case

m

k(x-y)c(x-y). .

tYyzmkz

mcz ωω sin2=−=++ &&&&&

yxz −=

Inertial sensor model

51

222

2

)2()1( rrrY

YZ

ς+−=

21

12tan

rr

−= − ζφ

22

2 onAccelerati)sin()(nn

tYtzωω

φωω=

−=

)sin()( φω −= tYtz

Frequency Response Function for Vibration Sensor

52

22

2 onAccelerati)sin()(nn

tYtzωω

φωω=

−=

If stiff spring, big k → high ωn

r=ω /ωn → small

a) If no damping (ζ = 0) → r < 0.1

r = .1 output value has 1% error

b) If ζ = 0.7, r < 0.25

If soft spring, small k → low ωn

r=ω /ωn → large

z(t) ~ y(t)

Frequency Response Function for Vibration Sensor

54

Example 10.2• A vibrometer having a natural frequency

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

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Solution• Amplitude of recorded motion:

• Amplitude of vibration of structure:• Y = Z/1.0093 = 3.9631 mm

( )( ) ( )( )[ ]

mm 40093.1102.02101

10 222

2

==+−

= YYZ

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10.3.2 Accelerometer• Measures acceleration

of a vibrating body.

( ) ( )( ) ( )

( ) ( )( ) ( )φωωω

ζ

ζ

φωωω

−−≈−

≈+−

+−

−−=−

tYtz

rr

rr

tYtz

n

n

sin

,121

1 If

21

sin

22

222

222

22

57

10.3.2 Accelerometer

• If 0.65< ζ < 0.7,

• Accelerometers are preferred due their small size.

( ) ( )6.00for 04.1

21

196.0222

≤≤≤+−

≤ rrr ζ

58

Example 10.3• An accelerometer has a suspended mass

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

59

Solution

( ) ( )( ) ( ) ( )

( )

( )

( ) (E.2) 14444.0ror 16667.0

6667.048.94232.628

11 Thus

rad/s 48.94221501

rad/s 32.62860

26000 speed Operating

(E.1) 0663.19684.0/121or

9684.081.95.9

valueTrue valueMeasured

21

1

222

22d

2d

2222

222

ζζ

ζωζω

ωω

πωζω

πω

ζ

ζ

−=−=∴

==−

=−

=

==−=

==

==+−

===+−

r

r

rr

rr

n

n

60

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

( )( )

( )( )( )s/m-N 8571.19

7253.08889.136801.022constant Damping

N/m 5628.187388889.136801.0

rad/s 8889.13687253.0148.942

1

22

22d

n

===

===

=−

=−

=

ζω

ω

ζωω

n

n

mc

mk

61

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

Machine fatique, wear, deformation, etc, lead to increase in the clearances between mating parts, misalignments in shaft, crack initiation in parts and unbalance in rotors

6

Machine Maintenance TechniquesBreakdown 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.

63

Machine Maintenance TechniquesPreventive maintenance:• Maintenance performed at fixed intervals• Intervals determined statistically from past experience• This method is uneconomicalCondition-based/Predictive maintenance:• Replace fixed-interval overhaul with fixed-interval

measurements• Can extrapolate measured vibration levels to predict

when they will reach unacceptable values

64

Machine Maintenance Techniques• Maintenance costs are greatly reduced

65

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.

66

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

67

Vibration Monitoring TechniquesTime domain analysis• E.g. following is an acceleration waveform of a

faulty 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.

68

Vibration Monitoring Techniques• Peak level, RMS level and crest factor may be

used as indices to identify damage• Changes in Lissajous figures can be used to

identify faults.

69

Each rotating element generates identifiable frequency. Thus changes in the spectrum at a given freq can be attributed to thecorresponding element.

Relationship between machine components and the vibration spectrum

70

Vibration Monitoring Techniques• 2nd gear was at

fault although 1st

gear was engaged.