Vibration Control B&K

7/14/2019 Vibration Control B&K

1/84

Modal Analysis 1

ExperimentalModal Analysis

Copyright 2003Brel & Kjr Sound & Vibration Measurement A/SAll Rights Reserved


2/84

Modal Analysis 2

SDOF and MDOF Models

Different Modal AnalysisTechniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis PostProcessing

f(t)

x(t)

kc

m

= ++ + +


3/84

Modal Analysis 3

Simplest Form of Vibrating System

D

d = D sinnt

m

k

T

Time

Displacement

Frequency1

T

Period, Tn in [sec]

Frequency, fn= in [Hz = 1/sec]1Tn

Displacement

k

mn= 2 fn =


4/84Modal Analysis 4

Mass and Spring

time

m1

m

Increasing massreduces frequency

1

nmm

k2

+== nf


5/84Modal Analysis 5

Mass, Spring and Damper

Increasing dampingreduces the amplitude

time

m

k c1 + c2


6/84

Modal Analysis 6

)()()()( tftKxtxCtxM =++ &&&

M = mass (force/acc.)

C = damping (force/vel.)

K = stiffness (force/disp.)

)t(x&&)t(x& )t(x

Acceleration Vector

Velocity Vector

Displacement Vector

)t(f Applied force Vector

Basic SDOF Model

f(t)

x(t)

kc

m


7/84

Modal Analysis 7

SDOF ModelsTime and Frequency Domain

kcjmF

XH

++==

2

1

)(

)()(

H()F() X()

|H()|

1k

H()

0

90

180

0 = k/m

12m

1c

)()()()( tkxtxctxmtf ++= &&&

f(t)

x(t)

kc

m


8/84

Modal Analysis 8

Modal Matrix

Modal Model

(Freq. Domain)

( )( )

( )

( )

( ) ( ) ( )

( )

( )

( ) ( )

( )

=

3

1

23

22

12111

3

2

1

F

HH

H

H

HHH

X

X

XX

nnn

n

n

X1

X2

X3X

4

F3H21

H22


9/84

Modal Analysis 9

MDOF Model

0

-90

-180

2 1

1+2

1 2

1+2

Frequency

FrequencyPhase

Magnitude

m

d1+ d2

d1

dF


10/84

Modal Analysis 10

Why Bother with Modal Models?

Physical Coordinates = CHAOS Modal Space = Simplicity

q1

1

1

20112

2

q21

20222

q31

20332

3

Bearing

Rotor

Bearing

Foundation


11/84

Modal Analysis 11

Definition of Frequency Response Function

F X

f f f

H(f) is the system Frequency Response Function

F(f) is the Fourier Transform of the Input f(t)

X(f) is the Fourier Transform of the Output x(t)

F(f) H(f) X(f)

)f(F

)f(X)f(H =

H

H


12/84

Modal Analysis 12

Benefits of Frequency Response Function

F(f) H(f) X(f)

Frequency Response Functions are properties of lineardynamic systems

They are independent of the Excitation Function

Excitation can be a Periodic, Random or Transient functionof time

The test result obtained with one type of excitation can beused for predicting the response of the system to any othertype of excitation


13/84

Modal Analysis 13

Compliance Dynamic stiffness

(displacement / force) (force / displacement)

Mobility Impedance

(velocity / force) (force / velocity)

Inertance or Receptance Dynamic mass(acceleration / force) (force /acceleration)

Different Forms of an FRF


14/84

Modal Analysis 14

Alternative Estimators

)()()(

fFfXfH =

)()(

)(1 fG

fGfH

FF

FX=

)()()(

2 fGfGfH

XF

XX=

213 )( HHG

G

G

G

fH FX

FX

FF

XX

==

2

1

2

2 *)(

H

H

G

G

G

G

GG

Gf

XX

FX

FF

FX

XXFF

FX ==

=

X(f)F(f) H(f)


15/84

Modal Analysis 15

Which FRF Estimator Should You Use?

Definitions:

User can choose H1, H2 or H3 after measurement

Accuracy for systems with: H1 H2 H3

Input noise - Best -

Output noise Best - -

Input + output noise - - Best

Peaks (leakage) - Best -

Valleys (leakage) Best - -

Accuracy

)()(

)(1 fG

fGfH

FF

FX=)()(

)(2 fG

fGfH

XF

XX=FX

FX

FF

XX

G

G

G

GfH =)(3


16/84

Modal Analysis 16






f(t)

x(t)

kc

m

= ++ + +


17/84

Modal Analysis 17

Three Types of Modal Analysis

1. Hammer Testing

Impact Hammer taps...serial or parallel measurements

Excites wide frequency range quickly

Most commonly used technique

2. Shaker Testing

Modal Exciter shakes product...serial or parallel measurements

Many types of excitation techniques Often used in more complex structures

3. Operational Modal Analysis

Uses natural excitation of structure...serial or parallel measurements Cutting edge technique


18/84

Modal Analysis 18

Different Types of Modal Analysis (Pros)

Hammer Testing Quick and easy

Typically Inexpensive

Can perform poor man modal as well as full modal

Shaker Testing

More repeatable than hammer testing

Many types of input available

Can be used for MIMO analysis Operational Modal Analysis

No need for special boundary conditions

Measure in-situ

Use natural excitation

Can perform other tests while taking OMA data


19/84

Modal Analysis 19

Different Types of Modal Analysis (Cons)

Hammer Testing Crest factors due impulsive measurement

Input force can be different from measurement to measurement(different operators, difficult location, etc.)

Calibrated elbow required (double hits, etc.) Tip performance often an overlooked issue

Shaker Testing

More difficult test setup (stingers, exciter, etc.)

Usually more equipment and channels needed

Skilled operator(s) needed

Operational Modal Analysis

Unscaled modal model Excitation assumed to cover frequency range of interest

Long time histories sometimes required

Computationally intensive


20/84

Modal Analysis 20

Frequency Response Function

Input

Output

Time(Response) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s] Time(Response) - InputWorking : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s]

Time(Excitation) - Input

Working : Input : Input : FFT Analyzer

0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N] Time(Excitation) - Input


0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N]

FFT

FFTFrequency Domain

Autospectrum(Excitation) - Input


0 200 400 600 800 1k 1,2k 1,4k 1 ,6k

100u

1m

10m

100m

1

[Hz]

[N] Autospectrum(Excitation) - Input


0 200 400 600 800 1k 1,2k 1,4k 1 ,6k

100u

1m

10m

100m

1

[Hz]

[N]

Autospectrum(Response) - Input


0 200 400 600 800 1k 1,2k 1,4k 1 ,6k

1m

10m

100m

1

10

[Hz]

[m/s] Autospectrum(Response) - Input


0 200 400 600 800 1k 1,2k 1,4k 1 ,6k

1m

10m

100m

1

10

[Hz]

[m/s]

Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s)/N]Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer

0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s)/N]

Time Domain

Impulse Response h1(Response,Excitation) - Input (Real Part)


0 40m 80m 120m 160m 200m 240m-2k

-1k

0

1k

2k

[s]

[(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)


0 40m 80m 120m 160m 200m 240m-2k

-1k

0

1k

2k

[s]

[(m/s)/N/s]Inverse

FFT

Excitation

Response

Force

Motion

Input

OutputH ===


21/84

Modal Analysis 21

Hammer Test on Free-free Beam

Frequency

Distanc

eAmplitude First

Mode

Beam

SecondModeThirdMode

ForceForceForceForceForceForceForceForceForceForceForce

Roving hammer method:

Response measured at one point

Excitation of the structure at a number of pointsby hammer with force transducer

FrequencyDomainView

Modes of structure identified

FRFs between excitation points and measurement point calculated

Press anywhereto advance animation

Force

Modal

Doma

in

View

Acceleration


22/84

Modal Analysis 22

Measurement of FRF Matrix (SISO)

One row

One Roving Excitation

One Fixed Response (reference)

SISO

=

X1 H11 H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2

X3 H31 H32 H33...H 3n F3: : :

Xn Hn1 Hn2 Hn3...Hnn Fn


23/84

Modal Analysis 23

Measurement of FRF Matrix (SIMO)

More rows

One Roving Excitation

Multiple Fixed Responses (references)

SIMO

=

X1 H11 H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2

X3 H31 H32H33...H 3n F3: : :

Xn Hn1 Hn2 Hn3...Hnn Fn


24/84

Modal Analysis 24

FRFs between excitation point and measurement points calculated

Whitenoiseexcitation

Shaker Test on Free-free Beam

Frequency

Distan

ceAmplitude First

Mode SecondMode

ThirdMode

Beam

Acceleration

Press anywhereto advance animation

Modal

Doma

in

View

FrequencyDomainView Force

Shaker method:

Excitation of the structure at one pointby shaker with force transducer

Response measured at a number of points

Modes of structure identified

f (S S O)


25/84

Modal Analysis 25

Measurement of FRF Matrix (Shaker SIMO)

One column Single Fixed Excitation (reference)

Single Roving Response SISO

or Multiple (Roving) Responses SIMO

Multiple-Output: Optimize data consistency

X1 H11H12 H13 ...H1n F1X2 H21H22 H23 ...H2n F2

X3 H31 H32 H33...H 3n F3: : :

Xn Hn1Hn2 Hn3...Hnn Fn

=


26/84

Modal Analysis 26

Why Multiple-Input and Multiple-Output ?

Multiple-Input: For large and/or complex structures moreshakers are required in order to:

get the excitation energy sufficiently distributed

and

avoid non-linear behaviour

Multiple-Output: Measure outputs at the same time inorder to optimize data consistency

i.e. MIMO


27/84

Modal Analysis 27

Situations needing MIMO

One row or one column is not sufficient for determination ofall modes in following situations:

More modes at the same frequency (repeated roots),

e.g. symmetrical structures

Complex structures having local modes, i.e. referenceDOF with modal deflection for all modes is not available

In both cases more columns or more rows have to bemeasured - i.e. polyreference.

Solutions:

Impact Hammerexcitation with more response DOFs One shakermoved to different reference DOFs

MIMO

M f FRF M i (MIMO)


28/84

Modal Analysis 28

Measurement of FRF Matrix (MIMO)

=

More columns Multiple Fixed Excitations (references)

Single Roving Response MISOor

Multiple (Roving) Responses MIMO

X1 H11H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2X

3 H31H

32H

33...H 3n F

3: : :

Xn Hn1Hn2Hn3...Hnn Fn

( ) /


29/84

Modal Analysis 29

Modal Analysis (classic): FRF = Response/Excitation

Output

Frequency Response H1(Response,Excitation) - Input (Magnitude)Frequency Response H1(Response,Excitation) - Input (Magnitude)


0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s)/N]


0 200 400 600 800 1k 1,2k 1,4k 1,6k

100m

10

[Hz]

[(m/s)/N]

Autospectrum(Excitation) -Input


0 200 400 600 800 1k 1,2k 1,4k 1,6k

100u

1m

10m

100m

1

[Hz]

[N] Autospectrum(Excitation) -Input


0 200 400 600 800 1k 1,2k 1,4k 1,6k

100u

1m

10m

100m

1

[Hz]

[N]

Autospectrum(Response) -Input


0 200 400 600 800 1k 1,2k 1,4k 1,6k

1m

10m

100m

1

10

[Hz]

[m/s] Autospectrum(Response) -Input


0 200 400 600 800 1k 1,2k 1,4k 1,6k

1m

10m

100m

1

10

[Hz]

[m/s]

Time(Excitation) - Input


0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N] Time(Excitation) - Input


0 40m 80m 120m 160m 200m 240m

-200

-100

0

100

200

[s]

[N]

Time(Response) - Input


0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s] Time(Response) - Input


0 40m 80m 120m 160m 200m 240m

-80

-40

0

40

80

[s]

[m/s]

FFT

FFT

Impulse Response h1(Response,Excitation) - Input (Real Part)


0 40m 80m 120m 160m 200m 240m

-2k

-1k

0

1k

2k

[s]

[(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)


0 40m 80m 120m 160m 200m 240m

-2k

-1k

0

1k

2k

[s]

[(m/s)/N/s]

Inverse

FFT

Input

Excitation

Response

Force

Vibration

Input

OutputH() = = =

FrequencyResponseFunction

Frequency Domain

ImpulseResponseFunction

Time Domain

Operational Modal Analysis (OMA): Response only!

Natural Excitation


30/84

Modal Analysis 30






f(t)

x(t)

kc

m

= ++ + +

Th Et l Q ti i M d l


31/84

Modal Analysis 31

The Eternal Question in Modal

F1a

F2

Impact Excitation


32/84

Modal Analysis 32

Impact Excitation

H11() H12() H15()

Measuring one row of the FRF matrix bymoving impact position

LAN

# 1

# 2

# 3# 4

# 5

Accelerometer

ForceTransducer Impact

Hammer

Impact Excitation


33/84

Modal Analysis 33

Impact Excitation

t

GAA(f)

Magnitude and pulse duration depends on: Weight of hammer

Hammer tip (steel, plastic or rubber)

Dynamic characteristics of surface

Velocity at impact

Frequency bandwidth inversely proportional to the pulse duration

a(t)

a(t)

ft

1

2

1

2

Weighting Functions for Impact Excitation


34/84

Modal Analysis 34

Weighting Functions for Impact Excitation

How to select shift and lengthfor transient and exponentialwindows:

Leakage due to exponential timeweighting on response signal is welldefined and therefore correction of themeasured damping value is often possible

Transient weighting of the input signal

Criteria

Exponential weightingof the output signal

Compensation for Exponential Weighting


35/84

Modal Analysis 35

Compensation for Exponential Weighting

With exponentialweighting of theoutput signal, themeasured time

constant will be tooshort and thecalculated decayconstant anddamping ratio

therefore too large shift Length =Record length, T

Original signal

Weighted signal

Window function

Time

1

b(t)

W

Correction of decay constant and damping ratio

Wm =

W

W

1

=

Wm

0

W

0

m

0

=

=

=

Correctvalue

Measuredvalue

Range of hammers


36/84

Modal Analysis 36

3 lb HandSledge

Building andbridges

12 lb Sledge

Large shaftsand largermachine tools

Car framed

and machinetools

1 lb hammer

ComponentsGeneralPurpose, 0.3

lbHard-drives,circuit boards,turbine blades

Mini Hammer

ApplicationDescription

Range of hammers

Impact hammer excitation


37/84

Modal Analysis 37

Impact hammer excitation

Advantages: Speed

No fixturing

No variable mass loading

Portable and highly suitable forfield work

relatively inexpensive

Disadvantages High crest factor means

possibility of driving structureinto non-linear behavior

High peak force needed forlarge structures meanspossibility of local damage!

Highly deterministic signalmeans no linear approximation

Conclusion

Best suited for field work

Useful for determining shakerand support locations

Shaker Excitation


38/84

Modal Analysis 38

Shaker Excitation

H11()

H21()

H51()

Measuring one column of the FRF matrix bymoving response transducer

# 1

# 2

# 3# 4

# 5Accelerometer

ForceTransducer

VibrationExciter

PowerAmplifier

LAN

Attachment of Transducers and Shaker


39/84

Modal Analysis 39

Attachment of Transducers and Shaker

Accelerometer mounting:

Stud

Cement

Wax

(Magnet)

Force Transducer and Shaker:

Stud

Stinger (Connection Rod)

Properties of Stinger

Axial Stiffness: High

Bending Stiffness: Low

Accelerometer

ShakerFa

ForceTransducer

Advantages of Stinger:

No Moment Excitation

No Rotational Inertia Loading Protection of Shaker

Protection of Transducer

Helps positioning of Shaker

Connection of Exciter and Structure


40/84

Modal Analysis 40

Connection of Exciter and Structure

ExciterSlenderstinger

Measured structure

Accelerometer

Force Transducer

Tip mass, m

Shaker/Hammermass, M

Fs

Fm

Piezoelectricmaterial

Structure

M

MmFF ms

+=

XMFm &&=

X)Mm(Fs&&+=

Force and acceleration measurements unaffected by stinger compliance, but ...

Minor mass correction required to determine actual excitation

Shaker Reaction Force


41/84

Modal Analysis 41

Shaker Reaction Force

Reaction byexternal support

Reaction byexciter inertia

Example of animproper

arrangement

Structure

Suspension

Exciter

Support

Structure

Suspension

Exciter

Suspension

Sine Excitation


42/84

Modal Analysis 42

Sine Excitation

a(t)

Time

A

Crest factor 2RMSA ==

For study of non-linearities,

e.g. harmonic distortion

For broadband excitation: Sine wave swept slowly through

the frequency range of interest

Quasi-stationary condition

A(f1)

B(f1)

RMS

Swept Sine Excitation


43/84

Modal Analysis 43

p

Advantages Low Crest Factor

High Signal/Noise ratio

Input force well controlled

Study of non-linearities

possibleDisadvantages

Very slow

No linear approximation ofnon-linear system

Random Excitation


44/84

Modal Analysis 44

A(f1)Freq.

Time

a(t)

B(f1)GAA(f), N = 1 GAA(f), N = 10

Random variation of amplitude and phaseAveraging will give optimum linear

estimate in case of non-linearities

Freq.

SystemOutput

System

Input

Random Excitation


45/84

Modal Analysis 45

Random signal:

Characterized by power spectral density (GAA) andamplitude probability density (p(a))

a(t)

p(a)

Can be band limited according to frequency range of interest

GAA(f) GAA(f)Baseband Zoom

Time

Signal not periodic in analysis time Leakage in spectral estimates

Frequencyrange

Frequencyrange

Freq. Freq.

Random Excitation


46/84

Modal Analysis 46

Advantages Best linear approximation of system

Zoom Fair Crest Factor

Fair Signal/Noise ratio

Disadvantages Leakage

Averaging needed (slower)

Burst Random


47/84

Modal Analysis 47

Characteristics of Burst Random signal :

Gives best linear approximation of nonlinear system

Works with zoom

a(t)

Time

Advantages Best linear approximation of system

No leakage (if rectangular time weighting can be used)

Relatively fast

Disadvantages Signal/noise and crest factor not optimum

Special time weighting might be required

Pseudo Random Excitation


48/84

Modal Analysis 48

a(t)

Pseudo random signal:

Block of a random signal repeated every T

Time

A(f1)

B(f1)

T T T T

Time period equal to record length T

Line spectrum coinciding with analyzer lines

No averaging of non-linearities

Freq.

GAA(f), N = 1 GAA(f), N = 10

Freq.

SystemOutput

System Input



49/84

Modal Analysis 49

T T T T

a(t)

Time

p(a)

Freq. range Freq. range

GAA(f) GAA(f)Baseband Zoom

Can be band limited according to frequency range of interest

Time period equal to T No leakage if Rectangular weighting is used

Pseudo random signal:

Characterized by power/RMS (GAA) and amplitude probability density (p(a))

Freq. Freq.



50/84

Modal Analysis 50

Advantages No leakage

Fast Zoom

Fair crest factor

Fair Signal/Noise ratioDisadvantages


Multisine (Chirp)


51/84

Modal Analysis 51

For sine sweep repeated every time record, Tr

Tr

A special type of pseudo random signal wherethe crest factor has been minimized (< 2)

It has the advantages and disadvantages of

the normal pseudo random signal but with alower crest factor

Additional Advantages:

Ideal shape of spectrum: The spectrum is a flatmagnitude spectrum, and the phase spectrum is smooth

Applications:

Measurement of structures with non-linear behaviour

Time

Periodic Random


52/84

Modal Analysis 52

Pseudo-random signalchanging with time:

Analysed time data:(steady-state response)

transient response

steady-state response

A B C

A A A B B B C CC

T T T

A combined random and pseudo-random signal givingan excitation signal featuring:

No leakage in analysis

Best linear approximation of system

Disadvantage:

The test time is longer than the test timeusing pseudo-random or random signal

Periodic Pulse


53/84

Modal Analysis 53

Leakage can be avoided using rectangular time weighting

Transient and exponential time weighting can be used to increase

Signal/Noise ratio Gating of reflections with transient time weighting

Effects of non-linearities are not averaged out

The signal has a high crest factor

Rectangular, Hanning, or Gaussian pulse with user definable t repeated with a user definable interval, T

The line spectrum for a

Rectangular pulse has a

shaped envelope curve

sin xx

t T Time

1/ t Frequency

Special case of pseudo random signal

Periodic Pulse


54/84

Modal Analysis 54

Advantages

Fast

No leakage(Only with rectangular weighting)

Gating of reflections

(Transient time weighting) Excitation spectrum follows

frequency span in baseband

Easy to implement

Disadvantages


High Crest Factor

High peak level might excite

non-linearities No Zoom

Special time weighting mightbe required to increase

Signal/Noise Ratio . This canalso introduce leakage

Guidelines for Choice of Excitation Technique


55/84

Modal Analysis 55

Swept sine excitation For study of non-linearities:

Random excitation For slightly non-linear system:

Pseudo random excitation For perfectly linear system:

Impact excitation For field measurements:

Random impact excitation For high resolution field measurements:


56/84

Modal Analysis 56


Different Modal Analysis

Techniques



Modal Analysis Post

Processing

f(t)

x(t)

kc

m

= ++ + +

Garbage In = Garbage Out!


57/84

Modal Analysis 57

A state-of-the Art Assessment of Mobility Measurement Techniques

Result for the Mid Range Structure (30 - 3000 Hz)

D.J. Ewins and J. GriffinFeb. 1981

Transfer MobilityCentral Decade

Transfer MobilityExpanded

Frequency Frequency

Plan Your Test Before Hand!


58/84

Modal Analysis 58

1. Select Appropriate Excitation Hammer, Shaker, or OMA?

2. Setup FFT Analyzer correctly Frequency Range, Resolution, Averaging, Windowing

Remember: FFT Analyzer is a BLOCK ANALYZER!

3. Good Distribution of Measurement Points Ensure enough points are measured to see all modes of interest

Beware of spatial aliasing

4. Physical Setup Accelerometer mounting is CRITICAL!

Uni-axial vs. Triaxial Make sure DOF orientation is correct

Mount device under test...mounting will affect measurement!

Calibrate system

Where Should Excitation Be Applied?


59/84

Modal Analysis 59

Driving PointMeasurement

TransferMeasurement

i = j

i j

i

j

j

i

"j"Excitation"i"sponseRe

FXH

j

iij ==

12

2121111 FHFHX +=

=

2

1

2221

1211

2

1

F

F

HH

HH

X

X

}{ [ ]{ }FHX =

2221212 FHFHX +=

Check of Driving Point Measurement

I [H ]


60/84

Modal Analysis 60

All peaks in

)f(F

)f(XImand

)f(F

)f(XRe,

)f(F

)f(XIm

&&&

An anti-resonance in Mag[Hij]

must be found between everypair of resonances

Phase fluctuations must bewithin 180

Im[Hij]

Mag[Hij]

Phase [Hij]

Driving Point (DP) Measurement


61/84

Modal Analysis 61

The quality of the DP-measurement is very important, as the

DP-residues are used for the scaling of the Modal Model

DP- Considerations:

Residues for all modesmust be estimatedaccurately from a singlemeasurement

DP- Problems:

Highest modal coupling, as allmodes are in phase

Highest residual effect fromrigid body modes

m Aij

Re Aij

Log MagAij

=

Without rigid

body modes

Measured

Tests for Validity of Data: Coherence


62/84

Modal Analysis 62

Coherence)f(G)f(G

)f(G)f(XXFF

2

FX2 =

Measures how much energy put in to thesystem caused the response

The closer to 1 the more coherent

Less than 0.75 is bordering on poor coherence

Reasons for Low Coherence


63/84

Modal Analysis 63

Difficult measurements:

Noise in measured output signal

Noise in measured input signal

Other inputs not correlated with measured input signal

Bad measurements:

Leakage Time varying systems

Non-linearities of system

DOF-jitter

Propagation time not compensated for

Tests for Validity of Data: Linearity


64/84

Modal Analysis 64

Linearity X1 = HF1X2 = HF2

X1+X2 = H(F1 + F2)aX1 = H(a F1)

More force going in to the system will equate to

F() X()

more response coming out

Since FRF is a ratio the magnitude should bethe same regardless of input force

H()

Tips and Tricks for Best Results


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Modal Analysis 65

Verify measurement chain integrity prior to test:

Transducer calibration

Mass Ratio calibration

Verify suitability of input and output transducers:

Operating ranges (frequency, dynamic range, phase response)

Mass loading of accelerometers

Accelerometer mounting

Sensitivity to environmental effects

Stability

Verify suitability of test set-up:

Transducer positioning and alignment

Pre-test: rattling, boundary conditions, rigid body modes, signal-to-noise ratio,linear approximation, excitation signal, repeated roots, Maxwell reciprocity,

force measurement, exciter-input transducer-stinger-structure connection

Quality FRF measurements are the foundation ofexperimental modal analysis!


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Modal Analysis 66


Different Modal Analysis

Techniques



Modal Analysis Post

Processing

f(t)

x(t)

kc

m

= ++ + +

From Testing to Analysis


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Modal Analysis 67

MeasuredFRF

Curve Fitting

(Pattern Recognition)

Modal Analysis

)f(H

Frequency

)f(H

Frequency

From Testing to Analysis


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Modal Analysis 68

Modal Analysis

)f(H

Frequency

f(t)

x(t)

kc

m

f(t)

x(t)

kc

m

f(t)

x(t)

kc

mSDOF Models

Mode Characterizations


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Modal Analysis 69

All Modes Can Be Characterized By:

1. Resonant Frequency

2. Modal Damping

3. Mode Shape

Modal Analysis Step by Step Process

1. Visually Inspect Data


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Modal Analysis 70

y p

Look for obvious modes in FRF InspectALL FRFssometimes modes will show up in one

FRF but not another (nodes)

Use Imaginary part and coherence for verification

Sum magnitudes of all measurements for clues2. Select Curve Fitter

Lightly coupled modes: SDOF techniques

Heavily coupled modes: MDOF techniques

Stable measurements: Global technique

Unstable measurements: Local technique

MIMO measurement: Poly reference techniques

3. Analysis Use more than 1 curve fitter to see if they agree

Pay attention to Residue calculations

Do mode shapes make sense?

Modal Analysis Inspect Data



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Modal Analysis 71













Modal Analysis Curve Fitting



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Modal Analysis 72













How Does Curve Fitting Work?

Curve Fitting is the process of estimating the Modal


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Modal Analysis 73

Curve Fitting is the process of estimating the ModalParameters from the measurements

|H|

2

R/

d

0

-180

Phase

Frequency

Find the resonant

frequency Frequency where smallexcitation causes alarge response

Find the damping What is the Q of the

peak?

Find the residue Essentially the area

under the curve

Residues are Directly Related to Mode Shapes!


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Modal Analysis 74

*

*

r

ijr

r r

ijr

rijrij

pj

R

pj

RH)(H

+

==

Residues express thestrength of a mode for eachmeasured FRF

Therefore they are relatedto mode shape at eachmeasured point!

Frequency

Dista

nce

Amplitude FirstMode

Beam

SecondMode

ForceForceForceForceForceForceForceForceForceForceForceForce

Acceleration

ThirdMode

SDOF vs. MDOF Curve Fitters


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Modal Analysis 75

Use SDOF methods on LIGHTLY COUPLED modes Use MDOF methods on HEAVILY COUPLED modes

You can combine SDOF and MDOF techniques!

SDOF

(light coupling)

|H|

MDOF

(heavy coupling)

Local vs. Global Curve Fitting


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Modal Analysis 76

Local means that resonances, damping, and residues arecalculated for each FRF firstthen combined for curvefitting

Global means that resonances, damping, and residues are

calculated across all FRFs

|H|

Modal Analysis Analyse Results



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Modal Analysis 77










3. Analysis

Use more than one curve fitter to see if they agree



Which Curve Fitter Should Be Used?



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Modal Analysis 78

q y p

Hij ()

Real Imaginary

Nyquist Log Magnitude

Which Curve Fitter Should Be Used?



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Modal Analysis 79

Real Imaginary

Nyquist Log Magnitude

Hij ()

Modal Analysis and Beyond


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Modal Analysis 80

Hardware ModificationResonance Specification

Simulate RealWorld Response

Dynamic Modelbased on

Modal Parameters

ExperimentalModal Analysis

Structural

Modification

Response

Simulation

F

Conclusion

All Physical Structures can be characterized by the simple


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Modal Analysis 81

SDOF model

Frequency Response Functions are the best way tomeasure resonances

There are three modal techniques available today:Hammer Testing, Shaker Testing, and Operational Modal

Planning and proper setup before you test can save timeand effortand ensure accuracy while minimizingerroneous results

There are many curve fitting techniques available, try touse the one that best fits your application

Literature for Further Reading

Structural Testing Part 1: Mechanical Mobility Measurements


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Modal Analysis 82

Brel & Kjr Primer

Structural Testing Part 2: Modal Analysis and Simulation

Brel & Kjr Primer

Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J.Ewin

Research Studies Press Ltd.

Dual Channel FFT Analysis (Part 1)

Brel & Kjr Technical Review # 1 1984

Dual Channel FFT Analysis (Part 1)

Brel & Kjr Technical Review # 2 1984

Appendix: Damping Parameters


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Modal Analysis 83

3 dB bandwidth=

22fdB3

Loss factor 0

dB3

0

db3

f

f

Q

1

=

==

Damping ratio

0

Bd3

0

dB3

2f2

f

2

=

=

=

Decay constant

Quality factordB3

0

dB3

0

f

fQ

=

=

2f dB3dB30

===

= 2dB3,

Appendix: Damping Parameters

t -t


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Modal Analysis 84

( )tsineR2)t(h d= , where the Decay constant is given by e

The Envelope is given by magnitude of analytic h(t):

The time constant, , isdetermined by the time it

takes for the amplitude to

decay a factor of e = 2,72

or

10 log (e2) = 8.7 dB

)t(h~

)t(h)t(h 22 +=

Decay constant

Time constant :

Damping ratio

Loss factor

Quality

= 1

=1

=

=00 f2

1

==

0f

12

=

= 0f1

Q

h(t)

Time

t

e

Vibration Control B&K

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