# Brüel and Kjaer

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Modal Analysis 1

ExperimentalModal Analysis

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

Modal Analysis 2

SDOF and MDOF Models

Different Modal Analysis Techniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis Post Processing

f(t)

x(t)

kc

m

= ++ + +

Modal Analysis 3

Simplest Form of Vibrating System

D

d = D sinnt

m

k

T

Time

Displacement

Frequency1T

Period, Tn in [sec]

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

Displacement

kmn= 2 fn =

Modal Analysis 4

Mass and Spring

time

m1

m

Increasing mass reduces frequency

1n mm

k2 +== nf

Modal Analysis 5

Mass, Spring and Damper

Increasing damping reduces the amplitude

time

m

k c1 + c2

Modal Analysis 6

)()()()( tftKxtxCtxM =++ &&&M = mass (force/acc.)C = damping (force/vel.)K = stiffness (force/disp.)

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

Acceleration VectorVelocity VectorDisplacement Vector

=)t(f Applied force Vector

Basic SDOF Model

f(t)x(t)

kc

m

Modal Analysis 7

SDOF Models Time and Frequency Domain

kcjmFXH ++==

2 1)()()(

H()F() X()

|H()|1k

H()

0 90

180

0 = k/m

12m

1c

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

f(t)x(t)

kc

m

Modal Analysis 8

Modal Matrix

Modal Model(Freq. Domain)

( )( )( )

( )

( ) ( ) ( )( )( )

( ) ( )

( )

=

3

1

23

22

12111

3

2

1

F

HH

HH

HHH

X

XXX

nnn

n

n

X1

X2X3

X4

F3H21

H22

Modal Analysis 9

MDOF Model

0

-90

-180

2 1

1+2

1 2

1+2

Frequency

FrequencyPhase

Magnitude

m

d1+ d2

d1

dF

Modal Analysis 10

Why Bother with Modal Models?

Physical Coordinates = CHAOS Modal Space = Simplicity

q1

11

20112 2

q21

20222

q31

20332

3

Bearing

Rotor

Bearing

Foundation

Modal Analysis 11

Definition of Frequency Response Function

F X

f f f

H(f) is the system Frequency Response FunctionF(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

Modal Analysis 12

Benefits of Frequency Response Function

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

z Frequency Response Functions are properties of linear dynamic systems

z They are independent of the Excitation Function

z Excitation can be a Periodic, Random or Transient function of time

z The test result obtained with one type of excitation can be used for predicting the response of the system to any other type of excitation

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

Modal Analysis 14

Alternative Estimators

)()()( fFfXfH =

)()()(1 fGfGfH

FF

FX=

)()()(2 fGfGfH

XF

XX=

213 )( HHGG

GGfH

FX

FX

FF

XX ==

2

1

22 *)( H

HGG

GG

GGG

fXX

FX

FF

FX

XXFF

FX ===

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

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 H3Input noise - Best -

Output noise Best - -

Input + output noise - - Best

Peaks (leakage) - Best -

Valleys (leakage) Best - -

Accuracy

)()()(1 fGfGfH

FF

FX= )()()(2 fGfGfH

XF

XX=FX

FX

FF

XX

GG

GGfH =)(3

Modal Analysis 16

SDOF and MDOF Models

Different Modal Analysis Techniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis Post Processing

f(t)

x(t)

kc

m

= ++ + +

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

Modal Analysis 18

Different Types of Modal Analysis (Pros)

z Hammer Testing Quick and easy Typically Inexpensive Can perform poor man modal as well as full modal

z Shaker Testing More repeatable than hammer testing Many types of input available Can be used for MIMO analysis

z Operational Modal Analysis No need for special boundary conditions Measure in-situ Use natural excitation Can perform other tests while taking OMA data

Modal Analysis 19

Different Types of Modal Analysis (Cons)

z 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

z Shaker Testing More difficult test setup (stingers, exciter, etc.) Usually more equipment and channels needed Skilled operator(s) needed

z Operational Modal Analysis Unscaled modal model Excitation assumed to cover frequency range of interest Long time histories sometimes required Computationally intensive

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) - InputWorking : Input : Input : FFT Analyzer

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

-200

-100

0

100

200

[s]

[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

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

-200

-100

0

100

200

[s]

[N]

FFT

FFTFrequency Domain

Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

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

100u

1m

10m

100m

1

[Hz]

[N] Autospectrum(Excitation) - InputWorking : Input : Input : FFT Analyzer

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

100u

1m

10m

100m

1

[Hz]

[N]

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

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

1m

10m

100m

1

10

[Hz]

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

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)Working : Input : Input : FFT Analyzer

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)Working : Input : Input : FFT Analyzer

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

-2k

-1k

0

1k

2k

[s]

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

ExcitationResponse

ForceMotion

InputOutputH ===

Modal Analysis 21

Hammer Test on Free-free Beam

Frequency

Distan

ceAmplitude First

Mode

Beam

SecondMode Third

Mode

ForceForceForceForceForceForceForceForceForceForceForce

Roving hammer method:z Response measured at one point z Excitation of the structure at a number of points

by hammer with force transducer

Frequency Domain View

z Modes of structure identifiedz FRFs between excitation points and measurement point calculated

Press anywhereto advance animation

ForceMo

dal

Domain

View

Acceleration

Modal Analysis 22

Measurement of FRF Matrix (SISO)

One rowz One Roving Excitationz One Fixed Response (reference)

SISO

=

X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

Modal Analysis 23

Measurement of FRF Matrix (SIMO)

More rowsz One Roving Excitationz Multiple Fixed Responses (references)

SIMO

=

X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

Modal Analysis 24

z FRFs between excitation point and measurement points calculated

Whitenoiseexcitation

Shaker Test on Free-free Beam

Frequency

Distan

ceAmplitude First

Mode SecondMode

ThirdMode

BeamAcceleration

Press anywhereto advance animation

Modal

Domain

View

Frequency Domain View Force

Shaker method:z Excitation of the structure at one point

by shaker with force transducerz Response measured at a number of points

z Modes of structure identified

Modal Analysis 25

Measurement of FRF Matrix (Shaker SIMO)

One columnz Single Fixed Excitation (reference)

z Single Roving Response SISOor

z Multiple (Roving) Responses SIMOMultiple-Output: Optimize data consistency

X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

=

Modal Analysis 26

Why Multiple-Input and Multiple-Output ?z Multiple-Input: For large and/or complex structures more

shakers are required in order to: get the excitation energy sufficiently distributedand avoid non-linear behaviour

z Multiple-Output: Measure outputs at the same time in order to optimize data consistency

i.e. MIMO

Modal Analysis 27

Situations needing MIMO

z One row or one column is not sufficient for determination of all modes in following situations:

More modes at the same frequency (repeated roots), e.g. symmetrical structures

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

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

Solutions: Impact Hammer excitation with more response DOFs One shaker moved to different reference DOFs MIMO

Modal Analysis 28

Measurement of FRF Matrix (MIMO)

=

More columnsz Multiple Fixed Excitations (references)

z Single Roving Response MISOor

z Multiple (Roving) Responses MIMO

X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn

Modal Analysis 29

Modal Analysis (classic): FRF = Response/Excitation

Output

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

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

100m

10

[Hz]

[(m/s)/N]

Autospectrum(Excitation) - Input

Working : Input : Input : FFT Analyzer

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

100u

1m

10m

100m

1

[Hz]

[N] Autospectrum(Excitation) - Input

Working : Input : Input : FFT Analyzer

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

100u

1m

10m

100m

1

[Hz]

[N]

Autospectrum(Response) - Input

Working : Input : Input : FFT Analyzer

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

1m

10m

100m

1

10

[Hz]

[m/s] Autospectrum(Response) - Input

Working : Input : Input : FFT Analyzer

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

1m

10m

100m

1

10

[Hz]

[m/s]

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

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

-200

-100

0

100

200

[s]

[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer

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

-200

-100

0

100

200

[s]

[N]

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]

FFT

FFT

Impulse Response h1(Response,Excitation) - Input (Real Part)Working : Input : Input : FFT Analyzer

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)Working : Input : Input : FFT Analyzer

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

-2k

-1k

0

1k

2k

[s]

[(m/s)/N/s]

InverseFFT

Input

ExcitationResponse

ForceVibration

InputOutput

H() = = =

FrequencyResponseFunction

Frequency Domain

ImpulseResponseFunction

Time Domain

Operational Modal Analysis (OMA): Response only!

Natural Excitation

Modal Analysis 30

SDOF and MDOF Models

Different Modal Analysis Techniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis Post Processing

f(t)

x(t)

kc

m

= ++ + +

Modal Analysis 31

The Eternal Question in Modal

F1a

F2

Modal Analysis 32

Impact Excitation

H11() H12() H15()

Measuring one row of the FRF matrix by moving impact position

LAN

# 1# 2

# 3# 4

# 5Accelerometer

ForceTransducer Impact

Hammer

Modal Analysis 33

Impact Excitation

t

GAA(f)

z Magnitude and pulse duration depends on: Weight of hammer Hammer tip (steel, plastic or rubber) Dynamic characteristics of surface Velocity at impact

z Frequency bandwidth inversely proportional to the pulse durationa(t)

a(t)

ft

12

12

Modal Analysis 34

Weighting Functions for Impact Excitation

z How to select shift and length for transient and exponential windows:

z Leakage due to exponential time weighting on response signal is well defined and therefore correction of the measured damping value is often possible

Transient weighting of the input signal

Criteria

Exponential weighting of the output signal

Modal Analysis 35

Compensation for Exponential Weighting

With exponential weighting of the output signal, the measured time constant will be too short and the calculated decay constant and damping ratio therefore too large shift Length =

Record length, T

Original signal

Weighted signal

Window function

Time

1b(t)

W

Correction of decay constant and damping ratio :

Wm =

WW

1=

Wm0

W

0

m

0

=

==

Correctvalue

Measuredvalue

Modal Analysis 36

3 lb Hand Sledge

Building and bridges

12 lb Sledge

Large shafts and larger machine toolsCar framed and machine tools

1 lb hammer

ComponentsGeneral Purpose, 0.3 lb

Hard-drives, circuit boards, turbine blades

Mini Hammer

ApplicationDescription

Range of hammers

Modal Analysis 37

Impact hammer excitation

z Advantages: Speed No fixturing No variable mass loading Portable and highly suitable for

field work relatively inexpensive

z Disadvantages High crest factor means

possibility of driving structure into non-linear behavior

High peak force needed for large structures means possibility of local damage!

Highly deterministic signal means no linear approximation

z Conclusion Best suited for field work Useful for determining shaker and support locations

Modal Analysis 38

Shaker Excitation

H11() H21() H51()

Measuring one column of the FRF matrix by moving response transducer

# 1# 2

# 3# 4

# 5Accelerometer

ForceTransducer

VibrationExciter PowerAmplifier

LAN

Modal Analysis 39

Attachment of Transducers and Shaker

Accelerometer mounting:z Studz Cementz Waxz (Magnet)

Force Transducer and Shaker:z Studz Stinger (Connection Rod)

Properties of StingerAxial Stiffness: HighBending Stiffness: Low

Accelerometer

ShakerFa

ForceTransducer

Advantages of Stinger: z No Moment Excitationz No Rotational Inertia Loadingz Protection of Shakerz Protection of Transducerz Helps positioning of Shaker

Modal Analysis 40

Connection of Exciter and Structure

ExciterSlenderstinger

Measured structure

Accelerometer

Force Transducer

Tip mass, mShaker/Hammer

mass, M

FsFm

Piezoelectricmaterial

Structure

MMmFF ms

+=

XMFm &&=X)Mm(Fs &&+=

Force and acceleration measurements unaffected by stinger compliance, but ...Minor mass correction required to determine actual excitation

Modal Analysis 41

Shaker Reaction Force

Reaction byexternal support

Reaction byexciter inertia

Example of animproper

arrangementStructure

Suspension

ExciterSupport

StructureSuspension

ExciterSuspension

Modal Analysis 42

Sine Excitationa(t)

Time

A

Crest factor 2RMSA ==

z For study of non-linearities, e.g. harmonic distortion

z For broadband excitation: Sine wave swept slowly through

the frequency range of interest Quasi-stationary condition

A(f1)

B(f1)

RMS

Modal Analysis 43

Swept Sine Excitation

Advantagesz Low Crest Factorz High Signal/Noise ratioz Input force well controlledz Study of non-linearities

possible

Disadvantagesz Very slowz No linear approximation of

non-linear system

Modal Analysis 44

Random Excitation

A(f1)Freq.

Time

a(t)

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

Random variation of amplitude and phase Averaging will give optimum linear

estimate in case of non-linearities

Freq.

SystemOutput

SystemInput

Modal Analysis 45

Random Excitationz Random signal:

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

a(t)

p(a)

z Can be band limited according to frequency range of interestGAA(f) GAA(f)Baseband Zoom

Time

z Signal not periodic in analysis time Leakage in spectral estimatesFrequency

rangeFrequency

range

Freq. Freq.

Modal Analysis 46

Random Excitation

Advantagesz Best linear approximation of systemz Zoomz Fair Crest Factorz Fair Signal/Noise ratio

Disadvantagesz Leakagez Averaging needed (slower)

Modal Analysis 47

Burst Randomz Characteristics of Burst Random signal :

Gives best linear approximation of nonlinear system Works with zoom

a(t)

Time

Advantagesz Best linear approximation of systemz No leakage (if rectangular time weighting can be used)z Relatively fast

Disadvantagesz Signal/noise and crest factor not optimumz Special time weighting might be required

Modal Analysis 48

Pseudo Random Excitation

a(t)

z Pseudo random signal: Block of a random signal repeated every T

Time

A(f1)

B(f1)

T T T T

z 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

Modal Analysis 49

Pseudo Random Excitation

T T T T

a(t)

Timep(a)

Freq. range Freq. range

GAA(f) GAA(f)Baseband Zoom

z Can be band limited according to frequency range of interest

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

z Pseudo random signal: Characterized by power/RMS (GAA) and amplitude probability density (p(a))

Freq. Freq.

Modal Analysis 50

Pseudo Random Excitation

Advantagesz No leakagez Fastz Zoomz Fair crest factorz Fair Signal/Noise ratio

Disadvantagesz No linear approximation of

non-linear system

Modal Analysis 51

Multisine (Chirp)For sine sweep repeated every time record, Tr

TrA special type of pseudo random signal where the crest factor has been minimized (< 2)

It has the advantages and disadvantages of the normal pseudo random signal but with a lower crest factor

Additional Advantages: Ideal shape of spectrum: The spectrum is a flat

magnitude spectrum, and the phase spectrum is smoothApplications:

Measurement of structures with non-linear behaviour

Time

Modal Analysis 52

Pseudo-random signal changing with time:

Analysed time data:(steady-state response)

transient responsesteady-state response

A B C

A A A B B B C CC

T T T

Periodic RandomA combined random and pseudo-random signal giving an excitation signal featuring:

No leakage in analysis Best linear approximation of system

Disadvantage: z The test time is longer than the test time

using pseudo-random or random signal

Modal Analysis 53

Periodic Pulse

z Leakage can be avoided using rectangular time weightingz Transient and exponential time weighting can be used to increase

Signal/Noise ratioz Gating of reflections with transient time weightingz Effects of non-linearities are not averaged outz 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

Modal Analysis 54

Periodic Pulse

Advantages

z Fastz No leakage

(Only with rectangular weighting)

z Gating of reflections(Transient time weighting)

z Excitation spectrum follows frequency span in baseband

z Easy to implement

Disadvantages

z No linear approximation of non-linear system

z High Crest Factorz High peak level might excite

non-linearitiesz No Zoomz Special time weighting might

be required to increase Signal/Noise Ratio . This can also introduce leakage

Modal Analysis 55

Guidelines for Choice of Excitation Technique

Swept sine excitationz For study of non-linearities:

Random excitationz For slightly non-linear system:

Pseudo random excitationz For perfectly linear system:

Impact excitationz For field measurements:

Random impact excitationz For high resolution field measurements:

Modal Analysis 56

SDOF and MDOF Models

Different Modal Analysis Techniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis Post Processing

f(t)

x(t)

kc

m

= ++ + +

Modal Analysis 57

Garbage In = Garbage Out!A state-of-the Art Assessment of Mobility Measurement Techniques

Result for the Mid Range Structure (30 - 3000 Hz) D.J. Ewins and J. Griffin

Feb. 1981

Transfer MobilityCentral Decade

Transfer MobilityExpanded

Frequency Frequency

Modal Analysis 58

Plan Your Test Before Hand!

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

Modal Analysis 59

Where Should Excitation Be Applied?

Driving Point Measurement

Transfer Measurement

i = j

i j

i

j

j

i

"j"Excitation"i"sponseRe

FXH

j

iij ==

12

2121111 FHFHX +=

=

2

1

2221

1211

2

1

FF

HHHH

XX

}{ [ ]{ }FHX =

2221212 FHFHX +=

Modal Analysis 60

Check of Driving Point Measurement

z All peaks in

)f(F)f(XImand

)f(F)f(XRe,

)f(F)f(XIm

&&&

z An anti-resonance in Mag [Hij]must be found between every pair of resonances

z Phase fluctuations must be within 180

Im [Hij]

Mag [Hij]

Phase [Hij]

Modal Analysis 61

Driving Point (DP) MeasurementThe 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 modes

must be estimated accurately from a single measurement

DP- Problems: Highest modal coupling, as all

modes are in phase Highest residual effect from

rigid body modes

m Aij

Re Aij

Log MagAij

=

Without rigidbody modes

Measured

Modal Analysis 62

Tests for Validity of Data: Coherence

Coherence)f(G)f(G

)f(G)f(

XXFF

2FX2 =

Measures how much energy put in to the system caused the response

The closer to 1 the more coherent Less than 0.75 is bordering on poor coherence

Modal Analysis 63

Reasons for Low Coherence

Difficult measurements:z Noise in measured output signalz Noise in measured input signalz Other inputs not correlated with measured input signal

Bad measurements:z Leakagez Time varying systemsz Non-linearities of systemz DOF-jitterz Propagation time not compensated for

Modal Analysis 64

Tests for Validity of Data: Linearity

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 be

the same regardless of input force

H()

Modal Analysis 65

Tips and Tricks for Best Resultsz Verify measurement chain integrity prior to test:

Transducer calibration Mass Ratio calibration

z 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

z 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 of experimental modal analysis!

Modal Analysis 66

SDOF and MDOF Models

Different Modal Analysis Techniques

Exciting a Structure

Measuring Data Correctly

Modal Analysis Post Processing

f(t)

x(t)

kc

m

= ++ + +

Modal Analysis 67

From Testing to Analysis

MeasuredFRF

Curve Fitting(Pattern Recognition)

Modal Analysis

)f(H

Frequency

)f(H

Frequency

Modal Analysis 68

From Testing to Analysis

Modal Analysis

)f(H

Frequency

f(t)x(t)

kc

m

f(t)x(t)

kc

m

f(t)x(t)

kc

mSDOF Models

Modal Analysis 69

Mode Characterizations

All Modes Can Be Characterized By:1. Resonant Frequency

2. Modal Damping

3. Mode Shape

Modal Analysis 70

Modal Analysis Step by Step Process1. Visually Inspect Data

Look for obvious modes in FRF Inspect ALL 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 clues

2. 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 71

Modal Analysis Inspect Data1. Visually Inspect Data

Look for obvious modes in FRF Inspect ALL 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 clues

2. 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 72

Modal Analysis Curve Fitting1. Visually Inspect Data

Look for obvious modes in FRF Inspect ALL 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 clues

2. 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 73

How Does Curve Fitting Work?z Curve Fitting is the process of estimating the Modal

Parameters from the measurements

|H|

2

R/

d0

-180Phase

Frequency

z Find the resonant frequency

Frequency where small excitation causes a large response

z Find the damping What is the Q of the

peak?

z Find the residue Essentially the area

under the curve

Modal Analysis 74

*

*

r

ijr

r r

ijr

rijrij pj

Rpj

RH)(H +==

Residues are Directly Related to Mode Shapes!

z Residues express the strength of a mode for each measured FRF

z Therefore they are related to mode shape at each measured point!

Frequency

Distan

ceAmplitude First

Mode

Beam

SecondMode

ForceForceForceForceForceForceForceForceForceForceForceForce

Acceleration

ThirdMode

Modal Analysis 75

SDOF vs. MDOF Curve Fitters

z Use SDOF methods on LIGHTLY COUPLED modes

z Use MDOF methods on HEAVILY COUPLED modes

z You can combine SDOF and MDOF techniques!

SDOF(light coupling)

|H|

MDOF(heavy coupling)

Modal Analysis 76

Local vs. Global Curve Fitting

z Local means that resonances, damping, and residues are calculated for each FRF firstthen combined for curve fitting

z Global means that resonances, damping, and residues are calculated across all FRFs

|H|

Modal Analysis 77

Modal Analysis Analyse Results1. Visually Inspect Data

Look for obvious modes in FRF Inspect ALL FRFssometimes modes will show up in one

3. Analysis Use more than one curve fitter to see if they agree Pay attention to Residue calculations Do mode shapes make sense?

Modal Analysis 78

Which Curve Fitter Should Be Used?Frequency Response Function

Hij ()

Real Imaginary

Nyquist Log Magnitude

Modal Analysis 79

Which Curve Fitter Should Be Used?

Real Imaginary

Nyquist Log Magnitude

Frequency Response FunctionHij ()

Modal Analysis 80

Modal Analysis and Beyond

Hardware ModificationResonance Specification

Simulate RealWorld Response

Dynamic Modelbased on

Modal Parameters

ExperimentalModal Analysis

StructuralModification

ResponseSimulation

F

Modal Analysis 81

Conclusionz All Physical Structures can be characterized by the simple

SDOF model

z Frequency Response Functions are the best way to measure resonances

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

z Planning and proper setup before you test can save time and effortand ensure accuracy while minimizing erroneous results

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

Modal Analysis 82

Literature for Further Readingz Structural Testing Part 1: Mechanical Mobility Measurements

Brel & Kjr Primer

z Structural Testing Part 2: Modal Analysis and SimulationBrel & Kjr Primer

z Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J. EwinResearch Studies Press Ltd.

z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 1 1984

z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 2 1984

Modal Analysis 83

Appendix: Damping Parameters

3 dB bandwidth =

22f dB3

Loss factor0

dB3

0

db3

ff

Q1

===

Damping ratio0

Bd3

0

dB3

2f2f

2 ===

Decay constant

Quality factordB3

0

dB3

0

ffQ

==

2f dB3dB30

===

= 2dB3,

Modal Analysis 84

Appendix: Damping Parameters

( )tsineR2)t(h dt = , where the Decay constant is given by e-tThe Envelope is given by magnitude of analytic h(t):

The time constant, , is determined by the time it takes for the amplitude to decay a factor of e = 2,72or 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 f21

== 0f12

== 0f1Q

h(t)

Time

te

ExperimentalModal AnalysisSimplest Form of Vibrating SystemMass and SpringMass, Spring and DamperBasic SDOF Model SDOF Models Time and Frequency DomainModal MatrixMDOF ModelWhy Bother with Modal Models?Definition of Frequency Response FunctionBenefits of Frequency Response FunctionDifferent Forms of an FRFAlternative EstimatorsWhich FRF Estimator Should You Use?Three Types of Modal AnalysisDifferent Types of Modal Analysis (Pros)Different Types of Modal Analysis (Cons)Hammer Test on Free-free BeamMeasurement of FRF Matrix (SISO)Measurement of FRF Matrix (SIMO)Shaker Test on Free-free BeamMeasurement of FRF Matrix (Shaker SIMO)Why Multiple-Input and Multiple-Output ?Situations needing MIMOMeasurement of FRF Matrix (MIMO)The Eternal Question in ModalImpact ExcitationImpact ExcitationWeighting Functions for Impact ExcitationCompensation for Exponential WeightingRange of hammersImpact hammer excitationShaker ExcitationAttachment of Transducers and ShakerConnection of Exciter and StructureShaker Reaction ForceSine ExcitationSwept Sine ExcitationRandom ExcitationRandom ExcitationRandom ExcitationBurst RandomPseudo Random ExcitationPseudo Random ExcitationPseudo Random ExcitationMultisine (Chirp)Periodic RandomPeriodic PulsePeriodic PulseGuidelines for Choice of Excitation TechniqueGarbage In = Garbage Out!Plan Your Test Before Hand!Where Should Excitation Be Applied?Check of Driving Point MeasurementDriving Point (DP) MeasurementTests for Validity of Data: CoherenceReasons for Low CoherenceTests for Validity of Data: LinearityTips and Tricks for Best ResultsFrom Testing to AnalysisFrom Testing to AnalysisMode CharacterizationsModal Analysis Step by Step ProcessModal Analysis Inspect DataModal Analysis Curve FittingHow Does Curve Fitting Work?Residues are Directly Related to Mode Shapes!SDOF vs. MDOF Curve FittersLocal vs. Global Curve FittingModal Analysis Analyse ResultsWhich Curve Fitter Should Be Used?Which Curve Fitter Should Be Used?Modal Analysis and BeyondConclusionLiterature for Further ReadingAppendix: Damping ParametersAppendix: Damping Parameters