Brüel and Kjaer

Modal Analysis 1

ExperimentalModal Analysis

Copyright© 2003Brüel & Kjær 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 sinωnt

m

k

T

Time

Displacement

Frequency1T

Period, Tn in [sec]

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

Displacement

kmωn= 2 π fn =

Modal Analysis 4

Mass and Spring

time

m1

m

Increasing mass reduces frequency

1n mm

k2+

=π=ω 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

++−==

ωωωωω 2

1)()()(

H(ω)F(ω) X(ω)

ω

|H(ω)|

1k

ω

∠ H(ω)0º

– 90º– 180º

ω0 = √ k/m

1ω2m

1ωc

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

f(t)x(t)

kc

m

Modal Analysis 8

Modal Matrix

Modal Model(Freq. Domain)

( )( )( )

( )

( ) ( ) ( )( )( )

( ) ( )

( )

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⋅⋅⋅

⋅⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅⋅⋅⋅⋅⋅⋅

⋅⋅

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⋅⋅

ω

ωω

ωω

ωωω

ω

ωωω

3

1

23

22

12111

3

2

1

F

HH

HH

HHH

X

XXX

nnn

n

n

X1

⋅⋅

X2

X3X4

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

Γ1

1

201ω12σ

Γ2

q21

202ω22σ

q31

203ω32σ

Γ3

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)

Frequency Response Functions are properties of linear dynamic systems

They are independent of the Excitation Function

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

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

)()()( fF

fXfH =

)()()(1 fG

fGfHFF

FX=

)()()(2 fG

fGfHXF

XX=

213 )( HHGG

GGfH

FX

FX

FF

XX ⋅=⋅=

2

1

22 *)( H

HGG

GG

GGG

fXX

FX

FF

FX

XXFF

FX =⋅=⋅=γ

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 H3

Input noise - Best -

Output noise Best - -

Input + output noise - - Best

Peaks (leakage) - Best -

Valleys (leakage) Best - -

Accuracy

)()()(1 fG

fGfHFF

FX= )()()(2 fG

fGfHXF

XX=FX

FX

FF

XX

GG

GGfH ⋅=)(3

Modal Analysis 16






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)

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

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

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/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer

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

-80

-40

0

40

80

[s]

[m/s²]

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/s²] 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/s²]

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/s²)/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/s²)/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/s²)/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/s²)/N/s]InverseFFT⇒

ExcitationResponse

ForceMotion

InputOutputH ===

Modal Analysis 21

Hammer Test on Free-free Beam

Frequency

DistanceAmplitude FirstMode

Beam

SecondMode Third

Mode

ForceForceForceForceForceForceForceForceForceForceForce

Roving hammer method:Response measured at one point Excitation of the structure at a number of pointsby hammer with force transducer

Frequency Domain View

Modes of structure identified

FRF’s between excitation points and measurement point calculated

Press anywhereto advance animation

ForceModal

Domain

View

Acceleration

Modal Analysis 22

Measurement of FRF Matrix (SISO)

One rowOne Roving ExcitationOne Fixed Response (reference)

SISO

=

X1 H11 H12 H13 ...H1n F1

X2 H21 H22 H23 ...H2n F2

X3 H31 H32 H33...H 3n F3

: : :Xn Hn1 Hn2 Hn3...Hnn Fn

Modal Analysis 23

Measurement of FRF Matrix (SIMO)

More rowsOne Roving ExcitationMultiple Fixed Responses (references)

SIMO

=

X1 H11 H12 H13 ...H1n F1

X2 H21 H22 H23 ...H2n F2

X3 H31 H32 H33...H 3n F3


Modal Analysis 24

FRF’s between excitation point and measurement points calculated

Whitenoiseexcitation

Shaker Test on Free-free Beam

Frequency

DistanceAmplitude FirstMode Second

ModeThirdMode

BeamAcceleration

Press anywhereto advance animation

Modal

Domain

ViewFrequency Domain View ∝ Force

Shaker method:Excitation of the structure at one pointby shaker with force transducerResponse measured at a number of points

Modes of structure identified

Modal Analysis 25

Measurement of FRF Matrix (Shaker SIMO)

One columnSingle Fixed Excitation (reference)

Single Roving Response SISOor

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

X1 H11 H12 H13 ...H1n F1

X2 H21 H22 H23 ...H2n F2

X3 H31 H32 H33...H 3n F3


=

Modal Analysis 26

Why Multiple-Input and Multiple-Output ?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

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

i.e. MIMO

Modal Analysis 27

Situations needing MIMO

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 DOF’s– One shaker moved to different reference DOF’s– MIMO

Modal Analysis 28

Measurement of FRF Matrix (MIMO)

=

More columnsMultiple Fixed Excitations (references)

Single Roving Response MISOor

Multiple (Roving) Responses MIMO

X1 H11 H12 H13 ...H1n F1

X2 H21 H22 H23 ...H2n F2

X3 H31 H32 H33...H 3n F3


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/s²)/N]Working : Input : Input : FFT Analyzer

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

100m

10

[Hz]

[(m/s²)/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/s²] Autospectrum(Response) - Input


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

1m

10m

100m

1

10

[Hz]

[m/s²]

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/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer

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

-80

-40

0

40

80

[s]

[m/s²]

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/s²)/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/s²)/N/s]

InverseFFT

⇒Input

ExcitationResponse

ForceVibration

InputOutput

H(ω) = = =

FrequencyResponseFunction

Frequency Domain

ImpulseResponseFunction

Time Domain

Operational Modal Analysis (OMA): Response only!

Natural Excitation

Modal Analysis 30






f(t)

x(t)

kc

m

= ++ + ⋅ ⋅ ⋅ +

Modal Analysis 31

The Eternal Question in Modal…

F1

a

F2

Modal Analysis 32

Impact Excitation

H11(ω) H12(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ H15(ω)⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅

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)

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 durationa(t)

a(t)

ft

12

12

Modal Analysis 34

Weighting Functions for Impact Excitation

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

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)

Wτ

Correction of decay constant σ and damping ratio ζ:

Wm σ−σ=σ

WW

1τ

=σ

Wm0

W

0

m

0

ζ−ζ=ωσ

−ωσ

=ωσ

=ζ

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

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

field work– relatively inexpensive

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

Conclusion– Best suited for field work

– Useful for determining shaker and support locations

Modal Analysis 38

Shaker Excitation

H11(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅H21(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

⋅⋅⋅

H51(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

Measuring one column of the FRF matrix by moving response transducer

# 1# 2

# 3# 4

# 5Accelerometer

ForceTransducer

VibrationExciter Power

Amplifier

LAN

Modal Analysis 39

Attachment of Transducers and Shaker

Accelerometer mounting:StudCementWax(Magnet)

Force Transducer and Shaker:StudStinger (Connection Rod)

Properties of StingerAxial Stiffness: HighBending Stiffness: Low

Accelerometer

ShakerFa

ForceTransducer

Advantages of Stinger: No Moment ExcitationNo Rotational Inertia LoadingProtection of ShakerProtection of TransducerHelps positioning of Shaker

Modal Analysis 40

Connection of Exciter and Structure

ExciterSlenderstinger

Measured structure

Accelerometer

Force Transducer

Tip mass, mShaker/Hammer

mass, M

Fs

Fm

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 ==

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

Modal Analysis 43

Swept Sine Excitation

AdvantagesLow Crest FactorHigh Signal/Noise ratioInput force well controlledStudy of non-linearitiespossible

DisadvantagesVery slowNo 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 ExcitationRandom signal:

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

a(t)

p(a)

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

Time

Signal not periodic in analysis time ⇒ Leakage in spectral estimates

Frequencyrange

Frequencyrange

Freq. Freq.

Modal Analysis 46

Random Excitation

AdvantagesBest linear approximation of systemZoomFair Crest FactorFair Signal/Noise ratio

DisadvantagesLeakageAveraging needed (slower)

Modal Analysis 47

Burst RandomCharacteristics of Burst Random signal :

– Gives best linear approximation of nonlinear system– Works with zoom

a(t)

Time

AdvantagesBest linear approximation of systemNo leakage (if rectangular time weighting can be used)

Relatively fast

DisadvantagesSignal/noise and crest factor not optimumSpecial time weighting might be required

Modal Analysis 48

Pseudo Random Excitation

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

Modal Analysis 49


T T T T

a(t)

Timep(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.

Modal Analysis 50


AdvantagesNo leakageFastZoomFair crest factorFair Signal/Noise ratio

DisadvantagesNo linear approximation of non-linear system

Modal Analysis 51

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

Tr

A 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: The test time is longer than the test time using pseudo-random or random signal

Modal Analysis 53

Periodic Pulse

Leakage can be avoided using rectangular time weightingTransient and exponential time weighting can be used to increaseSignal/Noise ratioGating of reflections with transient time weightingEffects of non-linearities are not averaged outThe signal has a high crest factor

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

The line spectrum for a Rectangular pulse has a shaped envelope curve

sin xx

Δ t Δ T Time

1/Δ t Frequency

Special case of pseudo random signal

Modal Analysis 54

Periodic Pulse

Advantages

FastNo leakage(Only with rectangular weighting)

Gating of reflections(Transient time weighting)

Excitation spectrum follows frequency span in basebandEasy to implement

Disadvantages

No linear approximation of non-linear systemHigh Crest FactorHigh peak level might excite non-linearitiesNo ZoomSpecial 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 excitationFor study of non-linearities:

Random excitationFor slightly non-linear system:

Pseudo random excitationFor perfectly linear system:

Impact excitationFor field measurements:

Random impact excitationFor high resolution field measurements:

Modal Analysis 56






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

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 every pair of resonances

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

Ιm ⏐Aij⏐

Re ⏐Aij⏐

ω

ω

Log Mag⏐Aij⏐

=

ω

Without rigidbody modes

Measured

Modal Analysis 62

Tests for Validity of Data: Coherence

Coherence)f(G)f(G

)f(G)f(

XXFF

2FX2 =γ

– 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:Noise in measured output signalNoise in measured input signalOther inputs not correlated with measured input signal

Bad measurements:LeakageTime varying systemsNon-linearities of systemDOF-jitterPropagation time not compensated for

Modal Analysis 64

Tests for Validity of Data: Linearity

Linearity X1 = H·F1

X2 = H·F2

X1+X2 = H·(F1 + F2)

a·X1 = H·(a· F1)⇒

– More force going in to the system will equate to

F(ω) X(ω)

more response coming out– Since FRF is a ratio the magnitude should be

the same regardless of input force

H(ω)

Modal Analysis 65

Tips and Tricks for Best ResultsVerify 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 of experimental modal analysis!

Modal Analysis 66






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 FRFs…sometimes 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





Modal Analysis 72

Modal Analysis – Curve Fitting1. Visually Inspect Data





Modal Analysis 73

How Does Curve Fitting Work?Curve Fitting is the process of estimating the Modal Parameters from the measurements

|H|

ω

2σ

R/σ

ωd

0

-180Phase

Frequency

Find the resonant frequency

– Frequency where small excitation causes a large response

Find the damping – What is the Q of the

peak?

Find the residue– Essentially the ‘area

under the curve’

Modal Analysis 74

*

*

r

ijr

r r

ijr

rijrij pj

Rpj

RH)(H

−ω+

−ω==ω ∑∑

Residues are Directly Related to Mode Shapes!

Residues express the strength of a mode for each measured FRF

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

Frequency

DistanceAmplitude FirstMode

Beam

SecondMode

ForceForceForceForceForceForceForceForceForceForceForceForce

Acceleration

ThirdMode

Modal Analysis 75

SDOF vs. MDOF Curve Fitters

ω

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)

Modal Analysis 76

Local vs. Global Curve Fitting

Local means that resonances, damping, and residues are calculated for each FRF first…then combined for curve fitting

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

|H|

ω

Modal Analysis 77

Modal Analysis – Analyse Results1. Visually Inspect Data




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 (ω)

Real Imaginary

Nyquist Log Magnitude

Modal Analysis 79

Which Curve Fitter Should Be Used?

Real Imaginary

Nyquist Log Magnitude

Frequency Response FunctionHij (ω)

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

ConclusionAll Physical Structures can be characterized by the simple SDOF model

Frequency Response Functions are the best way to measure resonances

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

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

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

Modal Analysis 82

Literature for Further ReadingStructural Testing Part 1: Mechanical Mobility MeasurementsBrüel & Kjær Primer

Structural Testing Part 2: Modal Analysis and SimulationBrüel & Kjær Primer

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

Dual Channel FFT Analysis (Part 1)Brüel & Kjær Technical Review # 1 – 1984

Dual Channel FFT Analysis (Part 1)Brüel & Kjær Technical Review # 2 – 1984

Modal Analysis 83

Appendix: Damping Parameters

3 dB bandwidthπσ

=Δ22f dB3

Loss factor0

dB3

0

db3

ff

Q1

ωωΔ

=Δ

==η

Damping ratio0

Bd3

0

dB3

2f2f

2 ωωΔ

=Δ

=η

=ζ

Decay constant

Quality factordB3

0

dB3

0

ffQ

ωΔω

=Δ

=

2f dB3

dB30ωΔ

=Δπ=ωζ=σ

σ=ωΔ 2dB3,

Modal Analysis 84

Appendix: Damping Parameters

( )tsineR2)t(h dt ω⋅⋅⋅= σ− , where the Decay constant is given by e-σt

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

The time constant, τ, is determined 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

τ=σ

1

σ=τ

1

τπ=

ωσ

=ζ00 f2

1

τπ=ζ⋅=η

0f12

τπ=η

= 0f1Q

h(t)

Time

te σ−

Brüel and Kjaer

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