modalBK

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Modal Analysis 1

ExperimentalModal Analysis

Copyright© 2003Brüel & Kjær Sound & Vibration Measurement A/SAll Rights Reserved

8/8/2019 modalBK


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

= ++ + ⋅ ⋅ ⋅+

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Modal Analysis 3

Simplest Form of Vibrating System

D

d = D sin ωnt

m

k

T

Time

Displacement

Frequency1T

Period, T n in [sec]

Frequency, f n= in [Hz = 1 / sec ]1Tn

Displacement

km

ωn= 2 π fn =

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Mass and Spring

time

m1

m

Increasing massreduces frequency

1n mm

k2 +=π=ω n f

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Mass, Spring and Damper

Increasing dampingreduces the amplitude

time

m

k c1 + c 2

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Modal Analysis 6

)()()()( t f t Kxt xC t x M =++ &&&

M= mass (force/acc.)

C = damping (force/vel.)K = stiffness (force/disp.)

=)t(x&&

=)t(x&=)t(x

Acceleration Vector

Velocity VectorDisplacement Vector

=)t(f Applied force Vector

Basic SDOF Model

f(t)

x(t)

kc

m

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Modal Analysis 7

SDOF Models — Time and Frequency Domain

k c jmF X

H ++−==ω ω ω

ω ω 2

1)()(

)(

H(ω)F(ω) X(ω)

ω

|H(ω)|

1k

ω∠H(ω)

0º– 90º

– 180º

ω0 = √ k/m

1ω2m

1ωc

)()()()( t kxt xct xmt f ++= &&&

f(t)x(t)

kc

m

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Modal Analysis 8

Modal Matrix

Modal Model(Freq. Domain)

( )( )( )

( )

( ) ( ) ( )( )( )

( ) ( )

( )

⎪⎪

⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪⎪

⎨

⎧

⋅⋅

⋅

⋅⋅

⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢⎢

⎣

⎡

⋅⋅⋅

⋅⋅

⋅⋅

⋅⋅

⋅⋅

=

⎪⎪

⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪⎪

⎨

⎧

⋅

⋅

ω

ω ω

ω

ω

ω ω ω

ω

ω

ω

ω

3

1

23

22

12111

3

2

1

F

H H

H

H

H H H

X

X

X X

nnn

n

n

X1

⋅

⋅

X2X3

X4

F3H21

H22

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Modal Analysis 9

MDOF Model

0°

-90°

-180°

2 1

1+2

1 2

1+2

Frequency

FrequencyPhase

Magnitude

m

d1+ d 2

d1

dF

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Modal Analysis 10

Why Bother with Modal Models?

Physical Coordinates = C HAOS Modal Space = Simplicity

q1

Γ1

1

201ω12σ

Γ2

q21

202ω22σ

q31

203ω32σ

Γ3

Bearing

Rotor

Bearing

Foundation

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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∠

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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

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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

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Modal Analysis 14

Alternative Estimators

)()()(

f F f X f H =

)()()(1 f G

f G f H FF

FX =

)()()(2 f G

f G f H XF

XX =

213 )( H H G

G

G

G

f H FX

FX

FF

XX

⋅=⋅=

2

1

2

2 *)( H

H

G

G

G

G

GG

G f

XX

FX

FF

FX

XX FF

FX =⋅=⋅

=γ

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

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Modal Analysis 15

Which FRF Estimator Should You Use?

Definitions:

User can choose H 1, H 2 or H 3 after measurement

Accuracy for systems with: H 1 H2 H3

Input noise - Best -

Output noise Best - -

Input + output noise - - Best

Peaks (leakage) - Best -

Valleys (leakage) Best - -

Accuracy

)()()(1 f G

f G f H FF

FX =)()()(2 f G

f G f H XF

XX =FX

FX

FF

XX

GG

GG

f H ⋅=)(3

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Modal Analysis 16






f(t)

x(t)

kc

m

= ++ + ⋅ ⋅ ⋅+

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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

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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 analysisOperational Modal Analysis

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

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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

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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]Inverse

FFT

⇒

Excitation Response

Force Motion

Input Output

H ===

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Modal Analysis 21

Hammer Test on Free-free Beam

F r e q u e n c y

D i s t a n c e

Amplitude FirstMode

Beam

SecondMode Third

Mode

ForceForceForceForceForceForceForceForceForceForceForce

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

F r e q u e n c y D o m a i n V i e w

Modes of structure identifiedFRF’s between excitation points and measurement point calculated

Press anywhereto advance animation

Force

M o d a l

D o m a i n

V i e w

Acceleration

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Modal Analysis 22

Measurement of FRF Matrix (SISO)

One rowOne Roving ExcitationOne Fixed Response (reference)

SISO

=

X1 H11 H12 H13 ...H 1n F 1X 2 H 21 H 22 H 23 ...H 2n F 2

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

Xn Hn1 Hn2 Hn3 ...H nn F n

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Modal Analysis 23

Measurement of FRF Matrix (SIMO)

More rowsOne Roving ExcitationMultiple Fixed Responses (references)

SIMO

=

X1 H11 H12 H13 ...H 1n F 1X 2 H 21 H 22 H 23 ...H 2n F 2

X 3 H 31 H 32 H 33 ...H 3n F 3 : : :

Xn Hn1 Hn2 Hn3 ...H nn F n

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Modal Analysis 25

Measurement of FRF Matrix (Shaker SIMO)

One columnSingle Fixed Excitation (reference)

Single Roving Response SISO

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

X 1 H11 H 12 H13 ...H 1n F1

X 2 H21 H 22 H23 ...H 2n F 2

X 3 H31 H 32 H33 ...H 3n F3: : :

X n Hn1 H n2 Hn3 ...H nn Fn

=

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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 distributedand

– avoid non-linear behaviour

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

i.e. MIMO

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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 Hammer excitation with more response DOF’s – One shaker moved to different reference DOF’s – MIMO

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Modal Analysis 28

Measurement of FRF Matrix (MIMO)

=

More columnsMultiple Fixed Excitations (references)

Single Roving Response MISOor

Multiple (Roving) Responses MIMO

X 1 H11 H 12 H 13 ...H 1n F1

X 2 H21 H 22 H 23 ...H 2n F 2

X 3

H31

H 32

H 33

...H3n

F 3

: : :X n Hn1 H n2 H n3 ...H nn Fn

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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]


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) - Input


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

-80

-40

0

40

80

[s]

[m/s²] Time(Response) - Input


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)


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)


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

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Modal Analysis 30






f(t)

x(t)

kc

m

= ++ + ⋅ ⋅ ⋅+

Th E l Q i i M d l

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Modal Analysis 31

The Eternal Question in Modal…

F1

a

F2

I E i i

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Modal Analysis 32

Impact Excitation

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

Measuring one row of the FRF matrix bymoving impact position

LAN

# 1# 2

# 3# 4

# 5Accelerometer

ForceTransducer Impact

Hammer

I t E it ti

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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

W ighti g F ti f I t E it ti

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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

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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 largeshift Length =

Record length, T

Original signal

Weighted signal

Window function

Time

1

b(t)

Wτ

Correction of decay constant σ and damping ratio ζ:

Wm σ−σ=σ

W

W1

τ=σ

Wm0

W

0

m

0

ζ−ζ=ωσ−ω

σ=ωσ=ζ

Correctvalue

Measuredvalue

Range of hammers

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Modal Analysis 36

3 lb HandSledge

Building andbridges

12 lb Sledge

Large shaftsand largermachine toolsCar framedand machinetools

1 lb hammer

ComponentsGeneralPurpose, 0.3

lb Hard-drives,circuit boards,turbine blades

Mini Hammer

ApplicationDescription

Range of hammers

Impact hammer excitation

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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 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

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Modal Analysis 38

Shaker Excitation

H11 (ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

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

H51(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

Measuring one column of the FRF matrix bymoving response transducer

# 1# 2

# 3# 4

# 5Accelerometer

ForceTransducer

VibrationExciter

PowerAmplifier

LAN

Attachment of Transducers and Shaker

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Modal Analysis 39

Attachment of Transducers and Shaker

Accelerometer mounting:StudCementWax(Magnet)

Force Transducer and Shaker:StudStinger (Connection Rod)

Properties of StingerAxial Stiffness: High

Bending Stiffness: Low

Accelerometer

ShakerFaForce

Transducer

Advantages of Stinger:No Moment Excitation

No Rotational Inertia LoadingProtection of ShakerProtection of TransducerHelps positioning of Shaker

Connection of Exciter and Structure

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Modal Analysis 40

Connection of Exciter and Structure

ExciterSlenderstinger

Measured structure

Accelerometer

Force Transducer

Tip mass, m

Shaker/Hammermass, M

Fs

Fm

Piezoelectricmaterial

Structure

MMm

FF 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

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Modal Analysis 41

Shaker Reaction Force

Reaction byexternal support

Reaction byexciter inertia

Example of animproper

arrangementStructure

Suspension

ExciterSupport

Structure

Suspension

ExciterSuspension

Sine Excitation

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Modal Analysis 42

Sine Excitation

a(t)

Time

A

Crest factor 2RMS

A ==

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

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Modal Analysis 43

Swept Sine Excitation

AdvantagesLow Crest Factor

High Signal/Noise ratioInput force well controlledStudy of non-linearities

possibleDisadvantages

Very slowNo linear approximation ofnon-linear system

Random Excitation

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Modal Analysis 44

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

System

Input

Random Excitation

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Modal Analysis 45

Random signal: – Characterized by power spectral density (G AA) and

amplitude 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

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Modal Analysis 46

AdvantagesBest linear approximation of system

ZoomFair Crest FactorFair Signal/Noise ratio

DisadvantagesLeakageAveraging needed (slower)

Burst Random

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Modal Analysis 47

Characteristics 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

Pseudo Random Excitation

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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 G AA(f), N = 10

Freq.

SystemOutput

System Input


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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 (G AA) and amplitude probability density (p(a))

Freq. Freq.


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Modal Analysis 50

AdvantagesNo leakage

FastZoomFair crest factor

Fair Signal/Noise ratioDisadvantages

No linear approximation ofnon-linear system

Multisine (Chirp)

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Modal Analysis 51

For sine sweep repeated every time record, T r

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

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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

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Modal Analysis 53

Leakage can be avoided using rectangular time weightingTransient and exponential time weighting can be used to increase

Signal/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 definableΔ t repeated with a user definable interval, ΔT

The line spectrum for aRectangular pulse has ashaped envelope curve

sin xx

Δt ΔT Time

1/ Δ t Frequency

Special case of pseudo random signal

Periodic Pulse

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Modal Analysis 54

Advantages

FastNo leakage(Only with rectangular weighting)

Gating of reflections

(Transient time weighting)Excitation spectrum followsfrequency span in basebandEasy to implement

Disadvantages

No linear approximation ofnon-linear systemHigh Crest FactorHigh peak level might excitenon-linearitiesNo ZoomSpecial time weighting mightbe required to increase

Signal/Noise Ratio . This canalso introduce leakage

Guidelines for Choice of Excitation Technique

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Modal Analysis 55

Swept sine excitatio nFor study of non-linearities:

Random excitationFor slightly non-linear system:

Pseudo random exc itationFor perfectly linear system:

Impact excitationFor field measurements:

Random impact exc itationFor high resolution field measureme nts:

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Modal Analysis 56






f(t)

x(t)

kc

m

= ++ + ⋅ ⋅ ⋅+

Garbage In = Garbage Out!

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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!

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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?

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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

FF

HHHH

XX

}{ [ ]{ }FHX =

2221212 FHFHX ⋅+⋅=

Check of Driving Point Measurement[ ]

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Modal Analysis 60

All peaks in

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

)f(F)f(X

Imand)f(F)f(X

Re,)f(F)f(X

Im&&&

An anti-resonance in Mag [Hij]must be found between everypair of resonances

Phase fluctuations must bewithin 180 °

Im [H ij]

Mag [H ij]

Phase [H ij]

Driving Point (DP) Measurement

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Modal Analysis 61

The quality of the DP-measurement is very important, as theDP-residues are used for the scaling of the Modal Model

DP- Considerations: – Residues for all modes

must be estimatedaccurately from a singlemeasurement

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

Tests for Validity of Data: Coherence

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Modal Analysis 62

Coherence)f(G)f(G

)f(G)f(XXFF

2

FX2 =γ

– 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

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Modal Analysis 63

Difficult measurements:Noise in measured output signalNoise in measured input signal

Other inputs not correlated with measured input signal

Bad measurements:

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

Tests for Validity of Data: Linearity

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Modal Analysis 64

Linearity X1 = H ·F1

X2 = H ·F2

X1+X2 = H ·(F 1 + F 2)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)

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




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 Frequency2. Modal Damping

3. Mode Shape

Modal Analysis – Step by Step Process1. Visually Inspect Data

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Modal Analysis 70

y p

– 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 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 Data1. Visually Inspect Data

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Modal Analysis 71

y p






Modal Analysis – Curve Fitting1. Visually Inspect Data

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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|

ω

2σ

R/ σ

ωd

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

p j

Rp j

RH)(H

−ω+−ω==ω ∑∑ Residues express thestrength of a mode for each

measured FRF

Therefore they are relatedto mode shape at eachmeasured point!

F r e q u e n c y

D i s t a n

c eAmplitude First

Mode

Beam

SecondMode

ForceForceForceForceForceForceForceForceForceForceForceForce

Acceleration

ThirdMode

SDOF vs. MDOF Curve Fitters

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Modal Analysis 75

ω

Use SDOF methods on LIGHTLY COUPLED modesUse 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 first…then combined for curvefitting

Global means that resonances, damping, and residues arecalculated across all FRFs

|H|

ω

Modal Analysis – Analyse Results1. Visually Inspect Data

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Modal Analysis 77





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

Which Curve Fitter Should Be Used?


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Modal Analysis 78

q y pHij (ω)

Real Imaginary

Nyquist Log Magnitude

Which Curve Fitter Should Be Used?


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Modal Analysis 79

Real Imaginary

Nyquist Log Magnitude

q y pHij (ω)

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 effort…and 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

Brüel & Kjær Primer

Structural Testing Part 2: Modal Analysis and Simulation

Brüel & Kjær Primer

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

Research 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

Appendix: Damping Parameters

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Modal Analysis 83

3 dB bandwidth πσ=Δ

22f dB3

Loss factor 0

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

ff

Q ωΔω=Δ=

2f dB3

dB30ωΔ=Δπ=ωζ=σ

σ=ωΔ 2dB3,

Appendix: Damping Parameters

tσ− -σt

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Modal Analysis 84

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

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

The time constant, τ, isdetermined by the time ittakes for the amplitude todecay a factor of e = 2,72…or10 log (e 2) = 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=ζ

00 f21

τπ=ζ⋅=η

0f

12

τπ=η= 0f1

Q

h(t)

Time

t

eσ−

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