Fundamentals of Vibration and Modal Analysis

Fundamentals of Vibration and Modal AnalysisMeasurement functions Excitation techniques Testing practiceMeasurement functions – Excitation techniques – Testing practice

Sales New Hires Training 2008Swen Vandenberk

Lecture objectives

Know how measurements are

By completing this lecture, you will:

x(t)

f(t)

x(t)

f(t)

performed for Experimental Modal Analysis

Understand what an FRF and aground

m

ck

ground

m

ck

Understand what an FRF and a coherence is

Have a feeling for the practicalities of structural testing

Be able to talk about excitation techniquestec ques

2 copyright LMS International - 2008

Modal Analysis – Understanding the Dynamic Properties of Structures

Seat Vibration

Wheel & TireSteering Wheel

Shake

Engine

Noise at Driver’s &

RoadRearview mirror

vibrationTurbomachinery

Passenger’s Ears

Gearbox and TransmissionRotor

Cockpit vibration & noise

Cabin comfort

Environmental

System TransferSystem Transfer ReceiverReceiverX =

Accessories

Structural Integritysources

SourceSource


TransferTransfer

Systematic approach to noise & vibration testing The “source – transmitter - receiver” approach

Receiver!

Response:• noise• vibrations == =

XX X

TransmitterSystem characteristics:• structural• acoustic

Source

XX X

Operating loads:

acoustic

criticaldynamics

criticalloads

worst casescenario

p g• structural• acoustic


dynamicsloads scenario

Structural Dynamics ModellingSDOF (Single degree of freedom) system

System TransferSystem Transfer ReceiverReceiverX =SourceSource

100 Frequency Response Function

e

xf H

( ) 1( ) x ω

2

10-1

Log-

Mag

nitu

de damping controlled region

stiffness controlled region

mass controlled region

2( ) 1( )( )

xHf m cj kω

ω = =ω − ω + ω+

0 2 4 6 8 10 12 14 16 18 2010

-2

Frequency Hz

-50

0mx(t)

f(t)

-200

-150

-100Phas

e

ground

ck

The simplest dynamic system


0 2 4 6 8 10 12 14 16 18 20200

Frequency Hz

Structural Dynamics ModellingMDOF system

und

k1

f1(t)

undkn+1k2

f2(t) fn(t)

More complex dynamic system

HHFF XX

InputInput SystemSystem OutputOutput

grou m1

c1

k1m2 mn gr

ou2

c2 cn+1x1(t) x2(t) xn(t)

HHFF XX

Eigenfrequency = Peak in FRF

abstract

0.10

As many peaks as masses

g q yDamping ratio = Width of FRF peak / decay in IRFMode shape = ± Deformation at eigenfrequency

Modal parameters =

( )pqH jω

10.0e-6

Log

( g/N

)

180 00

0.91

lN)

eigenfrequency

( )pqH j∠ ω-180.00

180.00

Phas

e°

0.00 6.00 s-1.07

Rea

l( g

/N


0.00 80.00 Hz

Freq. domain: Frequency Response Function (FRF) Time Domain: Impulse Response Function (IRF)

Mode Shapes

Mode 1

2Mode 21


Analytical Modal Analysis: only for simple cases

OK for

Distributed parameters Lumped parametersww

xx

024

4

=ω− wEIm

dxwd

But what about real-lifestructures?

We have to look for other approaches• Virtual prototype: Finite Element Modal Analysis• Physical prototype: Experimental Modal Analysis

See CAE technology lecture

See here!


• Physical prototype: Experimental Modal Analysis

Engineering for improved Noise & Vibration performance Experimental Modal Analysis

Excitation techniquesDSPDSPFrequency Response Functions (FRFs)Curve-fitting / (modal) parameter estimationest at oValidation

Modal Parameters:Modal Parameters:FrequencyFrequencyDampingDampingp gp gMode shapesMode shapes


Experimental Modal Analysis:Aircraft Test Setup Example

Responses

Ground Vibration Test

InputsInputs

F4

F3 Ground Vibration Test

(GVT) System

⎥⎥⎥⎤

⎢⎢⎢⎡

24232221

14131211

HHHHHHHHHHHH

pons

espo

nses

F2

F1 ⎥⎥

⎦⎢⎢

⎣ 44434241

34333231

HHHHHHHH

Res

pR

esp

Force Inputs

0 . 1 0

N)

1 row or column suffices to determine modal parametersReciprocity

0 . 0 0

Log

( (m/s

2 )/N

1 8 0 . 0 0

Phas

e

°

qppq HH =


0 . 0 0 8 0 . 0 0H z- 1 8 0 . 0 0

Experimental Modal Analysis

Required knowledge for a successful modal test

Test Setup Purpose of the test • Knowledge of expected modes of the

system• Expected results • Transducers and excitation devices

Make measurements • Knowledge of digital signal processing, parameters such as leakage, windows,parameters such as leakage, windows, time and frequency relationships, FFT, excitation techniques

Identify Parameters • Knowledge of modal theory • Knowledge of modal parameter estimation

techniquestechniquesVerify/document results • Knowledge of modal theory

• Synthesis. MAC


Experimental Modal Analysis vs.Finite Element Modal Analysis

Experimental Numerical

( )H ω , ,{ },i i i iQω ξ φ , ,M C K , ,{ },i i i iQω ξ φ

Requires prototypeVery fast (1-5 days)Very accurate for frequency

Requires FE modelMany days/weeksFast alternative evaluationVery accurate for frequency

More reliable for dampingLimited number of points

Fast alternative evaluationA lot of model uncertainties

(joints / damping / …)High number of points


Experimental Modal Analysis

5. Use modal parametersTroubleshooting

1. Measure FRFsTroubleshooting

• Check frequencies• Qualitative descriptions of

mode shapesSimulation and predictionDesign optimisationDiagnostics and health monitoringFinite Element model

2. Estimate poles

Finite Element model verification/improvementHybrid system model building

3. Estimate shapes

4. Validate


Experimental Modal Analysis Applications

Car body, fully equipped car, car interior cavitycavity, …Aircraft fuselage, full aircraft, interior cavity, …Components: engine block, suspension systems, brakes, antennasProcessing plants: piping systems, equipment mountingMechanical equipment: turbine blades, q p ,compressors, pumpsAudio & household: CD-drive, washing machine, loudspeakersInfrastructure: bridge off shore platformsInfrastructure: bridge, off-shore platforms


Digital Signal Processing for Structural TestingToC

Basic DSPFourier transform

See DSP lecture

Fourier transformQuantisationAliasingLeakageea age

Frequency Response Function(FRF) estimatorsCoherence functions

InputInput SystemSystem OutputOutput

HHFF XX


Frequency Response Function (FRF) Measurements –SISO

Frequency Response FunctionInputInput SystemSystem OutputOutput

( ) ( ) ( )X H Fω = ω ω

How to estimateIdeal world

HHFF XX( ) ( ) ( )X H Fω = ω ω

Real life: averaging required

“Naïve” averaging approaches

FXH =

∑ ==

N

i iXNH

1

1Random excitation: averaging of linear• Mechanical noise

• Non-linear behaviour• Electrical noise in the instrumentation

∑ =

=N

i iFN

H

1

1

∑N X1

averaging of linear spectra go to 0

∑ ==

N

ii

i

FX

NH

1

1May be 0 (very small) at some spectral lines

Use statistical noise modelling instead


Use statistical noise modelling instead

FRF measurements

Time signals Linear spectra Power andcross spectra

FRF andcoherence

DFT averaging calc.

f(t) F(ω) GFF(ω)

Inpu

t

FF

XF

GGH =1

GXF(ω)

H(ω)

ross

XFG 2

hX(ω) G (ω)

XF( )

coh(ω)

Cr

FFXX

XF

GGcoh =

x(t)

X(ω) GXX(ω)

Out

put


O

FRF estimators – graphical interpretation

X H1 X H2 X Hv

At single frequency ω: N measurements available

X 1 X 2 v

F F F

H1 estimateLeast squares

Output noise

H2 estimateLeast squares

Input noise

Hv estimateTotal least squares

Input noiseOutput noise

XF

N

N

i ii

GG

FF

FXNH ==∑

∑=

*

1*

1 1

1

FX

XX

N

N

i ii

GG

XF

XXNH ==∑

∑=

*

1*

2 1

1


FFi ii

GFFN ∑=1

FXi ii

GXFN ∑=1

Coherence

Cross spectrum inequalityNon coherent noise

8

6

7

(ms-2/N)

0 800100 200 300 400 500 600 700

01.2

1

Hz

linear amplitude

FRF8

6

7

(ms-2/N)

0 800100 200 300 400 500 600 700

01.2

1

Hz

linear amplitude

FRFNon-coherent noise

FFXXFX GGG ≤2

2

3

4

5

ar amplitude

0 800100 200 300 400 500 600 700

01.2

1

Hz

linear amplitude

2

3

4

5

ar amplitude

0 800100 200 300 400 500 600 700

01.2

1

Hz

linear amplitude

Coherence

FX

GGG 2

2 =γ0 800100 200 300 400 500 600 7000

1

Hz

linea1.2

0 800100 200 300 400 500 600 7000

1

Hz

linea1.2

Smaller than 1 when

FFXXGG

10 2 ≤γ≤

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz

linear amplitude (ms-2/N)

1amplitude

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz


1amplitude

Smaller than 1 when …Noise in the measurementsNonlinearitiesLeakage

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz


0

linear a

Coherence

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz


0

linear a

Coherence


g

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz


0 800100 200 300 400 500 600 7000

Hz

0 800100 200 300 400 500 600 700

08

12

34

56

7

Hz


0 800100 200 300 400 500 600 7000

Hz

FRF + CoherenceTypical Examples

Lo g

100

1

10

24

2040

1.05 10.0

g/N

0.001

0.01

0.1

1

0 0000.0004

0.002

0.004

0.020.04

0.20.4

Leakage

Hz

0 2047.5

1000

200 400 600 800 1200

1400

1600

1800

1e-05

0.0001

3e-05

0.00024

1

0.9

Ampl

itude

/ dB

( (m

/s2)

/N)

Ampl

itude

/

0 3

0.4

0.5

0.6

0.7

0.8

Hz

0 2047.5

1000

200 400 600 800 1200

1400

1600

1800

0

0.1

0.2

0.3

0.00 45.00 Hz

0.00 -70.08.38

F coherence DRV:1:+XB FRF DRV:1:+X / FOR:1:+XB FRF DRV:1:+X / FOR:2:+X


Coherence and FRF Variance

FRF Variance

Large coherence

2

222 1.

)1(2 γγ−

−=σ

NH

H

a ge co e e cecorresponds to a goodFRF estimationWhen the coherence is low take more averageslow, take more averages

90% confidence bounds on the estimated FRF magnitude and phase


p

Aircraft In-flight Testing “Noisy” FRFs

In-flight excitation, 2 wing-tip vanes, 2 min sine sweep9 accelerometersNoisy data (additional unmeasurable turbulence excitation)Noisy data (additional unmeasurable turbulence excitation)

1.00 1.00

Coherences FRFs

Log

(m/s

2)/N

)

Ampl

itude

/

0.00

( (180.00

0 00

F Coherence w ing:vvd:+Z/MultipleF Coherence back:vde:+Y/Multiple

-180 00

180.00

Phas

e°


Hz

0.00

Hz

180.00

Vehicle FRFs

Body-in-white Fully-trimmed vehicle

Lowly-damped structure, sharp peaks

Highly-damped structure, rounded peaks

0.10

)/N)

98.1e-3

)/N)

100e-6

Log

( (m

/s2 )

180 00

98.1e-6

Log

( (m

/s2

FRF moto:9:+Z/karo:25:+Z

180 00

0.00 80.00 Hz-180.00

180.00

Phas

e°

3.50 30.00 Hz-180.00

180.00

Phas

e°

FRF moto:9:+Z/karo:25:+Z


Demo_car Porsche

Radarsat Satellite

5 shaker excitation

1.00

Log

( (m

/s2)

/N)

100e-9180.00

10.00 64.00 Hz

-180.00

Phas

e°

Low contribution of “red” input to response


Low contribution of red input to response

Structural testing equipment

ExcitationShakers or hammerShakers or hammerForce cell

ResponseAccelerometerscce e o ete sLDV (laser)


Boundary Conditions

Fixed boundary conditionsDifficult to realise

• Flexibility of fixtures

Free-free suspensionIn practice: almost free-free

• Soft spring elastic cord• Flexibility of fixtures• Added damping• Environmental noise

• Soft spring, elastic cord• Soft cushion

Check if your Check if your suspension is soft suspension is soft

enough !enough !

Rigid bod modeRigid bod mode freq enc < 10 % of firstfreq enc < 10 % of first fle ible modefle ible mode


Rigid body modeRigid body mode frequency < 10 % of firstfrequency < 10 % of first flexible modeflexible mode

Boundary ConditionsPractical Examples

Fixed-free± Free-free (soft tires)


ATA Engineering, IMAC 05

GVT of Embraer 170Influence of Tire Pressure

Tires not soft enough


Embraer, IMAC 05

Boundary ConditionsPractical Examples

Pneumatic suspension

Courtesy Airbus France

Elastic cords Operational boundary conditionsElastic cords Operational boundary conditions


FRF measurements Impact Testing

AdvantagesLimited equipmentEasy and fast

TimeTime FrequencyFrequency

put

puty

Low costExcellent for troubleshooting

Disadvantages

Inp

Inp

nse

nse

DisadvantagesPoor Signal to Noise ratioPoor for non-linear structuresDouble impactsADC underload / overload

Res

pon

Res

pon

ADC underload / overload

Typically: fixed response accelerations -roving impact location

FRFFRF

roving impact location


Impact testingAbout Hammer Tips

Force spectrumSoft tipCoherence

FRF

Soft tip

Hard tip

Right tip


FRF measurements Shaker Testing

Fast & reliable

Best ratio quality/time tt

TimeTime FrequencyFrequency

Better energy distribution over structure

Excellent for trouble shooting &

modification simulation

Inpu

tIn

put

ee

Typically fixed excitation point, multiple

response points - measured in batches

Only way to characterize non-linearities

Res

pons

eR

espo

nse

FRFFRF


Shaker testingRequired instrumentation

An excitation device is attached to the structure using a rod (“stinger”)rod ( stinger )

Characteristics of the stinger to ensure that the only input is along the shaker excitation axis

• High axial stiffness• Low transverse and bending stiffness

Multiple shakers can be used Energy distribution over structure

• All responses are above the background noise• All responses are above the background noise• Exciting different parts of a real structure (e.g.

wing and tail plane of an aircraft)Exciting a 3D-structure in different directions (X,Y,Z)Multiple-reference measurements

• Mode multiplicity• Less risk to miss modes (“controllability”)


Shaker Excitation signals

Random

Burst Random

Stepped SineNormal mode excitation

ChirpSwept Sine


p

Random excitationAveraging

3 averages3.50

plitu

de/

2.70 litu

de/

10 averages0.00

Amp

0.00 1100.00 Hz

-180.00

180.00

Phas

e°

0.00

Ampl/

0.00 1100.00 Hz

-180.00 180.00

Phas°

2.20

e

20 averages

0.00 Am

plitu

de/

-180.00 180.00

hase°

2.10

40 averages

0.00 1100.00 Hz

Ph

0.00

Ampl

itude

/

180 00se35 copyright LMS International - 2008

0.00 1100.00 Hz

-180.00 180.00

Phas°

Random excitationWith Hanning window

Comparison of random with and without a Hanning window after 40 averages

CoherenceFRF2.80 1.00 1.00 1.00

FRF random with Hanning

FRF random without Hanning

Ampl

itude

/

Ampl

itude

Ampl

itude

/

Ampl

itude


0.00 1100.00 Hz

0.00 0.00

0.00 1100.00 Hz

0.00 0.00


FRF random without Hanning

The smearing of energy to neighboring spectral lines in the FRF’s is far less when a Hanning window is applied, which result in a far better approximation of the studied system and a vast improvement of the coherence


Shaker testing(MIMO) burst excitation

Advantages: applicable for lightly and heavily damped systemssyste sLeakage? Only if the responses do not die out within the observation period (“block”)Disadvantage wrt random: less energy in structure, less good Signal to Noise ratioSignal to Noise ratio


Shaker testingComparison random and burst random

Random with Hanning

B t R dBurst Random

FRF comparison

Conclusion:

Avoid Random on lightly


damped structures !

Shaker testing(MIMO) Sine Shaker Signals

Stepped sine Normal modes

Sinusoidal excitationCovering entire frequency rangeBuild FRF line by lineAll h k t i l f

Excite the structure at resonance frequency with “tuned” input force combination (several shakers) such that only one mode in resonance

pp

All shaker energy at single frequencyHigh qualityBest signal to characterize non-linear properties

resonanceOldest methodVery accurateFeel the mode directlyB t ti iBut ... time consuming

Traditionally preferred method by aircraft manufacturers


Shaker testingOverview Excitation Methods

Random Swept Sine Stepped Sine Normal Modes

t

AP

(F) [

N²]

AP

(F) [

N²]

AP

(F) [

N²]

AP

(F) [

N²]

t

ω [Hz] ω [Hz] ω [Hz] ω [Hz]ωresonance

f


Shaker testingOverview Processes

Random Swept Sine Stepped Sine Normal Modes

t

AP

(F) [

N²]

AP

(F) [

N²]

AP

(F) [

N²]

AP

(F) [

N²]

t

ω [Hz] ω [Hz] ω [Hz] ω [Hz]ωresonance

f

Phase Seperation or Frequency Response Function (FRF) based methods

Phase Resonance /Mode Appropriation

Measure Modal Parameter Experimental Modal Measure / IdentifyMeasure FRF

Modal Parameter Estimator

Experimental Modal Model [ ω, ξ, ψ ]

Measure / Identify Mode [ ω, ξ, ψ ]

-19.83

2)/N

)


-89.83

dB( (

m/s

2

FRF DRV :1:+X / FOR:1:+FRF DRV :2:+X / FOR:2:+

Data Verification

Excitation power spectraDriving point FRFsReciprocityReciprocityLinearityCoherences

Input 1 at right wing

Input 2 at the rear part of fuselage

100e+6

F AutoPow er FOR:1:+XF AutoPow er FOR:2:+X

Log

N2

10 0e 3


0.00 400.00 Hz

10.0e-3

Driving Point FRFs

Selection and verification of excitation locationsAll modes present in driving point FRF ? Bad quality driving Bad quality driving All modes present in driving point FRF ?Alternating resonances and anti-resonancesPhases between 0-180°

q y gq y gpoint FRF = Bad point FRF = Bad

quality modal model !quality modal model !

0.07

Log

( g/N

)

33.6e-6

FRF DRV:1:+X/FOR:1:+XFRF DRV:2:+X/FOR:2:+X

180.00

Phas

e°


0.00 100.00 Hz-180.00

Reciprocal FRFs

Alignment stinger – force cell – accelerometer

1.00

FRF DRV:1:+X / FOR:2:+XFRF DRV:2:+X / FOR:1:+X

Log

( (m

/s2)

/N)

100e-6

0.00 100.00 Hz

-180.00

180.00

Phas

e°


Linearity of FRFs

3 different excitation levels FHXFHXα=α

=

1.00 0.10

Log

N2

Log

( g/N

)

F AutoPow er FOR:1:+XF AutoPow er FOR:1:+XF AutoPow er FOR:1:+X 10.0e-6

FRF DRV:1:+X/FOR:1:+XFRF DRV:1:+X/FOR:1:+X (1)FRF DRV:1:+X/FOR:1:+X (2)

180 00

0.00 100.00 Hz

1.00e-6

0.00 100.00 Hz

-180.00

180.00

Phas

e°


Coherences

Coherence differs from 1 in case of:Non Linearity 1 00Non-LinearityLeakageUnmeasured sourcesOther noise

1.00

Ot e o se

Rea

l/

F Coherence DRV:1:+XF Coherence DRV:2:+XF Coherence ENG:1:+YF Coherence FUSL:5:+X

0.00 100.00 Hz

0.00


Course summary

Structural ExcitationBoundary conditions

testing in source –transfer –

Excitation techniques

transfer receiver model

DSP for Structural T ti FRF d

Introduction to Experimental

M d l Testing: FRFs and coherences

Modal Analysis


Thank you

Sales New Hires Training 2008Swen Vandenberk

Documents

Fundamentals of Vibration and Modal Analysis