Modal Testing - Iran University of Science and Technology · IUST ,Modal Testing Lab ,Dr H Ahmadian Response Function Measurement Techniques Loaded Support The structure is connected

Modal Testing(Lecture 11)

Dr. Hamid AhmadianSchool of Mechanical Engineering

Iran University of Science and [email protected]

Dr H Ahmadian ,Modal Testing Lab ,IUSTResponse Function Measurement Techniques

Response Function Measurement Techniques

IntroductionTest PlanningBasic Measurement SystemStructure PreparationExcitation of the structure


IntroductionThe measurements techniques used for modal testing are discussed:

Response measurement only Force and response measurement

The 2nd type of measurement techniques is of our concern:

Single-point excitation( SISO/SIMO)Multi-point excitation (MIMO)


Test PlanningObjective of the test

Levels according to Dynamic Testing Agency:UpdatingOut of range

residuesUsabe for validationMode ShapesDamping

ratioNatural FreqLevel

0

Only in few points

1

2

3

4


Test Planning

Extensive test planning is required before full-scale measurement:

Method of excitationSignal processing and data analysisProper selection of pickup pointsExcitation locationSuspension method


Quality of measured dataSignal quality

Sufficient strength and clarity/noise freeSignal fidelity

No cross sensitivityMeasurement repeatabilityMeasurement reliabilityMeasurement data consistency, including reciprocity


Basic Measurement SystemAn excitation mechanismA transduction mechanismAn Analyzer


Basic Measurement SystemSource of excitation signal:

SinusoidalPeriodic (with specific freq. content)RandomTransient

Power AmplifierExciter

TransducersCondition AmplifiersAnalyzers


Structure Preparation

Free SupportsGrounded SupportLoaded SupportPerturbed Support


Free SupportsTheoretically the structure will possess 6 rigid body modes @ 0 Hz.In practice this is provided by a soft supportRigid body modes are less then 10% of strain modes

Suspending from nodal points for minimum interference The suspension adds significant damping to the lightly damped structures


Free SupportsSuspension wires, should be normal to the primary vibration direction The mass and inertia properties can be determined from the RBMs.


Free Supports


Free Supports


Grounded SupportThe structure is fixed to the ground at selected points.The base must be sufficiently rigid to provide necessary grounding.Usually is employed for large structures

Parts of power generation stationCivil engineering structures

Another application is simulating the operational condition

Turbine BladeStatic stiffness can be obtained from low frequency mobility measurements.


Loaded SupportThe structure is connected to a simple component with known mobility

A specific mass

The effect of added mass can be removed analyticallyMore modes are excited in a certain

frequency range compared to free suspensionThe modes of structure are quite different


Perturbed SupportThe data base for the structure can be extended by repetition of modal tests for different boundary conditions


Perturbed Support


Perturbed Support


Excitation of the structure

Various devices are available for exciting the structure:

ContactingMechanical (Out-of-balance rotating masses)Electromagnetic (Moving coil in magnetic field)Electrohydraulic

Non-ContactingMagnetic excitation


Electromagnetic ExcitersSupplied input to the shaker is converted to an alternating magnetic field acting on a moving coil.


Electromagnetic ExcitersThere is a small difference between the force generated by the shaker and the applied force to the structure

The force required to accelerate the shaker moving

The force required to excite the structure sharply reduces near the resonance point,

Much smaller than the generated force in the shaker and the inertia of the drive rodVulnerable to noise or distortion


Attachment to the structurePush rod or stingers:

Applying force in only one directionFlexible drive rod/stinger introduces its own resonance into the measurement.


Support of shakers


Support of shakers


Hammer or ImpactorExcitation


Other excitation methods

Step Relaxation/sudden releaseCharge/Explosive impactor….


Moving SupportCorresponds to grounded modelOnly responses are measuredWhen the mass properties are known, the modal properties can be calculated from measured data


Moving Support




Dr H Ahmadian ,Modal Testing Lab ,IUSTFRF Measurement Techniques

Digital Signal ProcessingIntroductionBasics of Discrete Fourier Transform (DFT)AliasingLeakageWindowingFilteringImproving Resolution


IntroductionThe measured force or accelerometer signals are in time domain.The signals are digitized by an A/D converterAnd recorded as a set of N discrete values evenly spaced in the period T


Basics of DFT

The spectral properties of the recorded signal can be obtained using Discrete Fourier Transform/Series (DFT/DFS):

The DFT assumes the signal x(t) is periodicIn the DFT there are just a discrete number of items of data in either form

There are just N values xk

The Fourier Series is described by just N values


Basics of DFT

*

0

0

0

1

0

)(1

)(

)sin()(2

)cos()(2

,2

)sin()cos(2

)(

)()(

nn

tiT

n

n

tin

T

nn

T

nn

n

nnnnn

XX

dtetxT

X

eXtx

or

dtttxT

b

dtttxT

a

Tn

tbtaatx

Ttxtx

n

n

=

=

=

=

=

=

++=

+=

−

−

∞

−∞=

∞

=

∫

∑

∫

∫

∑

ω

ω

ω

ω

πω

ωω

∗−

−

=

−

−

=

−

=

−

=

−

=

==

=

=

=

++=

∑

∑

∑

∑

∑

rrN

N

k

Ninkkn

N

n

Ninknk

N

kkn

N

kkn

N

nnnk

XXexN

X

eXx

orNnkx

Nb

Nnkx

Na

Nnkb

Nnkaax

,1

)2sin(2

)2cos(2

)2sin()2cos(2

1

0

/2

1

0

/2

1

0

1

0

21

1

0

π

π

π

π

ππ


Basics of DFT

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

↑

↑↑↑↑

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

−−

−

=

−∑

1

1

0

1

1

0

1

0

/21

NN

N

k

Ninkkn

x

xx

X

XX

exN

X

M

NKK

MKKM

OLN

L

M

π


Basics of DFT

The sampling frequency:

The range of frequency spectrum:

The resolution of frequency spectrum:

TN

tTN

tf

ss

ss

ππω 221==⇒==

TNff ss πωω ===⇒=

22 maxmax

TTf πω 2,1

=Δ=Δ

Nyquist Frequency


Basics of DFT

There are a number of features of DF analysis which if not properly treated, can give rise to erroneous results:

AliasingMis-interoperating a high frequency component as a low frequency one

LeakagePeriodicity of the signal


AliasingDigitizing a ‘low’frequency signal produces exactly the same set of discrete values as result from the same process applied to a higher frequency signal

2sωω <

ωω −s


Aliasing

2

)2sin(

)22sin(

))(2sin()2sin(

:

NpNpk

Npkk

NkpN

Nkp

Compare

<

−

−

−⇔

π

ππ

ππ


Aliasing


AliasingThe solution to the problem is to use an anti-aliasing filter

Subjecting the original signal to low pass with sharp filterFilters have a finite cut-off rate; it is necessary to reject the spectral range near Nayquist frequency

2)0.108( sωω −>


LeakageA direct consequence of taking a finite length of time history coupled with assumption of periodicityEnergy is leaked into a number of spectral lines close to the true frequency.


Leakage


Leakage

To avoid the leakage there are number of scenarios:

Increasing the record time TWindowing

Multiply the time record by a function that is zero at the ends of the time record and large in the middle, the FFT content is concentrated on the middle of the time record


WindowingWindowing involves the imposition of a prescribed profile on the time signal prior to performing the FT

T

elsewhere

Tttata

tatataatw

txtwtx

πω

ωωωωω

20

,0)4cos()3cos(

)2cos()cos()cos()(

)()()(

0

0403

0201010

=

⎪⎩

⎪⎨⎧

<<+−

++−=

×=′


Windowing


Windowing


Windowing

a4a3a2a1a0Function

----1Rectangular

---11Hanning

-0.0030.2441.2981Kaser-Bessel

0.0320.3881.2861.9331Flat top


Windowing


Windowing


Improving Resolution (Zoom)There arises limitations of inadequate frequency resolution

at the lower end of the frequency rangeFor lightly-damped systems

A common solution is to concentrate all spectral lines into a narrow band

Within fmin-fmax

Instead of 0-fmax


ZoomMethod 1:

Shifting the frequency origin of the spectrum

The modified signal is then analysed in the range of 0-(fmax-fmin)

[ ]ttAttAtx

tAtx

)sin()sin(2

)cos()sin()()sin()(

minmin

min

ωωωω

ωωω

++−⇒

×=′=


ZoomMethod 2:

A controlled aliasing effect

Applying a band pass filterBecause of the aliasing phenomenon, the frequency component between f1 and f2 will appear aliased between 0-(f2-f1)





Use of Different Excitation Signals

IntroductionStepped-Sine TestingSlow Sine Sweep TestingPeriodic ExcitationRandom ExcitationTransient Excitation


Introduction

There are three different classes of excitation signals used:

PeriodicTransientRandom


IntroductionPeriodic:

Stepped sineSlow sine sweepPeriodicPseudo-random


IntroductionTransient:

Burst sineBurst randomChirpImpulse


IntroductionRandom:

(true) randomWhite noise


Stepped-Sine Testing

Classical method of FRF measurementTo encompass a frequency range of interest, the command signal frequency is stepped from one frequency to another

The excitation/response(s) are measured (amplitudes and phase(s)) .It is necessary to ensure that the steady-state condition is attained before the measurement.


Stepped-Sine Testing


Stepped-Sine TestingThe extent of unwanted transient response depends on:

Proximity of excitation frequency to a natural frequency,The abruptness of the changeover from the previous command signal to the new one,The lightness of the damping of nearby modes.


Stepped-Sine TestingAn advantage of stepped-sine testing is the facility of taking measurement where and as they are required.

Largest Error

dB%

No. point between

HPP’s3301

1102

0.553

0.225

0.118


Slow Sine Sweep Testing

Involves the use of a sweep oscillatorProvides a sinusoidal signalIts frequency is varied slowly but continuously

If an excessive sweep rate is used then distortions of FRF plot are introduced


Slow Sine Sweep TestingOne way of checking the suitability of a sweep rate is to make the measurement twice:

Once sweeping upAnd the 2nd time sweeping down


Slow Sine Sweep TestingIt is possible to prescribe an optimum sweep rate for a given structure taking into account its damping levels


Slow Sine Sweep TestingRecommended sweep rate:


Slow Sine Sweep TestingISO prescribes maximum linear and log sweep rate through a resonance as:

min/)(310

min/)(216

2max

2max

OctavesSLog

HzSLinear

rr

rr

ωζ

ωζ

×<

×<


Periodic ExcitationA natural extension of the sine wave test methods:

To use a complex periodic input signal which contains all the frequencies of interest, The DFT of both input and output signals are computed and the ratio of these gives the FRFBoth signal have the same frequency contents


Periodic ExcitationTwo types of periodic signals are used:

A deterministic signal (square wave)Some frequency components are inevitably weak.

Pseudo-Random type of signalThe frequency components may be adjusted to suit a particular requirements-such as equal energy at each frequency,Its period is exactly equal to the sampling time resulting zero leakage .


Random Excitation

)()(

)()()(

)()(

)(

)()()()()()(

)()()(

2

12

2

12

ωωγ

ωωω

ωω

ω

ωωω

ωωω

ωωω

HH

SSH

SS

H

SHSSHS

SHS

xf

xx

ff

fx

xfxx

fffx

ffxx

=

=

=

=

=

=


Random Excitation

There may be noise on one of the two signals

Near resonance this is likely to influence the force signalAt anti-resonances it is the response signal which will suffer


Random ExcitationH2 might be a better indication near resonances while H1 is a better indication near anti-resonances:

)()()()(,

)()()(

)( 21 ωωωω

ωωω

ωxf

mmxx

nnff

fx

SSSH

SSS

H +=

+=

Auto-spectra of noise on the input signal

Auto-spectra of noise on the output signal


Random ExcitationA closer optimum formula for the FRF is defined as the geometric mean of the two standard estimates

Phase is identical to that in the two basic estimates

)()()( 21 ωωω HHHv =


Random ExcitationTypical measurement made using random excitation:


Random ExcitationDetails from previous plot around a resonance:

H1

H2


Random ExcitationUse of zoom spectrum analysis:

Improving the resolution removes the major source of low coherence


Random ExcitationEffect of averaging:


Transient ExcitationThe excitation and the response are contained within the single measurement


Transient ExcitationBurst excitation signals:

A short section of a continuous signal (sin, random, …) followed by a period of zero wave.


Transient ExcitationChirp excitation:

The spectrum can be strictly controlled to be such within frequency range of interest


Transient ExcitationImpulsive excitation by Hammer:

Different impulsive excitationsSignals and spectra for double hit case


Transient ExcitationImpulsive excitation by Shaker:





RESPONSE FUNCTION MEASUREMENT TECHNIQUES

3.9 Calibration3.10 Mass Cancellation3.11 Rotational FRF Measurement3.12 Measurement on Nonlinear Structures

Effects of Different ExcitationsLevel Control in FRF Measurement


Calibration

In all measurement systems it is necessary to calibrate the equipment.There should be two levels of calibration:

Absolute calibration of individual transducersThe overall sensitivity of instrumentation system


CalibrationThe overall system calibration

The scale factor should be checked against computed factor using manufacturers stated sensitivityShould be carried out before & after each test


Mass CancellationNear resonance the actual applied force becomes very small and is thus very prone to inaccuracy.Some applied mass is used to move additional transducer mass

measuredFX

requiredFX

xmff

MM

TT

MT

→=

→=−= ∗

&&

&&

&&

α

α


Mass CancellationAdded mass to be cancelled and the typical analogue circuitAt deriving point a relation between measured and required FRF’s can be obtained


Mass Cancellation

)/1Im()/1Im()/1Re()/1Re(

)Im()Im()Im()Re()Re()Re(

MT

MT

MT

MT

mor

XmFFXmFF

αααα

=−=

−=

−=

∗

∗

∗

&&

&&


Rotational FRF MeasurementMeasurement of rotational FRFs using two or more transducers:

L

xxx

BAo

BAo

θθθ +=

+=

2


Rotational FRF MeasurementApplication of moment excitation

MFMX

FX θθ ,,,


Measurement on Nonlinear Structures

Many structures, especially in vicinity of resonances, behave in a nonlinear way:

Natural frequency varies with positionand strength of excitationDistorted frequency responses (near resonances)Unstable or unrepeatable data


Measurement on Nonlinear StructuresExamples of different nonlinear system response for different excitation levels

Softening effectIncrease in damping


Effects of Different ExcitationsFRF measurement on nonlinear system:

Sinusoidal ExcitationCompatible with theory

Random ExcitationLinearized system

Transient Excitation


Effects of Different Excitations

Most types of nonlinearity are amplitude dependent:

A linearized behaviour is observed when the response level is kept constantThe obtained linear model is valid for that particular vibration level


Level Control in FRF MeasurementResponse level control,

Best linear representation (nonlinearities are displacement dependent)

Force level controlOr no level control


Level Control in FRF MeasurementInverse FRF plots for a SDOF

Real part is expected to be liner wrt frequency squaredImaginary part should be linear/constantAny deviation from the expected behaviour can be detected as nonlinearity in the system


Level Control in FRF MeasurementUse of Hilbert transform to detect non-linearity

The Hilbert transform express the relations between real and imaginary parts of the Fourier Transform


Notes: Hilbert Transform

The Hilbert transform express the relations between real and imaginary parts of the Fourier Transform

Fourier Transform is considered to map functions of time to functions of frequency and vice versaHilbert transform map functions of time or frequency to the same domain


Notes: Hilbert TransformFor causal functions:

⎩⎨⎧

<−>

=

⎩⎨⎧

<>

=

+=

0,2/)(0,2/)(

)(

0,2/)(0,2/)(

)(

),()()(

ttgttg

tg

ttgttg

tg

tgtgtg

odd

even

oddeven


Notes: Hilbert Transform

{ } { }{ } { }

{ }

.)(Re)(Im

,)(Im)(Re

:)(

,)()()()(Im,)()()()(Re

πωωω

πωωω

πω

ωω

iGG

iGiG

theormnconvolutioonbaseditsignSince

tsigntgtgGtsigntgtgG

evenodd

oddeven

−∗=

−∗=

−=ℑ

×ℑ=ℑ=×ℑ=ℑ=




Dr H Ahmadian ,Modal Testing Lab ,IUSTModal Parameter Extraction Methods

Modal Parameter ExtractionIntroductionPreliminary checks of FRF data

Visual checksAssessment of multiple-FRF data set using SVDMode indicator functions

SDOF modal analysis methodsPeak amplitude methodCircle fit methodInverse or line fit method


IntroductionSome of the many available procedures for fitting a model to the measured data are discussed:

Their various advantages and limitations are explained,No single method is best for all cases.

This phase of the modal test procedure is often called modal parameter extractionor modal analysis


Introduction

Types of modal analysis:Frequency domain (of FRFs)Time domain (of Impulse Response Function)

The analysis will be performed usingSDOF methods, andMODF methods.


Introduction

Another classification of methods relates to the number of FRFs used in the analysis:

Single-FRF methods, andMulti-FRF methods:

Global methods which deals with SIMO data sets and Polyreference which deals with MIMO data


IntroductionDifficulty due to damping:

In practice we are obliged to make certain assumption about the damping model,Significant errors can be incurred in the modal parameter estimates as a result of conflict between assumed and actual damping effects.Decision on the issue of real and complex modes.


Preliminary checks of FRF dataLow-frequency asymptotes,

Stiffness-like characteristics for grounded structuresMass-line asymptotes for free structures

High-frequency asymptotes,Mass line or stiffness line

Incidence of antiresonancesFor a point FRF there must be a resonance after each antiresonance


Preliminary checks of FRF data

Mode Indicator Functions:The Peak-Picking Method

Sum of amplitudes of all measured FRFs to locate the resonance points

The frequency-domain decomposition method

Defined by the SVD of the FRF matrix


Case Study: MODES OF A RAILWAY VEHICLE


Case Study: Test set-up


Case Study: Sensor Locations


Case Study: Sensor Locations


Case Study: Excitation


Case Study: Excitation


Case Study: Measurements


The Peak-Picking MethodSum of amplitudes of all measured FRFs to locate the resonance points

987654321Mode#

24.6716.0014.0013.3312.338.335.334.672.67Frequency )Hz(


The frequency-domain decomposition method

A more advanced method consists of computing the Singular Value Decomposition of the spectrum matrix. The method is based on the fact that the transfer function or spectrum matrix evaluated at a certain frequency is only determined by neighboring modes.


The frequency-domain decomposition method [ ] { } { } { }[ ][ ] [ ][ ][ ][ ] [ ] [ ])()()(

)()()()(

)()()()( 2111

ωωω

ωωωω

ωωωω

ΣΣ=

Σ=

=

T

T

np

MIF

VUH

HHHH K


SDOF modal analysis methodsThe SDOF assumption

jkrrrr

jkrjk

N

rss sss

jks

rrr

jkrjk

N

s sss

jksjk

Bi

A

iA

iA

iA

++−

=

+−+

+−=

+−=

∑

∑

≠=

=

22

12222

122

)(

)(

)(

ωηωωωα

ωηωωωηωωωα

ωηωωωα


SDOF modal analysis methods

SDOF modal analysis methodsPeak amplitude methodCircle fit methodInverse or line fit method


SDOF modal analysis methods: Peak Amplitude

Individual resonance peaks are detected from the FRF

The frequency of the maximum responses is takes as the natural frequency of that mode,The peak amplitude and the half power points are determined,


SDOF modal analysis methods: Peak Amplitude

⎪⎪⎩

⎪⎪⎨

⎧

==

=−

=−

=⇒

⎪⎩

⎪⎨

⎧⇒

HAAHthen

Hknowns

rrrrr

r

rrr

ba

r

bar

ba

r

ˆ,ˆ

2,2

,,

ˆ

22

2

22

ωηωη

ηζωωω

ωωωη

ωω

ω


SDOF modal analysis methods: Peak AmplitudeAnother estimate for modal residue:

( )( )min(Re)max(Re)

min(Re)max(Re)ˆ

2 +=

+=

rrrA

H

ωη





Modal Parameter Extraction

Circle-fit methodProperties of the modal circleCircle-fit analysis procedureInterpretation of damping plots


Properties of the modal circleAssuming a system with structural damping the basic function to deal with is:

Since the effect of modal constant is to scale the size and rotate the circle, we consider:

))/(1()( 222

rrr

jkr

iA

ηωωωωα

+−=

))/(1(1)( 222

rrr iηωωωωα

+−=


Properties of the modal circleFinding the natural frequency:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=⇒−=⇒

−==−

−=

222222

2

2

)/(112

))2

tan(1(

)/(1)2

tan()90tan(,)/(1

tan

r

rrrrr

r

r

r

r

dd

ηωωωη

θωθηωω

ηωωθγ

ωωηγ


Properties of the modal circle

Dampingdd

frequencyNaturaldd

dd

ratesweepdd

rr

r

r

rrr

r

⇒−=⎟⎠⎞

⎜⎝⎛

⇒==⎟⎟⎠

⎞⎜⎜⎝

⎛

⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=

=22

2

2222

2

@.0

)(1)/(11

2

ωηωθ

ωωθω

ω

ηωωωη

θω

ωω


Properties of the modal circle

r

bar

ba

bar

bar

r

bar

bar

r

raa

r

rbb

when

for

ωωωη

θθ

θθω

ωωη

η

θθω

ωωη

ηωωθη

ωωθ

−=⇒

==

+

−=⇒

≤

+

−=⇒

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=

o

K

90

))2

tan()2

(tan(

)(2%3%2

))2

tan()2

(tan(1)/()2

tan(

)/(1)2

tan(

2

22

2

2


Properties of the modal circleThe final property relates to the diameter of the circle (D):

2rr

jkrjkr

AD

ωη=


Circle-fit analysis procedureSelect points to be usedFit circle, calculate quality of fitLocate natural frequency, …


Circle-fit analysis procedureObtain damping estimates

Calculate multiple damping estimate and scatter

Determine modal constant module and argument.


Interpretation of damping plots

Noise may contribute to the roughness of the surface.Systematic distortions due to:

LeakageErroneous estimates for natural frequencyNonlinearity


Circle-fit Minimizing the algebraic distance:

.0

1

1

11

:.0

)()(

22

2222

22

1121

21

22

222

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

++

=++++

=+++

CBA

yxyx

yxyxyxyx

SolutionSquaresLeastCByAxyxRbyax

nnnn

MMMM


Circle-fit Minimizing the geometric distance:

2

1

3

2

12

32

12

222

21

)(min

,

)()(

Ud

uuu

ULetuuu

Xd

Rbxax

n

ii

ii

∑=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=⎟

⎟⎠

⎞⎜⎜⎝

⎛−

⎭⎬⎫

⎩⎨⎧

−=

=+++


Circle-fit

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−+−

−

−+−

−

−−+−

−

−+−

−

=∂∂

Δ∂∂

+≈

+Δ∂∂

+=

1)()()()(

1)()()()(

minmin

222

211

222

222

11

11

2122

2111

1222

1222

111

111

0

00

00

mm

m

mm

m

xuxuxu

xuxuxu

xuxuxu

xuxuxu

pd

ppddd

ppddd

MMM

L


Circle-fit

Least-Squares Fitting of Circles and Ellipses By: Walter Gander, Gene H. Golub, and Rolf StrebelYou may find it in ftpmech.iust.ac.ir


Home work 1

Determine the modal properties of the beam tested in the lab

Frequency range of 0-400HzNatural frequenciesDamping (carpet plots)Mode Shapes

Due time 87/2/22


Importing the ASCII files


Importing the ASCII files





MDOF Modal Analysis in the Frequency Domain (SISO)

In some cases the SDOF approach to modal analysis is simply inadequate or inappropriate:

closely-coupled modes,the natural frequencies are very closely spaced, orwhich have relatively heavy damping,

those with extremely light damping


MDOF Modal Analysis in the Frequency Domain (SISO)One step MDOF curve fitting methods:

Non-linear Least Squares MethodRational Fraction Polynomial MethodA method particularly suited to very lightly damped structures

Global Modal Analysis in Frequency Domain

Global Rational Fraction Polynomial MethodGlobal SVD Method


Non-linear Least Squares Method

etcAAAqdqdEwE

HH

MKiA

HH

jkjkjk

p

lll

lmll

rlr

m

mr rrlr

jkrlljk

,,,,,.,0,

11)(

13211

2

2222

2

1

ωε

ε

ωωηωωω

L===

−=

+++−

==

∑

∑

=

=

The difference between measurement and analytical model


Non-linear Least Squares Method

The set of obtained equations are nonlinear

No direct solution (iterative procedures)Non-uniqueness of solutionHuge computational load


Rational Fraction Polynomial Method

NN

NN

N

r rrr

jkr

iaiaiaaibibibbH

iA

H

22

2210

1212

2210

122

)()()()()()()(

2)(

ωωωωωωω

ζωωωωω

++++++++

=

+−=

−−

=∑

L

L



( )( )m

kmkkk

mkmkkk

kmkmkk

mkmkk

k

iaiaiaaH

ibibibbeor

Hiaiaiaaibibibbe

22

2210

1212

2210

22

2210

1212

2210

)()()(

)()()(

)()()()()()(

ωωω

ωωω

ωωωωωω

++++−

++++=′

−++++

++++=

−−

−−

L

L

L

L

Order of model is selected



{ }

{ } mkkm

m

mkkkk

m

mkkkk

iHa

a

aaa

iiiH

b

bbb

iiie

22

12

2

1

0

122

12

2

1

0

122

)()()()(1

)()()(1

ωωωω

ωωω

−

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=′

−

−

−

−

M

L

M

L


Rational Fraction Polynomial MethodA set of linear equations using each individual measured FRF is formed.The unknowns ai and bi are obtained using a least square solution.The modal properties are extracted from obtained coefficients ai and bi.The analysis may repeat for a different model order.


Lightly Damped StructuresIn these structures it is easy to locate the natural frequencies,

Its accuracy is equal to the frequency resolution of the analyzer

The damping ratio is assumed to be zero.The modal constants are obtained using curve fittings.


Lightly Damped Structures

( ) ( )( ) ( )

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

Ω−Ω−Ω−Ω−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧ΩΩ

−=

−−

−−

=∑

M

M

MMM

MMM

L

L

M

Mjk

jk

N

r r

jkr

AA

HH

AH

2

112

222

122

21

121

22

121

21

2

1

122

)()(

)(

ωωωω

ωωω

The natural frequencies are known


Global Modal Analysis in Frequency Domain

So far each measured FRF is curve fitted individually,

Multi-estimates for global parameters (natural frequencies and damping)

Another way is to use measured FRF curves collectively.

Frequency and damping characteristics appear explicitly.


Global Rational Fraction Polynomial Method

If we take several FRF’s from the same structure then the denominator polynomial will be the same in every case.A natural extension of RFP method is to fit all n FRFs simultaneously

2m-1 values of ai and,and n(2m-1) values of bi


Global SVD Method

( ){ }

( )( )

( )[ ] ( )[ ] { } { } 11

11

2

1

)( ××−××

×

+−Φ=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

NkNkNNrNn

nnk

k

k

k

Rsi

H

HH

H

ωφω

ω

ωω

ωM


Global SVD Method

( ){ } [ ] ( )[ ] { } { }( ){ } [ ] [ ] ( )[ ] { } { }kNkNNrrNnk

kNkNNrNnk

RsisH

RsiH

+−Φ=

+−Φ=

×−××

×−××

11

11

φωω

φωω&

( ){ } ( )[ ] { }( ){ } [ ] ( ){ } { }( ){ } [ ] [ ] ( ){ } { }kkrNnk

kkNnk

NkNNrk

RgsH

RgHsig

+Φ=

+Φ=

−=

×

×

×−×

ωω

ωωφωω

&

11


Global SVD Method

( ){ } ( ){ } ( ){ }kcikiki HHH +−=Δ ωωω

( ){ } [ ] ( ){ }( ){ } [ ] [ ] ( ){ }ikrNnki

ikNnki

gsH

gH

ωω

ωω

ΔΦ=Δ

ΔΦ=Δ

×

×

&


Global SVD MethodConsider data from several different frequencies to obtain frequencies and damping:

( )[ ] [ ] ( )[ ]( )[ ] [ ] [ ] ( )[ ]

[ ] [ ]( ){ } [ ] [ ] Tr

Tkr

Tk

LNkrNnLnk

LNkNnLnk

zzHsH

gsH

gH

+

×××

×××

Φ==Δ−Δ

ΔΦ=Δ

ΔΦ=Δ

,0&

& ωω

ωω


Global SVD MethodThe eigen-problem is solved using the SVD.The rank of the FRF matrices and eigenvalues are obtained.Then the modal constants can be recovered from:

Modal Testing(Lecture 18-1)




MDOF Modal Analysis in the Time DomainThe basic concept: Any Impulse Response Function can be expressed by a series of Complex Exponentials

The Complex Exponential Series contain the eigenvalues and eigenvectors information.The IRF is obtained by taking inverse Fourier transform of the measured FRF.

( )22

1

1;)( rrrr

N

r

tsjkrjk iseAth r ζζω −+−==∑

=


Complex Exponential Method

∑

∑

∑

=

=

∗

∗

=

=⇒

−=

−+

−=⇒

N

r

tsjkrjk

N

r r

jkrjk

r

jkrN

r r

jkrjk

reAthIRF

siA

or

siA

siA

FRF

2

1

2

1

1

)(

)(:

)(

ωωα

ωωωα


Complex Exponential Method (Single FRF)

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

==⇒= ∑∑∑==

Δ

=

NqN

qq

N

N

q

N

r

lrr

N

r

tlsrl

N

r

tsr

A

AA

VVV

VVVVVV

h

hhh

VAeAheAth rr

2

2

1

221

22

22

21

221

2

1

0

2

1

2

1

2

1

111

)(

M

M

LL

MLLMM

KK

LL

LL

M



⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

NqN

qq

N

N

T

qq

T

q A

AA

VVV

VVVVVV

h

hhh

2

2

1

221

22

22

21

221

2

1

0

2

1

0

2

1

0 111

M

M

LL

MLLMM

KK

LL

LL

MMM

β

βββ

β

βββ

∑ ∑∑= ==

⎟⎟⎠

⎞⎜⎜⎝

⎛=

N

j

q

i

ijij

q

iii VAh

2

1 00ββ


Complex Exponential Method (Single FRF)The are selected to be coefficients of the polynomial:

N

N

iii

N

iii

N

i

ijiN

j

q

i

ijij

q

iii

qq

hh

h

VVAhNqSet

VVV

2

12

0

2

0

2

02

1 00

2210

.0

,02:

.0

−=

⎪⎪⎩

⎪⎪⎨

⎧

=

=⇒⎟⎟

⎠

⎞⎜⎜⎝

⎛=⇒=

=++++

∑

∑

∑∑ ∑∑

−

=

=

=

= ==

β

β

βββ

ββββ L

iβ



⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

−

+

−−+−

−

−

=∑

14

12

2

12

1

0

2412212

2321

12210

2

12

0

N

N

N

NNNNN

N

N

N

N

iii

h

hh

hhhh

hhhhhhhh

hh

M

M

M

M

L

MLMMM

MLMMM

L

L

β

ββ

β


Complex Exponential Method (Single FRF)The values and Ai are obtained from:

tsi

reV Δ=

.022

2210 =++++ N

NVVV ββββ L

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−−−− N

NN

NN

N

N

N A

AA

VVV

VVVVVV

h

hhh

2

2

1

122

122

121

22

22

21

221

12

2

1

0 111

M

M

LL

MLLMM

KK

LL

LL

M


Complex Exponential Method (Single FRF)Implementation Procedure:

Order of modal model is selected,Modal model is identified using the defined steps in previous slides,FRF is regenerated from modal information and compared with the measured FRFThe procedure repeated using another order for the modal model until stable results are obtained.


Stabilization Diagram


Global Analysis in Time Domain (Ibrahim Time Domain Method)

The basic concept is to obtain a unique set of modal parameters from a set of vibration measurements:

Scaled (mass normalized) mode shapes when the force is known,Un-scaled mode shapes when the force is not measured.


Ibrahim Time Domain Method

∑=

=m

r

tsiri

retx2

1)( ψ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

qmm

q

q

tsts

tsts

tsts

mnnn

m

m

qnnn

q

q

ee

eeee

txtxtx

txtxtxtxtxtx

212

212

111

2,21

2,22221

2,11211

21

22212

12111

)()()(

)()()()()()(

LL

MLLM

LL

LL

L

MLMM

L

L

L

MLMM

L

L

ψψψ

ψψψψψψ

[ ] [ ] [ ]Λ×Ψ=X


Ibrahim Time Domain MethodA 2nd set of eqns:

( ) ∑∑

∑

==

Δ

=

Δ+

==

=Δ+

m

r

tsir

m

r

tstsir

m

r

ttsirli

lrlrr

lr

eee

ettx

2

1

2

1

2

1

)(

ˆ

)(

ψψ

ψ

[ ] [ ] [ ]Λ×Ψ= ˆX



[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ]+×=

=×⇒⎭⎬⎫

Λ×Ψ=Λ×Ψ=

Ψ=Ψ×

XXA

XXAXX

A

ˆ

ˆˆˆ

ˆ



[ ]{ } { }rts

rreA ψψ Δ=

Eigenvectors of matrix [A] are the mode shapes,The natural frequencies and damping ratios are obtained from eigenvalues of [A].

Modal Testingoda est g(MDOF Modal Analysis in the Time Domain)

Dr. Hamid AhmadianDr. Hamid AhmadianSchool of Mechanical Engineering

Iran University of Science and Technologyy [email protected]

MDOF Modal Analysis in theMDOF Modal Analysis in the Time Domain The basic concept: Any Impulse Response

fFunction can be expressed by a series of Complex Exponentials

22

11;)( rrrr

N

r

tsjkrjk iseAth r

The Complex Exponential Series contain the eigenvalues and eigenvectors information

1r

eigenvalues and eigenvectors information. The IRF is obtained by taking inverse Fourier

transform of the measured FRFDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

transform of the measured FRF.

Complex Exponential h dMethod

(CE)(CE)

Dr H Ahmadian, Modal Testing Lab, IUST

Modal Parameter Extraction Methods

Complex Exponential Method

jkr

Njkr AA

FRF )(

r

jkr

r r

jkrjk sisi

FRF1

)(

N

jkrjk

Aor

2

)(:

N

r rjk si

or

2

1)(:

N

tsjkrjk

reAthIRF2

)(Dr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

r 1

Complex Exponential MethodComplex Exponential Method (Single FRF)

N

lN

tlsN

ts VAeAheAth rr

222

)(

r

rrr

rlr

r

Ah

VAeAheAth111

111

)(

N AA

VVVhh

2

1

2211

0 111

N

N

VVVh2

22

22

21

221

2

1

NqN

qq AVVVh 2221

Dr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

NNq AVVVh 2221


TT

Ah 111

N AA

VVVVVV

hhh

2

1

2222211

0

1

0

1

0 111

NVVVh 2

22

22

1222

NqN

qqqqq AVVVh 2221

N qq 2

N

j

q

i

ijij

q

iii VAh

2

1 00


j ii 1 00

Complex Exponential MethodComplex Exponential Method (Single FRF) The are selected to be coefficients of the i

polynomial:q

qVVV 2210 .0

NijiN q

iq

q

V2

2

210

,0

N

iii

i

j i

ijij

iii

hVAhNqSet 2

0

0

1 00 .02:

N

N

ii

i

hh 2

12

0


i 0


12N

hh

2

0N

iii hh

12

2

1

0

2321

12210

N

N

N

N

hh

hhhhhhhh

1212321 NN hhhhh

hhhhh


14122412212 NNNNNN hhhhh

Complex Exponential MethodComplex Exponential Method (Single FRF) The values and Ai are obtained

f

tsi

reV from:

.022

2210 N

NVVV

AA

VVVhh

2

1

2211

0 111

N

N AVVVVVV

hh 2

22

22

21

221

2

1

NN

NNN

N AVVVh 212

212

212

112


NNN

Complex Exponential MethodComplex Exponential Method (Single FRF) Implementation Procedure:

Order of modal model is selected, Modal model is identified using the defined g

steps in previous slides, FRF is regenerated from modal information g

and compared with the measured FRF The procedure repeated using another p p g

order for the modal model until stable results are obtained.


The Least Squares qComplex Exponential

h dMethod

(LSCE)(LSCE)



The Least Squares ComplexThe Least Squares Complex Exponential Method (LSCE)

The LSCE is the extension of CE to a global procedure.

It processes several IRF’s obtained It processes several IRF s obtained using SIMO method.

The coefficients β that provide the solution of characteristic polynomial are p yglobal quantities.


The Least Squares ComplexThe Least Squares Complex Exponential Method (LSCE)

NN hhhhh

2012210 One Typical IRF

qq

NN

hhorhhhhh

1212321

,

NNNNNN hhhhh

14122412212

hh

11

Extending to all measured IRFs

GT

GGT

GGG hhhhhhor

h

h

h

h

122 ,


qp hh

The PolyReference yComplex Exponential

h d ( C )Method (PRCE)



The PolyReference ComplexThe PolyReference Complex Exponential Method (PRCE)

Constitutes the extension of LSCE to MIMO.

A general and automatic way of A general and automatic way of analyzing dynamics of a structure.

MIMO test method overcomes the problem of not exciting some modes as p gusually happens in SIMO.



Considering q input reference points:

N

r

tsjrj

reAth2

111 )(

ll QA

N

r

tsjrj

reAth2

111 )(

N

r

tsjrj

reAth2

122 )( jlrklrjkr

lrjrrjlr

AWA

QA

N

r

tsjrrj

reAWth2

11212 )(

N

ts

r

reAth2

1

)(

lr

krklrW

N

ts

r

reAWth2

1

)(

r

jqrjq eAth1

)(

r

jrqrjq eAWth1

11)(


Modal Participation factor


2

111 )(

N

r

tsjrj eAth r

1

2

11212

1

)()(j

tj

N

r

tsjrrj

r

AeWtheAWth r

2

11)(N

tsjrqrjq eAWth r

111

1

2

1

0000111

)()( j

ts

tsj

r

AAe

WWWthth

1221221221122 00

)(

)( j

ts

tsNj

A

Ae

WWW

WWW

th

th


12121211

200)( jNts

qNqqjq AeWWWth N


jj AWh )0( 1 tjj

jj

VAVWth

AWh

)()0(

1

1

t

L

jj eV ,

jL

j AVWtLh )( 1

NqLVWVWVW LL 2,02

210



100 )0( jj AWh

111

100

)()(

jj

jj

AVWth

1222 )( jj AVWth

1)( jL

LjL AVWtLh

L L

jk

kjk AVWtkh 1)(


k k0 0


0

,0)( L

L

kjk Itkh

1

0

0

)()(L

kjjk

k

tLhtkh0k

tjjj

tNhththtNhthh

)()2()(

))1(()()0(

tjjjL

tNLhtLhtLh

tNhthth

))2(()())1((

)()2()(110

tjjj

tjjj

hhB

tNLhtLhtLh

tNLhtLhtLh

))1(())1(()(

))2(()())1((


jjT hhB


hhB jjT hhBConsidering for each response location j=1,…,p:

2121

ppT

hhBhhhhhhB

g p j , ,p

1

TT

TTT

hhhhB

hhB

TTT

TTTT hhhhB

Knowing the coefficient matrix [B], we must now determine [V]



L ,02

210 LL VWVWVW

L

k

kk VW

00

01

LL

0

0

0

0

01

0

1

L

k

ktsk

L

k

kk WeVW

L

0

0

10

02

0

2

L

k

ktsk

L

k

kk WeVW

0

0

r

L

k

krk WV

010

22

L

Nkts

k

Lk

k WeVW N

Dr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods0

00

kk


0

r

Lk

rk WV 0

rk

rk

LLL WVWVVV 1

12

210 rrL WVWVVV 1210

0 rz W

V W V

1 02

2 1

r r r

r r r

z V W V zz V W V z

0 0 1 1z z

V

11 2

LL r r r L

L

z V W V z

1 1 1L L r Lz V z


1LL r r r Lz V W V z


An standard eigenvalue problem to obtain Vr

1 11 2 1 0

L LL L

z zz z

g p r

2 20 0 0 L L

r

z zI

V

1 1

0 0

0 0 0z z

Iz z

0 0

The eigenvetors z0 correspond to Wr


The PolyReference ComplexThe PolyReference Complex Exponential Method (PRCE) LkAVWtkh j

kj ,,1,0,1

tkhtkhj1

tkhtkh j

j 2

1

0

j

j

VWW

thh

tkhjq

11 jVjj

L

j AWHorA

VWtLh

j VWtLh

HWA jVj HWA 1 The residue calculation is repeated for

all meas ed points j 1 2 pDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

all measured points, j=1,2,…,p.

The PolyReference ComplexThe PolyReference Complex Exponential Method (PRCE) The method provide more accurate modal

frepresentation of the structure. It can determine multiple roots or closely

spaced modes. Shortcomings:Shortcomings:

Sensitive to nonlinearities and any lack of reciprocity in frequency responsesreciprocity in frequency responses,

Some difficulties in analyzing structures with more than 5% viscous dampingwith more than 5% viscous damping.


Global Analysis in Time iDomain

(Ibrahim Time Domain(Ibrahim Time Domain Method)



Global Analysis in Time DomainGlobal Analysis in Time Domain (Ibrahim Time Domain Method)

The basic concept is to obtain a unique set of modal parameters from a set of vibration measurements:vibration measurements: Scaled (mass normalized) mode shapes

when the force is knownwhen the force is known, Un-scaled mode shapes when the force is

not measurednot measured.



m

tsrt2

)(

r

tsiri

retx1

)(

q

q

tsts

tsts

m

m

q

q

eeee

txtxtxtxtxtx

212

111

2,22221

2,11211

22212

12111

)()()()()()(

qmm tstsmnnnqnnn

q

eetxtxtx 2122,21

,

21 )()()(

XDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

Ibrahim Time Domain Method A 2nd set of eqns:

m

ttsirli

lrettx2

)()(

mts

mtsts

r

lrlrr eee22

1

r

irr

irlrlrr eee

11

ˆXDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods


A ˆ

XA

XXA

X ˆˆˆ

X

XXA ˆDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods


tsA rts

rreA

f [ ] h d Eigenvectors of matrix [A] are the mode shapes,

The natural frequencies and damping ratios are obtained from eigenvalues of [A].





Dr H Ahmadian ,Modal Testing Lab ,IUSTDerivation of Mathematical Models

Derivation of Mathematical Models

Spatial Models (mass, stiffness, damping)Needs measurement of most of the modesRequires measurement in many DOFs

Response Models (FRF) Needs measurement in frequency range of interestRequires measurement in selected DOFs

Modal Models (natural frequencies and mode shapes)

Needs measurement of only one modeRequires measurement in handful of DOFs


Derivation of Mathematical ModelsModal Models

Requirements to construct Modal ModelsRefinement of Modal Model

Conversion to real modesCompatibility of DOFs

ReductionExpansion

Response ModelsFRFTransmissibilityBase Excitation


Requirement to construct Modal Models

Minimum requirements One column in case of fixed excitation orOne row when response is measured at a fixed point.


Requirement to construct Modal ModelsProof:

ii

nimi

i

i

n

i

i

m

n

mmn

XF

FX

FX

FX

ααα

ωω

ωω

ωω

ωωα

=

××=

=

)()(

)()(

)()(

)()(


Requirement to construct Modal ModelsSeveral additional elements of FRF or even columns are measured to:

Replace poor data,To provide checksModes have not been missed


Refinement of Modal ModelsComplex to real conversion:

Taking the modulus of each element and assigning a phase of 0 or 180.Finding a real mode with maximum projection to the measured one:

Multi point excitation (Asher’s method)

CR

CTR

φφ

φφmax


Compatibility of DOFs

Employment of the measured modes in updating/modification of analytical models requires the compatibility of DOFs.There are two approaches in compatibility excursive:

Analytical model reductionExpansion of measured modes


Reduction of Analytical Model(Guyan Reduction)

{ } [ ]{ }

{ } { }112212

1211

12

11

121

222

1

1121

222

1

2

1

2212

1211

,

0

fxTKKKK

T

xTxx

xKK

Ixx

xKKx

fxx

KKKK

TT

T

T

T

=⎥⎦

⎤⎢⎣

⎡

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−

−


Dynamic Model Reduction

( ) ( )

( ) ( ) { } [ ]{ }

{ } .0

,

.0

12212

12112

2212

1211

12

11

122

121

222

222

1

1122

121

222

222

2

1

2212

12112

2212

1211

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−−−

=⎭⎬⎫

⎩⎨⎧

−−−=

=⎭⎬⎫

⎩⎨⎧⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

−

−

φω

φφφ

φωωφ

φφωωφ

φφ

ω

TMMMM

KKKK

T

TMKMK

IMKMK

MMMM

KKKK

TTT

TT

TT

TT


Expansion of Models

In order to compare analytical model with the measured modal data on may expand the measured data by:

Geometric interpolation using splinefunctionsUsing analytical model spatial modelUsing analytical model modal model


Expansion in Spatial Domain

.02

1

2212

12112

2212

1211 =⎭⎬⎫

⎩⎨⎧⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡φφ

ωMMMM

KKKK

TT

( ) ( ) 1122

121

222

222 φωωφ TT MKMK −−−=−


Expansion in Modal Domain

,~~.:~~min

:

0

0

0

TTT

T

R

T

VURVU

IRRstR

SolR

=⇒Σ=ΦΦ

=Φ−Φ

Φ=ΦAssuming Mass Matrix is correct


Response ModelFrequency response functions

Transmissibilities

[ ] [ ]( )[ ][ ]TrH Φ−Φ=−122 ωλ

( ) ( ) ( )( )ωω

ωω ω

ω

ki

jijkiti

k

tij

jk HH

TeXeX

T == ,


Transmissibility Plots


Response ModelThe amplitude’s peaks of the transmissibilitiesdo not correspond with the resonant frequencies. Transmissibilities cross each other at the resonant frequencies (becomes independent of the location of the input)

∑

∑

−

−==

r r

irkr

r r

irjr

ki

jijki H

HT

22

22

)()(

)(

ωωφφωωφφ

ωω

ω

kr

jrjki

rT

φφ

ωωω≈

≈)(


Base ExcitationAn application area of transmissibility.Input is measured as response at the drive point.

{ } { }

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

+=

1

11

Mrefrel xxx


Base Excitation

[ ]{ } [ ]{ } [ ]{ } { }

( )[ ] { } { }( ) [ ]{ }{ } { }( ) ( )[ ][ ]{ }.

,

.

1

11

,

2

21

gMHx

gxXor

gMxgxXH

ggMxxKxM

ref

ref

refref

refrelrel

ωω

ωω

=−

=−

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=−=+

−

M&&&&


Base Excitation

{ } { }( ) ( )[ ]( ) [ ]{ }( )

{ } { } { }( ) { } [ ]{ }

( ) jj r r

jriri

ref

ref

ref

ref

uQ

gMux

gxXQ

gMHx

gxX

∑∑ −=

=−

=

=−

22

2

2

,,

,

ωωφφ

ω

ω

ωω


Spatial Models

[ ] [ ] [ ][ ] [ ] [ ][ ] 1

1

−−

−−

ΦΦ=

ΦΦ=

rT

T

K

M

λ




Dr H Ahmadian, Modal Testing Lab, IUSTDerivation of Mathematical Models

Derivation of Mathematical ModelsIntroductionEquation Error Method (Sec. 6.3.6 page 456)

Identification of Rod FE ModelParameter Identification

Solution of Over-determined set of EquationsSolution of Under-determined set of EquationsError Analysis


IntroductionConstruction of Spatial Model from modal data:

111 ,, −−−−−− ΓΦΦ=ΦΦ=ΛΦΦ= TTT CMKModal model must be complete:

All modes must be presentMode shapes are measured in all DOF’s

Measurement of complete Modal Model is impractical.


IntroductionAlternative methods are required to construct the spatial model from

incomplete and noisy measured modes.

The difficulty with incompleteness is removed by reducing the number of unknowns in spatial model.The noise effects are removed by averaging.


Equation Error MethodWe have some information regarding the spatial model format:

SymmetryPattern of zeros…

We may incorporate these information into the identification procedure and reconstruct the spatial model.


Equation Error MethodIn this method the eigen problem is rearranged to obtain the spatial model:

[ ] 0

.0

=⎭⎬⎫

⎩⎨⎧

ΛΦΦ⇒

=ΦΛ−Φ

MK

MK

TT

The DOF’s of measured modes must be compatible with the DOF’s of spatial model.


Rearrangement Example:

.000

0

:

,00

0

2

1

2

1

22

12

12

211

2

1

2

12

22

221

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡

−−−−

=⎭⎬⎫

⎩⎨⎧⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−

−+

mmkk

or

mm

kkkkk

r

r

r

φωφφφωφφφ

φφ

ω


Identification of A Rod FE ModelConsider a fixed-free rod with n elements.The mass and stiffness matrices are:

),,,,( 121

11

3322

221

nn

nn

nnnn

mmmmdiagMkkkkkk

kkkkkkk

K

−

−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−+−

−+−−+

=

L

LLL


Identification of A Rod FE ModelThe equilibrium state at modes r and s are:

( )( ) .0

.0=−=−

ss

rr

MKMK

φλφλ

The last rows of equilibrium state equations are: ( )

( ) .0,0

,1,

,1,

=−+−

=−+−

−

−

nsnrnnsn

nrnrnnrn

mkkmkk

φλφφλφ


Identification of A Rod FE Model

( )( )

⎪⎪⎩

⎪⎪⎨

⎧

−=

−−

=⇒≠≠

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−−−−

−

−

−

−

−

1,,

,

1,,,

1,,,

,1,,

,1,,

0,0

.0

nrnr

nrr

n

n

nrnrns

nsnsnrrs

nn

n

n

nssnsns

nrrnrnr

mk

mk

mk

φφφλ

φφφφφφ

λλ

φλφφφλφφ


Identification of A Rod FE ModelFrom other rows one obtains:

n

n

nnn

n

nn mm

mm

mm

mk

mk

mk 121121 ,,,,,,, −− KK

Using total mass information mm is obtained:

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑

−

=

1

1

1n

l n

lntotal m

mmm


Identification of A Rod FE Model

Only two modes and one natural frequency are required to construct the mass and stiffness matrices.More details can be found in:

GML Gladwell, YM Ram, ”Constructing Finite Element Model of a Vibrating Rod”, Journal of Sound and Vibration, 169,229-237,1994.


Parameter Identification

In a general case the mass and stiffness matrices are parameterized and are obtained by rearranging:

Equation of motion in modal domainOrthogonality requirements,etc.



K

KK

,,,,0:

.,,0:

).,,,(),,,,(:

2121

xyxxRTRR

TT

nn

IImMKExtras

KIMMKEOM

mmmMMkkkKKzationParameteri

⇒ΦΦ=Φ

Λ=ΦΦ=ΦΦ=ΦΛ−Φ

==



⎪⎩

⎪⎨

⎧

→<→>

=→=⇒×⇒

=Λ=ΛΦΦ=

=−

−

ederOvernmederUndernm

bAxnmmnrankfullA

mmmkkkxxImbbAA

bAxtarrangemen

nn

xxtotalR

mindetmindet)(

),,,,,,,(),,,(),,,(

:Re

1

2121 KK

K


Solution of underdetermined case

( )( )( ) ( ) bAAAxbAAbAA

AxbAxxxSolution

bAxbxASTxor

bAxSTxx

TTTT

TTTTT

11

0

0

.0222min:

:,min

:,min

−−=∆⇒=⇒=⇒

=−∆⇒−∆−∆∆

=−=∆∆

=−

λλ

λλ


Solution of Over-determined set of Equations

[ ] [ ]

( ) ( ) bAAAxbAAxAx

bbbAxAxAx

Solution

EEAssumebAx

nmbAx

TTTTT

TTTTTT

T

ijji

122

2

minmin

.,0

,

−=⇒−=

∂∂

+−=

=⇒

==⇒=−

<=

εε

εε

εεε

σδεεεε


Error Analysis

( )[ ] [ ] ( )[ ][ ] [ ] ( )[ ]

[ ] [ ] [ ][ ] [ ] [ ]xExEAEnoisyAIf

xExEEmasAAAExExE

AAAExEbAAAE

Axb

T

TT

TTTT

≠⇒≠→→

=⇒→∞→+=

+=

+=

−

−−

00,

1

11

ε

εε

ε

ε


Example:The parameters to be updated are the 10 stiffness and 6 massesThe measured data consists of the 1st three natural frequencies and mode shapes (added with uniformly distributed random noise)


Example:Eigenvalue equations arearrangment:

31 equations ( 3*6 equations for each eigenvector term, 2*6 symmetric orthogonality equations, and 1 total mass equation)16 parameters and The terms in A and b contain noisy data.

{ }

.:.,,0:

,,,,,,,,: 6211021

mMExtrasKIMMKEOM

mmmkkkxParameters

RTR

TT

=

Λ=ΦΦ=ΦΦ=ΦΛ−Φ

=

φφ

KK


Example

0.28-0.020.960.10.210.972323-8321352-102410111160984415021243954S/N=20

-0.010.20.990.10.21-632035-41499141008100610008151812661041S/N=100

0.10.110.10.2110001000700050001000100010000150012501000Exact

m6m5m4m3m2m1k10k9k8k7k6k5k4k3k2k1


Regularized SolutionAhmadian, Mottershead, and Friswell, REGULARISATION METHODS FOR FINITE ELEMENT MODEL UPDATING, Mechanical Systems and Signal Processing (1998) 12(1),47-64


Home Work 3

Develop a procedure to construct the FE model of a fixed-free beam from minimum modes.

How many modes are required to obtain EI an m of each element?Add some noise to the modes and try to reconstruct the model. Investigate the correlated noise effects?

Documents

Modal Testing - Iran University of Science and Technology · IUST ,Modal Testing Lab ,Dr H Ahmadian Response Function Measurement Techniques Loaded Support The structure is connected