Modal Testing

Modal Testing(Lecture 1)

Dr. Hamid AhmadianSchool of Mechanical Engineering

Iran University of Science and [email protected]

Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing

Overview

Introduction to Modal TestingApplications of Modal TestingPhilosophy of Modal TestingSummary of TheorySummary of Measurement MethodsSummary of Modal Analysis ProcessesReview of Test Procedures and Levels


Introduction to Modal TestingExperimental Structural Dynamics

To understand and to control the many vibration phenomenon in practice

Structural integrity (Turbine blades- Suspension Bridges)Performance ( malfunction, disturbance, discomfort)


Introduction to Modal Testing (continued)

Necessities for experimental observations

Nature and extend of vibration in operationVerifying theoretical modelsMaterial properties under dynamic loading (damping capacity, friction,…)



Test types corresponding to objectives:Operational Force/Response measurements

Response measurement of PZL Mielec Skytruck Mode Shapes (3.17 Hz, 1.62 %), (8.39 Hz, 1.93 %)



Modal Testing in a controlled environment/ Resonance Testing/ Mechanical Impedance Method

Testing a component or a structure with the objective of obtaining mathematical model of dynamical/vibration behaviorStructural Analysis of ULTRA Mirror



Milestones in the development:Kennedy and Pancu (1947)

Natural frequencies and damping of aircrafts

Bishop and Gladwell (1962)Theory of resonance testing

ISMA (bi-annual since 1975)IMAC (annual since 1982)


Applications of Modal Testing

Model Validation/Correlation:Producing major test modes validates the model

Natural frequenciesMode shapesDamping information are not available in FE models


Applications of Modal Testing (continued)

Model UpdatingCorrelation of experimental/analytical modelAdjust/correct the analytical modelOptimization procedures are used for updating.



Component Model Identification

Substructure processThe component model is incorporated into the structural assembly



Force DeterminationKnowledge of dynamic force is requiredDirect force measurement is not possibleMeasurement of response + Analytical Model results the external force

[ ] [ ]( ){ } { }fxMK =− 2ω


Philosophy of Modal TestingIntegration of three components:

Theory of vibrationAccurate vibration measurementRealistic and detailed data analysis

Examples:Quality and suitability of data for processExcitation typeUnderstanding of forms and trends of plotsChoice of curve fittingAveraging


Summary of Theory (SDOF)


Summary of Theory (MDOF)


Summary of TheoryDefinition of FRF:

[ ] [ ] [ ]( ).

)()(

)(

)(

122

12

∑=

−

−==

+−=N

r r

krjr

k

jjk f

xh

DiMKH

ωωφφ

ωω

ω

ωω

Curve-fitting the measured FRF:

Modal Model is obtainedSpatial Model is obtained


Summary of Measurement MethodsBasic measurement system:

Single point excitation


Summary of Modal Analysis ProcessesAnalysis of measured FRF data

Appropriate type of model (SDOF,MDOF,…)Appropriate parameters for chosen model


Review of Test Procedures and Levels

The procedure consists of:FRF measurementCurve-FittingConstruct the required model

Different level of details and accuracy in above procedure is required depending on the application.


Review of Test Procedures and LevelsLevels according to Dynamic Testing Agency:

UpdatingOut of range residues

Usable for validationMode ShapesDamping

ratioNatural FreqLevel

0

Only in few points

1

2

3

4


Text BooksEwins, D.J. , 2000, “Modal Testing; theory, practice and application”, 2nd edition, Research studies press Ltd.McConnell, K.G., 1995, “Vibration testing; theory and practice”, John Wiley & Sons.Maia, et. al. , 1997, “Theoretical and Experimental Modal Analysis”, Research studies press Ltd.


Evaluation Scheme

Home Works (20%)Mid-term Exam (20%)Course Project (30%)Final Exam (30%)




Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis

Theoretical Basis

Analysis of weakly nonlinear structuresApproximate analysis of nonlinear structuresCubic stiffness nonlinearityCoulomb friction nonlinearityOther nonlinearities and other descriptions


Analysis of weakly nonlinear structuresThe whole bases of modal testing assumes linearity:

Response linearly related to the excitationResponse to simultaneous application of several forces can be obtained by superposition of responses to individual forces

An introduction to characteristics of weakly nonlinear systems is given to detect if any nonlinearity is involved during modal test.


Cubic stiffness nonlinearity

)sin()3sin(41)sin(

43

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

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

)sin(

33

2

333

2

33

φωωω

ωωωωω

φωωωωωωω

ωφω

−=⎟⎠⎞

⎜⎝⎛ −

+++−⇒

−=+++−⇒

=⇒−=+++

tFttXk

tkXtXctXmtF

tXktkXtXctXmtXtx

tFxkkxxcxm &&&



⎪⎩

⎪⎨⎧

−=

=++−⇒

−

=⎟⎠⎞

⎜⎝⎛ −

+++−

)sin(

)cos(43

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

)3sin(41)sin(

43

)sin()cos()sin(

33

2

33

2

φω

φω

φωφω

ωω

ωωωωω

FXc

FXkkXXm

tFtF

ttXk

tkXtXctXm



( )

23

22

23

2

43

43

1

Xkkk

cXkkmFX

eq +=

+⎟⎠⎞

⎜⎝⎛ ++−

=

ωω



Softening effect Hardening effect


Softening-stiffness effect








Coulomb friction nonlinearity

( )

⎟⎠⎞

⎜⎝⎛ ++−

=

=Δ

=⇒=Δ

+=

∫

XccimkF

X

Xc

dttx

EcXcE

txtxctxctf

F

FeqF

Fd

πωωω

πωωπ

41

44

)()()()(

2

/2

0

2&

&

&&


Coulomb friction nonlinearity

X increasing

⎟⎠⎞

⎜⎝⎛ ++−

=

XccimkF

XF

πωωω 4

12


Other nonlinearities and other descriptions

BacklashBilinear StiffnessMicroslip friction dampingQuadratic (and other power law damping)…..





MODAL ANALYSIS THEORYUnderstanding of how the structural parameters of mass, damping, and stiffness relate to

the impulse response function (time domain), the frequency response function (Fourier, or frequency domain), andthe transfer function (Laplace domain)

for single and multiple degree of freedom systems.


Theoretical Basis

SDOF systemTime Domain: Impulse Response FunctionPresentation of FRFProperties of FRF

Undamped MDOF systemMDOF system with proportional damping


SDOF SystemThree classes of system:

UndampedViscously-dampedStructurally Damped

Response Models:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+−

+−

−

==

idmk

icmk

mk

FXH

2

2

2

1

1

1

)()()(

ω

ωω

ω

ωωω


Time Domain: Impulse Response Function


Frequency Domain: Frequency Response Function


Alternative Forms of FRFReceptance

Inverse is “Dynamic Stiffness”

MobilityInverse is “Dynamic Impedance”

InertanceInverse is “Apparent mass”

)()(

ωω

FX

)()(

)()(

ωωω

ωω

FXi

FV

=

)()(

)()( 2

ωωω

ωω

FX

FA

−=


Graphical Display of FRF


Graphical Display of FRF

The magnitude of the three mobility functions (accelerance, mobility and compliance)


Stiffness and Mass Lines


Reciprocal PlotsThe “inverse” or “reciprocal” plots

Real part Imaginary part ⎪

⎪⎩

⎪⎪⎨

⎧

=

−=⇒

ωωω

ωωω

cXF

mkXF

))()(Im(

))()(Re( 2


Nyquist PlotFor viscous damping the Mobility plot is a circle.

For structural damping the Receptance and Inertance plots are circles.


3D FRF Plot (SDOF)


Properties of SDOF FRF PlotsNyquist Mobility for viscose damping

( )( )

2

22222

222222

222

2

222

2

2

21

)()(4)()(

)Im(,21)Re(

)()()()Im(

)()()Re(

)(

⎟⎠⎞

⎜⎝⎛=

+−

+−=+

=⎟⎠⎞

⎜⎝⎛ −=

+−−

=+−

=

+−=

ccmkccmkVU

YVc

YU

cmkmkY

cmkcY

icmkiY

ωωωω

ωωωω

ωωωωω

ωω


Properties of SDOF FRF PlotsNyquist Receptance for structural damping

( )( )

( )( ) ( )

222

222222

2

222

2

2

21

21

,

1)(

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛ ++

+−=

+−

−=

+−

−−=

−+=

ddVU

dmkdV

dmkmkU

dmkidmk

midkH

ωωω

ωω

ωω


A DemoBasic Assumptions

The structure is assumed to be linearThe structure is time invariantThe structure obeys Maxwell’s reciprocityThe structure is observable

loose components, or degrees-of-freedom of motion that are not measured, are not completely observable.





Theoretical Basis

Undamped MDOF SystemsMDOF Systems with Proportional DampingMDOF Systems with General Structural DampingGeneral Force VectorUndamped Normal Mode


Undamped MDOF SystemsThe equation of motion:

The modal model:The orthogonality:

Forced response solution:

[ ]{ } [ ]{ } { })()()( tftxKtxM =+&&

[ ] ),,,(, 222

21 Ndiag ωωω K=ΓΦ

[ ] [ ][ ] [ ] [ ] [ ][ ] [ ]., Γ=ΦΦ=ΦΦ KIM TT

[ ] [ ]( ){ } { } titi eFeXMK ωωω =− 2

{ } [ ] [ ]( ) { } { } [ ]{ }FXFMKX )(12 ωαω =⇒−=−


Undamped MDOF Systems (continued)

Response Model

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

T

T

TT

I

I

I

MK

MK

Φ−ΓΦ=

Φ−ΓΦ=

ΦΦ=−Γ

ΦΦ=Φ−Φ

=−

−

−−−

−

−

−

12

121

12

12

12

)(

)(

)(

)(

)(

ωωα

ωωα

ωαω

ωαω

ωαω


Undamped MDOF Systems (continued)

The receptance matrix is symmetric.

∑∑== −

=−

=

===

N

r r

jkrN

r r

krjrjk

j

kkj

k

jjk

AFX

FX

122

122)(

,

ωωωωφφ

ωα

αα Modal Constant/Modal Residue

Single Input


Example:

[ ] [ ]

[ ] [ ]

.64.2118

62.16205

545.0)(

1111

21

6254

/2.18.08.02.1

11

42

2

2211

2

ωωω

ωωωα

ω

+−−

=−

+−

=

⎥⎦

⎤⎢⎣

⎡−

=Φ⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡=

eee

ee

ee

mMNKkgM

r


.64.2118

62.16205

545.0)( 42

2

2211 ωωω

ωωωα

+−−

=−

+−

=ee

eee

Poles

Example (continued):

Zero


Example (continued):

.64.211848

6205

545.0)( 422212 ωωωω

ωα+−

=−

−−

=eee

ee


MDOF Systems with Proportional DampingA proportionally damped matrix is diagonalized by normal modes of the corresponding undamped system

[ ] [ ][ ] ),,,( 21 NT ddddiagD L=ΦΦ

Special cases:[ ] [ ][ ] [ ][ ] [ ] [ ].

,,

MKDMDKD

δβδβ

+===


MDOF Systems with Structurally Proportional Damping

Response Model[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ]( ) [ ]

∑=

−

−−−

−

−

−

−+=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r rr

krjrjk

Trr

rrT

Trr

TT

i

Ii

Ii

Ii

MDiK

MDiK

1222

1222

12221

1222

12

12

)1()(

)1()(

)1()(

)()1(

)(

)(

ωηωφφ

ωα

ωηωωα

ωηωωα

ωαωηω

ωαω

ωαω

Real Residue

Complex Pole


MDOF Systems with Viscously Proportional DampingResponse Model[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ] [ ]( ) [ ]

∑=

−

−−−

−

−

−

+−=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r rrr

krjrjk

Trrr

rrrT

Trrr

TT

Ii

Ii

Ii

MCiK

MCiK

122

122

1221

122

12

12

2)(

2)(

2)(

)(2

)(

)(

ωωζωωφφ

ωα

ωωζωωωα

ωωζωωωα

ωαωωζωω

ωαωω

ωαωω


MDOF Systems with General Structural DampingThe equation of motion:

The orthogonality:

Forced response solution:

[ ]{ } [ ] [ ]( ){ } { })()()( tftxDiKtxM =++&&

[ ] [ ][ ] [ ] [ ] [ ][ ] [ ]., Γ=Φ+Φ=ΦΦ iDKIM TT

[ ] [ ] [ ]( ){ } { } titi eFeXMDiK ωωω =−+ 2

{ } [ ] [ ] [ ]( ) { } { } [ ]{ }FXFMDiKX )(12 ωαω =⇒−+=−

Complex Eigen-valuesComplex Mode Shapes


Example:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Γ

=====

142.0493.0635.0318.0782.0536.0318.1218.0464.0

,6698

3352950

6,...,1,/315.1,1,5.0

1

321

UndampedjmNek

kgmkgmkgmModel

j


Example:[ ] [ ]

[ ]

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

+=Γ

===−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+=Γ

=

)1.3(142.0)3.1(492.0)0.1(636.0)7.6(318.0)181(784.0)0.0(537.0)181(318.1)173(217.0)5.5(463.0

,)078.01(6690

)042.01(3354)067.01(957

6,...,2,0.0,3.0Pr

142.0493.0635.0318.0782.0536.0318.1218.0464.0

,6698

3352950

)05.01(

05.0Pr

11

ooo

ooo

ooo

ii

i

jdkdoportionalNon

i

KDoportional

j

Almost real modes


Example:

[ ]

[ ] [ ]

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+=Γ

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Γ

=====

207.0752.0587.0827.0215.0567.0

552.0602.0577.0,

41243892

999)05.01(

,05.0Pr

207.0752.0587.0827.0215.0567.0

552.0602.0577.0,

41243892

999

6,...,1,/3105.1,95.0,1

2

321

i

KDoportional

UndampedjmNek

kgmkgmkgmModel

j


Example:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

++

+=Γ

===−

)50(560.0)12(848.0)2(588.0)176(019.1)101(570.0)2(569.0)40(685.0)162(851.0)4(578.0

,)019.01(4067

)031.01(3942)1.01(1006

6,...,2,0.0,3.0Pr

11

ooo

ooo

ooo

ii

i

jdkdoportionalNon

j

Heavily complex modes


MDOF Systems with General Structural Damping[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ]( ) [ ]

∑=

−

−−−

−

−

−

−+=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r rr

krjrjk

Trr

rrT

Trr

TT

i

Ii

Ii

Ii

MDiK

MDiK

1222

1222

12221

1222

12

12

)1()(

)1()(

)1()(

)()1(

)(

)(

ωηωφφ

ωα

ωηωωα

ωηωωα

ωαωηω

ωαω

ωαω

Complex Residues

Complex Poles


General Force VectorIn many situations the system is excited at several points.


General Force Vector (continued)

The response is governed by:

The solution:

All forces have the same frequency but may vary in magnitude and phase.

[ ] [ ]( ){ } { } titi eFeXMiDK ωωω =−+ 2

{ } { } { }{ }∑= −+

=N

r rr

rTr

iFX

1222 )1( ωηω

φφ



The response vector is referred to:Forced Vibration Modeor Operating Deflection Shape (ODS)

When the excitation frequency is close to the natural frequency:

ODS reflects the shape of nearby modeBut not identical due to contributions of other modes.



Damped system normal mode:By carefully tuning the force vector the response can be controlled by a single mode.The is attained ifDepending upon damping condition the force vector entries may well be complex (they have different phases)

{ } { } rssTr F δφ =


Undamped Normal Mode

Special Case of interest:Harmonic excitation of mono-phased forces

Same frequencySame phaseMagnitudes may vary

Is it possible to obtain mono-phased response?


Undamped Normal Mode (continued)

The real force response amplitudes:

Real and imaginary parts:

The 2nd equation is an eigen-value problem; its solutions leads to real

{ } { }{ } { } )(ˆ)(

ˆ)(θω

ω

−=

=ti

ti

eXtx

eFtf[ ] [ ]( ){ } { } titi eFeXMiDK ωωω ˆˆ2 =−+

[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ) [ ]( ){ } { }0ˆcossin

ˆˆsincos2

2

=+−

=+−

XDMK

FXDMK

θθω

θθω

{ }F̂

N solutions



At a frequency that the phase lag between all forces and all responses is 90 degree then

Results Undamped normal modesNatural frequencies of undamped system

[ ] [ ]( ) [ ]( ){ } { }0ˆcossin2 =+− XDMK θθω



The base for multi-shaker test procedures.Modal Analysis of Large Structures: Multiple Exciter Systems By: M. Phil. K. Zaveri





Theoretical Basis

General Force VectorUndamped Normal ModeMDOF System with General Viscous DampingForce Response Solution/ General Viscous Damping


General Force VectorIn many situations the system is excited at several points.


General Force VectorOtherwise you end up damaging the structure!!!!



The response is governed by:

All forces have the same frequency but may vary in magnitude and phase.The solution:

[ ] [ ]( ){ } { } titi eFeXMiDK ωωω =−+ 2

{ } { } { }{ }∑= −+

=N

r rr

rTr

iFX

1222 )1( ωηω

φφ



The response vector is referred to:Forced Vibration Modeor Operating Deflection Shape (ODS)

When the excitation frequency is close to the natural frequency:

ODS reflects the shape of nearby modeBut not identical due to contributions of other modes.



Damped system normal mode:By carefully tuning the force vector the response can be controlled by a single mode.This is attained ifDepending upon damping condition the force vector entries may well be complex (they have different phases)

{ } { } rssTr F δφ =



Special Case of interest:Harmonic excitation of mono-phased forces

Same frequencySame phaseMagnitudes may vary

Is it possible to obtain mono-phased response?



512 channel37 Shakers



The real force response amplitudes:

Real and imaginary parts:

The 2nd equation is an eigen-value problem; its solutions leads to real

{ } { }{ } { } )(ˆ)(

ˆ)(θω

ω

−=

=ti

ti

eXtx

eFtf[ ] [ ]( ){ } { } titi eFeXMiDK ωθωω ˆˆ )(2 =−+ −

[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ) [ ]( ){ } { }0ˆcossin

ˆˆsincos2

2

=+−

=+−

XDMK

FXDMK

θθω

θθω

{ }F̂

N solutions



At a frequency that the phase lag between all forces and all responses is 90 degree then

Results Undamped normal modesNatural frequencies of undamped system

[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ){ } { }0ˆ

0ˆcossin2

2

=−⇒

=+−

XMK

XDMK

ω

θθω



The base for multi-shaker test procedures.Modal Analysis of Large Structures: Multiple Exciter Systems By: M. Phil. K. Zaveri


MDOF System with General Viscous Damping

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

eXtxeFtffxKxCxMMOE

titi

12

)()(...

−+−=

=⇒=

=++⇒

ωω

ωω

&&&

Next the orthogonality properties of the system in 2N space is used for force response solution.


Force Response Solution

{ } { } { }

{ } { }

).,,,(0

0,

0

00

00

00

00

.

000

0

221 NTT

rr

sssdiagUM

KUIU

MMC

U

uM

KM

MCssolutionEigen

uM

Ku

MMC

VibFree

fxx

MK

xx

MMC

EOM

L

&

&&&

&

=⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡⇒−

=⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡⇒

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡⇒


Force Response Solution

*1

2

1

000

r

rHrN

r r

rTrN

r r

rTr

si

uF

u

si

uF

u

si

uF

u

XiX

−⎭⎬⎫

⎩⎨⎧

+−⎭⎬⎫

⎩⎨⎧

=−⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

∗

==∑∑ ωωωω

The above simplification is due to the fact that eigen-values and eigen-vectors occur in complex conjugate pairs.


Force Response SolutionSingle point excitation:

∗

∗∗

= −+

−= ∑

r

krjrN

r r

krjrjk si

uusi

uuωω

ωα1

)(





Modal Analysis of Rotating StructuresNon-symmetry in system matricesModes of undamped rotating system

Symmetric StatorNon-Symmetric Stator

FRF’s of rotating systemOut-of-balance excitation

Synchronous excitationNon-Synchronous excitation


Non-symmetry in System Matrices The rotating structures are subject to additional forces:

Gyroscopic forcesRotor-stator rub forcesElectrodynamic forcesUnsteady aerodynamic forcesTime varying fluid forces

These forces can destroy the symmetry of the system matrices.


Non-rotating system properties

A rigid disc mounted on the free end of a rigid shaft of length L, The other end of is effectively pin-jointed.


Modes of Undamped Rotating System

.00

00

0//0

/00/

0

0

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡yx

LkLk

yx

LJLJ

yx

LILI

y

x

z

z

&

&

&&

&&


Symmetric stator

( ) ( )( ) ( )

.02

,00

////

,,

,

2

0

22

2

00

24

20

22

220

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω+−

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−Ω−Ω−

=

=

==

IkL

IJ

IkL

YX

LIkLJiLJiLIk

YeyXex

kkk

z

z

z

ti

ti

yx

ωω

ωωωω

ω

ω

Support is symmetric

Simple harmonic motion


Natural Frequencies

⎪⎪⎩

⎪⎪⎨

⎧

==

Ω+Ω±

Ω+=

⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω+−

00

220

2220

2220

22,1

2

0

22

2

00

24

,

42

.02

IJ

IkL

IkL

IJ

IkL z

γω

γωγγωω

ωω


Mode Shapes( ) ( )( ) ( )

( ) ( )( ) ( ) .

00

1////

,001

////

20

22

22

22

20

22

20

21

21

21

20

21

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−Ω−Ω−

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

−Ω−Ω− i

LIkLJiLJiLIk

iLIkLJiLJiLIk

z

z

z

z

ωωωω

ωωωω


Non-symmetric Stator

yx kk ≠


FRF of the Rotating Structure

,0

0/

//00/

2

2

20

20

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−Ω

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

y

x

z

z

ff

yx

kk

yx

cLJLJc

yx

LILI

&

&

&&

&&

[ ] ( ) ( )( ) ( )

⎩⎨⎧

−==

⇒

⎥⎦

⎤⎢⎣

⎡

+−Ω−Ω+−

=−

)()()()(

////

)(1

20

22

220

2

ωαωαωαωα

ωωωωωω

ωα

yxxy

yyxx

z

z

icLIkLJiLJiicLIk

Loss of Reciprocity

External Damping

Coupling Effect


FRF of the Rotating Structurewith External Damping

Complex Mode Shapes

due to significant imaginary part


Out-of-balance excitationResponse analysis for the particular case of excitation provided by out-of-balance forces is investigated:

When the force results from an out-of-balance mass on the rotor, it is of a synchronous natureWhen the force results from an out-of-balance mass on a co/counter rotating shaft, it is of a non-synchronous nature


Synchronous OOB Excitation

{ }

( ))1(

:

1)sin()cos(

2200

2

2

γω −Ω−=

⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

⇒

−⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

ΩΩ

Ω=

ΩΩ

Ω

ILA

eiAA

FeYX

StatorSymmetric

ei

Ftt

mrF

tiOOB

ti

tiOOB


Synchronous OOB Excitation


Non-Synchronous OOB ExcitationForce is generated by another rotor at different speed

( ))(2200

2

γββω

β

ββ

−Ω−=

⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧

Ω⇒

ΩΩ

ILA

eiAA

FeYX

Excitation

tiOOB

ti

The essential results are the same as for synchronous case.





Theoretical Basis

Analysis using rotating frameDamping in rotating and stationary framesDynamic analysis of general rotor-stator systems

Linear Time Invariant Rotor-Stator SystemsLTI Rotor-Stator Viscous Damp SystemLTI Systems Eigen-Properties


Analysis using rotating frame

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩΩ−Ω

=⎭⎬⎫

⎩⎨⎧

r

r

yx

tttt

yx

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

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

yx

tttt

yx

r

r


Y

X

Yr

Xr

tΩ



[ ]

[ ] [ ]

[ ] [ ] [ ]⎭⎬⎫

⎩⎨⎧

Ω+⎭⎬⎫

⎩⎨⎧

Ω+⎭⎬⎫

⎩⎨⎧

=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

Ω−⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−Ω−ΩΩ−

Ω+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

Ω+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−Ω−ΩΩ−

Ω+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

xx

Tyx

Tyx

T

yx

tttt

yx

tttt

yx

tttt

yx

yx

Tyx

Tyx

tttt

yx

tttt

yx

yx

Tyx

tttt

yx

r

r

r

r

r

r

12

21

2

21

1

2


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

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

,)sin()cos(

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

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

&

&

&&

&&

&

&

&&

&&

&&

&&

&

&

&

&

&

&

Transformation Matrices



zzzzzz Ω−Ω+Ω+=Ω+Ω−Ω+= 2/)2/(2/)2/( 2202

2201 γγωωγγωω

.00

//)()(//

0//2//20

/00/

222220

2

222220

2

220

220

20

20

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

++Ω+Ω−−−++Ω+Ω−

+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−ΩΩ+Ω−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

r

r

yxzzxy

xyyxzz

r

r

zz

zz

r

r

yx

skckLJLIkkcskkcsskckLJLI

yx

LJLILJLI

yx

LILI

&

&

&&

&&

Equation of Motion in Stationary Coordinates

Equation of Motion in Rotating Coordinates

2/)2/(2/)2/(

,00

00

0//0

/00/

2202

2201

2

2

20

20

zzzz

y

x

z

z

yx

kk

yx

LJLJ

yx

LILI

Ω+Ω+=Ω−Ω+=

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−Ω

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

γγωωγγωω

&

&

&&

&&

Note: Eigenvectors remain unchanged



⎭⎬⎫

⎩⎨⎧

Ω++Ω−Ω++Ω−

=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ΩΩ−ΩΩ

=⎭⎬⎫

⎩⎨⎧

ttttF

tF

tttt

FF

FF

tttt

FF

yr

xr

y

x

yr

xr

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

2

)cos(0)cos()sin(

)sin()cos(

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

0

0

ωωωω

ω

For Example: Response harmonies not present in the excitation


Internal Damping in rotating and stationary frames

.00

//00//

//2//2

/00/

2220

2

2220

2

220

220

20

20

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

+Ω+Ω−+Ω+Ω−

+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−ΩΩ+Ω−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

r

r

zz

zz

r

r

Izz

zzI

r

r

yx

kLJLIkLJLI

yx

cLJLILJLIc

yx

LILI

&

&

&&

&&

,00

//

/00/

2

2

20

20

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−Ω

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡yx

kcck

yx

cLJLJc

yx

LILI

yIz

Izx

Iz

zI

&

&

&&

&&

Equation of Motion in Rotating Coordinates

Equation of Motion in Stationary Coordinates


Internal/External Damping in 2DOF System

,00

//

/00/

2

2

20

20

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

+Ω−Ω+

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

yx

kcck

yx

ccLJLJcc

yx

LILI

yIz

Izx

IEz

zIE

&

&

&&

&&

At super critical speeds the real parts of eigen-values may become positive, i.e. unstable system


Dynamic Analysis of General Rotor-Stator SystemsThe rotating machines and their modal testing is much more complex

Non-symmetric bearing supportFixed/Rotating observation frameNon-axisymmetric rotorsInternal/External damping

These lead to:Time-varying modal propertiesResponse harmonies not present in the excitationInstabilites (negative modal damping)


Dynamic Analysis of General Rotor-Stator Systems

Equation of motion of rotating systems are prone:

to lose the symmetry to generate complex eigen-values/vectors from velocity/displacement related non-symmetryto include time varying coefficients as appose to conventional Linear Time Invariant (LTI) systems


Dynamic Analysis of General Rotor-Stator Systems

Rotating Coord.Stationary Coord.System Type

LTILTIR-symm;S-symm

L(t)LTIR-symm;S-nonsymm

LTIL(t)R-nonsymm;S-symm

L(t)L(t)R-nonsymm;S-nonsymm

LTI: Linear Time InvariantL(t): Linear Time Dependent


Linear Time Invariant Rotor-Stator Systems

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

.,,,)()(

)(

symmSkewEGSymmDKCM

tfxEDiKxGCxM

−⇒ΩΩ⇒

=Ω+++Ω++ &&&

Solution of equations will follow different routs depending upon the specific features.


LTI Rotor-Stator Systems (Viscous Damping Only)[ ]{ } [ ]{ } { }

[ ]

[ ]

{ }⎭⎬⎫

⎩⎨⎧

=

⎥⎦

⎤⎢⎣

⎡ Ω+=

⎥⎦

⎤⎢⎣

⎡−

Ω+=

=+

xx

u

MEK

B

MMGC

A

uBuA

&

&

00)(

0)(

,0

The system matrices are non-symmetricComplex eigenvalsTwo eigenvect sets:

RH; mode shapesLH; normal excitation shapes


LTI Rotor-Stator Systems (Viscous Damping Only)Symmetric Rotor/ Non-symmetric Support

Backwards Whirl

Forwards Whirl


FRF of LTI Rotor-Stator Systems

[ ] [ ] ( )[ ] [ ]HLHrRH ViV 1)( −−= ωλωα


LTI Systems Eigen-PropertiesSkew-symmetry in damping Matrix

[ ] [ ]

[ ] ⎥⎦

⎤⎢⎣

⎡−

−Δ−+⎥

⎦

⎤⎢⎣

⎡−

Δ=

⎥⎦

⎤⎢⎣

⎡−

−=⎥

⎦

⎤⎢⎣

⎡=

15.05.01

)1(05.05.00

,3113

,1

1

CCC

KM


LTI Systems Eigen-Properties

-1.001-0.75+1.85i0.0-1.05+0.08i1-0.68+1.88i0.1-1.08+0.28i1-0.52+1.94i0.3-1.03+0.49i1-0.37+1.99i0.5-0.90+0.63i1-0.23+2.04i0.7-0.76+0.71i1-0.07+2.08i0.9-0.69+0.73i12.11i1.0

CΔ 1X 2X2λ


LTI Systems Eigen-PropertiesSkew-symmetry in stiffness Matrix

[ ] [ ]

[ ] ⎥⎦

⎤⎢⎣

⎡−

−Δ−+⎥

⎦

⎤⎢⎣

⎡−

Δ=

=⎥⎦

⎤⎢⎣

⎡=

3113

)1(0110

,0,1

1

KKK

CM


LTI Systems Eigen-Properties

-1.0012.00i0.0-1.1211.90i0.1-1.5811.65i0.3

Infinity11.23i0.51.58i10.32+1.00i0.71.12i10.57+0.79i0.9

i10.70+0.70i1.0

KΔ 1X 2X2λ





Complex Measured Modes


Complex Measured Modes


Display of Mode Complexity


Analytical Real Modes


H Ahmadian, GML Gladwell - Proceedings of the 13th International Modal Analysis (1995):

The optimum real mode is the one with maximum correlation with the complex measured one:

Extracting real modes fromcomplex measured modes


Extracting real modes

Normalizing the complex measured mode shape:

The problem is rewritten as:



Skew-symmetricSymmetric

Rank 2 matrices



Since V is skew symmetric,

Therefore the problem is equivalent to:












Follow-ups:E. Foltete, J. Piranda, “Transforming Complex Eigenmodes into Real Ones Based on an Appropriation Technique”, Journal of Vibration and Acoustics, JANUARY 2001, Vol. 123S.D. GARVEY, J.E.T. PENNY, “THE RELATIONSHIP BETWEEN THE REAL AND IMAGINARY PARTS OF COMPLEX MODES”, Journal of Sound and Vibration 1998,212(1),75-83





Theoretical Basis

Non-sinusoidal Vibration and FRF Properties:

Periodic VibrationTransient VibrationRandom Vibration

Violation of Dirichlet’s conditionsAutocorrelation and PSD functionsH1 and H2

Incomplete Response Models


Non-sinusoidal Vibration and FRF PropertiesWith the FRF data, response of a MDOF system to a set of harmonic loads:

{ } [ ]{ } titi eFeX ωω ωα )(=Different amplitudes and phases

The same frequency

We shall now turn our attention to a range of other excitation/response situvations.


Periodic VibrationExcitation is not simply sinusoidal but retain periodicity.The easiest way of computing the response is by means of Fourier Series,

ti

nnknjkj

nti

nnkk

n

n

eFtx

TeFtf

ω

ω

ωα

πω

∑

∑∞

=

∞

=

=

==

1

1

)()(

2)(


Periodic VibrationTo derive FRF from periodic vibration signals:

Determine the Fourier Series components of the input force and the relevant responseBoth series contain components at the same set of discrete frequenciesThe FRF can be defined at the same set of frequency points by computing the ratio of response to input components.


Transient VibrationAnalysis via Fourier Transform

∫

∫

∞+

∞−

+∞

∞−

−

=

=

=

ωωω

ωωωπ

ω

ω

ω

deFHtx

FHX

dtetfF

ti

ti

)()()(

)()()(

)(21)(


Transient VibrationResponse via time domain (superposition)

)()(21)(

21)()0()(

)()()(

thdeHtxThen

FtfLet

dfthtx

ti ==→

=⇒=→

−=

∫

∫

∞+

∞−

+∞

∞−

ωωπ

πωδ

τττ

ω


Transient Vibration

To derive FRF from transient vibration signals:

Calculation of the Fourier Transforms of both excitation and response signalsComputing the ratio of both signals at the same frequency

In practice it is common to compute a DFT of the signals.


Random Vibration

Neither excitation nor response signal can be subject to a valid Fourier Transform:

Violation of Dirichlet ConditionsFinite number of isolated min and maxFinite number of points of finite discontinuity

Here we assume the random signals to be ergodic


Random Vibration

∫

∫∞+

∞−

−

∞+

∞−

=

+=

ττπ

ω

ττ

ωτdeRS

dttftfR

tf

iffff

ff

)(21)(

)()()(

)( Time Signal

Autocorrelation Function

Power Spectral Density


Random Vibration

Time Signal

Autocorrelation



Random Vibration

Sinusoidal Signal

Autocorrelation



Random Vibration

Random Signal

Autocorrelation



Random Vibration

Autocorrelation

Noisy Signal


Random VibrationThe autocorrelation function is real and even:

τ

ττ

ττ

+=

−=−=

+=

∫

∫∞+

∞−

+∞

∞−

tu

Rduufuf

dttftfR

ff

ff

)()()(

)()()(

The Auto/Power Spectral Density function is real and even.


Random Vibration

Cross Correlation / Spectral Densities

Cross Correlation functions are real but not always even.Cross Spectral Densities are complex functions.

∫∫+∞

∞−

−+∞

∞−

=+= ττπ

ωττ ωτdeRSdttftxR ixfxfxf )(

21)()()()(


Random Vibration

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

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

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

*

*

*

ωωωττ

ωωωττ

ωωωττ

XXSdttxtxR

FXSdttftxR

FFSdttftfR

xxxx

xfxf

ffff

=⇒+=

=⇒+=

=⇒+=

∫

∫

∫

∞+

∞−

∞+

∞−

+∞

∞−

Time Domain Frequency Domain


Random VibrationTo derive FRF from random vibration signals:

)()(

)()()()()(

)()(

)()()()()(

)()()(

*

*

2

*

*

1

ωω

ωωωωω

ωω

ωωωωω

ωωω

ff

fx

xf

xx

SS

FFXFH

SS

FXXXH

FXH

==

==

=


Complete/ Incomplete Models

It is not possible to measure the response at all DOF or all modes of structure (N by N)Different incomplete models:

Reduced size (from N to n) by deleting some DOFsNumber of modes are a reduced as well (from N to m, usually m<n)



[ ][ ]⎩

⎨⎧

Φ⇒

+−=

×

×

<

=∑

mn

mmr

Nm

r rrr

jkrjk i

A

2

1222)(

ω

ωηωωωα







Theoretical Basis

Sensitivity of ModelsModal Sensitivity

SDOF eigen sensitivityMDOF system natural frequency sensitivityMDOF system mode shape sensitivity

FRF SensitivitySDOF FRF sensitivityMDOF FRF sensitivity


Sensitivity of Models

The sensitivity analysis are required:to help locate errors in models in updatingto guide design optimization proceduresthey are used in the course of curve fitting

A short summery on deducing sensitivities from experimental and analytical models is given.


Modal Sensitivities (SDOF)

.21

21

,21

21

00

03

0

0

kmkk

mmk

m

mk

ωω

ωω

ω

==∂∂

−=−=∂∂

=


Modal Sensitivities (MDOF)

[ ] [ ]( ){ } { }

[ ] [ ]( ){ } { }

[ ] [ ]( ) { } [ ] [ ] [ ] { } { },0

,0

,0

22

2

2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+∂∂

−

=−∂∂

=−

rrrr

r

rr

rr

pMM

ppK

pMK

MKp

MK

φωωφω

φω

φω


Eigenvalue Sensitivity (MDOF){ }

{ } [ ] [ ]( ) { }

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

{ } [ ] [ ] { }

{ } [ ]{ }rTr

rrT

rr

rrrT

r

rr

Tr

Tr

MpM

pK

presults

pMM

ppK

pMK

byMultiply

φφ

φωφω

φωωφ

φωφ

φ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=∂∂

⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+∂∂

−

22

22

2

,0


Eigenvector Sensitivity (MDOF)

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

{ } { }

[ ] [ ]( ) { } [ ] [ ] [ ] { } { }0

,0

:

22

1

2

1

22

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+−⇒

=∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+∂∂

−

∑

∑

≠=

≠=

rrr

N

rjj

jrjr

N

rjj

jjr

rrrr

r

pMM

ppKMK

ptakingand

pMM

ppK

pMK

fromStarting

φωωφγω

φγφ

φωωφω



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

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

( ) { } [ ] [ ] { } { }0

0

0

222

22

1

2

22

1

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+−⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+−⇒

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+−

∑

∑

≠=

≠=

rrT

srsrs

rrrT

s

N

rjj

jrjrT

s

rrr

N

rjj

jrjr

pM

pK

pMM

ppKMK

pMM

ppKMK

φωφγωω

φωωφφγωφ

φωωφγω



( ) { } [ ] [ ] { }

{ } [ ] [ ] { }

( )

{ } { } [ ] [ ] { }

( ) { }∑≠= −

⎟⎠

⎞⎜⎝

⎛∂∂−

∂∂

=∂∂

⇒

−

⎟⎠

⎞⎜⎝

⎛∂∂

−∂∂

=⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+−⇒

N

rss

ssr

rrT

sr

sr

rrT

s

rs

rrT

srsrs

pM

pK

p

pM

pK

pM

pK

122

2

22

2

222

φωω

φωφφ

ωω

φωφγ

φωφγωω


Updating, Redesign, Reanalysis

{ }{ }

{ } { } { }

{ } { } { }⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

ΔΔΔ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

ΔΔ

ΔΔ

M

OMMM

K

K

KMMM

K

K

M

M

3

2

1

3

2

2

2

1

2

3

1

2

1

1

1

3

22

2

22

1

22

3

21

2

21

1

21

2

1

22

21

ppp

ppp

ppp

ppp

ppp

φφφ

φφφ

ωωω

ωωω

φφ

ωω


Updating, Redesign, Reanalysis

The change in parameters must be very small for accurate analysisWhen the change in parameters is not small:

Higher order sensitivity analysis Iterative linear sensitivity analysis


FRF Sensitivities (SDOF)

( )

( )

( )22

2

22

22

2

)(

)(

1)(

1)(

mcikm

mciki

c

mcikk

mcik

ωωωωα

ωωωωα

ωωωα

ωωωα

−+=

∂∂

−+

−=

∂∂

−+

−=

∂∂

−+=


FRF Sensitivities (MDOF)( )[ ] [ ] [ ] [ ]

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

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

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

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

AAxxAx

xA

Z

ZZZZZZthen

ZBAZAtake

ABBAABA

MCiKZ

ωαωωαωαωα

ωωωωωω

ωω

ωωω

Δ−=−

−−=⇒

⇒+⇒

+−=+⇒

−+=

−−−−−

−−−−

11111

1111

2

,

,


FRF Sensitivities (MDOF)

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

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

TjAx

AxAx

Z

Z

ωαωωαωαωα

ωαωωαωαωα

Δ=−

Δ−=− ,


FRF Sensitivities (MDOF)


FRF Sensitivities (MDOF)( )[ ] ( )[ ]( ) ( )[ ] ( )[ ] ( )[ ]

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

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

ωαωωαωα

ωωωωωα

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

−=∂

∂

∂∂

−=∂

∂

∂∂

−=∂

∂=

∂∂ −−

−

pM

pCi

pK

p

pZ

p

Zp

ZZp

Zp

2

111

Documents

Modal Testing