correlation topics 122901 1

7/24/2019 Correlation Topics 122901 1

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1 Dr. Peter AvitabileModal Analysis & Controls Laboratory

Test/Analysis Correlation/Updating Considerations

Test-Analysis

Correlation-UpdatingConsiderations

Peter Avitabile

Modal Analysis and Controls LaboratoryUniversity of Massachusetts Lowell


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FINITE ELEMENT MODEL

EXPERIMENTAL MODAL MODEL

[M] , [K] [U ] , [ ]n 2

[T ] = [U ] [U ]nu ag

[E ] = [T ] [E ]un a

COMBINING ANALYTICAL AND EXPERIMENTAL DATA

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MAC

0

0.2

0.4

0.6

0.8

1

1.2

GUYAN

0

0.2

0.4

0.6

0.8

1

1.2

IRS

0

0.2

0.4

0.6

0.8

1

1.2

SEREP

FINITE ELEMENT EXPERIMENTAL

Experimental Analytical

CoMAC CORTHOG

DOF CORRELATION

DOF CORRELATION

EXP1 EXP 2

EXP 3EXP 4EXP 5

FEM 1

FEM 2

FEM 3

FEM 4

FEM 5

0

0.2

0.4

0.6

0.8

1

FINITE ELEMENT

EXPERIMENTAL

OR

MODESWITCHING

M A C

P O C

VECTOR CORRELATION

VECTOR CORRELATION

F R A C

EXPERIMENTALFINITE ELEMENT

DOF CORRELATION

VECTOR CORRELATION

R V A C

MAC AND ORTHOGONALITY

MODEL

IMPROVEMENT

REGIONS

The Overall Correlation and Updating Process


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Test-Analysis Correlation-Updating Considerations

Briefly describe the different correlation tools

available

Conceptually, overview the correlation process

Briefly overview the model updating process

A significant amount of effort is required tocompletely describe all the techniques and toolsavailable

Objectives of this lecture:


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Correlation Techniques

Correlation between analytical and experimentaldata is an important part of the structuraldynamic characterization and updating of systems

RESPONSE

ASSURANCE

CRITERIA

F R A C


FREQUENCY

DOF CORRELATION

VECTOR CORRELATION

VECTOR

ASSURANCE

CRITERIA

R V A CRESPONSE


MODAL

ASSURANCE

CRITERIA

ORTHOGONALITY

CRITERIA

OR

Exp er imen ta l Ana ly ti ca l

COORDINATE COORDINATE

CoMAC CORTHOG

DOF CORRELATION

EXP1EXP 2

EXP 3EXP 4

EXP 5

FEM1

FEM2

FEM3

FEM4

FEM5

0

0.2

0.4

0.6

0.8

1

FINITE ELEMENT

EXPERIMENTAL

MODAL

ASSURANCE

CRITERIA

MATRIX

PSEUDO

ORTHOGONALITY

CRITERIA

MATRIX

OR

MODE

SWITCHING

M A C

P O C

VECTOR CORRELATION


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Overview of Correlation Techniques

Modal Assurance Criteria

Orthogonality Checks

Vector correlation provides global indicator:

Coordinate Modal Assurance Criteria

Coordinate Orthogonality Check

Frequency Response Assurance Criteria

DOF correlation provides spatial indicator:

MAC Contribution

Force Unbalance

Other tools:


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Modal vector correlation provides a

global indicator of the level ofcorrelation achieved

Degree of freedom (dof) correlationprovides an indicator as to how the

individual dofs contribute to the overallmodal vector correlation

Two basic levels of correlation are considered:


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Vector Correlation Techniques:

Simple dot product independent of mass weighting

Modal Assurance Criteria (MAC):

Performed at n space or a space

mass reduced for a space calc

shape expanded for n space calc reduction/expansion has an effect

Orthogonality Checks (POC):


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DOF Correlation Techniques:

Simple dot product correlation on dof basis for correlated

mode pairs

independent of mass weighting

Coordinate Modal Assurance Criteria (CoMAC):

Extension of CoMAC

Enhanced Coordinate Modal Assurance Criteria:


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DOF Correlation Techniques:

simple dot product correlation of FEM and Test FRFs

Frequency Response Assurance Criteria (FRAC):

Identified correlation on a dof basis

mass matrix used for weighting

similar to CoMAC in concept exceptcorrelated mode pairs not required

Coordinate Orthogonality Check (CORTHOG)


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Modal Assurance Criteria - MAC

Originally formulated for the test engineer todetermine the degree of correlation betweenvectors from different tests, MAC between two

vectors is defined as:

values range between 0 and 1

approaching zero indicates no similarity

approaching one indicates high similarity

{ }{ }( )

{ } { }( ){ } { }( )jT

ji

T

i

2

j

T

i

ij

VVVV

VVMAC =


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Modal Assurance Criteria - MAC

MAC was extended to allow an assessmentbetween analytical and experimental modalvectors:

low values - not similar

high values - very similar

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

2

j

T

i

ijeeuu

euMAC =

EXP1EXP 2

EXP 3EXP 4

EXP 5

FEM 1

FEM 2

FEM 3

FEM 4

FEM 5

0

0.2

0.4

0.6

0.8

1

FINITE ELEMENT

EXPERIMENTAL

MODAL

ASSURANCE

CRITERIA

MATRIX

MODE

SWITCHING

M A C

VECTOR CORRELATION


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Orthogonality Check

For modal vectors scaled to unit modal mass, thevectors must satisfy the orthogonality condition:

[ ]

[ ] ][]U[K]U[

]I[]U[M]U[

2T

T

=

=


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Pseudo Orthogonality Check - POC

The Pseudo Orthogonality Check relating thecorrelation between the analytical andexperimental modal vectors with the analytical

mass matrix is[ ] [ ] [ ] [ ]IUMEPOC

?T ==

Typically, most people feel the smaller the POCoff-diagonal terms the better correlation thatexists. However, these terms may be small

and vectors may still be relatively uncorrelated

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MAC

0

0.2

0.4

0.6

0.8

1

1.2

GUYAN

0

0.2

0.4

0.6

0.8

1

1.2

IRS

0

0.2

0.4

0.6

0.8

1

1.2

SEREP


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The Pseudo Orthogonality Check is an assessmentas to how close the experimental vectors arealigned with the analytical vectors

[ ] [ ] ][]U[K]E[]I[]U[M]E[ 2?T?T ==

FEM Space - requires expansion

Reduced Space - requires reduction Intermediate space - requires both

These equations can be evaluated at:

Substantial numerical advantages using SEREP!


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Expansion toFull Space

may smearand distortmode shapes

Reduction toTest Spacemay resultin distorted

mass andstiffnessmatrices


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Cross Orthogonality Check

The Cross Orthogonality Check is also used forcorrelation purposes

[ ] [ ] ][]E[K]E[]I[]E[M]E[ 2?T?T ==

FEM Space - requires expansion

Reduced Space - requires reduction

These equations can be evaluated at:

Similar to POC (off-diagonal terms are squared)


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Coordinate Modal Assurance Criteria - CoMAC

The CoMAC gives an indication of the contributionof each dof to the MAC for a given mode pair

Low values of CoMACindicate little correlation

whereas high values ofCoMAC indicate veryhigh correlation

( ) ( )

= =

=

=m

1c

m

1c

2)c(

k

2)c(

k

2m

1c

)c(k

)c(k

eu

eu

)k(CoMAC


MODAL

ASSURANCE

CRITERIA Exper imen tal Analy ti cal

COORDINATE

CoMAC

DOF CORRELATION


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Modulus Difference

The Modulus Difference was developed tosupplement the results from CoMAC

Assists in identifyingdiscrepanciesbetween analyticaland experimentalvectors

)c(k)c(k eu)k(DifferenceModulus =


DOF CORRELATION


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Enhanced CoMAC - ECoMAC

The CoMAC gives an indication of the contributionof each dof to the MAC for a given mode pair

Low values of ECoMAC indicate high correlationwhereas high values of CoMAC indicate very low

correlationVery sensitive to phasing of vectors - whichmakes it more sensitive

m2

eu

)k(ECoMAC

m

1c

)c(k

)c(k

==


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Frequency Response Assurance Criteria - FRAC

The FRAC is used to identify similarity between ameasured and analytical FRF - formed like MAC

Low values of FRACindicate littlecorrelation whereashigh values of FRACindicate very highcorrelation

( ) ( ){ } ( ){ }( ){ } ( ){ }( ) ( ){ } ( ){ }( )*x

ji

x

ji

*a

ji

a

ji

2*x

ji

a

ji

HHHH

HHjFRAC

=

RESPONSE

ASSURANCE

CRITERIA

F R A C


FREQUENCY

DOF CORRELATION


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Response Vector Assurance Criteria - RVAC

The RVAC is used to identify the degree ofsimilarity that exists at a particular frequency

Low values of RVAC

indicate littlecorrelation whereashigh values of RVAC

indicate very highcorrelation

{ } { }( ))(U,)(EMAC)(RVAC

femtest =


VECTOR CORRELATION

VECTORASSURANCE

CRITERIA

R V A CRESPONSE


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Coordinate Orthogonality Check - CORTHOG

The Coordinate Orthogonality Check helps toidentify the contribution of individual dofs toeach of the off-diagonal terms of the POC

matrixIdentifies which dof are most discrepant betweenthe analytical and experimental vectors on a mass

weighted basis

POC Orthogonality

=p

pjkpki

k

ij umePOC =p

pjkpki

k

ij umuORTHOG


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Coordinate Orthogonality Check - CORTHOG

The Coordinate Orthogonality Check is simply thecomparison of what should have been obtained

analytically for each dof in an orthogonality check

to what was actually obtained for each dof in apseudo-orthogonality check from test

Variety of differentformulations with

different scalingapproaches

==

p

pjkpkipjkpki

k

ij umuumeCORTHOGSD

-4 -3 -2 -1 0 1 2 3

emu

emu

emu

umu

umu

umu

Experimental

Analytical

dof 1

dof 2

dof 3


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MAC Contribution

The MAC Contribution is a relatively simple andstraightforward technique to determine the degreeof contribution of each dof to the MAC value

achieved

pick a mode pair of interest

select a target MAC value

delete dof until target MAC value achieved


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Force Unbalance

The Force Balance is a simple calculation todetermine the inequality that exists in theequation of motion

uses the FEM mass and stiffness matrices

uses experimental frequencies and mode shapes compute the inequality that exists

[ ] [ ][ ]{ } { }0xMK?

=


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Model Updating Topics

Model Updating techniques can be broken downinto two categories:

Direct Techniques

Indirect Techniques (Sensitivity based)

Modal Based TechniquesResponse Based Techniques

Some basic theory of analytical modelimprovement and localization of model change aredescribed


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Model Improvement Terminology


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Analytical Model Improvement - AMI


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Analytical Model Improvement - SSO


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Analytical Model Improvement - MSSO


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Model Updating - Sensitivity Approaches


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Comments on Direct Techniques

Usually a one step process that does not require iterationto obtain a solution

Usually based on equation of motion and orthogonalityconditions

Exact results obtained (in the sense that the target modes

are reproduced Generally updated matrices are difficult to interpret and

smearing of results occurs

Skyline approaches attempt to retain the original topologyof the system assembly

Reduction and expansion have a dramatic effect on results

Direct Techniques


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Comments on Direct Techniques

Matrix smearing Skyline containment


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Model Updating - Sensitivity Approaches

Frequency differences

Mode shape differences

Frequency response differences

Differences that are typically minimized:

mass/stiffness of individual elements

mass/stiffness of groups of elements

parameters associated with individual elements

parameters associated with groups of elements

Parameters that may be updated:


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Comments on Indirect Techniques

Indirect Techniques - Sensitivity approach

Modal Based Techniques


Shape differences

Response differences


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Indirect Techniques -Sensitivity -Modal Approach

Likely to be the most accurate parametermeasured

No spatial information needed

Relatively simple calculations No reduction/expansion problems



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Indirect Techniques -Sensitivity -Modal Approach

Less accurate on a dof basis Spatial information included

Mode pairing necessary

Calculations more complicated Reduction/expansion is a problem

Shape differences


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Indirect Techniques - Sensitivity approach

Response Based Techniques

Contains complete information in frequency range No need to estimate modal parameters

FRFs are more accurate than modal parameters

Response may be item of interest

Damping may be difficult to determine

Selection of certain spectral lines may causenumerical difficulties

Using only a few FRFs may distort the results Difficult to identify parameters for change

Measured FRFs must be acquired with high accuracy


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General Comments


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General Comments

Use of all the correlation tools necessary tointerpret the data available

Both modal and response based techniques should beused together for the updating

One technique alone may not be sufficient toadequately update the model

Once updated, the model should be perturbed bothanalytically and experimentally and the correlation

process repeated to assure that meaningfulparameters have been obtained from the updatingprocess


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D P t A it bil

General Comments

Model Updating requires extreme care in order toobtain reliable results

A firm understanding of the modeling techniques

employed are necessary in order to adequately adjustthe finite element model

A thorough understanding of the experimental dataused for the updating process is critical

A clear definition of what is meant by an improvedmodel is necessary

The analyst has a tremendous responsibility in

identifying which areas of the model are to beupdated and which sets of modes are the best modesto use in the updating process

correlation topics 122901 1

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