Comparison of Test and Analysis

Modal Analysis and Testing

S. Ziaei-Rad

S. Ziaei-Rad

Objectives

Objectives of this lecture:

• to review some of the different types of structural models which are derived from modal tests;

• to discuss some of the applications to which the model obtained from a modal test can be put;

• to prepare the way for some of the more advanced applications of test-derived models.

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Applications Of Test-derived Models

•comparison with theoretical model

•correlation with theoretical model

•correction of theoretical model

•structural modification analysis

•structural assembly analysis

•structural optimisation

•operating response predictions

•excitation force determination

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Strategy For Model Validation

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Types Of Mathematical Model

Spatial model Modal model Response model

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Derivation Of Model From Modal Test

Step 1 - measure Step2 - modal analysis

Step 3 - model

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Theory/Experiment ComparisonComparisons possible:

(a) FRFs b) Modal Properties

Modal Properties-Natural Frequencies-Mode Shapes

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Comparison of Modal Properties1- Comparison of Natural Frequencies

Natural Frequencies

Standard Comparison ji

jijiNFD

,min

),(

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Comparison of Modal Properties

Mode shapes

2- Mode Shapes (Graphical)

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Comparison of Modal Properties

Modes 1,2 & 3(systematic error)

Modes 1,2 & 3(remeasured)

2- Mode Shapes (Graphical)

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Correlation Of Modal Properties2- Mode Shapes (numerical correlation)

Modal scale factor (MSF) - slope of best-fit line from {}1 vs {}2 plot

Or if we take the experimental mode as reference

If

If

n

jjXjX

n

jjXjA

XAMSF

1

*

1

*

)()(

)()(

),(

n

jjAjA

n

jjAjX

XAMSF

1

*

1

*

)()(

)()(

),(

1),(),(}{}{ AXMSFXAMSFAX

/1),(,   ),(}{}{ AXMSFXAMSFAX

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Correlation Of Modal Properties2- Mode Shapes (numerical correlation)

Mode Shape Correlation Coefficient, or Modal Assurance Criterion (MAC)-scatter of points about best fit line:

Or

If

If

MAC A X

X j A jj

n

X j X jj

n

A j A jj

n( , )

.

*

* *

1

2

1 1

}{}{}{}{

}{}{),(

2

AT

AXT

X

AT

XXAMAC

1),(),(}{}{ AXMACXAMACAX

1),(),(}{}{ AXMACXAMACAX

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MAC Correlation Between Two Sets Of Modes

1 2 3 4 5 6 7

S1

S3

S5

S7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Experimental Mode Number

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Natural Frequency Plot For CorrelatedModels

.. paired by frequencies .. paired by CMPs

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Data for Correlated ModesFREQUENCY MATCHED CORRELATED UPDATED

Exp. Exp freq. FE FE. freq. diff. FE FE freq. diff. Updated Updat. FE diff.mode (Hz) mode (Hz) (%) mode (Hz) (%) FE mode freq. (Hz) (%) no. no. no. no.

1 9.2 7 10.5 14 7 10.5 15 7 8.9 -32 14.5 8 9 18.3 26 8 14.7 13 16.1 9 17.1 6 10 20.3 26 10 17 44 17.0 10 18.3 8 8 17.1 0 9 15.8 -75 21.5 11 20.3 -5 11 26.5 23 11 22.6 56 27.0 12 26.5 -2 14 43.4 61 12 26 -47 30.2 13 13 38.8 28 13 32.8 98 35.3 14 38.8 10 17 71.7 103 14 33.9 -49 40.8 15 43.4 6 16 60.4 48 15 45.9 13

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Effectiveness Of The Correlation Process

Some features of the MAC (which affect its effectiveness):• lack of scaling (so not a true orthogonality measure)• inadequate selection of DOFs• inappropriate selection of DOFs

Modified versions of the MAC:• the AutoMAC• the Mass-Normalised MAC• the Selected-DOF MAC

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Inadequate Selection of Dofs in Mac

MAC using all DOFs MAC using subset of DOFs

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Use of Automac to Check Adequacy of DOFs

AUTOMAC is the MAC computed from the correlation ofa set of vectors with themselves

AIUTOMAC using all DOFs AIUTOMAC using subset of DOFs

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Use of Automac to Check Adequacy of DOFs

a- Automac(A) for full set of 102 DOFsb- Automac(A) for reduced set of 72 DOFsc- Automac(A) for reduced set of 30 selected DOFsd- Automac(X) for reduced set of 30 selected DOFse- MAC for reduced set of 30 DOFs

A

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Normalised Version Of The MacMass-normalised MAC can be computed using theanalytical mass matrix from:

NCO A X

W

W W

X

T

A

X

T

X A

T

A

( , ).

2

-Weighting matrix W, can be provided either by mass or stiffness matrices of the system.-The difficulty is the reduction of the mass or stiffness matrices to the size of the measured DOF-A Guyan type or a SEREP-based reduction can be used. If SEREP used then a pseudo-mass matrix of the correct size can be calculated as ][][][ TRM

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Normalised Version Of The Mac

Approximate mass-normalised MAC (SCO) can be

computed using the active modal properties only:

SCO A X

X

T T

A

X

T T

X A

T T

A

( , ).

2

][][][ TRM

SCO = SEREP-Cross-Orthogonality

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Normalised Mac - Features

AUTOMAC for test case AUTOSCO for test case

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Error Location - The COMAC-COMAC is a means of identifying which DOFs display the

best or the worse correlation across the structure.

-COMAC uses the same data as is used to compute the MAC

but it performs the summation of all contributions (one from

each DOF for each mode pair) across all the mode pairs

instead of across all the DOFs (as is done in the MAC)

-COMAC is defined as:

COMAC A Xi i

X il A ill

L

X ill

L

A ill

L( , ).

2

1

2

1

2

1

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COMAC - Example 1

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COMAC - Example 2

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Correlation Of Other Parameters:Frequency Response Functions

The Assurance Criterion concept can be applied toany pairs of corresponding vectors (not only modeshape vectors) including FRFs - to give the FRAC -

and also to vectors of Operating Deflection Shapes, insituations where modal properties are difficult to obtain

FRAC A XH H

H H H Hj i

X i

T

A j

X i

T

X i A j

T

A j

( ( ), ( )).

2

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Correlation Of Other Parameters:Frequency Response Functions

L

i

L

iijkAijkX

L

iijkAijkX

k

HH

HHjFRAC

1 1

22

1

2*

)()(

)()()(

Frequency Response Assurance Criterion:

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Example Of FRAC Plot