Transcript Comparison of Test and Analysis - Saeed Ziaei-Rad

Comparison of Test and Analysis
Modal Analysis and Testing
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
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Response model
Derivation Of Model From Modal Test
Step 1 - measure
Step2 - modal analysis

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Step 3 - model
Theory/Experiment Comparison
Comparisons possible:
(a) FRFs
b) Modal Properties
Modal Properties
-Natural Frequencies
-Mode Shapes
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Comparison of Modal Properties
1- Comparison of Natural Frequencies
Standard Comparison
i   j
NFD(i, j ) 
mini ,  j 
Natural Frequencies
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Comparison of Modal Properties
2- Mode Shapes (Graphical)
Mode shapes
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Comparison of Modal Properties
2- Mode Shapes (Graphical)
Modes 1,2 & 3
(remeasured)
Modes 1,2 & 3
(systematic error)
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Correlation Of Modal Properties
2- Mode Shapes (numerical correlation)
n
Modal scale factor (MSF)
MSF( A, X ) 
- slope of best-fit line from
{f}1 vs {f}2 plot
*
(

)
(

)
 X j Aj
j 1
n
*
(

)
(

)
 Aj Aj
j 1
n
Or if we take the experimental mode MSF( A, X ) 
as reference
 (
j 1
n
) j ( X )*j
*
(

)
(

)
 X j X j
j 1
If { X }  { A}  MSF( A, X )  MSF( X , A)  1
If { X }   { A}  MSF( A, X )   , MSF( X , A)  1 / 
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A
Correlation Of Modal Properties
2- Mode Shapes (numerical correlation)
2
n
Mode Shape Correlation
Coefficient, or Modal
MAC( A, X ) 

Assurance Criterion (MAC)

-scatter of points about best fit line:

    
j 1
X j
n



*
*
j1  X  j  X  j  . j1  A  j  A  j 
n
{ X } { A }
T
Or
MAC( A, X ) 
{

2

T
T
}
{

}
{

}
{ A }
X
X
A
If { X }  { A}  MAC( A, X )  MAC( X , A)  1
If { X }   { A}  MAC( A, X )  MAC( X , A)  1
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*
A j
MAC Correlation Between Two Sets Of
Modes
Experimental Mode Number
N
1
2
3
4
5
6
7
1
0.97
0
0
0
0
0
0
2
0
0.97
0
0
0
0
0
3
0
0
0.95
0.02
0
0
0
4
0
0
0
0.97
0
0
0
5
0
0
0
0
0.56
0.38
0
6
0
0
0
0.01
0.06
0.87
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
S7
S5
0.1
0
0
S3
1
7
0
0
0
0
0
0
0.95
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2
3
4
5
S1
6
7
Natural Frequency Plot For Correlated
Models
.. paired by frequencies
.. paired by CMPs
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Data for Correlated Modes
Exp.
mode
no.
Exp freq.
(Hz)
1
2
3
4
5
6
7
8
9
9.2
14.5
16.1
17.0
21.5
27.0
30.2
35.3
40.8
FREQUENCY MATCHED
FE
FE. freq. diff.
(%)
mode (Hz)
no.
7
8
9
10
11
12
13
14
15
10.5
14
17.1
18.3
20.3
26.5
6
8
-5
-2
38.8
43.4
10
6
CORRELATED
FE
FE freq. diff.
(%)
mode (Hz)
no.
7
9
10
8
11
14
13
17
16
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10.5
18.3
20.3
17.1
26.5
43.4
38.8
71.7
60.4
15
26
26
0
23
61
28
103
48
UPDATED
Updated Updat. FE diff.
FE mode freq. (Hz) (%)
no.
7
8
10
9
11
12
13
14
15
8.9
14.7
17
15.8
22.6
26
32.8
33.9
45.9
-3
1
4
-7
5
-4
9
-4
13
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 of
a 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 DOFs
b- Automac(A) for reduced set
of 72 DOFs
c- Automac(A) for reduced set
of 30 selected DOFs
d- Automac(X) for reduced set
of 30 selected DOFs
e- MAC for reduced set of 30
DOFs
A
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Normalised Version Of The Mac
Mass-normalised MAC can be computed using the
analytical mass matrix from:
 X  W  A
NCO( A, X ) 
T
T

W

.









W  A
 X
X  
A
T
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 [M R ]  []T []
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Normalised Version Of The Mac
Approximate mass-normalised MAC (SCO) can be
computed using the active modal properties only:
 X     A
SCO( A, X ) 
T
T

T
T





.



 X       X   A      A
T
T
[M R ]  []T []
SCO = SEREP-Cross-Orthogonality
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
2
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:
L
COMAC( Ai , X i ) 
    
l 1
L
A il
L
  X il .   A il
l 1
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X il
2
2
l 1
2
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 to
any pairs of corresponding vectors (not only mode
shape vectors) including FRFs - to give the FRAC -
 H   H  
), X ( )) 
H   H  . H   H  
2
T
FRAC( A( j
X
i
i
A
j
T
T
X
i
X
i
A
j
and also to vectors of Operating Deflection Shapes, in
situations where modal properties are difficult to obtain
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A
j
Correlation Of Other Parameters:
Frequency Response Functions
Frequency Response Assurance Criterion:

L
FRAC( j ) k 
i 1

L
i 1
X
X

H jk ( i )  A H
H jk ( i ) 
2

*
jk
( i )
L
i 1
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A
H
jk

2
( i )

2
Example Of FRAC Plot
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