Data Presentation Schemes for Selection and Identification of Modal
Parameters
Allyn W. Phillips, PhD, Research Associate Professor
Randall J. Allemang, PhD, Professor
P.O. Box 210072, University of Cincinnati, Cincinnati, OH 45221-0072
Nomenclature
Eigen solution state vector
φ
ψ
λ
n
MAC
pwMAC
pw
N
XH
Mode shape (vector)
System pole or modal frequency
Left model order
Modal Assurance Criteria
Pole Weighted MAC
Pole weight
Pole weight order
Hermitian (complex conjugate) transpose of vector X
ABSTRACT
Although the theories behind various methods of estimating modal parameters normally receive the most
attention, it is the presentation of the results that is fundamental to the selection of an optimal set of modal
parameters from the large number of solutions represented by the process of allowing the model order to vary in
the problem solution (iteration over model order). Many software implementations provide only a limited set of
presentation tools, but access to more tools may be very beneficial. This paper reviews several of the various
data presentation tools, and their relative effectiveness, for selection of a set of modal parameters. These include:
coefficient normalization, stability and consistency diagrams, pole (LaPlace) surface plots, pole surface density
plots, and a new companion matrix state vector (phi) MAC, called pwMAC, method.
1.0 Background
The history of developing visualization tools for evaluating modal parameters began at least 25 years ago. When
least squares and nonlinear (iterative) mathematical tools were first used to aid in the estimation of modal
parameters, the development of an error chart was utilized to visualize how the error between the synthesized
frequency response function (FRF) or impulse response function (IRF) was reduced as a function of increasing
model order (number of modes) used in the identification model [1]. Due to the limited availability of graphical
presentations, these plots were generally bar graphs utilizing printed characters to represent linear or logarithmic
changes of the error. Today, the error chart would be presented as in Figure 1.
As numerical tools such as singular value decomposition, condition number and matrix rank computation became
available in the early 1980s, these tools were used to present a rank chart. The rank chart looked similar to the
error chart but was a presentation of the number of independent parameters (effectively modes) in a given matrix
equation developed for a specific, maximum model order [2]. Once again, if the rank indicated that the matrix was
dominated by a limited number of independent parameters less than the maximum model order, this was a good
indication of the number of modes that could be found in the measured data. Singular values were used to
calculate the condition number as a more sensitive indicator of the number of dominating parameters (effectively
modes). Due to the limited availability of graphical presentations, these plots were generally bar graphs utilizing
printed characters to represent linear or logarithmic changes of the error. Today, the rank chart would be
presented as in Figure 2.
Relative Rank and Reciprocal Condition Number
0
10
Error Chart
0
Relative Rank
1/Condition
10
-2
10
-2
10
-4
-4
10
10
-6
10
-6
10
-8
10
-8
10
-10
10
0
10
20
30
Number of Modes
40
50
-10
10
Figure 1: Error Chart
0
5
10
15
20
25
Model Iteration
30
35
40
Figure 2: Rank Chart
By the end of the 1980s, more advanced graphical presentation capabilities were becoming routinely available to
the users of modal parameter estimation software and the development of more useful visual presentation
methods were being evaluated. In particular, the stability diagram was developed which is still in wide use
today. The stability diagram is a visual presentation of the frequency location of the estimated modal frequencies,
as a function of increasing model order (number of modes), plotted on a background of an FRF, an auto-moment
of FRFs, a complex mode indication function (CMIF) [3], or a multi-variate mode indication function (MvMIF) [4].
Symbols are used to represent whether successive estimates of modal parameters, using the next higher model
order, give a result for a modal frequency that is realistic (reasonable damping), stable frequency (damped natural
frequency within a tolerance, for example 1 percent), stable pole (stable frequency plus a stable fraction of critical
damping, for example 5 percent) and stable modal vector (stable modal frequency plus a stable estimate of modal
vector, for example modal assurance criterion of 0.95). An example of a modern stability diagram is given in
Figure 3. The computation of stability is generally successive in that modal vector stability is not evaluated unless
the pole is stable and the pole stability is not evaluated unless the modal frequency is stable, etc. Note that the
usefulness of the stability diagram will depend upon actual values of tolerance that is used for each level of
stability.
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
Consistency Diagram
Model Iteration
Model Iteration
Stability Diagram
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
cluster
pole & vector
pole
frequency
conjugate
0
500
Figure 3: Stability Diagram
1000
1500
Frequency (Hz)
2000
2500
non realistic
1/condition
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 4: Consistency Diagram
An extension of the stability diagram, the consistency diagram, was developed shortly after the stability diagram
was in wide use. It has long been recognized that the solutions for the modal frequencies were sensitive to many
numerical issues, including the normalization of the linear, least squares equation that is fundamental to the
estimation of modal parameters [2]. This issue has generally been considered a trade secret by commercial
vendors until recently where several researchers have attempted to theoretically explain why different
normalizations yielded different stabilization diagrams [5-8]. The consistency diagram looks very similar to the
stability diagram but includes variations in solution method (different normalizations or inclusions of different
residuals, for example) in addition to increasing model order. Figure 4 is an example of a consistency diagram
for which several different solution parameters are varying.
More recently [9], a more discriminating visual presentation, the pole surface consistency plot or pole surface
density plot, is being successfully utilized. The pole surface consistency plot involves plotting all of the modal
frequencies found in a stability or consistency diagram in the complex frequency plane (s-plane). Primarily, the
modal frequencies that are in the second quadrant of the complex frequency plane are those that exhibit positive
frequency with appropriate damping. Clusters of modal frequencies in this plot, together with the same symbols
used in the consistency diagram give clear indications of modal frequencies that are of interest. The statistical
distribution of the cluster can also yield error bound information associated with each modal frequency chosen.
The pole surface consistency plot is shown in Figure 5. In order to clear up this plot, modal frequencies can be
included only when they represent a certain level of stability. To utilize the pole surface information in a more
discriminating manner, the density of the poles in a given region can be plotted in a pole surface density plot.
This plot, particularly when limited to densities greater than one, gives a very clear visualization of modal
frequencies that have been found consistently through all solution conditions evaluated. Figure 6 is an example
of a pole surface density plot.
Pole Surface Density
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
Zeta (%)
Zeta (%)
Pole Surface Consistency
2
1
0.8
0.6
cluster
pole & vector
0.6
79
frequency
0.4
Density
0.4
pole
0.2
conjugate
0
non realistic
1/condition
1
0.8
0
500
1000
1500
Imaginary (Hz)
2000
2500
Figure 5: Pole Surface Consistency
40
0.2
20
0
1
0
500
1000
1500
Imaginary (Hz)
2000
2500
Figure 6: Pole Surface Density
1.1 Application Issues
The critical use of tolerances and floors in the visual presentations is very important in selecting modal
frequencies correctly. In general, in the stability and consistency diagrams, changing the tolerances can yield
stability and consistency diagrams that are much clearer and/or cleaner. The risk is that modal frequencies
associated with realistic structural modes may be excluded when the tolerances are manipulated. This happens
when realistic structural modes are not well represented in the measured FRF data. This can easily happen when
a minimum number of references are utilized where the included references are near nodes of certain modes or
when insufficient response information (spatially) is included to weight the realistic structural modes in the least
squares solution methods utilized. The estimates of these modal frequencies will often be poor or erratic and will
confuse the visualization methods utilized. This raises the philosophical question of whether it is desirable to
have no information about these modes or to have inaccurate information about these modes.
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
Consistency Diagram
Model Iteration
Model Iteration
Consistency Diagram
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
cluster
pole & vector
pole
frequency
conjugate
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 7: Consistency Diagram – Standard Tolerances
non realistic
1/condition
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
500
1000
1500
Frequency (Hz)
2000
2500
Figure 8: Consistency Diagram – Relaxed Tolerances
Figures 7 and 8 represent two consistency diagrams for the same data. Figure 7 represents a standard choice of
tolerances (1,5,95 percent) and Figure 8 represents a relaxed set of tolerances (5,20,80 percent) for a relatively
simple structure with repeated roots (circular plate) that is often used to evaluate modal parameter estimation
methodology. In Figure 8, only those modal frequencies that are stable in a modal vector sense are included in
the consistency diagram yielding a consistency diagram that is very easy to utilize when selecting modal
frequencies. Figures 9 and 10 show Figures 7 and 8 expanded in the region of 2300 Hertz. From these two
plots, it is clear that the repeated root around 2320 Hertz is poorly identified in Figure 10 compared to Figure 9.
cluster
pole & vector
pole
frequency
conjugate
non realistic
cluster
pole & vector
pole
frequency
conjugate
2280
1/condition
Consistency Diagram
Model Iteration
Model Iteration
Consistency Diagram
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
2290
2300
2310
2320
Frequency (Hz)
2330
2340
2350
Figure 9: Consistency Diagram – Standard Tolerances
non realistic
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
2280
1/condition
2290
2300
2310
2320
Frequency (Hz)
2330
2340
2350
Figure 10: Consistency Diagram – Relaxed Tolerances
Similarly, the floor that is used in the pole surface density plots can affect the ability to quickly and easily use the
pole surface density plot to identify pole clusters. Figure 11 shows a default plot with the floor set to show any
pole surface density (1+). Figure 12 is the same information but utilizing a floor of three (3) which will only show
pole surface densities of three or greater (3+). This plot includes all of the default stability tolerances but quickly
identifies the clusters of modal frequency that should be examined.
79
Pole Surface Density
2
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
Zeta (%)
Zeta (%)
Pole Surface Density
2
1
0.8
0.6
79
0.2
20
0
1
0.6
0.4
Density
Density
0.4
40
1
0.8
0
500
1000
1500
Imaginary (Hz)
Figure 11: Pole Surface Density – All Poles
2000
2500
40
0.2
20
0
3
0
500
1000
1500
Imaginary (Hz)
2000
2500
Figure 12: Pole Surface Density – Floor = 3
2.0 New Visualization/Selection Method ( pwMAC )
A new visualization method has been developed and evaluated that combines the characteristics of frequency,
pole and modal vector consistency. This method is known as the pole weighted modal assurance criterion
( pwMAC ) and was initially evaluated as part of a new, autonomous modal identification method [10]. The
following presentation starts with the fundamental idea of computing the modal assurance criterion (MAC) [11-12]
between associated state vectors and then proceeds to separating the pole weighting computation form the MAC
computation. The state vectors are the actual eigenvectors found in the UMPA formulation [13] of most modal
identification algorithms. Each state vector involves a stacked repetition of a modal vector weighted by the
associated modal frequency raised to successively higher powers.
2.1 Theory
The equation for evaluating the pole weighted MAC (
standard MAC:
MAC (ψ 1 ,ψ 2
pwMAC ) can be derived from the expression for the
(ψ
)=
(ψ
Replacing the vectors, ψ 1 & ψ 2 , with the state vectors,
φ1
pwMAC (φ1 , φ2 ) =
By recognizing that the state vector,
modal frequency,
ψ 2 )(ψ 2Hψ 1 )
H
1
&
φ2 , results in:
(φ φ )(φ φ )
(φ φ )(φ φ )
H
1
2
H
2 1
H
1 1
H
2 2
φ , is simply the modal vector, ψ
λ , the expression for the pwMAC
(1)
ψ 1 )(ψ 2Hψ 2 )
H
1
(2)
, multiplied by successive powers of the
can be significantly reduced. Substituting for
φ:
 λ n {ψ } 
 n −1

λ {ψ }
{φ} =  # 
 λ 1 {ψ } 


 {ψ } 
(3)
And recognizing that:

n

{φ1} {φ2 } =  ∑ ( λ1*λ2 )  {ψ 1} {ψ 2 }
H
 k =0
k
H

(4)
The expression for pwMAC reduces to:
k  n
k 
 n
*
λ
λ
λ2*λ1 ) 
H
H
(
)
(
∑
∑
1 2



φ φ )(φ φ )  k =0
(
ψ 1} {ψ 2 }{ψ 2 } {ψ 1}
{
k =0



=
pwMAC (φ1 , φ2 ) = H
(φ1 φ1 )(φ2H φ2 )  ∑n ( λ1*λ1 )k   ∑n ( λ2*λ2 )k  {ψ 1}H {ψ 1}{ψ 2}H {ψ 2}
 k =0
  k =0

H
1
2
H
2 1
(5)
It is now observed that the pwMAC is a pole weighting function multiplied by the standard MAC. Since the
function is separable, it should be recognized that the pole weight order does not need to be identical to the actual
model order. Therefore, the pole weighting can be defined in terms of the actual pole and the desired effective
model order, as follows:
 N * k  N * k 
 ∑ ( λ1 λ2 )   ∑ ( λ2 λ1 ) 
  k =0

pw ( λ1 , λ2 , N ) =  kN=0
N
k
k



*
*
 ∑ ( λ1 λ1 )  ∑ ( λ2 λ2 ) 
 k =0
 k =0

Using this definition, the pwMAC reduces to:
pwMAC (φ1 , φ2 ) = pw ( λ1 , λ2 , N ) ⋅ MAC (ψ 1 ,ψ 2 )
(6)
(7)
This formulation makes it clear that the original complete (full) state vector ( φ ) is not actually needed for the
pwMAC calculation and further, that the calculation can be applied to the results of any modal parameter
estimation scheme. (Note also that the special case of N = 0 results in simply the ordinary MAC calculation.)
For numerical reasons, the pole weight ( pw ) will rarely be calculated using the frequency domain poles directly,
(except for small values of N ). Rather, the true system poles are converted to the Z domain using:
λz = exp ( λ ⋅ ∆t )
(8)
Where: ∆t = f max and
f max is a real value greater than the largest absolute frequency in the set of λ ' s .
(ie. f max > max( abs (imag (λ )))
This results in a significantly numerically better conditioned solution and effectively synthesizes the state vector
that would have resulted had a time domain parameter estimation method been used.
2.1 Pole Weighting Characteristic
The effectiveness of this variable pole weighting at identifying the comparable poles and vectors between two
independent sets can be seen in the following figures. The figures on the left (Figures 13, 15, 17) are the standard
MAC , the pole weighted MAC for λ 5 , and the pole weighted MAC for λ 15 . The figures on the right (Figures 14,
16, 18) are the same figures with a MAC floor of 0.95 . In all cases, each plot has used the same set of poles
and vectors for presentation, only the pole weighting has changed.
MAC - (>0.95)
60
55
55
50
50
45
45
40
40
Left Pole Index
Left Pole Index
MAC
60
35
30
25
35
30
25
20
20
15
15
10
10
5
5
5
10
15
20
25
30
35
Right Pole Index
Figure 13: Standard MAC
40
45
50
55
60
5
10
15
20
25
30
35
Right Pole Index
40
45
Figure 14: Standard MAC – Floor = 0.95
50
55
60
Pole Weighted MAC - λ 5
55
55
50
50
45
45
40
40
35
30
25
30
25
20
15
15
10
10
5
5
10
15
20
25
30
35
Right Pole Index
40
45
50
55
60
Figure 15: : Pole Weighted MAC – N = 5
5
50
45
45
40
40
Left Pole Index
55
50
35
30
25
15
10
10
5
5
25
30
35
Right Pole Index
40
45
Figure 17: : Pole Weighted MAC – N = 15
40
45
50
55
60
Pole Weighted MAC - λ 15 - (>0.95)
25
20
20
25
30
35
Right Pole Index
30
15
15
20
35
20
10
15
60
55
5
10
Figure 16: Pole Weighted MAC – N = 5, Floor = 0.95
Pole Weighted MAC - λ 15
60
Left Pole Index
35
20
5
Pole Weighted MAC - λ 5 - (>0.95)
60
Left Pole Index
Left Pole Index
60
50
55
60
5
10
15
20
25
30
35
Right Pole Index
40
45
50
55
60
Figure 18: : Pole Weighted MAC – N = 15, Floor = 0.95
Although the MAC plots clearly show the effect and value of the pole weighting, the direct usefulness of such plots
is limited.
2.2 Pole Weighting MAC Consistency
By using the pole weighted MAC calculation to identify comparable poles and vectors and presenting those
pwMAC results (over a specified threshold) as a consistency diagram, a clearer indication of global (ie.
simultaneous pole and vector) consistency is provided (Figures 19-22).
cluster
pole & vector
pole
frequency
conjugate
non realistic
Consistency Diagram
40
35
30
Model Iteration
Model Iteration
Consistency Diagram
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
pweight = λ 15
pwMAC > 0.99
pwMAC > 0.90
1000
1500
Frequency (Hz)
2000
2500
Figure 19: Traditional - Consistency
15
5
pwMAC > 0.94
500
20
10
pwMAC > 0.97
0
1/condition
25
0
pwMAC > 0.85
0
500
1.8
1.8
1.6
1.6
1.4
1.4
1.2
1.2
Zeta (%)
Zeta (%)
2
1
0.8
frequency
0.4
pwMAC > 0.97
0.2
pwMAC > 0.94
0.2
pwMAC > 0.90
conjugate
0
non realistic
1/condition
0
500
1000
1500
Imaginary (Hz)
2000
2500
Figure 21: Traditional – Pole Surface Consistency
1
0.8
0.6
pweight = λ 15
pwMAC > 0.99
0.4
pole
2500
Pole Surface Consistency
2
0.6
pole & vector
2000
Figure 20: Pole Weighted MAC - Consistency
Pole Surface Consistency
cluster
1000
1500
Frequency (Hz)
pwMAC > 0.79
0
pwMAC > 0.85
0
500
1000
1500
Imaginary (Hz)
pwMAC > 0.79
2000
2500
Figure 22: Pole Weighted MAC – Pole Surface
Consistency
The following figures (Figures 23-28) demonstrate the effectiveness of this presentation scheme. While the first
two figures (Figures 23 & 24) both give a clear indication of the two poles, the pwMAC consistency shows the
degradation in the estimated poles at higher model order. Contrariwise, the second two figures (Figures 25 & 26)
show the other issue. Whereas both figures provide clear indication of three poles, the traditional consistency
diagram indicates that the vector never attains consistency; however, the pwMAC consistency clearly indicates
that the higher model order estimates are more consistent and are to be preferred. This result is also shown in the
pole surface plots (Figures 27 & 28), where the pwMAC density more clearly indicates the pole location.
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
Consistency Diagram
40
2000
35
30
Model Iteration
Model Iteration
Consistency Diagram
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
pweight = λ 15
pwMAC > 0.99
pwMAC > 0.97
pwMAC > 0.94
pwMAC > 0.90
2005
2010
2015
2020 2025 2030
Frequency (Hz)
Figure 23: Traditional - Consistency
2035
2040
2045
2050
pwMAC > 0.85
pwMAC > 0.79
25
20
15
10
5
0
2000
2005
2010
2015
2020 2025 2030
Frequency (Hz)
2035
2040
Figure 24: Pole Weighted MAC - Consistency
2045
2050
cluster
pole & vector
pole
frequency
conjugate
non realistic
2300
1/condition
Consistency Diagram
40
35
30
Model Iteration
Model Iteration
Consistency Diagram
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
pweight = λ 15
pwMAC > 0.99
pwMAC > 0.90
2315
2320 2325 2330
Frequency (Hz)
2335
2340
2345
2350
Figure 25: Traditional - Consistency
15
5
pwMAC > 0.94
2310
20
10
pwMAC > 0.97
2305
25
0
2300
pwMAC > 0.85
2305
2310
pole
frequency
conjugate
non realistic
1/condition
0.18
0.178
0.178
0.176
0.176
0.174
0.174
0.172
0.172
0.17
0.168
0.166
pweight = λ 15
pwMAC > 0.99
0.164
pwMAC > 0.97
pwMAC > 0.94
0.162
0.16
2322
pwMAC > 0.90
2322.1
2322.2
2322.3
2322.4
Imaginary (Hz)
2335
2340
2345
2350
Pole Surface Consistency
0.18
Zeta (%)
Zeta (%)
pole & vector
2320 2325 2330
Frequency (Hz)
Figure 26: Pole Weighted MAC - Consistency
Pole Surface Consistency
cluster
2315
pwMAC > 0.79
2322.5
2322.6
Figure 27: Traditional – Pole Surface Consistency
2322.7
pwMAC > 0.85
pwMAC > 0.79
0.17
0.168
0.166
0.164
0.162
0.16
2322
2322.1
2322.2
2322.3
2322.4
Imaginary (Hz)
2322.5
2322.6
2322.7
Figure 28: Pole Weighted MAC – Pole Surface
Consistency
3.0 Conclusions
Historically, as computational and graphical computing power has increased, new calculation and presentation
techniques have been developed for identifying the true system model. In this paper, a new companion matrix
state vector (phi) MAC, called pwMAC (pole weighted MAC) has been developed. This approach has been
shown to be extremely effective at identifying consistent modal parameter (pole and vector) estimates from
different model iterations and/or solutions. As formulated, this technique represents a natural progression in
parameter presentation/identification schemes, and as such, is easily integrated into existing consistency diagram
presentations. It thus provides a powerful new tool for the user to identify the estimates that truly represent the
underlying system model.
4.0 References
1. Brown, D.L., Allemang, R.J., Zimmerman, R.D., Mergeay, M. "Parameter Estimation Techniques for
Modal Analysis", SAE Paper Number 790221, SAE Transactions, Volume 88, pp. 828-846, 1979.
2. Allemang, R.J., Brown, D.L., "Modal Parameter Estimation" Experimental Modal Analysis and Dynamic
Component Synthesis, USAF Technical Report, Contract No. F33615-83-C-3218, AFWAL-TR-87-3069,
Vol. 3, 1987, 130 pp.
3. Shih, C.Y., Tsuei, Y.G., Allemang, R.J., Brown, D.L., "Complex Mode Indication Function and Its
Application to Spatial Domain Parameter Estimation", Mechanical System and Signal Processing, Vol.
2, No. 4, pp. 367-377, 1988.
4. Williams, R., Crowley, J., Vold, H., "The Multivariable Mode Indicator Function in Modal Analysis",
Proceedings, International Modal Analysis Conference, 1985, pp. 66-70.
5. Verboven, P., Guillaume, P., Cauberghe, B., Parloo, E., Vanlanduit, S., "Stabilization Charts and
Uncertainty Bounds for Frequency Domain Linear Least Squares Estimators", Proceedings, International
Modal Analysis Conference, 10 pp., 2003.
6. Hung, Chen-Far, Ko, Wen-Jiunn, Peng, Ken-Tung, “Identification of Modal Parameters from Measured
Input and Output Data Using a Vector Backward Auto-Regressive with Exogeneous Model”, Journal of
Sound and Vibration, Vol. 276, 2004, pp. 1043-1063.
7. Verboven, P., Cauberghe, B., Vanlanduit, S., Parloo, E., Guillaume, P., "The Secret Behind Clear
Stabilization Diagrams: The Influence of the Parameter Constraint on the Stability of the Poles",
Proceedings, Society of Experimental Mechanics (SEM) Annual Conference, 17 pp., 2004.
8. Cauberghe, B., "Application of Frequency Domain System Identification for Experimental and Operational
Modal Analysis", PhD Dissertation, Department of Mechanical Engineering, Vrije Universiteit Brussel,
Belgium, 259 pp., 2004.
9. Phillips, A.W., Allemang, R.J., Pickrel, C.R., "Clustering of Modal Frequency Estimates from Different
Solution Sets", Proceedings, International Modal Analysis Conference, pp. 1053-1063, 1997.
10. Brown, D.L., Phillips, A.W., Allemang, R.J., “A First Order, Extended State Vector Expansion Approach to
Experimental Modal Parameter Estimation”, Proceedings, International Modal Analysis Conference, 11
pp., 2005.
11. R.J. Allemang, D.L. Brown, "A Correlation Coefficient for Modal Vector Analysis", Proceedings,
International Modal Analysis Conference, pp.110-116, 1982.
12. Allemang, R.J., “The Modal Assurance Criterion (MAC): Twenty Years of Use and Abuse”, Proceedings,
International Modal Analysis Conference, 9 pp., 2002.
13. Allemang, R.J., Phillips, A.W., “The Unified Matrix Polynomial Approach to Understanding Modal
Parameter Estimation: An Update”, Proceedings, International Seminar on Modal Analysis (ISMA), 36
pp, 2004.
14. Strang, G., Linear Algebra and Its Applications, Third Edition, Harcourt Brace Jovanovich Publishers,
San Diego, 1988, 505 pp.
15. Lawson, C.L., Hanson, R.J., Solving Least Squares Problems, Prentice-Hall, Inc., Englewood Cliffs,
New Jersey, 1974, 340 pp.
16. Jolliffe, I.T., Principal Component Analysis, Springer-Verlag New York, Inc., 1986, 271 pp.
17. Ljung, Lennart, System Identification: Theory for the User, Prentice-Hall, Inc., Englewood Cliffs, New
Jersey, 1987, 519 pp.
18. Wilkinson, J.H., The Algebraic Eigenvalue Problem, Oxford University Press, 1965, pp. 12-13.