The Impact of Measurement Condensation and Modal Participation Vector
Normalization on the Estimation of Modal Vectors and Scaling
Randall J. Allemang, PhD, Professor
Allyn W. Phillips, PhD, Research Associate Professor
P.O. Box 210072, University of Cincinnati, Cincinnati, OH 45221-0072
Nomenclature
Frequency [Hz]
f
ω
λr
Circular frequency [rad/sec]
System pole or modal frequency for mode r
[ A]r
{ψ }r
†
{ψ }r
Residue matrix for mode r
 L  r
 H (ω ) 
participation vector for mode r
 H (ω ) 
 H 
[T ]
H
[X ]
ℜ( X )
ℑ( X )
Mode shape for mode r
Mode shape for condensed, or virtual, mode r
Frequency response function (FRF) matrix
†
Condensed, or virtual, frequency response function matrix
Rectangular (2d) frequency response function matrix
FRF condensation matrix
Hermitian (complex conjugate transpose) of matrix X
Real part of complex matrix X
Imaginary part of complex matrix X
ABSTRACT
Whenever multiple input, multiple output (MIMO) frequency response function (FRF) data is utilized in a multiple
reference modal parameter estimation algorithm, measurement condensation may be utilized to efficiently
process the FRF data relative to the modal parameter estimation (MPE) algorithm. Additionally, the modal
participation vectors, estimated in the eigenvalue portion of the MPE process, are used to weight the solution for
the modal vectors and scaling. Both of these procedures may have an effect on the final estimation of the modal
vector and scaling, depending upon how the condensation vectors or modal participation vectors are normalized.
When elements of the condensation matrix and/or the modal participation vector(s) are not/are normalized to a
real number, the impact on the modal vector and scaling estimates can be severe. This paper presents a study of
the impact of how changes to the condensation vectors and modal participation vectors can impact the modal
vectors and scaling. Specific changes to the condensation vectors that are studied include several methods of
real vector normalization. Specific changes to modal participation vector coefficients that are studied include
truncating small coefficients and several methods of real vector normalization.
1.0 Background
Modern modal parameter estimation algorithms that involve multiple input, multiple output (MIMO) frequency
response function (FRF) data all have similar characteristics in the way in which the FRF data is processed,
regardless of the algorithm involved [1-3]. Two common aspects of every algorithm are measurement
condensation before the FRF data is processed by the algorithm and modal participation vector weighting as part
of the modal vector and modal scaling estimation process within the algorithm. The estimation of the
condensation matrix and the modal participation matrix both occur as a result of a numerical eigenvalue (or
equivalent) decomposition procedure that yield complex valued eigenvectors as a general result. The important
characteristic of the eigenvalue decomposition is that the eigenvectors are unique to within an arbitrary complex
constant. This means that the eigenvectors used in any transformation, if normalized by a consistent magnitude
and phase approach, will not introduce any arbitraryness in the transformation process. In the condensation case,
there is no expectation concerning the nature of the eigenvectors (real or complex valued). However, in the
modal participation case, if the FRF data represents a proportionally damped system, the eigenvectors should be
real valued vectors. For the modal participation case, if the FRF data represents a non-proportionally damped
system or when errors are present in the data, the eigenvectors should be complex valued. Understanding how
these two eigenvalue decompositions impact the resultant modal vectors and scaling estimation is very important.
The evaluation conducted in this study involves alternate eigenvalue decompositions that yield real valued
eigenvectors and/or real-normalization of the eigenvectors as well as truncation of the eigenvectors to eliminate
small contributions.
1.1 Measurement Condensation Matrix
The application of a formal measurement condensation matrix via eigenvalue decomposition methods to modal
parameter estimation was first noted by Lembregts [4] and was subsequently documented and evaluated by a
number of other authors [2-8]. Since the size of the matrices in the modal parameter estimation algorithms is a
function of the number of inputs or outputs in the FRF data matrix, the size of the matrices can easily become
much larger than the number of physical modes that are desired. In this situation, the modal parameter
estimation algorithm implementation becomes unwieldy in terms of memory usage and solution time when either
the number of inputs (Ni) and/or outputs (No) becomes very large. For example, it is very common to evaluate a
frequency range of interest for up to 10-20 modes of vibration but, in many testing situations, the number of inputs
and/or outputs may be in the 100-500 range. The actual number of FRFs used in the procedure can be sieved to
eliminate some of the inputs or outputs but this takes considerable user interaction and may cause modes present
in the FRF data to be eliminated. If the number of inputs or outputs (larger of the two) cannot be reduced in some
intelligent or automated procedure, the method becomes unattractive.
The basic concept of measurement condensation is to use an automated procedure (eigenvalue or singular value
decomposition) to transform the FRF measurements made in physical space to an alternate, transformed, virtual
space that is smaller but preserves the essential dynamics of the original, physical FRF measurements. The
transformation matrix is made up of the eigenvectors associated with the major contributions while the smaller
contributions are eliminated. Hopefully these small contributions represent noise, modes that were not well
excited in the data set and the redundancy that occurs in the data set due to the limited number of structural
modes present in the frequency band of interest and the large number of physical degrees of freedom involved in
the input and output measurement space.
The basic measurement transformation equation is represented by Equation (1) where the transformation matrix
[T] is formulated by the eigenvectors from the eigenvalue decomposition given in Equation (2). For clarity, the
format of the basic FRF matrix layout is given in Equation (3). Since the matrix that is being decomposed
(  H   H   )
H
is hermitian, the eigenvalues are real valued and the eigenvectors associated with the Ne largest
eigenvalues are retained in the transformation matrix.
 H (ω )  = [T ] *  H (ω ) 
†
T
[T ] = eigenvectors of (  H   H 
(1)
H
)
(2)
( )
 H 
≡   H (ω1 )   H (ω2 )  "  H ω N s  
N o × Ni N s

  N o × Ni N s

(3)
Note that, in general, the transformation matrix that is generated by this process will be complex valued. Most of
the literature discussing this transformation does not clearly discuss the complex valued nature of the
transformation matrix nor the potential impact of the complex valued transformation vectors on the resultant
estimation of modal vectors and scaling.
{ψ }r = [T ]{ψ }r
†
(4)
In order to evaluate the changes in the modal vectors for the measurement condensation cases, the condensation
matrix ([T]) must be utilized as shown in Equation (4) to transform the eigenvectors found in condensed or virtual
space back to eigenvectors that are in the original, physical measurement space.
1.2 Modal Participation Matrix
The implementation of most modern modal parameter estimation algorithms approach the estimation of modal
parameters in two stages [1-2]. In the first stage, the complex modal frequencies (frequency and damping) are
found as the eigenvalues of an eigenvalue problem with the associated eigenvectors yielding the modal
participation vectors of spatial dimension equal to either the number of inputs (Ni) or outputs (No). These modal
participation vectors should represent the modal vectors at Ni or No measurement locations but are normalized
with out any modal scaling. In the second stage, a weighted least squares approach is often used to estimate the
complete modal vectors, at the largest dimension, with scaling. The relationship between the modal participation
vectors, the modal vectors and the scaled residue vector/matrix for each mode is given by Equation (5).
[ A]r = {ψ }r  L  r
(5)
{ψ }r  L  r
[ A]r
 H (ω )  = ∑
=∑
jω − λr
r jω − λr
r
(6)
Equation (6) gives the relationship between the FRF matrix and the modal frequencies, residues, modal
participation vectors and modal vectors.
Note that other forms of modal scaling could be defined (modal A or modal mass) and included as an alternate,
equivalent form of Equation (6) but for the purpose of this study, the scaled residue vector will suffice as a
combined measure of both modal vector and scaling.
Once again, the modal participation vectors result from the eigenvalue decomposition of a general matrix
involving the FRF (or impulse response function (IRF)) data. This means that the modal participation vectors are
again, in general, complex valued. However, in this case, because of the relationship of the modal participation
vectors to the modal vectors for a reciprocal system, the expectation is that the modal participation vectors will be
essentially real valued for a proportionally damped system and complex valued when errors in the data occur or
the system is non-proportionally damped. Also, since the modal participation vectors are used as the weighting
vectors, mode by mode, in the second stage estimation of the scaled modal vectors, the small values in the modal
participation vectors represent the physical situation describing when a particular mode is not well excited or
represented from a specific reference location in the FRF (or IRF) data. This means that, rather than adding this
data into the weighted least squares solution it may be preferable to zero this weighting coefficient to prevent any
data from being utilized from poor references.
2.0 Case Study Overview
To investigate the effects of changes in the FRF condensation and participation vector normalizations on the
resulting residues, a 12 degree-of-freedom (DOF) proportionally damped, lumped parameter model was
developed with nominally 1-2% critical damping, 6 distinct modes and 3 pseudo-repeated modes. This model was
perturbed to provide three additional non-proportional models to serve as reference system cases, one lightly
non-proportional, one mildly non-proportional, and one severely non-proportional. Figure 1 shows a typical
column of data from the FRF matrix for the Base system. The modal parameters for each case (Base, Light, Mild
and Severe) are given in Appendix A.1.
For each system case, the effects of FRF measurement condensation, modal participation vector normalization
and modal participation vector truncation were evaluated. The effects of FRF measurement condensation were
evaluated by constructing both real and complex measurement condensation transformations. The complex
condensation matrix is found by following Equations (1) through (3) as documented in the literature. This will in
general result in a complex valued transformation matrix ([T]). The real condensation matrix was developed by
using only the imaginary part of the FRF data matrix in the measurement condensation equation as is shown by
Equation (7).

 H  N × N N ≡ ℑ    H (ω1 )   H (ω2 )  "  H ω N s  





o
i s
N o × Ni N s 

( )
(7)
This will in general result in a real valued transformation matrix ([T]). Appendix A.2 tabulates the results of this
real measurement condensation matrix in the measurement condensation process. Similar results were obtained

when the real part of the FRF matrix, ℜ    H (ω1 ) 

( 
)

, or the real part of
 H (ω2 )  "  H ω N s  
 No × Ni N s 
( )
H
the hermitian product, ℜ   H   H   , was used as the basis for the FRF measurement condensation.

Phase [deg]
Appendix A.3 tabulates the results for the complex condensation matrix solution.
180
90
0
-90
-2
10
-3
10
-4
Magnitude
10
-5
10
-6
10
-7
10
-8
10
0
10
20
30
40
50
60
Frequency [Hz]
70
80
Figure 1 – FRF’s for Base System (Reference DOF #1)
90
100
Modal participation truncation is evaluated by scaling the largest component of each modal participation vector to
unity and then setting any component in each modal participation vector with a magnitude below a threshold
equal to zero. Several threshold levels were evaluated and the results for a 20 percent threshold are tabulated in
Appendix A.4. Modal participation normalization is evaluated by scaling the largest component of each modal
participation vector to unity, adjusting the vector to have a mean phase deviation of zero degrees (removing any
rotation in the complex plane) and preserving either the amplitude or the real part of each vector component.
These results are tabulated in Appendix A.5.
2.1 Case Study – Measurement Condensation
The comparison of real measurement condensation vectors (based upon the imaginary part of the FRF matrix) vs.
complex measurement condensation vectors in Figure 2 shows that using complex measurement condensation
vectors distort the resulting participations and residues regardless of whether the underlying modal vectors are
real or complex. While some complex condensation results show little distortion for this theoretical evaluation (as
evaluated by the modal assurance criterion (MAC)), some are quite bad and completely skew the answers. In
general, the most distortion is seen with the cases that have the most complex valued modal vectors.
1]
mode shape [x] participation [+] residue [o] mac res [0.71371
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
mode shape [x] participation [+] residue [o] mac res [1
1
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-0.5
0
0.5
-1
1
-1
-0.5
Real
Imaginary condensation
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-0.5
0
Real
0.5
Imaginary condensation
1
1
0.98236]
0
-0.2
-1
0.5
mode shape [x] participation [+] residue [o] mac res [0.98341
1
-1
0
Real
Complex condensation
1]
Imaginary
Imaginary
mode shape [x] participation [+] residue [o] mac res [1
0.96354]
-1
-1
-0.5
0
Real
0.5
1
Complex condensation
Figure 2 – Base and Mild System Cases for Mode 9 Example
Specific numerical results that support these generalizations are tabulated in the Appendix. Appendix A.1 vs. A.2
shows that, when real measurement condensation (based upon the imaginary part of the FRF matrix)is used, the
calculated system has the same frequency, damping, shape (MAC) and complexity as the analytical system.
Appendix A.1 vs. A.3 shows that, when complex condensation is used, small distortions in frequency, damping,
shape and complexity occur for the Base system. For increasing modal complexity, distortion increases. For the
severe system, modes are estimated where none exist anywhere near in the data (E.g. 73 & 98 Hz) and the
modal frequencies associated with the calculated shapes are mismatched with respect to the analytical results.
2.3 Case Study – Participation Vector Truncation
The use of modal participation vector truncation/zero-floor (i.e. zeroing any element in the complex participation
vector with a magnitude less than some specific value, in this case 20% of maximum) has minimal impact upon
the resulting scaled modal vector (residue), see Figure 3 (Compare Figure 3:Imaginary Condensation to Figure
3:20% Floor). Appendix A.4 shows that truncation/flooring produces only minor differences between the
computed residues and the analytical solution.
1]
mode shape [x] participation [+] residue [o] mac res [0.97998
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
Imaginary
Imaginary
mode shape [x] participation [+] residue [o] mac res [1
1
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-0.5
0
Real
0.5
-1
1
-1
Imaginary Condensation
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-0.5
0
Real
0.5
20% Floor
0.5
1
1
0.9912]
0
-0.2
-1
0
Real
mode shape [x] participation [+] residue [o] mac res [0.99272
1
-1
-0.5
Complex condensation
0.9984]
Imaginary
Imaginary
mode shape [x] participation [+] residue [o] mac res [0.99887
0.91461]
-1
-1
-0.5
0
Real
0.5
1
Real Normalization
Figure 3 – Mild System Case for Mode 7 Example
2.4 Case Study – Participation Vector Normalization
The use of real vector normalization of the modal participation vector (i.e. forcing the modal participation vector to
be purely real by rotating the complex vector to be dominantly real and then either using the real part or the
signed magnitude) introduces a distortion comparable to that of the complex FRF condensation when the actual
mode shape is mildly complex, see Figure 3 (Compare Figure 3:Imaginary Condensation to Figure 3:Real
Normalization and contrast with Figure 3:Complex Condensation). Appendix A.2 vs. A.5 shows that normalizing
the modal participation vector to a real vector results in scaled modal vector (residue) estimates that are more
REAL than the actual analytical mode shape, but with generally good agreement on shape (MAC).
3.0 Conclusions
The results of this study show that:
• It is essential to use real measurement condensation vectors to condense the FRF matrix if the
introduction of complex distortions is to be avoided.
• The use of elimination/zeroing of small modal participation vector values doesn’t have much influence on
final scaled modal vectors (residues).
• Unless the system modes actually are real valued, normal modes, the use of real modal participation
vector normalization introduces distortions that make the scaled modal vectors (residues) appear more
normal than they really are.
Historically, these distortion effects have not been noticeable due to the focus on lightly damped, nearly
proportionally damped systems. For these systems, the distortions were dismissed as simply “noise in the data.”
However, the recent application of modal analysis techniques to systems of increasingly higher damping with
complex modes has meant that these distortions are more pronounced. Unless properly dealt with, the
identification of the correct modal parameters is made much more difficult and the results are compromised.
4.0 References
1. Allemang, R.J., Brown, D.L.,Fladung, W., "Modal Parameter Estimation: A Unified Matrix Polynomial
Approach", Proceedings, International Modal Analysis Conference, Society of Experimental Mechanics
(SEM), 1994, pp. 501-514.
2. Allemang, R.J., “Vibrations: Experimental Modal Analysis”, UC-SDRL-CN-20-263-663/664, University of
Cincinnati, 1999, 319 pp.
3. Heylen, W., Lammens, S., Sas, P., “Modal Analysis Theory and Testing”, Katholieke Universiteit Leuven –
Departement Werktuigkunde, ISBN 90-73802-61-X, pp. A.3.22-25, 1997.
4. Lembregts, F., "Frequency Domain Identification Techniques for Experimental Multiple Input Modal
Analysis", Ph. D. Dissertation, Katholieke Universiteit Leuven, Leuven, Belgium, pp. 62-72, 1988.
5. Lembregts, F., Leuridan, J.L., Van Brussel, H., "Frequency Domain Direct Parameter Identification for
Modal Analysis: State Space Formulation", Mechanical Systems and Signal Processing, Vol. 4, No. 1,
pp. 65-76, 1989.
6. Lembregts, F., Snoeys, R., Leuridan, J., "Application and Evaluation of Multiple Input Modal Parameter
Estimation", Journal of Analytical and Experimental Modal Analysis, Vol. 2, No. 1, pp.19-31,1987.
7. Dippery, K. D., Phillips, A. W., Allemang, R. J., "An SVD Condensation of the Spatial Domain in Modal
Parameter Estimation", Proceedings, International Modal Analysis Conference, 1994, 7 pp.
8. Dippery, K.D., Phillips, A.W., Allemang, R.J., "Condensation of the Spatial Domain in Modal Parameter
Estimation", Modal Analysis: International Journal of Analytical and Experimental Modal Analysis,
Volume 11, Number 3/4, 6 pp., 1996.
Appendix
A.1 Analytical Systems
Base System – Positive Frequencies only
Mode #
Frequency [Hz]
1
4.49169
2
17.344
3
17.4967
4
26.1054
5
28.1129
6
33.4369
7
33.5199
8
35.7105
9
36.021
10
46.7743
11
46.8913
12
56.0012
Damping %
1.91247
1.0037
1.00449
1.12499
1.1663
1.28851
1.29053
1.3448
1.35263
1.63976
1.64302
1.90172
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Conjugate MAC
1
1
1
1
1
1
1
1
1
1
1
1
Light Non-proportional System
Mode #
Frequency [Hz]
1
4.49138
2
17.3442
3
17.4952
4
26.1028
5
28.1111
6
33.4325
7
33.5164
8
35.7314
9
35.9943
10
46.7739
11
46.8908
12
56.001
Damping %
2.26382
1.25027
1.43275
1.78077
1.66081
2.08296
1.97024
1.42178
2.11385
1.66606
1.67339
1.91582
Complexity %
0.0895183
6.03326
6.11644
0.643553
0.92392
0.931167
1.60931
15.7914
27.1417
1.44093
1.43426
0.126011
Conjugate MAC
0.999998
0.98557
0.985225
0.999921
0.999896
0.999653
0.998981
0.906307
0.747377
0.999332
0.999331
0.999994
Mild Non-proportional System
Mode #
Frequency [Hz]
1
4.48806
2
17.3485
3
17.4655
4
26.023
5
28.0773
6
33.3055
7
33.4381
8
35.7486
9
35.7759
10
46.755
11
46.8645
12
55.9933
Damping %
5.42715
3.3425
5.41395
7.68764
6.10635
9.24512
8.07828
1.47242
9.62989
1.89619
1.93681
2.042
Complexity %
0.886809
10.1591
11.1128
6.45523
9.13485
5.28868
14.3469
3.40877
19.3954
14.4756
14.4056
1.25434
Conjugate MAC
0.999795
0.962849
0.956494
0.992166
0.989741
0.990382
0.936045
0.995432
0.935542
0.935125
0.935113
0.999389
Severe Non-proportional System
Mode #
Frequency [Hz]
1
4.41679
2
16.0757
3
16.9704
4
17.19
5
17.2092
6
17.6218
7
22.8766
8
25.2773
9
35.6497
10
46.3019
11
46.3889
12
55.5982
Damping %
37.6025
45.2939
87.8779
23.4732
84.4708
71.9017
71.9374
47.1791
2.24081
2.84554
2.58236
2.75325
Complexity %
17.5939
39.6919
7.37259
11.3089
33.7186
39.0384
26.4167
26.5134
7.89968
27.0938
26.9687
7.53008
Conjugate MAC
0.974158
0.606118
0.983825
0.952771
0.641759
0.562441
0.851185
0.779496
0.987274
0.75024
0.754358
0.977849
A.2 Measurement Condensation – Imaginary (Real)
Base System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
1.91247
1.0037
1.00449
1.12499
1.1663
1.28851
1.29053
1.3448
1.35263
1.63976
1.64302
1.90172
4.49169
17.344
17.4967
26.1054
28.1129
33.4369
33.5199
35.7105
36.021
46.7743
46.8913
56.0012
Light Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
2.26382
1.25027
1.43275
1.78077
1.66081
2.08296
1.97024
1.42178
2.11385
1.66606
1.67339
1.91582
4.49138
17.3442
17.4952
26.1028
28.1111
33.4325
33.5164
35.7314
35.9943
46.7739
46.8908
56.001
Mild Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
5.42715
3.3425
5.41395
7.68764
6.10635
9.24512
8.07828
1.47242
9.62989
1.89619
1.93681
2.042
4.48806
17.3485
17.4655
26.023
28.0773
33.3055
33.4381
35.7486
35.7759
46.755
46.8645
55.9933
Severe Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
37.6025
45.2939
87.8779
23.4732
84.4708
71.9017
71.9374
47.1791
2.24081
2.84554
2.58236
2.75325
4.41679
16.0757
16.9704
17.19
17.2092
17.6218
22.8766
25.2773
35.6497
46.3019
46.3889
55.5982
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
MAC
with
Analytical
1
1
1
1
1
1
1
1
1
1
1
1
respect
to
Participation
1
1
1
1
1
1
1
1
1
1
1
1
Residue
Participation
Complexity %
0.0895183
6.03326
6.11644
0.643553
0.92392
0.931167
1.60931
15.7914
27.1417
1.44093
1.43426
0.126011
Residue
Complexity %
0.0895183
6.03326
6.11644
0.643553
0.92392
0.931167
1.60931
15.7914
27.1417
1.44093
1.43426
0.126011
MAC
with
Analytical
1
1
1
1
1
1
1
1
1
1
1
1
respect
to
Participation
1
1
1
1
1
1
1
1
1
1
1
1
Residue
Participation
Complexity %
0.886809
10.1591
11.1128
6.45523
9.13485
5.28868
14.3469
3.40877
19.3954
14.4756
14.4056
1.25434
Residue
Complexity %
0.886809
10.1591
11.1128
6.45523
9.13485
5.28868
14.3469
3.40877
19.3954
14.4756
14.4056
1.25434
MAC
with
Analytical
1
1
1
1
1
1
1
1
1
1
1
1
respect
to
Participation
1
1
1
1
1
1
1
1
1
1
1
1
Residue
Participation
Complexity %
17.5939
39.6919
7.37259
11.3089
33.7186
39.0384
26.4167
26.5134
7.89968
27.0938
26.9687
7.53008
Residue
Complexity %
17.5939
39.6919
7.37259
11.3089
33.7186
39.0384
26.4167
26.5134
7.89968
27.0938
26.9687
7.53008
MAC
with
Analytical
1
1
1
1
1
1
1
1
1
1
1
1
respect
to
Participation
1
1
1
1
1
1
1
1
1
1
1
1
Residue
A.3 Measurement Condensation – Complex
Base System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
1.91208
0.999376
0.998246
1.11669
-0.142738
1.21214
1.20509
1.34256
0.317656
1.63885
1.64423
1.90141
Participation
Complexity %
0.0956523
0.345853
0.356826
0.625618
2.12589
3.0674
3.43544
4.04615
10.0397
1.60878
2.23965
0.612071
Residue
Complexity %
0.0183846
0.114745
0.0444482
0.0295055
2.19708
0.901914
0.306619
5.22091
10.2
1.61384
0.839498
0.08423
MAC
with
Analytical
1
0.999998
0.999999
1
0.999484
0.999912
0.999953
0.919836
0.713713
0.999059
0.999203
0.999998
respect
to
Participation
1
0.999986
0.999986
0.999967
0.995594
0.999045
0.998676
0.989693
0.963542
0.998596
0.998868
0.999703
Residue
Participation
Complexity %
0.186655
7.40812
7.83005
3.21393
5.6829
5.71235
21.6627
11.256
37.2403
14.5263
28.0399
2.9734
Residue
Complexity %
0.212364
2.88688
6.14956
0.522384
2.25089
2.03564
3.54269
8.98108
36.7286
18.9352
2.73419
0.467474
MAC
with
Analytical
0.999999
0.938789
0.90313
0.999987
0.99724
0.998461
0.998061
0.901011
0.84557
0.95439
0.97313
0.999972
respect
to
Participation
0.999998
0.990289
0.983275
0.999601
0.995098
0.995524
0.96715
0.957559
0.821489
0.938425
0.923241
0.997911
Residue
Participation
Complexity %
2.03436
3.41962
15.5156
22.5402
19.291
14.4172
21.1365
4.98685
5.57851
122.622
115.739
9.03058
Residue
Complexity %
1.19982
3.27146
6.65747
9.3691
4.81039
7.36614
14.4868
2.63676
6.63704
112.051
86.0793
1.9319
MAC
with
Analytical
0.99999
0.995207
0.974385
0.996661
0.991585
0.984008
0.979981
0.999822
0.983405
0.0991766
0.336978
0.999531
respect
to
Participation
0.999696
0.997405
0.919405
0.943427
0.956295
0.972329
0.914612
0.995122
0.982355
0.584785
0.919494
0.985506
Residue
Severe Non-proportional System
Mode
# Frequency [Hz]
Damping %
Participation
Residue
(Analytic)
Complexity %
Complexity %
1 (1)
4.53442
31.1229
8.14357
11.5003
2 (4)
17.5708
15.0261
29.9083
21.4433
3 (2)
21.5118
73.6189
299.177
44.6961
4 (8)
28.4711
19.763
23.6558
21.4051
5 (9)
35.5727
2.4458
20.2676
14.5323
6 (9)
35.8518
7.04253
86.7618
155.087
7 (6)
40.0741
54.036
48.112
107.135
8 (10)
46.4039
0.666952
51.845
60.7251
9 (11)
46.426
2.62791
139.303
155.579
10 (12)
55.8579
2.672
32.0683
4.36105
11 (7)
73.1884
13.9612
89.676
89.7613
12 (3)
98.2008
42.4407
125.108
45.9788
NOTE: Analytic mode #9 is duplicated and analytic mode #5 is lost (not represented).
MAC
with
Analytical
0.99909
0.99632
0.58144
0.79733
0.99298
0.68782
0.73225
0.89402
0.91858
0.99616
0.74830
0.98729
respect
to
Participation
0.983182
0.923961
0.215795
0.800945
0.938266
0.131059
0.651014
0.969145
0.94855
0.87204
0.786121
0.992749
Residue
4.49167
17.3434
17.4957
26.1029
28.5819
33.4073
33.4877
35.7348
36.7938
46.7743
46.8935
56.0012
Light Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
2.26247
1.22315
-0.107507
1.72848
0.979239
1.64905
1.2252
1.33016
0.350566
1.64386
1.63705
1.91471
4.49134
17.3514
17.8522
26.0885
28.0875
33.3408
33.4587
35.7416
36.2705
46.763
46.8647
56.0018
Mild Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
5.39687
3.24208
-0.0597987
4.2143
6.38221
9.74198
8.0257
1.46109
3.20654
1.39558
1.84069
2.03305
4.4882
17.3376
20.4648
26.0722
28.2117
33.8818
35.4212
35.7422
41.3494
46.7921
46.7981
55.9965
A.4 Participation Truncation (20%)
Base System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
1.91247
1.0037
1.00449
1.12499
1.1663
1.28851
1.29053
1.3448
1.35263
1.63976
1.64302
1.90172
4.49169
17.344
17.4967
26.1054
28.1129
33.4369
33.5199
35.7105
36.021
46.7743
46.8913
56.0012
Light Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
2.26382
1.25027
1.43275
1.78077
1.66081
2.08296
1.97024
1.42178
2.11385
1.66606
1.67339
1.91582
4.49138
17.3442
17.4952
26.1028
28.1111
33.4325
33.5164
35.7314
35.9943
46.7739
46.8908
56.001
Mild Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
5.42715
3.3425
5.41395
7.68764
6.10635
9.24512
8.07828
1.47242
9.62989
1.89619
1.93681
2.042
4.48806
17.3485
17.4655
26.023
28.0773
33.3055
33.4381
35.7486
35.7759
46.755
46.8645
55.9933
Severe Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
37.6025
45.2939
87.8779
23.4732
84.4708
71.9017
71.9374
47.1791
2.24081
2.84554
2.58236
2.75325
4.41679
16.0757
16.9704
17.19
17.2092
17.6218
22.8766
25.2773
35.6497
46.3019
46.3889
55.5982
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
0.101183
0.0783316
0.122342
0.24694
0.26679
1.11735
1.19265
4.59934
3.50621
3.22428
3.76522
0.811141
MAC
with
Analytical
0.999999
0.999939
0.999824
0.999991
0.999989
0.999886
0.999884
0.998112
0.998082
0.998893
0.998332
0.99993
respect
to
Participation
0.999999
0.989054
0.98909
0.999991
0.999795
0.999822
0.998881
0.997136
0.998082
0.949758
0.948722
0.977366
Residue
Participation
Complexity %
0.0895183
3.99165
4.03726
0.643553
0.915809
0.912691
1.5677
12.676
27.1417
1.43587
1.4177
0.01321
Residue
Complexity %
0.426121
3.36776
1.52812
0.507602
1.58706
2.17598
3.53531
16.1007
12.8669
3.56753
4.02867
0.695119
MAC
with
Analytical
0.999996
0.999279
0.997875
0.999984
0.999691
0.999567
0.999131
0.999024
0.969757
0.99871
0.998188
0.999958
respect
to
Participation
0.999996
0.988177
0.988096
0.999984
0.999114
0.999514
0.998668
0.979109
0.969757
0.94846
0.947475
0.977404
Residue
Participation
Complexity %
0.886809
9.55545
10.3489
6.45523
9.05204
5.28522
14.2477
3.40877
18.7195
14.435
14.2571
0.130426
Residue
Complexity %
1.01694
9.21648
9.33888
6.14384
9.1144
2.97451
13.3521
2.82557
19.3913
14.3949
14.3412
1.268
MAC
with
Analytical
0.999999
0.999916
0.999574
0.999923
0.999998
0.998459
0.998868
0.999775
0.999979
0.999948
0.999946
0.999993
respect
to
Participation
0.999999
0.998778
0.998333
0.999923
0.999625
0.998434
0.9984
0.999775
0.99746
0.949743
0.950112
0.977553
Residue
Participation
Complexity %
17.5939
39.6919
7.17187
11.3025
33.7764
38.5407
25.5573
24.7635
6.31016
25.7795
26.0289
0.412074
Residue
Complexity %
17.6891
38.4855
3.75113
10.8827
32.138
37.7355
27.1267
23.7859
7.75736
27.0899
26.9449
8.45152
MAC
with
Analytical
0.999989
0.997605
0.992939
0.999618
0.976125
0.973671
0.991242
0.997368
0.999972
0.999949
0.999833
0.999851
respect
to
Participation
0.999989
0.997605
0.990307
0.999605
0.973114
0.950682
0.968013
0.991281
0.981567
0.96753
0.964939
0.983565
Residue
A.5 Participation Normalization
Base System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
1.91247
1.0037
1.00449
1.12499
1.1663
1.28851
1.29053
1.3448
1.35263
1.63976
1.64302
1.90172
4.49169
17.344
17.4967
26.1054
28.1129
33.4369
33.5199
35.7105
36.021
46.7743
46.8913
56.0012
Light Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
2.26382
1.25027
1.43275
1.78077
1.66081
2.08296
1.97024
1.42178
2.11385
1.66606
1.67339
1.91582
4.49138
17.3442
17.4952
26.1028
28.1111
33.4325
33.5164
35.7314
35.9943
46.7739
46.8908
56.001
Mild Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
5.42715
3.3425
5.41395
7.68764
6.10635
9.24512
8.07828
1.47242
9.62989
1.89619
1.93681
2.042
4.48806
17.3485
17.4655
26.023
28.0773
33.3055
33.4381
35.7486
35.7759
46.755
46.8645
55.9933
Severe Non-proportional System
Mode #
Frequency [Hz]
Damping %
1
2
3
4
5
6
7
8
9
10
11
12
37.6025
45.2939
87.8779
23.4732
84.4708
71.9017
71.9374
47.1791
2.24081
2.84554
2.58236
2.75325
4.41679
16.0757
16.9704
17.19
17.2092
17.6218
22.8766
25.2773
35.6497
46.3019
46.3889
55.5982
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
MAC
with
Analytical
1
1
1
1
1
1
1
1
1
1
1
1
respect
to
Participation
1
1
1
1
1
1
1
1
1
1
1
1
Residue
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
0.200816
1.43289
1.90355
0.647848
0.377819
0.586294
1.41353
1.93043
3.41442
0.364736
0.389089
0.115237
MAC
with
Analytical
0.999999
0.997885
0.9937
1
0.99992
0.999928
0.999934
0.980835
0.913474
0.999836
0.999838
1
respect
to
Participation
0.999997
0.999789
0.999622
0.999979
0.999928
0.999972
0.9998
0.998922
0.995935
0.999982
0.99998
0.999999
Residue
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
1.23571
3.65951
6.48554
6.24362
4.96739
5.16867
7.95818
2.329
4.40834
2.86233
3.02761
1.25739
MAC
with
Analytical
0.999989
0.99589
0.97907
0.999653
0.998751
0.999728
0.992724
0.99963
0.974307
0.9841
0.984349
0.999998
respect
to
Participation
0.999902
0.998707
0.995617
0.997734
0.998522
0.997509
0.991202
0.999316
0.989397
0.999076
0.99898
0.999843
Residue
Participation
Complexity %
0
0
0
0
0
0
0
0
0
0
0
0
Residue
Complexity %
12.6526
37.3152
3.59381
9.08266
7.06133
17.1895
13.441
11.6231
9.00292
7.81107
6.57511
7.55113
MAC
with
Analytical
0.999414
0.833205
0.981792
0.999075
0.817796
0.727713
0.862097
0.952269
0.999779
0.937077
0.938078
0.999974
respect
to
Participation
0.99613
0.763264
0.989948
0.992505
0.934168
0.756871
0.9396
0.965184
0.995377
0.993532
0.995318
0.994364
Residue