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 f Frequency [Hz] ω Circular frequency [rad/sec]

rλ System pole or modal frequency for mode r

[ ]rA Residue matrix for mode r

{ }rψ Mode shape for mode r

{ }†

rψ Mode shape for condensed, or virtual, mode r

rL participation vector for mode r

( )H ω Frequency response function (FRF) matrix

( ) †H ω

Condensed, or virtual, frequency response function matrix

H Rectangular (2d) frequency response function matrix

[ ]T FRF condensation matrix

[ ]HX Hermitian (complex conjugate transpose) of matrix X

( )Xℜ Real part of complex matrix X

( )Xℑ 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

( )HH H

is hermitian, the eigenvalues are real valued and the eigenvectors associated with the Ne largest

eigenvalues are retained in the transformation matrix.

( ) [ ] ( )†*TH T Hω ω= (1)

[ ] ( )HT eigenvectors of H H = (2)

( ) ( ) ( )1 2 so i s o i sNN N N N N N

H H H Hω ω ω× ×

≡ (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.

{ } [ ]{ }†

r rTψ ψ= (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). [ ] { }r rr

A Lψ= (5)

( ) [ ] { }rr r

r rr r

A LH

j jψ

ωω λ ω λ

= = − −∑ ∑ (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).

( ) ( ) ( )1 2 so i s o i sNN N N N N N

H H H Hω ω ω× ×

≡ ℑ (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, ( ) ( ) ( )1 2 so i s

NN N N

H H Hω ω ω×

ℜ , or the real part of

the hermitian product, ( )HH H ℜ

, was used as the basis for the FRF measurement condensation.

Appendix A.3 tabulates the results for the complex condensation matrix solution.

0 10 20 30 40 50 60 70 80 90 10010-8

10-7

10-6

10-5

10-4

10-3

10-2

Frequency [Hz]

Mag

nitu

de

-900

90180

Pha

se [d

eg]

Figure 1 – FRF’s for Base System (Reference DOF #1)

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.

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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.

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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] Damping % Complexity % Conjugate MAC 1 4.49169 1.91247 0 1 2 17.344 1.0037 0 1 3 17.4967 1.00449 0 1 4 26.1054 1.12499 0 1 5 28.1129 1.1663 0 1 6 33.4369 1.28851 0 1 7 33.5199 1.29053 0 1 8 35.7105 1.3448 0 1 9 36.021 1.35263 0 1 10 46.7743 1.63976 0 1 11 46.8913 1.64302 0 1 12 56.0012 1.90172 0 1 Light Non-proportional System Mode # Frequency [Hz] Damping % Complexity % Conjugate MAC 1 4.49138 2.26382 0.0895183 0.999998 2 17.3442 1.25027 6.03326 0.98557 3 17.4952 1.43275 6.11644 0.985225 4 26.1028 1.78077 0.643553 0.999921 5 28.1111 1.66081 0.92392 0.999896 6 33.4325 2.08296 0.931167 0.999653 7 33.5164 1.97024 1.60931 0.998981 8 35.7314 1.42178 15.7914 0.906307 9 35.9943 2.11385 27.1417 0.747377 10 46.7739 1.66606 1.44093 0.999332 11 46.8908 1.67339 1.43426 0.999331 12 56.001 1.91582 0.126011 0.999994 Mild Non-proportional System Mode # Frequency [Hz] Damping % Complexity % Conjugate MAC 1 4.48806 5.42715 0.886809 0.999795 2 17.3485 3.3425 10.1591 0.962849 3 17.4655 5.41395 11.1128 0.956494 4 26.023 7.68764 6.45523 0.992166 5 28.0773 6.10635 9.13485 0.989741 6 33.3055 9.24512 5.28868 0.990382 7 33.4381 8.07828 14.3469 0.936045 8 35.7486 1.47242 3.40877 0.995432 9 35.7759 9.62989 19.3954 0.935542 10 46.755 1.89619 14.4756 0.935125 11 46.8645 1.93681 14.4056 0.935113 12 55.9933 2.042 1.25434 0.999389 Severe Non-proportional System Mode # Frequency [Hz] Damping % Complexity % Conjugate MAC 1 4.41679 37.6025 17.5939 0.974158 2 16.0757 45.2939 39.6919 0.606118 3 16.9704 87.8779 7.37259 0.983825 4 17.19 23.4732 11.3089 0.952771 5 17.2092 84.4708 33.7186 0.641759 6 17.6218 71.9017 39.0384 0.562441 7 22.8766 71.9374 26.4167 0.851185 8 25.2773 47.1791 26.5134 0.779496 9 35.6497 2.24081 7.89968 0.987274 10 46.3019 2.84554 27.0938 0.75024 11 46.3889 2.58236 26.9687 0.754358 12 55.5982 2.75325 7.53008 0.977849

A.2 Measurement Condensation – Imaginary (Real) Base System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49169 1.91247 0 0 1 1 2 17.344 1.0037 0 0 1 1 3 17.4967 1.00449 0 0 1 1 4 26.1054 1.12499 0 0 1 1 5 28.1129 1.1663 0 0 1 1 6 33.4369 1.28851 0 0 1 1 7 33.5199 1.29053 0 0 1 1 8 35.7105 1.3448 0 0 1 1 9 36.021 1.35263 0 0 1 1 10 46.7743 1.63976 0 0 1 1 11 46.8913 1.64302 0 0 1 1 12 56.0012 1.90172 0 0 1 1 Light Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49138 2.26382 0.0895183 0.0895183 1 1 2 17.3442 1.25027 6.03326 6.03326 1 1 3 17.4952 1.43275 6.11644 6.11644 1 1 4 26.1028 1.78077 0.643553 0.643553 1 1 5 28.1111 1.66081 0.92392 0.92392 1 1 6 33.4325 2.08296 0.931167 0.931167 1 1 7 33.5164 1.97024 1.60931 1.60931 1 1 8 35.7314 1.42178 15.7914 15.7914 1 1 9 35.9943 2.11385 27.1417 27.1417 1 1 10 46.7739 1.66606 1.44093 1.44093 1 1 11 46.8908 1.67339 1.43426 1.43426 1 1 12 56.001 1.91582 0.126011 0.126011 1 1 Mild Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.48806 5.42715 0.886809 0.886809 1 1 2 17.3485 3.3425 10.1591 10.1591 1 1 3 17.4655 5.41395 11.1128 11.1128 1 1 4 26.023 7.68764 6.45523 6.45523 1 1 5 28.0773 6.10635 9.13485 9.13485 1 1 6 33.3055 9.24512 5.28868 5.28868 1 1 7 33.4381 8.07828 14.3469 14.3469 1 1 8 35.7486 1.47242 3.40877 3.40877 1 1 9 35.7759 9.62989 19.3954 19.3954 1 1 10 46.755 1.89619 14.4756 14.4756 1 1 11 46.8645 1.93681 14.4056 14.4056 1 1 12 55.9933 2.042 1.25434 1.25434 1 1 Severe Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.41679 37.6025 17.5939 17.5939 1 1 2 16.0757 45.2939 39.6919 39.6919 1 1 3 16.9704 87.8779 7.37259 7.37259 1 1 4 17.19 23.4732 11.3089 11.3089 1 1 5 17.2092 84.4708 33.7186 33.7186 1 1 6 17.6218 71.9017 39.0384 39.0384 1 1 7 22.8766 71.9374 26.4167 26.4167 1 1 8 25.2773 47.1791 26.5134 26.5134 1 1 9 35.6497 2.24081 7.89968 7.89968 1 1 10 46.3019 2.84554 27.0938 27.0938 1 1 11 46.3889 2.58236 26.9687 26.9687 1 1 12 55.5982 2.75325 7.53008 7.53008 1 1

A.3 Measurement Condensation – Complex Base System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49167 1.91208 0.0956523 0.0183846 1 1 2 17.3434 0.999376 0.345853 0.114745 0.999998 0.999986 3 17.4957 0.998246 0.356826 0.0444482 0.999999 0.999986 4 26.1029 1.11669 0.625618 0.0295055 1 0.999967 5 28.5819 -0.142738 2.12589 2.19708 0.999484 0.995594 6 33.4073 1.21214 3.0674 0.901914 0.999912 0.999045 7 33.4877 1.20509 3.43544 0.306619 0.999953 0.998676 8 35.7348 1.34256 4.04615 5.22091 0.919836 0.989693 9 36.7938 0.317656 10.0397 10.2 0.713713 0.963542 10 46.7743 1.63885 1.60878 1.61384 0.999059 0.998596 11 46.8935 1.64423 2.23965 0.839498 0.999203 0.998868 12 56.0012 1.90141 0.612071 0.08423 0.999998 0.999703 Light Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49134 2.26247 0.186655 0.212364 0.999999 0.999998 2 17.3514 1.22315 7.40812 2.88688 0.938789 0.990289 3 17.8522 -0.107507 7.83005 6.14956 0.90313 0.983275 4 26.0885 1.72848 3.21393 0.522384 0.999987 0.999601 5 28.0875 0.979239 5.6829 2.25089 0.99724 0.995098 6 33.3408 1.64905 5.71235 2.03564 0.998461 0.995524 7 33.4587 1.2252 21.6627 3.54269 0.998061 0.96715 8 35.7416 1.33016 11.256 8.98108 0.901011 0.957559 9 36.2705 0.350566 37.2403 36.7286 0.84557 0.821489 10 46.763 1.64386 14.5263 18.9352 0.95439 0.938425 11 46.8647 1.63705 28.0399 2.73419 0.97313 0.923241 12 56.0018 1.91471 2.9734 0.467474 0.999972 0.997911 Mild Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.4882 5.39687 2.03436 1.19982 0.99999 0.999696 2 17.3376 3.24208 3.41962 3.27146 0.995207 0.997405 3 20.4648 -0.0597987 15.5156 6.65747 0.974385 0.919405 4 26.0722 4.2143 22.5402 9.3691 0.996661 0.943427 5 28.2117 6.38221 19.291 4.81039 0.991585 0.956295 6 33.8818 9.74198 14.4172 7.36614 0.984008 0.972329 7 35.4212 8.0257 21.1365 14.4868 0.979981 0.914612 8 35.7422 1.46109 4.98685 2.63676 0.999822 0.995122 9 41.3494 3.20654 5.57851 6.63704 0.983405 0.982355 10 46.7921 1.39558 122.622 112.051 0.0991766 0.584785 11 46.7981 1.84069 115.739 86.0793 0.336978 0.919494 12 55.9965 2.03305 9.03058 1.9319 0.999531 0.985506 Severe Non-proportional System Mode # (Analytic)

Frequency [Hz] Damping % Participation Complexity %

Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 (1) 4.53442 31.1229 8.14357 11.5003 0.99909 0.983182 2 (4) 17.5708 15.0261 29.9083 21.4433 0.99632 0.923961 3 (2) 21.5118 73.6189 299.177 44.6961 0.58144 0.215795 4 (8) 28.4711 19.763 23.6558 21.4051 0.79733 0.800945 5 (9) 35.5727 2.4458 20.2676 14.5323 0.99298 0.938266 6 (9) 35.8518 7.04253 86.7618 155.087 0.68782 0.131059 7 (6) 40.0741 54.036 48.112 107.135 0.73225 0.651014 8 (10) 46.4039 0.666952 51.845 60.7251 0.89402 0.969145 9 (11) 46.426 2.62791 139.303 155.579 0.91858 0.94855 10 (12) 55.8579 2.672 32.0683 4.36105 0.99616 0.87204 11 (7) 73.1884 13.9612 89.676 89.7613 0.74830 0.786121 12 (3) 98.2008 42.4407 125.108 45.9788 0.98729 0.992749 NOTE: Analytic mode #9 is duplicated and analytic mode #5 is lost (not represented).

A.4 Participation Truncation (20%) Base System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49169 1.91247 0 0.101183 0.999999 0.999999 2 17.344 1.0037 0 0.0783316 0.999939 0.989054 3 17.4967 1.00449 0 0.122342 0.999824 0.98909 4 26.1054 1.12499 0 0.24694 0.999991 0.999991 5 28.1129 1.1663 0 0.26679 0.999989 0.999795 6 33.4369 1.28851 0 1.11735 0.999886 0.999822 7 33.5199 1.29053 0 1.19265 0.999884 0.998881 8 35.7105 1.3448 0 4.59934 0.998112 0.997136 9 36.021 1.35263 0 3.50621 0.998082 0.998082 10 46.7743 1.63976 0 3.22428 0.998893 0.949758 11 46.8913 1.64302 0 3.76522 0.998332 0.948722 12 56.0012 1.90172 0 0.811141 0.99993 0.977366 Light Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49138 2.26382 0.0895183 0.426121 0.999996 0.999996 2 17.3442 1.25027 3.99165 3.36776 0.999279 0.988177 3 17.4952 1.43275 4.03726 1.52812 0.997875 0.988096 4 26.1028 1.78077 0.643553 0.507602 0.999984 0.999984 5 28.1111 1.66081 0.915809 1.58706 0.999691 0.999114 6 33.4325 2.08296 0.912691 2.17598 0.999567 0.999514 7 33.5164 1.97024 1.5677 3.53531 0.999131 0.998668 8 35.7314 1.42178 12.676 16.1007 0.999024 0.979109 9 35.9943 2.11385 27.1417 12.8669 0.969757 0.969757 10 46.7739 1.66606 1.43587 3.56753 0.99871 0.94846 11 46.8908 1.67339 1.4177 4.02867 0.998188 0.947475 12 56.001 1.91582 0.01321 0.695119 0.999958 0.977404 Mild Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.48806 5.42715 0.886809 1.01694 0.999999 0.999999 2 17.3485 3.3425 9.55545 9.21648 0.999916 0.998778 3 17.4655 5.41395 10.3489 9.33888 0.999574 0.998333 4 26.023 7.68764 6.45523 6.14384 0.999923 0.999923 5 28.0773 6.10635 9.05204 9.1144 0.999998 0.999625 6 33.3055 9.24512 5.28522 2.97451 0.998459 0.998434 7 33.4381 8.07828 14.2477 13.3521 0.998868 0.9984 8 35.7486 1.47242 3.40877 2.82557 0.999775 0.999775 9 35.7759 9.62989 18.7195 19.3913 0.999979 0.99746 10 46.755 1.89619 14.435 14.3949 0.999948 0.949743 11 46.8645 1.93681 14.2571 14.3412 0.999946 0.950112 12 55.9933 2.042 0.130426 1.268 0.999993 0.977553 Severe Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.41679 37.6025 17.5939 17.6891 0.999989 0.999989 2 16.0757 45.2939 39.6919 38.4855 0.997605 0.997605 3 16.9704 87.8779 7.17187 3.75113 0.992939 0.990307 4 17.19 23.4732 11.3025 10.8827 0.999618 0.999605 5 17.2092 84.4708 33.7764 32.138 0.976125 0.973114 6 17.6218 71.9017 38.5407 37.7355 0.973671 0.950682 7 22.8766 71.9374 25.5573 27.1267 0.991242 0.968013 8 25.2773 47.1791 24.7635 23.7859 0.997368 0.991281 9 35.6497 2.24081 6.31016 7.75736 0.999972 0.981567 10 46.3019 2.84554 25.7795 27.0899 0.999949 0.96753 11 46.3889 2.58236 26.0289 26.9449 0.999833 0.964939 12 55.5982 2.75325 0.412074 8.45152 0.999851 0.983565

A.5 Participation Normalization Base System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49169 1.91247 0 0 1 1 2 17.344 1.0037 0 0 1 1 3 17.4967 1.00449 0 0 1 1 4 26.1054 1.12499 0 0 1 1 5 28.1129 1.1663 0 0 1 1 6 33.4369 1.28851 0 0 1 1 7 33.5199 1.29053 0 0 1 1 8 35.7105 1.3448 0 0 1 1 9 36.021 1.35263 0 0 1 1 10 46.7743 1.63976 0 0 1 1 11 46.8913 1.64302 0 0 1 1 12 56.0012 1.90172 0 0 1 1 Light Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.49138 2.26382 0 0.200816 0.999999 0.999997 2 17.3442 1.25027 0 1.43289 0.997885 0.999789 3 17.4952 1.43275 0 1.90355 0.9937 0.999622 4 26.1028 1.78077 0 0.647848 1 0.999979 5 28.1111 1.66081 0 0.377819 0.99992 0.999928 6 33.4325 2.08296 0 0.586294 0.999928 0.999972 7 33.5164 1.97024 0 1.41353 0.999934 0.9998 8 35.7314 1.42178 0 1.93043 0.980835 0.998922 9 35.9943 2.11385 0 3.41442 0.913474 0.995935 10 46.7739 1.66606 0 0.364736 0.999836 0.999982 11 46.8908 1.67339 0 0.389089 0.999838 0.99998 12 56.001 1.91582 0 0.115237 1 0.999999 Mild Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.48806 5.42715 0 1.23571 0.999989 0.999902 2 17.3485 3.3425 0 3.65951 0.99589 0.998707 3 17.4655 5.41395 0 6.48554 0.97907 0.995617 4 26.023 7.68764 0 6.24362 0.999653 0.997734 5 28.0773 6.10635 0 4.96739 0.998751 0.998522 6 33.3055 9.24512 0 5.16867 0.999728 0.997509 7 33.4381 8.07828 0 7.95818 0.992724 0.991202 8 35.7486 1.47242 0 2.329 0.99963 0.999316 9 35.7759 9.62989 0 4.40834 0.974307 0.989397 10 46.755 1.89619 0 2.86233 0.9841 0.999076 11 46.8645 1.93681 0 3.02761 0.984349 0.99898 12 55.9933 2.042 0 1.25739 0.999998 0.999843 Severe Non-proportional System Mode # Frequency [Hz] Damping % Participation

Complexity % Residue Complexity %

MAC with respect to ResidueAnalytical Participation

1 4.41679 37.6025 0 12.6526 0.999414 0.99613 2 16.0757 45.2939 0 37.3152 0.833205 0.763264 3 16.9704 87.8779 0 3.59381 0.981792 0.989948 4 17.19 23.4732 0 9.08266 0.999075 0.992505 5 17.2092 84.4708 0 7.06133 0.817796 0.934168 6 17.6218 71.9017 0 17.1895 0.727713 0.756871 7 22.8766 71.9374 0 13.441 0.862097 0.9396 8 25.2773 47.1791 0 11.6231 0.952269 0.965184 9 35.6497 2.24081 0 9.00292 0.999779 0.995377 10 46.3019 2.84554 0 7.81107 0.937077 0.993532 11 46.3889 2.58236 0 6.57511 0.938078 0.995318 12 55.5982 2.75325 0 7.55113 0.999974 0.994364