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2-1 레이라이 지수법(Rayleigh Quotient Method)
2-2 섭동법(Perturbation Method)

2-3 민감도 해석법(Sensitivity Analysis Method)
2-3 Original Formulations
2-3 Higher Order Sensitivity
2-3 Eigenvector Sensitivity
2-3
Repeated Eigenvalue Sensitivity
2-3 Applications
2-4 모드 합성법 (Modal Synthesis Method)
2-3 Eigenvalue Upper Bound and Lower Bound
2-3
Applications
2-3 Beam Applications
2-5 Dual Modal Space Method (DMSM)
2-3
Applications
2-3
Modal Truncation Error
2-6 국부 구조 변경법(Localized Modification Method)
2-7 라그랑지 승수법(Lagrange Multiplier Method)


 

 
 

 

2-1 레이라이 지수법(Rayleigh Quotient Method)

[2-1-1] Irons, B. and Ahmad, S., 1980, Techniques of Finite Elements, Ellis Horwood Ltd., Chichester, England, [1-1-4] pp. 360~361.
[2-1-2] Wang, B. P., and Pilkey, W. D., June 1986, "Eigenvalue Reanalysis of Locally Modified Structures Using a [1-1-4] Generalized Rayleigh's Method," AIAA Journal, Vol. 24, No. 6, pp. 983~990.
[2-1-3] Pilkey, W. D., and Schramm, U., Jan.1995, "Reanalysis of Free Vibrations Considering the Frequency-
[1-1-4] Dependency of the Structural Matrices," The International Journal of Analytical and Experimental Modal
[1-1-4] Analysis, Vol.19, No. 1, pp. 53~68.
[2-1-4] Carne, T. G. and Dohrmann, C. R., 1998, "Support Conditions, Their Effect on Measured Modal
[1-1-4] Parameters," Proceedings of the 16th International Modal Analysis Conference, pp. 477~483.
[2-1-5] W. M. To and D. J. Ewins, 1990, "Structural Modification Analysis Using Rayleigh Quotient Iteration," Int. J.
[1-1-4] Mech. Sci., Vol. 32, No. 3, pp. 169-179.

 


 

 
 

 

2-2 섭동법(Perturbation Method)

[2-2-1] Rayleigh, B., 1945, The Theory of Sound, Vol.1, Dover, New York, pp.109-118, pp. 214-217.
[2-2-2] Chen, J. C. and Wada, B. K., Aug.1977, "Matrix Perturbation for Structural Dynamic Analysis," AIAA
[1-1-4] Journal, Vol. 15, No. 8, pp. 1095~1100.
[2-2-2.5] Chon Su-Huan, Liu Ying-Li, Huang Dun-Pu, "A Matrix Perturbation of Vibration Modal Analysis," IMAC
[2-2-2.6] M. S. Eldred, P. B. Lerner and W. J. Anderson, 1992, "Higher Order Eigenpair Perturbations," AIAA
[1-1-4] Journal, Vol. 30, No 7, pp.1870-1875.
[2-2-3] Ravi, S. S. A., Kundra, T. K. and Nakra, B. C., 1998, "Reanalysis of Damped Structures Using the Single
[1-1-4] Step Perturbation Method," Journal of Sound and Vibration, Vol. 211, No. 3, pp. 355~363.
[2-2-4] Eldred, M. S., Lerner, P. B., and Anderson, W. J., July.1992, "Higher Order Eigenpair Perturbations," AIAA
[1-1-4] Journal, Vol. 30, No. 7, pp.1870~1876.
[2-2-5] Chen, J. Y., Liu, J. K., and Zhao, L. C., 1995, "An Improved Perturbation Method for Free Vibration
[1-1-4] Analysis," Journal of Sound and Vibration, Vol. 180, No. 3, pp. 519~523.
[2-2-6] Zhenhua, L. and Zhendong, F., 1988, "The Optimal Structural Modification Principle for The Maximum
[1-1-4] Shifts of Natural Vibration Frequencies," Proceedings of the 6th International Modal Analysis Conference,
[1-1-4] pp.1107~1111.
[2-2-7] Kelly, W. J., and Stevens, K. K., 1989, "Application of Perturbation Techniques to the Modal Analysis of a
[1-1-4] Shaft With Added Viscoelastic Damping," Proceedings of the 7th International Modal Analysis
[1-1-4] Conference, pp. 45~51.
[2-2-8] Kim, K. O., Anderson, W. J. and Sandstrom, R. E., 1983, "Nonlinear Inverse Perturbation Method in
[1-1-4] Dynamic Analysis," AIAA Journal, Vol. 21, No. 9, pp. 1310~1316. See [8-1-5].
[2-2-9] Paulo C. Kaminski, 1993, "Determination of Eigenvalues of Modified Structures Using the First Order
[1-1-4] Perturbation Method and the Rayleigh's Quotient: - A Comparative Study," IMAC 11, pp. 1181-1187.

 


 

 
 

 

2-3 민감도 해석법(Sensitivity Analysis Method)

Original Formulations

[2-3-1] Fox, R. L. and Kapoor, M. P., 1968, "Rates of Change of Eigenvalues and Eigenvectors," AIAA Journal,
[1-1-4] Vol.6, No.12, pp. 2426~2429.
[2-3-2] Rogers, L. C., 1970, "Derivatives of Eigenvalues and Eigenvectors," AIAA Journal, Vol. 8, No. 5, pp.
[1-1-4] 943~944.
[2-3-3] Rudisill, C. S., 1974, "Derivatives of Eigenvalues and Eigenvectors for a General Matrix," AIAA Journal,
[1-1-4] Vol. 12, No. 5, pp. 721~722.
[2-3-4] Haug, E. J. and Choi, K. K., 1984, "Structural Design Sensitivity Analysis with Generalized Global Stiffness
[1-1-4] and Mass Matrices," AIAA Journal, Vol. 22, No. 9, pp.1299~1303.
[2-3-5] Haug, E. J., Choi, K. K. and Komkov, V., 1986, Design Sensitivity Analysis of Structural System, Academic
[1-1-4] Press.
[2-3-6] Yoon, B. G. and Belegundu, A. D., 1988, "Iterative Methods for Design Sensitivity Analysis," AIAA Journal,
[1-1-4] Vol. 26, No. 11, pp. 1413~1415.

 


 

 
 

 

Higher Order Sensitivity

[2-3-7] Belle, H. V., 1982, "Higher Order Sensitivities in Structural Systems," AIAA Journal , Vol. 20, No. 2, pp.
[1-1-4] 286~288.
[2-3-7.5] Peng Zemin, Song Jianwei, 1986, "Invariant Theorem of Relative Sensitivity Summation and Higher
[1-1-4.5] Order Sensitivity Analysis Applied to Structural Modification," IMAC 4, pp. 66-74.
[2-3-8] Jankovic, M. S., 1988, "Analytical Solutions for the Nth Derivatives of Eigenvalues and Eigenvectors for a
[1-1-4] Nonlinear Eigenvalue Problem," AIAA Journal, Vol. 26, No. 2, pp. 204~205.
[2-3-9] Brandon, J. A., 1991, "Second-Order Design Sensitivities to Assess the Applicability of Sensitivity
[1-1-4] Analysis," AIAA Journal, Vol. 29, No. 1, pp. 135~139.
[2-3-10] Zeng, X., Zhang, K., and Yang, J., 1998, "A New Dynamic Reanalysis Method for Large Structure
[1-1-4] Modification," Proceedings of the 16th International Modal Analysis Conference, pp. 1598~1601.

 


 

 
 

 

Eigenvector Sensitivity

[2-3-11] Nelson, R. B., 1976, "Simplified Calculation of Eigenvector Derivatives," AIAA Journal, Vol. 14, No. 9, pp.
[1-1-14] 1201~1205.
[2-3-12] Lim, K. B., Juang, J. N. and Ghaemmaghami, P., 1989, "Eigenvector Derivatives of Repeated Eigenvalues
[1-1-14] Using Singular Value Decomposition," Journal of Guidance, Vol. 12, No. 2, pp. 282~283.
[2-3-13] Wang, B. P., 1991, "Improved Approximate Methods for Computing Eigenvector Derivatives in Structural
[1-1-14] Dynamics," AIAA Journal, Vol. 29, No. 6, pp.1018~1020.
[2-3-14] Sutter, T.R., Camarda, C. J., Walsh, J. L. and Adelman, H. M., 1988, "Comparison of Several Methods for
[1-1-14] Calculation Vibration Mode Shape Derivatives," AIAA Journal, Vol. 26, No. 12, pp. 1506~1511.
[2-3-15] Alvin, K. F., 1997, "Efficient Computation of Eigenvector Sensitivities for Structural Dynamics," AIAA
[1-1-14] Journal, Vol. 35, No. 11, pp. 1760~1766.
[2-3-16] Zhang, D. W. and Wei, F, S., 1997, "Efficient Computation of Many Eigenvector Derivatives Using
[1-1-14] Dynamic Flexibility Method," AIAA Journal, Vol. 35, No. 4, pp.712~718.
[2-3-17] Zhang, O. and Zerva, A., 1997, "Accelerated Iterative Procedure for Calculating Eigenvector Derivatives,"
[1-1-14] AIAA Journal, Vol. 35, No. 2, pp. 340~348.

 


 

 
 

 

Repeated Eigenvalue Sensitivity

[2-3-18] Chen, T. Y., 1993, "Design Sensitivity Analysis for Repeated Eigenvalues in Structural Design," AIAA
[1-1-14] Journal, Vol. 31, No.12, pp. 2347~2350.
[2-8-18.5] Ki-Ook Kim, 1987, "Modal Design Sensitivities for Multiple Eigenvalues," AIAA Journal Review
[1-1-14.5] Manuscript.
[2-8-18.6] Gene J. W. Hou, Sean P. Kenny, September 1992, "Eigenvalue and Eigenvector Approximate Analysis
[1-1-14.6] for Repeated Eigenvalue Problems," AIAA Journal, Vol. 30, No. 9, pp. 2317-2324.

 


 

 
 

 

Applications

[2-3-19] Raju, K. K., Rao, G. V. and Venugopal, N., 1989, "Use of Design Sensitivity Coefficients for Prediction of
[1-1-14]
Structural Behavior after Design Modification," Computers & Structures, Vol. 33, No. 5, pp. 1329~1331.
[2-3-20] Rizai, M. N. and Bernard, J. E., 1987, "An Efficient Method to Predict the Effect of Design Modifications,"
[1-1-14]
Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 109, pp. 373~384.
[2-3-21] Patrick, G. B. and Wei, T. S., 1990, "Practical Sensitivity Analysis," Proceedings of the 5th International
[1-1-14]
Modal Analysis Conference, pp. 21-25.
[2-3-22] Ai-Qun, L., and De-bao, L., 1990, "Using Experimental Modal Analysis to A Shell Structural Dynamic
[1-1-14]
Modification," Proceedings of the 8th International Modal Analysis Conference, pp. 972~978.
[2-3-23] Wang, J., Heylen, W. and Sas, P., 1987, "Accuracy of Structural Modification Techniques," Proceedings
[1-1-14]
of the 5th International Modal Analysis Conference, pp. 65~71.
[2-3-24] Heo, J. H. and Ehmann, K. F., 1991, "A Method for Substructural Sensitivity Synthesis," Journal of
[1-1-14]
Vibration and Acoustics, Vol. 113, pp. 201~208.
[2-3-25] Yuan-Fang Chou, Jeng-Shyong Chen, 1985, "Structural Dynamics Modification Via Sensitivity Analysis,"
[1-1-14]
IMAC 3, pp. 483-489.
[2-3-26] She Shaohua, LiuRuiyan, 1988, "Sensitivity Analysis and Optimization for the Modal Parameters of
[1-1-14]
Structural Vibration," IMAC 6, pp. 966-972.
[2-3-27] Ju, J. and Xu, Z., 1989, "The Multi-terms Sensitivity Analysis of Structure Modification," IMAC 7, pp. 584-
[1-1-14]
588.
[2-3-28] Tang Xiao Bing, Zhang Kai Yin, 1993, "Sensitivity Analysis of Structural Dynamic Behavior," IMAC 11, pp.
[1-1-14]
913-916.
[2-3-29] Chang-Zhi Wu, Xia Xu and Hua Yan, 1993, "A Research for Analytical Method of Dynamic Optimization
[1-1-14]
Design in the Structure System of Machine Tools," IMAC 11, pp. 1212-1216.
[2-3-30] Itsurou Kajiwara, Akio Nagamatsu, 1993, "Optimum Design of Structure by Sensitivity Analysis," IMAC 11,
[1-1-14]
pp. 883-889.
[2-3-31] Chien-Huan Chu, John A. Cafeo, 1994, "Design Modification with Experimental Based Structural
[1-1-14]
Models," IMAC 12, pp. 1666-1672.
[2-3-misc] Miscellaneous papers on eigenvalue and eigenvector sensitivities.

 


 

 
 

 

2-4 모드 합성법 (Modal Synthesis Method)

Eigenvalue Upper Bound and Lower Bound

[2-4-1] Ram, Y. M., Braun, S. G. and Blech, J., 1988, "Structural Modifications in Truncated Systems by the
[1-1-14 Rayleigh-Ritz Method," Journal of Sound and Vibration, Vol. 125, No. 2, pp. 203~209.
[2-4-2] Ram, Y. M. and Braun, S. G., 1990, "Upper and Lower Bounds for the Natural Frequencies of Modified
[1-1-14 Structures Based on Truncated Modal Testing Results," Journal of Sound and Vibration, Vol. 137, No. 1,
[1-1-14 pp. 69~81.
[2-4-2.5] Ram, Y. M., Braun, S. G. and Blech, J., 1990, "Structural Dynamic Modifications Using Truncated Data:
[1-1-1.54 Bounds for the Eigenvalues," Mechanical Systems and Signal Processing, Vol. 4, No. 1, pp. 39~52.
[2-4-3] Tse, C. C. and Liao, J. Y., 1993, "A Comparison of Error Bounds for Eigenvalues in Structural Modification
[1-1-14 Problems," Mechanical Systems and Signal Processing, Vol. 7, No. 3, pp. 273~279.
[2-4-4] Ram, Y. M. and Braun, S. G., 1993, "Eigenvector Error Bounds and Their Applications to Structural
[1-1-14 Modification," AIAA Journal, Vol. 31, No. 4, pp. 759~764.

 


 

 
 

 

Applications

[2-4-5] Wang, B. P., 1987, "Structural Dynamic Optimization Using Reanalysis Technique," The International
[1-1-14 Journal of Analytical and Experimental Modal Analysis, Vol. 2, No. 1, pp. 50~58.
[2-4-6] Wang, W., Zhang, Q., Allemang, R. J. and Brown, D. L., 1990, "Optimization of Structural Redesign With
[1-1-14 Dynamic Constraints," Proceedings of the 8th International Modal Analysis Conference, pp. 946~953.
[2-4-6.5] Hiroshi Kanda, Max L. Wei, Randall J. Allemant and David L. Brown, 1986, "Structural Dynamic
[1-1-1.64 Modifications Using Mass Additive Technique," IMAC 4, pp. 691-699.
[2-4-6.6] Lec. Zhang Jie, Chen Kechen and Xu Zhandi, 1988, "Structural Dynamic Optimal Design Using Modal
[1-1-1.64 Model," IMAC 6, pp. 1525-1529.
[2-4-7] Ram, Y. M. and Braun, S., 1987, "On Structural Modifications in Truncated System," Proceedings of the
[1-1-14 5th International Modal Analysis Conference, pp. 1550~1556.
[2-4-8] Xiuqing, H. and Ku, C. H., 1992, "Modal Analysis and Structural Dynamics Modification for Vibration of
[1-1-14 Trial-Produce Vehicle," Proceedings of the 10th International Modal Analysis Conference, pp. 863~869.
[2-4-8.5] Y. Gallo, P. Level and Y. Ravalard, 1992, "Modal Reanalysis with Improvement of Initial Modelization. A
[1-1-1.54 New Development in Structural Modification," IMAC 10, pp. 1167-1172.

 


 

 
 

 

Beam Applications

[2-4-9] John C. O'Callahan, Chaur-Ming Chou, 1985, "Structural Dynamics Modification Using Generalized Beam
[1-1-14
Mass and Stiffness Matrices," IMAC 3, pp. 477-482.
[2-4-10] K. B. Elliott, L. D. Mitchell, 1987, "Structural Modification Utilizing Beam Elements," IMAC 5, pp. 956-965.
[2-4-11] T. K. Kundra, B. C. Nakra, 1987, "System Modification via Identified Dynamic Models," IMAC 5, pp. 79-85.
[2-4-12] R. R. Lin, A. B. Palazzolo, R. M. Alexander, C. H. Gerhold, 1987, "Beam Modifications of Structural
[1-1-124
Systems Utilizing the Receptance Approach with Static Flexibility," IMAC 5, pp. 1557-1564.
[2-4-13] K Jayaraman, Robert Reiss, 1987, "Frequency Optimization of Repetitive Lattice Beam-Like Structure
[1-1-124
Using a Continuum Model," IMAC 5, pp. 1565-1572.
[2-4-14] M. M. Tayeb, E. J. Williams, 1986, "Rib Stiffeners for Use in Structural Dynamics Modification," IMAC 4,
[1-1-124
pp. 1094-1099.
[2-4-15] J. Zeischka, 1989, "A Modification of the Classical Beam Element for Improved Dynamic Analysis," IMAC
[1-1-124
7, pp. 243-245.
[2-4-16] Wallack, P., Skoog, P., and Richardson, M., 1989, "Comparison of Analytical and Experimental Rib
[1-1-124
Stiffener Modifications to A Structure," Proceedings of the 7th International Modal Analysis Conference,
[1-1-124
pp. 965~973.
[2-4-17] Cafeo, J. A., Trethewey, M. W. and Sommer III, H. J., 1995, "Beam Element Structural Dynamics
[1-1-124
Modification Using Experimental Modal Rotational Data," Journal of Vibration and Acoustics, Vol. 117,
[1-1-124
pp. 265~271.
[2-4-18] Brian J. Schwarz, Mark H. Richardson, 1997, "Structural Modifications Using Higher Order Elements,"
[1-1-124
IMAC 15, pp. 313-318
[2-4-19] Keng C. Yap, David C. Zimmerman, 1998, "A Comparison of Structural Dynamic Modification and
[1-1-124
Sensitivity Method Approximation," IMAC 16, pp. 1409-1415.

 


 

 
 

 

2-5 Dual Modal Space Method (DMSM)

Applications


[2-5-1] Luk, Y. W. and Mitchell, L. D., 1983, "System Modeling and Modification Via Modal Analysis," Proceedings
[1-1-14
of the 1st International Modal Analysis Conference, pp. 423~429.
[2-5-2] Luk, Y. W. and Mitchell, L. D., 1984, "Implementation of the Dual Modal Space Structural Modification
[1-1-14
Method," Proceedings of the 2nd International Modal Analysis Conference, pp. 930~936.
[2-5-3] Luk, Y. W. and Mitchell, L. D., 1984, "Analytical and Geometric Substructuring via Dual Modal Space
[1-1-14
Structural Modification," Proceedings of the 2nd International Modal Analysis Conference2, pp. 50~57.
[2-5-3.5] Peter Wallack, Paul Skoog, and Mark Richafdson, 1988, "Simultaneous Structural Dynamics
[1-1-1.54
Modification (S2DM)," IMAC 6, pp. 1033-1038.
[2-5-4] Deel, J, C. and Luk, Y. W., 1985, "Modal Testing Considerations For Structural Modification Applications,"
[1-1-14
Proceedings of the 3rd International Modal Analysis Conference, pp. 46~52.
[2-5-5] Elliot, K. B., and Mitchell, L. D., 1985, "Realistic Structural Modification: Part I Theoretical Development,"
[1-1-14
Proceedings of the 3rd International Modal Analysis Conference, pp. 471~476.

 


 

 
 

 

Modal Truncation Error
[2-5-6] Elliott, K. B. and Mitchell, L. D., 1987, "The Effect of Modal Truncation on Modal Modification,"
[1-1-14 Proceedings of the 5th International Modal Analysis Conference, pp. 72~78.
[2-5-7] Avitabile, P. and O'Callahan, J. C., 1991, "Understanding Structural Dynamic Modification and the Effects
[1-1-14 of Truncation," The International Journal of Analytical and Experimental Modal Analysis, Vol. 6, No. 4, pp.
[1-1-14 215~235.
[2-5-8] Stephan C. Ulm, 1986, "Investigations into the Effective Use of Structural Modification," IMAC 4, pp. 1279-
[1-1-14 1286.

 


 

 

 

 

2-6 국부 구조 변경법(Localized Modification Method)

[2-6-1] Weissenburger, J. T., 1968, "Effect of Local Modifications on the Vibration Characteristics of Linear
[1-1-14 Systems," Journal of Applied Mechanics, Vol. 35, pp.327~332.
[2-6-2] Pomazal, R. J. and Snyder, V. W., 1971, "Local Modifications of Damped Linear Systems," AIAA Journal,
[1-1-14 Vol. 9, No. 11, pp. 2216~2221.
[2-6-2.5] Moraux D., Level P. and Ravalard Y., 1993, "Efficient Methods for Predicting the Effects of Modification
[1-1-1.54 on the Dynamic Behavior of Structures," IMAC 11, pp. 1175-1180.
[2-6-3] Hallquist, J. and Snyder, V. W., 1973, "Synthesis of Two Discrete Vibratory Systems Using Eigenvalue
[1-1-14 Modification," AIAA Journal, Vol. 11, No. 2, pp. 247~249.
[2-6-4] Hallquist, J. O. and Snyder, V. W., 1974, "On the Connection of Viscously Damped Continuous Vibratory
[1-1-14 Systems," Journal of Sound and Vibration, Vol. 32, No. 1, pp.131~142.
[2-6-5] Hallquist, J. and Snyder, V., 1973, "Synthesis of Two Discrete Vibratory Systems Using Eigenvalue
[1-1-14 Modification," AIAA Journal, Vol. 11, No. 2, pp. 247~249.
[2-6-6] Hallquist, J. O., 1976, "An Efficient Method for Determining the Effects of Mass Modifications in Damped
[1-1-14 Systems," Journal of Sound and Vibration, Vol. 44, No. 3, pp. 449~459.
[2-6-7] O'Callahan, J. C. and Chou, C. M., 1984, "Study of A Structural modification Procedure with Three
[1-1-14 Dimensional Beam Elements using A Local Eigenvalue Modification Procedure," Proceedings of the 2nd
[1-1-14 International Modal Analysis Conference, pp. 945~952.
[2-6-8] Ram, Y. M. and Blech, J. J., 1991, "The Dynamic Behavior of a Vibratory System after Modification,"
[1-1-14 Journal of Sound and Vibration, Vol. 150, No. 3, pp. 357~370.
[2-6-9] Wang, B. P., 1985, "The Limitation of Local Structural Dynamic Modification," Proceedings of the 3rd
[1-1-14 International Modal Analysis Conference, pp. 53~58.
[2-6-9.5] Sun Zhijian , Wang Xiaogang and Xu Xia, 1992, "An effective Solution Method For Local Modifications of
[1-1-1.54 Damped Linear System," IMAC 10, pp.1108-1112.
[2-6-10] Kashiwagi, M., Hirai, I., Katayama, T. and Pilkey, W. D., 1993, "Sturm Sequence Recurrence Formula for
[1-1-104 Eigensolution of Locally Modified Systems," Finite Elements in Analysis and Design , Vol.12, pp. 31~36.
[2-6-11] Kashiwagi, M., Hirai, I., Ohwaki, S. I. and Pilkey, W. D., 1991, "Stable Eigensolution of Locally Modified
[1-1-104 Systems Based on the Sturm Sequence Property," Finite Elements in Analysis and Design, Vol.9, pp.
[1-1-104 133~139.

 


 

 
 

 

2-7 라그랑지 승수법(Lagrange Multiplier Method)

[2-7-1] Dowell, E. H., 1971, "Free Vibrations of a Linear Structure with Arbitrary Support Conditions," Journal of Applied Mechanics, pp. 595~600.
[2-7-2] Dowell, E. H., 1972, "Free Vibrations of an Arbitrary Structure in Terms of Component Modes," Journal of Applied Mechanics, pp. 727~732.
[2-7-3] Dowell, E. H., 1979, "On Some General Properties of Combined Dynamical Systems," Journal of Applied Mechanics, Vol. 46, pp. 206~209.
[2-7-4] Dowell, E. H., 1980, "Bounds on Modal Damping by a Component Modes Method Using Lagrange Multipliers," Journal of Applied Mechanics, Vol. 47, pp. 211~213.
[2-7-5] Tongue, B. H. and Dowell, E. H., 1983, "Component Mode Analysis of Nonlinear, Nonconservative Systems," Journal of Applied Mechanics, Vol. 50, pp. 204~209.
[2-7-5.5] Dowell, E. H., 1982, "Component Mode Analysis of a Simple Nonlinear, Nonconservative Systems," Journal of Sound and Vibration, Vol. 80, No.2, pp. 204~209.
[2-7-6] Dowell, E. H., 1987, "Theory and Application of Dynamic Decoupling in Structural Analysis: Another View," Finite Elements in Analysis and Design, Vol. 3, pp. 119~125.
[2-7-7] Hallquist, J. and Snyder, V. W., 1973, "Linear Damped Vibratory Structures with Arbitrary Support Conditions," Journal of Applied Mechanics, pp. 312~313.
[2-7-8] Howell, L. J., 1975, "Effects of Constraint Modification on the Random Vibration of Damped Linear Structures," General Motors Corporation, Warren, Mich., pp. 685~687.
[2-7-9] Hu, L. J., Smith, I., and Chui, Y. H., 1994, "Vibration Analysis of Ribbed Plates with a Rigid Intermediate Line Support," Journal of Sound and Vibration, Vol. 178, No. 2, pp. 163~175.
[2-7-10] Zeng, Z., Huang, T. and Hamilton, J. F., 1990, "An Effective Approach to Determine Natural Frequencies and Mode Shapes of Constrained Beams Using Lagrange Multipliers," The International Journal of Analytical and Experimental Modal Analysis, Vol. 5, No. 2, pp. 109~114.
[2-7-10.5] Zeng Ziping, Huang Tian, James F. Hamilton, 1988, "An Effective Approach to Determine Natural Frequencies and Mode Shapes of Constrained Beams Using Lagrange Multipliers," IMAC 6, pp. 1509-1512.