Clarification about Component Mode Synthesis Methods for Substructures with Physical Flexible Interfaces

  • Ohayon, R. (Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Metiers (CNAM)) ;
  • Soize, C. (Laboratoire Modelisation et Simulation Multi-Echelle (MSME UMR 8208 CNRS), Universite Paris-Est)
  • Received : 2014.05.08
  • Accepted : 2014.05.16
  • Published : 2014.06.30


The objective of the paper is to clarify a methodology based on the use of the existing component mode synthesis methods for the case of two damped substructures which are coupled through a linking viscoelastic flexible substructure and for which the structural modes with free geometrical interface are used for each main substructure. The proposed methodology corresponds to a convenient alternative to the direct use either of the Craig-Bampton method applied to the three substructures (using the fixed geometric interface modes) or of the flexibility residual approaches initiated by MacNeal (using the free geometric interface modes). In opposite to a geometrical interface which is a topological interface on which there is a direct linkage between the degrees of freedom of substructures, we consider a physical flexible interface which exists in certain present technologies and for which the general framework linear viscoelasticity is used and yields a frequency-dependent damping and stiffness matrices of the physical flexible interface.


  1. Kuhar, E. J. and Stahle, C. V., "Dynamic transformation method for modal synthesis," AIAA Journal, Vol. 12, No. 5, 1974, pp. 672-678. DOI: 10.2514/3.49318
  2. Craig, R. R., "A Review of Time Domain and Frequency Domain Compo-nent Mode Synthesis Method," Combined Experimental-Analytical Modeling of Dynamic Structural Systems, edited by D.R. Martinez and A.K. Miller, Vol. 67, ASME-AMD, New York, 1985.
  3. de Klerk, D., Rixen, D. J. and Voormeeren, S. N., "General framework for dy-namic substructuring: History, review, and classification of techniques," AIAA Journal, Vol. 46, No. 5, 2008, pp. 1169-1181. DOI: 10.2514/1.33274
  4. Argyris, J. H. and Kelsey, S., "The analysis of fuselages of arbitrary cross-section and taper: A DSIR sponsored reserach program on the development and application of the matrix force method and the digital computer," Aircraft Engineering and Aerospace Technology, Vol. 31, No. 9, 1959, pp. 272-283. DOI: 10.1108/eb033156
  5. Przemieniecki, J. S., "Matrix structural analysis of substructures," AIAA Journal, Vol. 1, No. 1, 1963, pp. 138-147. DOI: 10.2514/3.1483
  6. Guyan, R. J., "Reduction of stiffness and mass matrices," AIAA Journal, Vol. 3, No. 2, 1965, pp. 380-380. DOI: 10.2514/3.2874
  7. Irons, B., "Structural eigenvalue problems elimination of unwanted variables," AIAA Journal, Vol. 3, No. 5, 1965, pp. 961-962. DOI: 10.2514/3.3027
  8. Hurty, W. C., "Vibrations of structural systems by component mode synthesis," Journal of Engineering Mechanics / American Society of Civil Engineers, Vol. 86, No. 4, 1960, pp. 51-69.
  9. Hurty, W. C., "Dynamic analysis of structural systems using component modes," AIAA Journal, Vol. 3, No. 4, 1965, pp. 678-685. DOI: 10.2514/3.2947
  10. Craig, R. R. and Bampton, M. C. C., "Coupling of substructures for dynamic analyses," AIAA Journal, Vol. 6, No. 7, 1968, pp. 1313-1322. DOI: 10.2514/3.4741
  11. Benfield, W. A., and Hruda, R. F., "Vibration analysis of structures by component mode substitution," AIAA Journal, Vol. 9, No. 7, 1971, pp. 1255-1261. DOI: 10.2514/3.49936
  12. Hintz, R. M., "Analytical methods in component modal synthesis," AIAA Journal, Vol. 13, No. 8, 1975, pp. 1007-1016. DOI: 10.2514/3.60498
  13. Agrawal, B. N., "Mode synthesis technique for dynamic analysis of structures," Journal of the Acoustical Soiety of America, Vol. 59, No. 6, 1976, pp. 1329-1338. DOI: 10.1121/1.381019
  14. Bathe, E. and Gracewski, S., "On non-linear dynamic analysis using substructuring and mode superposition," Computers and Structures, Vol. 13, No. 5-6, 1981, pp. 699-707. DOI: 10.1016/0045-7949(81)90032-8
  15. Meirovitch, L. and Hale, A. L., "On the substructure synthesis method," AIAA Journal, Vol. 19, No. 7, 1981, pp. 940-947. DOI: 10.2514/3.51023
  16. Hale, A. L. and Meirovitch, L., "A procedure for improving discrete sub-structure representation in dynamic synthesis," AIAA Journal, Vol. 20, No. 8, 1982, pp. 1128-1136. DOI: 10.2514/3.51173
  17. Meirovitch, L., and Kwak, M. K., "Rayleigh-Ritz based substructure synthesis for flexible multibody systems," AIAA Journal, Vol. 29, No. 10, 1991, pp. 1709-1719. DOI: 10.2514/3.10794
  18. Bourquin, F. and d'Hennezel, F., "Numerical study of an intrinsic component mode synthesis method," Computer Methods in Applied Mechanics and Engineering, Vol. 97, No. 1, 1992, pp. 49-76. DOI: 10.1016/0045-7825(92)90107-U
  19. Suarez, L. E., and Singh, M. P., "Improved fixed interface method for modal synthesis," AIAA Journal, Vol. 30, No. 12, 1992, pp. 2952-2958. DOI: 10.2514/3.11642
  20. Farhat, C. and Geradin, M., "On a component mode method and its applica-tion to incompatible substructures," Computers and Structures, Vol. 51, No. 5, 1994, pp. 459-473. DOI: 10.1016/0045-7949(94)90053-1
  21. Balmes, E., "Optimal Ritz vectors for component mode synthesis using the singular value decomposition," AIAA Journal, Vol. 34, No. 6, 1996, pp. 1256-1260. DOI: 10.2514/3.13221
  22. Castanier, M. P., Tan, Y. C. and Pierre, C., "Characteristic constraint modes for component mode synthesis," AIAA Journal, Vol. 39, No. 6, 2001, pp. 1182-1187. DOI: 10.2514/2.1433
  23. Rixen, D. J., "A dual Craig-Bampton method for dynamic substructuring," Journal of Computational and Applied Mathematics, Vol. 168, 2004, pp. 383-391. DOI: 10.1016/
  24. Voormeeren, S. N., van der Valk, P. L. and Rixen, D. J., "Generalized methodology for assembly and reduction of component models for dynamic substructuring," AIAA Journal, Vol. 49, No. 5, 2011, pp. 1010-1020. DOI: 10.2514/1.J050724
  25. Lindberg, E., Horlin, N. E. and Goransson, P., "Component mode synthesis using undeformed interface coupling modes to connect soft and stiff substructures," Shock and Vibration, Vol. 20, No. 1, 2013, pp. 157-170. DOI: 10.3233/SAV-2012-0734
  26. MacNeal, R.H., "A hybrid method of component mode synthesis," Computers and Structures, Vol. 1, No. 4, 1971, pp. 581-601.
  27. Rubin, S., "Improved component-mode representation for structural dynamic analysis," AIAA Journal, Vol. 13, No. 8, 1975, pp. 995-1006. DOI: 10.2514/3.60497
  28. Craig, R. R. and Chang, C. J., "Free-interface methods of substructure coupling for dynamic analysis," AIAA Journal, Vol. 14, No. 11, 1976, pp. 1633-1635. DOI: 10.2514/3.7264
  29. Kammer, D. C. and Baker, M., "Comparison of the Craig-Bampton and Residual Flexibility Methods of Substructure Representation," Journal of Aircraft, Vol. 24, No. 4, 1987, pp. 262-267. DOI: 10.2514/3.45435
  30. Admire, J. R., Tinker, M. L. and Ivey, E. W., "Residual Flexibility test Method for Verification of Constrained Structural Modes," AIAA Journal, Vol. 32, No. 1, 1994, pp. 170-175. DOI: 10.2514/3.11963
  31. Majed, A., and Spanos, P. D., "Nonlinear dynamics of structures via residual flexibility of components," Journal of Aerospace Engineering, Vol. 10, No. 4, 1997, pp. 173-178. DOI: 10.1061/(ASCE)0893-1321(1997)10:4(173)
  32. Majed, A., and Henkel, E. E., "Improved method of mixed-boundary component-mode representation for structural dynamic analysis," Journal of Spacecraft and Rockets, Vol. 42, No. 5, 2005, pp. 825,831. DOI: 10.2514/1.8334
  33. Tran, D. M., "Component mode synthesis methods using partial interface modes: Application to tuned and mistuned structures with cyclic symmetry," Computers and Structures, Vol. 87, No. 17-18, 2009, pp. 1141-1153. DOI: 10.1016/j.compstruc.2009.04.009
  34. Ohayon, R., Sampaio, R. and Soize, C., "Dynamic substructuring of damped structures using singular value decomposition," Journal of Applied Mechanics-Transactions of the ASME, Vol. 64, No. 2, 1997, pp. 292-298. DOI: 10.1115/1.2787306
  35. Park, K. C. and Park, Y. H., "Partitioned component mode synthesis via a flexibility approach," AIAA Journal, Vol. 42, No. 6, 2004, pp. 1236-1245. DOI: 10.2514/1.10423
  36. Markovic, D., Park, K. C. and Ibrahimbegovic, A., "Reduction of substructural interface degrees of freedom in flexibility-based component mode synthesis," International Journal for Numerical Methods in Engineering, Vol. 70, No. 2, 2007, pp. 163-180. DOI: 10.1002/nme.1878
  37. Klein, L. R. and Dowell, E. H., "Analysis of modal damping by component modes method using Lagrange multipliers," Journal of Applied Mechanics, Vol. 41, No. 2, 1974, pp. 527-528. DOI: 10.1115/1.3423328
  38. Craig, R. R. and Yung-Tsen, Y. T., "Generalized substructure coupling procedure for damped systems," AIAA Journal, Vol. 20, No. 3, 1982, pp. 442-444. DOI: 10.2514/3.51089
  39. Qian, D. A., and Hansen, J. S., "Substructure synthesis method for frequency-response of viscoelastic structures," AIAA Journal, Vol. 33, No. 3, 1995, pp. 520-527. DOI: 10.2514/3.12607
  40. Wang, W. and Kirkhope, J., "Complex component mode synthesis for damped systems," Journal of Sound and Vibration, Vol. 181, No. 5, 1995, pp. 781-800. DOI: 10.1006/jsvi.1995.0171
  41. Fenander, A., "Modal synthesis when modeling damping by use of fractional derivatives," AIAA Journal, Vol. 34, No. 5, 1996, pp. 1051-1058. DOI: 10.2514/3.13186
  42. Ohayon, R., and Soize, C., Structural Acoustics and Vibration, Academic Press, London, 1998.
  43. Friswell, M. I., and Inman, D. J., "Reduced-order models of structures with viscoelastic components," AIAA Journal, Vol. 37, No. 10, 1999, pp. 1318-1325. DOI: 10.2514/2.603
  44. Morgan, J. A., and Pierre, C., and Hulbert, G. M., "Baseband methods of component mode synthesis for nonproportionally damped systems," Mechanical Systems and Signal Processing, Vol. 17, No. 3, 2003, pp. 589-598. DOI: 10.1006/mssp.2001.1467
  45. Liu, M. H., and Zheng, G. T., "Improved componentmode synthesis for nonclassically damped systems," AIAA Journal, Vol. 46, No. 5, 2008, pp. 1160-1168. DOI: 10.2514/1.32869
  46. Dieker, S., Abdoly, K., and Rittweger, A., "Flexible boundary method in dynamic substructure techniques including different component damping," AIAA Journal, Vol. 48, No. 11, 2010, pp. 2631-2638. DOI: 10.2514/1.J050484
  47. Liu, Z.-S., and Wu Z.-G., "Iterative-order-reduction substructuring method for dynamic condensation of finite element models," AIAA Journal, Vol. 49, No. 1, 2011, pp. 87-96. DOI: 10.2514/1.J050184
  48. Soize, C. and Mziou, S., "Dynamic substructuring in the medium-frequency range," AIAA Journal, Vol. 41, No. 6, 2003, pp. 1113-1118. DOI: 10.2514/2.2052
  49. Sarkar, A. and Ghanem, R., "A substructure approach for the midfrequency vibration of stochastic systems," Journal of the Acoustical Society of America, Vol. 113, No. 4, 2003, pp. 1922-1934. DOI: 10.1121/1.1558374
  50. Herran, M., Nelias, D., Combescure, A. and Chalons, H., "Optimal component mode synthesis for medium frequency problem ," International Journal For Numerical Methods In Engineering, Vol. 86, No. 3, 2011, pp. 301-315. DOI: 10.1002/nme.3064
  51. Kassem, M., Soize, C., and Gagliardini, L., "Structural partitioning of complex structures in the medium-frequency range. An application to an automotive vehicle," Journal of Sound and Vibration, Vol. 330, No. 5, 2011, pp. 937-946. DOI: 10.1016/j.jsv.2010.09.008
  52. Brown, A. M. and Ferri, A. A., "Probabilistic component mode synthesis of nondeterministic substructures," AIAA Journal, Vol. 34, No. 4, 1996, pp. 830-834. DOI: 10.2514/3.13146
  53. Soize, C., and Chebli, H., "Random uncertainties model in dynamic substructuring using a nonparametric probabilistic model," Journal of Engineering Mechanics, Vol. 129, No. 4, 2003, pp. 449,457. DOI: 10.1061/(ASCE)0733-9399(2003)129:4(449)
  54. Hinke, L., Dohnal, F., Mace, B. R., Waters, T. P., and Ferguson, N. S., "Component mode synthesis as a framework for uncertainty analysis," Journal of Sound and Vibration, Vol. 324, No. 1-2, 2009, pp. 161-178. DOI: 10.1016/j.jsv.2009.01.056
  55. Hong, S.-K., Epureanu, B. I., Castanier, M. P., and Gorsich, D. J.,"Parametric reduced-order models for predicting the vibration response of complex structures with component damage and uncertainties," Journal of Sound and Vibration, Vol. 330, No. 6, 2011, pp. 1091-1110. DOI: 10.1016/j.jsv.2010.09.022
  56. Soize, C., and Batou, A., "Stochastic reduced-order model in low-frequency dynamics in presence of numerous local elastic modes," Journal of Applied Mechanics - Transactions of the ASME, Vol. 78, No. 6, 2011, pp. 061003-1,061003-9. DOI: 10.1115/1.4002593
  57. Ohayon, R., and Soize, C., "Advanced computational dissipative structural acoustics and fluid-structure interaction in low and medium-frequency domains. Reduced-order models and uncertainty quantification," International Journal of Aeronautical and Space Sciences, Vol. 13, No. 2, 2012, pp. 127-153. DOI: 10.5139/IJASS.2012.13.2.127
  58. Mignolet, M. P., Soize, C., and Avalos, J.,"Nonparametric stochastic modeling of structures with uncertain boundary conditions / coupling between substructures," AIAA Journal, Vol. 51, No. 6, 2013, pp. 1296-1308. DOI: 10.2514/1.J051555
  59. Jezequel, L., "A hybrid method of modal synthesis using vibration tests," Journal of Sound and Vibration, Vol. 100, No. 2, 1985, pp. 191-210. DOI: 10.1016/0022-460X(85)90415-8
  60. Urgueira, A. P. V., Dynamic Analysis of Coupled Structures Using Experimental Data, Thesis of the University of London for the Diploma of Imperial College of Science, Technology and Medecine, October 1989.
  61. Nobari, A. S., Robb, D. A., and Ewins, D. J., "A new approach to modal-based structural dynamic-model updating and joint identification," Mechanical Systems and Signal Processing, Vol. 9, No. 1, 1995, pp. 85,100. DOI: 10.1006/mssp.1995.0007
  62. Morgan, J. A., Pierre, C., and Hulbert, G. M.,"Calculation of component mode synthesis matrices from measured frequency response functions, part 1: Theory," Journal of Vibration and Acoustics-Transactions of the ASME, Vol. 120, No. 2, 1998, pp. 503-508. DOI: 10.1115/1.2893858
  63. Morgan, J. A., Pierre, C., and Hulbert, G. M.,"Calculation of component made synthesis matrices from measured frequency response functions, part 2: Application," Journal of Vibration and Acoustics-Transactions of the ASME, Vol. 120, No. 2, 1998, pp. 509-516. DOI: 10.1115/1.2893859
  64. Liu, W., and Ewins, D. J., "Substructure synthesis via elastic media Part I: Joint identification," Proceedings of the 18th IMAC Conference on Computational Challenges in Structural Dynamics (IMAC-XVIII), Book Series: Proceedings of The Society of Photo-Optical Instrumentation Engineers (SPIE), Bellingham, WA, Vol. 4062, 2000, pp. 1153-1159.
  65. Allen, M. S., Mayes, R. L., and Bergman, E. J., "Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multipoint connections," Journal of Sound and Vibration, Vol. 329, No. 23, 2010, pp. 4891-4906. DOI: 10.1016/j.jsv.2010.06.007
  66. Allen, M. S., Gindlin, H. M., and Mayes, R. L., "Experimental modal substructuring to estimate fixed-base modes from tests on a flexible fixture", Journal of Sound and Vibration, Vol. 330, No. 18-19, 2011, pp. 4413-4428. DOI: 10.1016/j.jsv.2011.04.010
  67. Saigal, S., and Kane, J. H., "Design sensitivity analysis of boundary-element substructures," AIAA Journal, Vol. 28, No. 7, 1990, pp. 1277-1284. DOI: 10.2514/3.25205
  68. Hou, G., Maroju, V., and Yang, R. J., "Component mode synthesis-based design optimization method for local structural modification," Structural Optimization, Vol. 10, No. 2, 1995, pp. 128-136. DOI: 10.1007/BF01743541
  69. Tu, J.-Y., Yang, H.-T., Lin, P.-Y., and Chen, P. C., "Dynamics, control and real-time issues related to substructuring techniques: application to the testing of isolated structure systems," Journal of Systems and Control Engineering, Vol. 227, No. 6, 2013, pp. 507-522. DOI: 10.1177/0959651813491743
  70. Sunar, M., and Rao, S. S., "Optimal structural control by substructure synthesis," AIAA Journal, Vol. 38, No. 11, 2000, pp. 2191-2194. DOI: 10.2514/2.885
  71. Masson, G., Brik, B. A., Cogan, S., and Bouhaddi, N., "Component mode synthesis (CMS) based on an enriched Ritz approach for efficient structural optimization," Journal of Sound and Vibration, Vol. 296, No. 4-5, 2006, pp. 845-860. DOI: 10.1016/j.jsv.2006.03.024
  72. Lim, C. N., Neild, S. A., Stoten, D. P., Drury, D., and Taylor, C. A., "Adaptive control strategy for dynamic substructuring tests," Journal of Engineering Mechanics- ASCE, Vol. 133, No. 8, 2007, pp. 864-873. DOI: 10.1061/(ASCE)0733-9399(2007)133:8(864)
  73. Amsallem, D., and Farhat, C., "An online method for interpolationg linear parametric reduced-order models," SIAM Journal on Scientific Computing, Vol. 33, No. 5, 2011, pp. 2169-2198. DOI: 10.1137/100813051
  74. Perdahcioglu, D. A., Geijselaers, H. J. M., Ellenbroek, M. H. M., and de Boer, A., "Dynamic substructuring and reanalysis methods in a surrogate-based design optimization environment," Structural and Multidisciplinary Optimization, Vol. 45, No. 1, 2012, pp. 129-138. DOI: 10.1007/s00158-011-0681-4
  75. Voormeeren, S. N., and Rixen, D. J., "Updating component reduction bases of static and vibration modes using preconditioned iterative techniques," Computer Methods in Applied Mechanics and Engineering, Vol. 253, 2013, pp. 39-59. DOI: 10.1016/j.cma.2012.10.007
  76. Morand, H. J. P., and Ohayon, R., "Substructure variational analysis for the vibrations of coupled," International Journal for Numerical Methods in Engineering, Vol. 14, No. 5, 1979, pp. 741-755. DOI: 10.1002/nme.1620140508
  77. Morand, H. J. P. and Ohayon, R., Fluid Structure Interaction, Wiley, New York, 1995.
  78. Clough, R. W. and Penzien, J., Dynamics of Structures, McGraw-Hill, New York, 1975.
  79. Bathe, K. J. and Wilson, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, New York, 1976.
  80. Meirovitch, L., Computational Methods in Structural Dynamics, Sijthoff and Noordhoff, The Netherlands, 1980.
  81. Sanchez-Hubert, J., and Sanchez-Palencia, E., Vibration and Coupling of Continuous Systems. Asymptotic Methods, Springer-Verlag, Berlin, 1989.
  82. Meirovitch, L., Dynamics and Control of Structures, Wiley, New York, 1990.
  83. Argyris, J. and Mlejnek, H. P., Dynamics of Structures, North-Holland, Amsterdam, 1991.
  84. Dautray, R. and Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, Springer-Verlag, Berlin, 1992.
  85. Leung, A. Y. T., Dynamic Stiffness and Substructures, Springer-Verlag, New York, 1993.
  86. Geradin, M. and Rixen, D., Mechanical Vibrations, Wiley, Chichester, 1997.
  87. Zienkiewicz, O.C., and Taylor, R.L., The Finite Element Method For Solid And Structural Mechanics, 6th ed., Elsevier, Butterworth-Heinemann, Amsterdam, 2005.
  88. Craig, R. R. and Kurdila, A., Fundamentals of Structural Dynamics, Wiley, New York, 2006.
  89. Saad Y., Numerical Methods for Large Eigenvalue Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2011.
  90. Chatelin, F., Eigenvalues of Matrices, Revised Edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2012.
  91. Soize, C., Stochastic Models of Uncertainties in Computational Mechanics, American Society of Civil Engineers, Reston, 2012.
  92. Ohayon, R., and Soize, C., Advanced Computational Vibroacoustics, Cambridge University Press, Cambridge, 2014.
  93. Ohayon, R., and Soize, C., "Variational-based reduced-order model in dynamic substructuring of coupled structures through a dissipative physical interface: Recent advances", Archives of Computational Methods in Engineering, in press 2014. DOI: 10.1007/s11831-014-9107-y
  94. Bland, D. R., The Theory of Linear Viscoelasticity, Pergamon, London, 1960.
  95. Truesdell, C. (ed.), Mechanics of Solids, Vol III, Theory of Viscoelasticity, Plasticity, Elastic Waves and Elastic Stability, Springer-Verlag, Berlin, 1984.

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