Efficient Dynamic Response Analysis Using Substructuring Reduction Method for Discrete Linear System with Proportional and Nonproportional Damping

Choi, Dong-Soo;Cho, Maeng-Hyo;Kim, Hyun-Gi

  • Published : 2008.05.10


The dynamic response analysis for large structures using finite element method requires a large amount of computational resources. This paper presents an efficient vibration analysis procedure by combining node-based substructuring reduction method with a response analysis scheme for structures with undamped, proportional or nonproportional damping. The iterative form of substructuring reduction scheme is derived to reduce the full eigenproblem and to calculate the dynamic responses. In calculating the time response, direct integration scheme is used because it can be applied directly to the reduced model. Especially for the non proportional damping matrix, the transformation matrices defined in the displacement space are used to reduce the system. The efficiency and the effectiveness of the present method are demonstrated through the numerical examples.


Substructuring reduction method;Nonproportional damping;Newmark's scheme


  1. Zu-Qing Qu and Wenjun Chang, 2000, 'Dynamic condensation method for viscously damped vibration systems in engineering', Engineering Structures, Vol. 22, No. 11, pp. 1426-1432
  2. Bathe KJ., 1996, 'Finite Element Procedure', Prentice Hall, pp. 887-888
  3. Robert J. Guyan, 1965, 'Reduction of stiffness and mass matrices,' AIAA Journal, Vol. 3, No. 2, pp. 380
  4. Bruce Irons, 1965, 'Structural eigenvalue problems : elimination of unwanted variables', AIAA Journal, Vol. 3, No. 5, pp. 961-962
  5. Zu-Qing Qu and Panneer Selvam, 2005, 'Model Order Reduction of Viscously Damped Vibration Systems Using Accelerated Iterative Dynamic Condensation', Journal of Applied Mechanics, Vol. 72, No. 5, pp. 761-771
  6. Hyungi Kim and Maenghyo Cho, 2005, 'Two-level scheme for selection of degrees of freedom and semi-analytic sensitivity based on the reduced system', Computer Methods in Applied Mechanics and Engineering, Vol. 195, No. 33-36, pp. 4244-4268
  7. J.C.O'Cahhahan, 1989, 'A new procedure for an improved reduced system (IRS) model', Proceedings of the 7th International Modal Analysis Conference, pp. 17-21
  8. Gordis and Joshua H., 1992, 'An analysis of the improved reduced system (IRS) model reduction procedure', Proceedings of the 10th International Modal Analysis Conference, Vol. 1 (A94-12476 02-39), pp. 471-479
  9. M.I.Friswell, S.D.Garvey and J.E.T.Penny, 1995, 'Model reduction using dynamic and iterated IRS techniques', Journal of Sound and Vibration, Vol. 186, No. 2, pp. 311-323
  10. M.I.Friswell, S.D.Garvey and J.E.T.Penny, 1995, 'The convergence of the iterated IRS method', Journal of Sound and Vibration, Vol. 211, No. 1, pp. 123-132
  11. K.A. Foss, 1958, 'Co-ordinates which uncoupled the equations of motion of damped linear dynamic system', Journal of Applied Mechanics, Vol. 25, pp. 361-364
  12. Kolja Elssel and Heinrich Voss, 2006, 'An a priori bound for automated multilevel substructuring', SIAM J. Matrix anal. Appl., Vol. 28, No. 2, pp. 386-397
  13. Hyungi Kim and Maenghyo Cho, 2007, 'Improvement of reduction method combined with sub-domain scheme in large-scale problem', Int. J. of Num. Meth. Eng., Vol. 70, No. 2, pp. 206-251
  14. Dongsoo Choi and Maenghyo Cho, 2007, 'Iterative Method for Dynamic Condensation Combined with Substructuring Scheme', 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 23-26 April
  15. Craig, R. R., Jr., 1968, 'Coupling of substructures for dynamic analyses', AIAA Journal, Vol. 6, No. 7, pp. 1313-1319
  16. N.Bouhaddi and R.Fillod, 1992, 'Substructuring using a linearized dynamic condensation method', Computer & Structures, Vol. 45, No. 4, pp. 679-683
  17. Maenghyo and Hyungi Kim, 2004, 'Element-based node selection method for reduction of eigenvalue problems', AIAA Journal, Vol. 42, No. 8, pp. 1677-1684
  18. N.Bouhaddi and R.Fillod, 1996, 'Substructuring by a two-level dynamic condensation method', Computer & Structures, Vol. 60, No. 3, pp. 403-409
  19. Daniel J. Rixen, 2004, 'A dual Craig-Bampton method for dynamic substructuring', J. Compt. Appl. Math., Vol. 168, No. 1-2, pp. 383-391
  20. Jeffrey K. Bennighof and R. B. Lehoucq., 2004, 'An automated multilevel substructuring method for eigenspace computation in linear elastodynamics', SIAM J. Sci. Comput. Vol. 25, No. 6, pp. 2084-2106