Advertisement

Enhanced Order Reduction of Forced Nonlinear Systems Using New Ritz Vectors

  • Mohammad A. AL-Shudeifat
  • Eric A. Butcher
  • Thomas D. Burton
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

Enhanced modal-based order reduction of forced structural dynamic systems with isolated nonlinearity has been performed using the iterated LELSM (Local equivalent linear stiffness method) modes and new type of Ritz vectors. The iterated LELSM modes have been found via iteration of the modes of the mass normalized local equivalent linear stiffness matrix of the nonlinear systems. The optimal basis vector of principal orthogonal modes (POMs) is found for such system via simulation and used for POD-based order reduction for comparison. Two new Ritz vectors are defined as a static load vectors where one of them gives a static displacement to the mass connected to the periodic forcing load and the other gives a static displacement to the mass connected to the nonlinear element. It is found that the use of these vectors, which are augmented to the iterated LELSM modes in the order reduction modal matrix, reduces the number of modes used in order reduction and considerably enhances the accuracy of order reduction. The combination of the new Ritz vectors with the iterated LELSM modes in the order reduction matrix yields more accurate reduced models than POD-based order reduction of forced and nonlinear systems. Hence, the LELSM modal-based order reduction is essentially enhanced over POD-based and linear-based order reductions by using these new Ritz vectors. In addition, the main advantage of using the iterated LELSM modes for order reduction is that, unlike POMs, they do not require a priori simulation and thus they can be combined with new Ritz vectors and applied directly to the system.

Keywords

Proper Orthogonal Decomposition Invariant Manifold Reduce Order Model Order Reduction Nonlinear Spring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Guyan R. J., Reduction of stiffness and mass matrices, AIAA Journal, 2, 380, 1965.CrossRefGoogle Scholar
  2. 2.
    Burton T. D. and Young M.E., Model Reduction and Nonlinear Normal Modes in Structural Dynamics, Nonlinear and Stochastic Dynamics Symposium, AMD-Vol. 192, DE-Vol. 78, pp. 9 - 16, ASME Winter Ann. Mtg., Chicago, Nov. 6-11,1994.Google Scholar
  3. 3.
    Friswell M. I., Penny J. E. T. and Garvey S. D., Using linear model reduction to investigate the dynamics of structures with local nonlinearities, Mechanical Systems and Signal Processing, 9(3), 317-328, 1995.CrossRefGoogle Scholar
  4. 4.
    Burton T. D. and Rhee W., On the reduction of nonlinear structural dynamics models, J. of Vibration and Control, 6, 531-556, 2000.CrossRefGoogle Scholar
  5. 5.
    Kim J. and Burton T. D., Reduction of structural dynamics models having nonlinear damping, J. Vibration and Control, 4,147-169, 2006.Google Scholar
  6. 6.
    Butcher E. A. and Lu R., Order reduction of structural dynamic systems with static piecewise linear nonlinearities, Nonlinear Dynamics, 49, 375-399, 2007.MATHCrossRefGoogle Scholar
  7. 7.
    Shaw S.W and Pierre C., Non-linear normal modes and invariant manifolds, J. Sound and Vibration, 150, 170-173, 1991.Google Scholar
  8. 8.
    Jiang D., Pierre C. and Shaw S., Large amplitude nonlinear normal modes of piecewise linear systems, J. Sound and Vibration,272, 869-891, 2004.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Peschek E., Boivin N. and Pierre C., Nonlinear modal analysis of structural systems using multi-mode invariant manifolds,Nonlinear Dynamics, 25, 183-205, 2001.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Burton T.D., Numerical Calculation of Nonlinear Normal Modes in Structural Systems, Nonlinear Dynamics, 49, 425 – 441, 2007.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Shaw S. W., Pierre C. and Pesheck E., Modal analysis-based reduced-order models for nonlinear structures – An invariant manifold approach, Sound and Vibration Digest, 31, 3-16, 1999.CrossRefGoogle Scholar
  12. 12.
    Pesheck E., Pierre C. and Shaw S. W., Modal reduction of a nonlinear rotating beam through nonlinear modes, Journal of vibration and Acoustics, 124, 229-236, 2002.CrossRefGoogle Scholar
  13. 13.
    Sinha S. C., Redkar S. and Butcher E. A., Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds, J. Sound & Vibration, 284, 985-1002, 2005.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Sinha, S. C., Butcher E. A. and Dávid A., Construction of dynamically equivalent time invariant forms for time periodic systems, Nonlinear Dynamics, 16, 203-221, 1998.MATHCrossRefGoogle Scholar
  15. 15.
    Feeny B. F. and Kappagantu R., On the physical interpretation of proper orthogonal modes in vibrations, Journal of Sound and Vibration, 211 (4), 607-616, 1998.CrossRefGoogle Scholar
  16. 16.
    Feeny B. F., On proper orthogonal co-ordinates as indicators of modal activity, Journal of Sound and Vibration, 255 (5), 805-817, 2002.CrossRefGoogle Scholar
  17. 17.
    Han S. and Feeny B. F., Enhanced proper orthogonal decomposition for the modal analysis of homogeneous structures, Journal of Vibration and Control, 8(1), 19-40, 2002.MATHCrossRefGoogle Scholar
  18. 18.
    Lenaerts V., Kerschen G., Golinval J. C., Proper orthogonal decomposition for model updating of nonlinear mechanical systems, Mechanical Systems and Signal Processing, 15(1), 31-43, (2001).CrossRefGoogle Scholar
  19. 19.
    Kappagantu R. and Feeny B. F., An" optimal" modal reduction of a system with frictional excitation, Journal of Sound and Vibration, 224 (5), 863-877, 1999.CrossRefGoogle Scholar
  20. 20.
    Kerschen G., Golival J., Vakakis A. and Bergman L., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dynamics, 41,147-169, 2005.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Kumar N. and Burton T.D., Use of random excitation to develop POD based reduced order models for nonlinear structural dynamics, Proc. ASME IDETC, Paper DETC2007/VIB-35539, Las Vegas, NV, September 4-7 (2007).Google Scholar
  22. 22.
    Kumar N. and Burton T.D., On Combined Use of POD modes and Ritz vectors for model reduction in nonlinear structural dynamics, Proc. ASME IDETC, Paper DETC2009-87416, San Diego, CA, September (2009).Google Scholar
  23. 23.
    Segalman J. D., Model reduction of systems with localized nonlinearities, Journal of Computational and Nonlinear Dynamics, 2, 249-266, 2007.CrossRefGoogle Scholar
  24. 24.
    Kline K. A., Dynamic analysis using a reduced basis of exact modes and Ritz vectors, AIAA Journal, 24 (12), 2022-2029, 1986MATHCrossRefGoogle Scholar
  25. 25.
    Balmès E., Optimal Ritz vectors for component mode synthesis using the singular value decomposition, AIAA Journal, 34 (6), 1256-1260, 1996.MATHCrossRefGoogle Scholar
  26. 26.
    Wilson E. L., Yuan M. W. and Dickens J. M., Dynamic analysis by direct superposition of Ritz vectors, Earthquake Engineering and Structural dynamic, 10, 813-821, 1982.CrossRefGoogle Scholar
  27. 27.
    Lèger P., Application of load dependent vectors bases for dynamic substructure analysis, AIAA Journal, 28 (1), 177-179, 1990.CrossRefGoogle Scholar
  28. 28.
    Butcher E. A., Clearance effects on bilinear normal mode frequencies, J. Sound & Vibration, 224, 305-328, 1999.CrossRefGoogle Scholar
  29. 29.
    AL-Shudeifat M. A., Butcher E. A. and Burton T. D., Comparison of order reduction methodologies and identification of NNMs in structural dynamic systems with isolated nonlinearities, Proc. of IMAC-XXVII, Orlando, FL, Feb. 2009.Google Scholar
  30. 30.
    Belendez A., Hernandez A., Marquez A. and Neip C., Analytical approximation for the period of a nonlinear pendulum, Eur. J. Phys., 27, 539-551, 2006.CrossRefGoogle Scholar

Copyright information

© The Society for Experimental Mechanics, Inc. 2011 2011

Authors and Affiliations

  • Mohammad A. AL-Shudeifat
    • 1
  • Eric A. Butcher
    • 1
  • Thomas D. Burton
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA

Personalised recommendations