Abstract
Methods for the study of weakly nonlinear continuous (distributed-parameter) systems are discussed. Approximate solution procedures based on reduced-order models via the Galerkin method are contrasted with direct application of the method of multiple scales to the governing partial-differential equations and boundary conditions. By means of several examples and an experiment, Nayfeh and co-worker had shown that reduced-order models of nonlinear continuous systems obtained via the Galerkin procedure can lead to erroneous results. A method is developed for producing reduced-order models that overcomes the shortcomings of the Galerkin procedure. Treatment of these models yields results in agreement with those obtained experimentally and those obtained by directly attacking the continuous system.
Similar content being viewed by others
References
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Nayfeh, A. H., Nayfeh, J. F., and Mook, D. T., ‘On methods for continuous systems with quadratic and cubic nonlinearities’, Nonlinear Dynamics 3, 1992, 145‐162.
Nayfeh, A. H., Nayfeh, S. A., and Pakdemirli, M., ‘On the discretization of weakly nonlinear spatially continuous systems’, in Nonlinear Dynamics and Stochastic Mechanics, W. Kliemann and N. Sri Namachchivaya (eds.), CRC Press, Boca Raton, FL, 1995, pp. 175‐200.
Nayfeh, A. H. and Lacarbonara, W., ‘On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities’, Nonlinear Dynamics 13, 1997, 203‐220.
Pakdemirli, M., Nayfeh, S. A., and Nayfeh, A. H., ‘Analysis of one-to-one autoparametric resonances in cables. Discretization vs. direct treatment’, Nonlinear Dynamics 8, 1995, 65‐83.
Chin, C. and Nayfeh, A. H., ‘Bifurcation and chaos in externally excited circular cylindrical shells’, Journal of Applied Mechanics 63, 1996, 565‐574.
Lacarbonara, W., Nayfeh, A. H., and Kreider, W., ‘On approximate methods for weakly nonlinear distributedparameter systems’, Nonlinear Dynamics, 1998 (accepted).
Troger, H. and Steindl, A., Nonlinear Stability and Bifurcation Theory, Springer-Verlag, Wien, 1991.
Nayfeh, A. H., Perturbation Methods, Wiley, New York, 1973.
Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley, New York, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nayfeh, A.H. Reduced-Order Models of Weakly Nonlinear Spatially Continuous Systems. Nonlinear Dynamics 16, 105–125 (1998). https://doi.org/10.1023/A:1008281121523
Issue Date:
DOI: https://doi.org/10.1023/A:1008281121523