Abstract
Reduction methods and the resulting models for studying nonlinear vibrations of shallow monodimensional continuous systems are discussed. Primary resonances of the first mode of buckled beams and suspended cables are investigated. The convergence of the relevant Galerkin-reduced models with variation of the nondi-mensional buckling level (buckled beam) and the elasto-geometric parameter (cable) is analyzed. For low values of the control parameter, one-dof models (with the first relevant linear eigenmode) are sufficiently accurate, whereas, for higher values of the parameter (above the first crossover), three- or higher-dof models (with the lowest relevant symmetric eigenmodes) are the minimum reduced-order models that can capture qualitatively the symmetric planar dynamics of the original systems. The major modification of the mode shapes of the lowest symmetric modes with respect to the initial nonlinear static shape, due to crossover and snap-through-type bifurcations, is highlighted as the key mechanism for the breakdown of low-dimensional models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Nayfeh, A. H., Nayfeh, S. A., and Pakdemirli, M. (1995) On the discretization of weakly nonlinear spatially continuous systems, in W. Kliemann and N. Sri Namachchivaya (eds.), Nonlinear Dynamics and Stochastic Mechanics, 175–200.
Pakdemirli, M. and Boyaci, H. (1995) Comparison of direct-perturbation methods with discretization-perturbation methods for non-linear vibrations, Journal of Sound and Vibration 186, 837–845.
Lacarbonara, W. (1999) Direct treatment and discretizations of non-linear spatially continuous systems, Journal of Sound and Vibration 221, 849–866.
Lacarbonara, W., Nayfeh, A. H., and Kreider, W. (1998) Experimental validation of reduction methods for weakly nonlinear distributed-parameter systems: Analysis of a buckled beam, Nonlinear Dynamics 17, 95–117.
Rega, G., Lacarbonara, W., Nayfeh, A. H., and Chin, C-M (1999) Multiple resonances in suspended cables: direct versus reduced-order models, International Journal of Non-Linear Mechanics 34, 901–924.
Nayfeh, A. H. (1998) Reduced-order models of weakly nonlinear spatially continuous systems, Nonlinear Dynamics 16, 105–125.
Benedettini, F., Rega, G., and Alaggio, R. (1995) Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions, Journal of Sound and Vibration 182, 775–798.
Irvine, H. M. and Caughey, T. K. (1974) The linear theory of free vibrations of a suspended cable, Proceedings of the Royal Society London A341, 299–315.
Nayfeh, A. H. (1973) Perturbation Methods, Wiley-Interscience, New York.
Nayfeh, A. H. (1981) Introduction to Perturbation Techniques, Wiley-lnterscience, New York.
Rega, G., Alaggio, R., and Benedettini, F. (1997) Experimental investigation of the nonlinear response of a hanging cable. Part I: Local analysis, Nonlinear Dynamics 14, 89–117.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Rega, G., Lacarbonara, W., Nayfeh, A.H. (2000). Reduction Methods for Nonlinear Vibrations of Spatially Continuous Systems with Initial Curvature. In: Van Dao, N., Kreuzer, E.J. (eds) IUTAM Symposium on Recent Developments in Non-linear Oscillations of Mechanical Systems. Solid Mechanics and Its Applications, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4150-5_24
Download citation
DOI: https://doi.org/10.1007/978-94-011-4150-5_24
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-5809-4
Online ISBN: 978-94-011-4150-5
eBook Packages: Springer Book Archive