Abstract
In the present paper we study freely propagating inertial, i.e., unforced, waves, in an elastic beam supported on a flat, rigid, inelastic surface, subject to a gravitational force and a compressive longitudinal load. We consider both the case wherein the potential energy of the system is quadratic, so that the equations of motion are linear in regions where the support constraint is inactive, and some quasilinear extensions of that model. In the linear case we obtain closed form solutions and we provide a partial stability analysis. The nonlinear models arise from a quartic extension of the potential energy functional and are treated analytically and numerically; existence is shown in some cases by means of a perturbation argument.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brush, D.O. (1975). Buckling of bars, plates, and shells, McGraw-Hill, New York.
Lagnese, J. E. (1991). Recent progress in exact boundary controllability and uniform stabilizability of thin beams and plates, in Distributed parameter control systems, G. Chen, E. B. Lee, W. Littman and L. Markus, Eds., Marcel Dekker, New York,, pp. 61–111
Leipholz, H. (1970). Stability theory, Academic Press, New York and London, (translation of the original German Stabilitätstheorie, B. G. Teubner, Stuttgart, 1968); Chapter 1, Sect. 1,2.
Russell, D.L., and White, L.W. (2000a). An elementary nonlinear beam theory with finite buckling deformation properties, to appear in SIAM Journal of Applied Mathematics.
Russell, D.L., and White, L.W. (2000b). Static buckling in a supported nonlinear elastic beam, accepted for publication in Proc. Conf On Control of Nonlinear Distributed Parameter Systems, College Stn., TX, October, 1999, G. Chen and I. Lasiecka, Eds. To be published by Marcel Dekker.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to the memory of Professor P.D. Panagiotopoulos.
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this chapter
Cite this chapter
Russell, D.L. (2001). Freely Propagating Waves in a Supported Nonlinear Elastic Beam. In: Gao, D.Y., Ogden, R.W., Stavroulakis, G.E. (eds) Nonsmooth/Nonconvex Mechanics. Nonconvex Optimization and Its Applications, vol 50. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0275-9_16
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0275-9_16
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7973-7
Online ISBN: 978-1-4613-0275-9
eBook Packages: Springer Book Archive