Parametric Excitation of a Non-Homogeneous Bernoulli-Euler Beam

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Abstract
In this paper is considered the problem of the stability of the parametric response of a simply supported Bernoulli-Euler beam having an elastic modulus that varies continuously and monotonically throughout its length. The beam is excited by axial harmonic forces applied to the ends. The Galerkin procedure is used, which, in the first approximation, leads to a single Mathieu equation representing the stability regions for an equivalent uniform beam having averaged properties. For the second and higher approximations, the co-ordinate functions used in the Galerkin procedure couple, leading to a coupled system of Mathieu equations. Results from the first and second approximations are compared with, a view toward establishing the degree of non-homogeneity for which the first approximation predicts the instability regions with acceptable accuracy. It is shown that for moderate non-homogeneities, such as might be introduced by thermal sources, the first approximation leads to results of quite tolerable accuracy. In an Appendix are presented some computed data for the free vibrational frequencies of the non-homogeneous beam under static end forces.

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