A semi-analytical approach to the non-linear dynamic response problem of S-S and C-C beams at large vibration amplitudes part I: General theory and application to the single mode approach to free and forced vibration analysis

In a previous series of papers [1-3], a general model based on Hamilton's principle and spectral analysis was developed for non-linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates. As an introduction to the present work, concerned with the forced non-linear response of C-C and S-S beams, the above model has been derived using spectral analysis, Lagrange's equations and the harmonic balance method. Then, the forced case has been examined and the analysis led to a set of non-linear partial differential equations which reduces to the classical modal analysis forced response matrix equation when the non-linear terms are neglected. On the other hand, if only one mode is assumed, this set reduces to the Duffing equation, very well known in one mode analyses of non-linear systems having cubic non-linearities. So, it appeared sensible to consider such a formulation as the multidimensional Duffing equation. In order to solve the multidimensional Duffing equation in the case of harmonic excitation of beam like structures, a method is proposed, based on the harmonic balance method, and a set of non-linear algebraic equations is obtained whose numerical solution leads in each case to the basic function contribution coefficients to the displacement response function. These coefficients depend on the excitation frequency and the distribution of the applied forces. The frequency response curve obtained here exhibits qualitatively a classical non-linear behaviour, with multivalued regions in which the jump phenomenon could occur. Quantitatively, the analytical result obtained here, without assuming any limitation to the scale of the excitation, is identical to that obtained by the multiple scale method which assumes small values of the scaling parameter. Attention was focused on the assumed one mode in order to improve the results obtained. In the case of free vibrations, the analytical solution obtained by elliptic functions has been expanded into power series of higher orders using the symbolic manipulation program "Maple'". It has been shown that extreme care must be taken in the choice of the polynomial approximation which is valid only in a zone limited by a radius of convergence. The use of Pade approximants permitted considerable increase in the zone of validity of the solution obtained for Very large vibration amplitudes. (C) 1998 Academic Press.