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 p rinciple and spectral analysis was developed for non-linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rect angular 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 equati ons and the harmonic balance method. Then, the forced case has been examine d and the analysis led to a set of non-linear partial differential equation s which reduces to the classical modal analysis forced response matrix equa tion when the non-linear terms are neglected. On the other hand, if only on e mode is assumed, this set reduces to the Duffing equation, very well know n in one mode analyses of non-linear systems having cubic non-linearities. So, it appeared sensible to consider such a formulation as the multidimensi onal Duffing equation. In order to solve the multidimensional Duffing equation in the case of harm onic excitation of beam like structures, a method is proposed, based on the harmonic balance method, and a set of non-linear algebraic equations is ob tained whose numerical solution leads in each case to the basic function co ntribution coefficients to the displacement response function. These coeffi cients depend on the excitation frequency and the distribution of the appli ed forces. The frequency response curve obtained here exhibits qualitativel y a classical non-linear behaviour, with multivalued regions in which the j ump phenomenon could occur. Quantitatively, the analytical result obtained here, without assuming any limitation to the scale of the excitation, is id entical to that obtained by the multiple scale method which assumes small v alues of the scaling parameter. Attention was focused on the assumed one mode in order to improve the resul ts obtained. In the case of free vibrations, the analytical solution obtain ed by elliptic functions has been expanded into power series of higher orde rs using the symbolic manipulation program "Maple'". It has been shown that extreme care must be taken in the choice of the polynomial approximation w hich 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 Acad emic Press.