THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC STRUCTURES .1. SIMPLY SUPPORTED AND CLAMPED CLAMPED BEAMS

A method for calculating the non-linear mode shapes and natural frequencies of fully clamped beams at large vibration amplitudes is presented, and results are compared with those of previous studies and of experimental measurements. First, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions. Then, the non-linear deformation energy is expressed by taking into account the non-linear terms due to the axial strain induced by large deflections. A set of non-linear algebraic equations, which reduces to the classical linear eigenvalue problem when non-linear terms are neglected, is determined through Hamilton's principle. It is also shown that unless a condition is imposed on the contribution of one mode, the solution of this set leads to the linear case. Consequently, in order to obtain a numerical solution for the non-linear problem in the neighbourhood of a given mode, the contribution of this mode is chosen and those of other modes are calculated. In this paper, part I of a series of three papers, this method is applied to obtain the first three non-linear mode shapes of clamped-clamped and simply supported beams. The results obtained corresponding to the fundamental non-linear mode shape are in good agreement with those of a previous theoretical and experimental study. In particular, high values of increase of beam curvatures are noticed near the clamps, causing a highly non-linear increase in bending strain with increasing deflection. © 1991.