Fully clamped rectangular plates are very often used as structural components in various engineering fields. However, it is almost impossible to achieve in real life the perfect theoretical clamped boundary conditions because of the residual flexibility which makes it very difficult to prevent completely the rotations; especially in the non-linear regime induced by large vibration amplitudes. The purpose of the present work is the development of a semi-analytical method, based on Hamilton's principal and spectral analysis, to deal with the linear and geometrically non-linear free and forced vibrations of simply supported rectangular plates restrained at two opposite edges by two distributions of rotational springs. The functions used in the plate displacement series expansions are obtained as products of beam functions, with appropriate end conditions in each direction. In the x-direction, the functions used are those of a beam simply supported at both ends. In the y-direction, the simply supported beam is assumed to be in addition restrained by two rotational springs. Using the Rayleigh-Ritz method, the plate mode shapes are obtained after a careful convergence study and compared with the solutions derived analytically, based on the plate double Laplacien vibration equation. Using the calculated plate linear mode shapes, Benamar's method is used to determine the non-linear free and forced response of the plate in the neighborhood of the fundamental mode. The results, presented in terms of the backbone curves obtained for various plate aspect ratios, in addition to the non-linear frequency response functions, based on the single mode approach for various levels of the excitation, give the expected non-linear amplitude dependence of the hardening type.
