Large Amplitude Vibrations of Completely Free Rectangular Plates

Nonlinear forced vibrations of completely free rectangular plates are studied using multi-modal Lagrangian approach. Nonlinear higher-order shear deformation theory is used and the nonlinear response to transverse harmonic excitation in the frequency neighborhood of the fundamental mode is investigated. Geometric imperfections are taken into account. The analysis is carried out in two steps. First, the plate displacements and rotations are expanded in terms of Chebyshev polynomials and a linear analysis is conducted to obtain the natural frequencies and mode shapes. Then, the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with cubic and quadratic nonlinear terms is obtained. A pseudo arc-length continuation and collocation scheme is employed to carry out a bifurcation analysis. The effect of number of modes retained in the approximation, thickness ratio and geometric imperfections on the trend of nonlinearity is discussed.