Nonlinear vibrations of laminated and sandwich rectangular plates with free edges. Part 1: Theory and numerical simulations

Nonlinear vibrations of completely free laminated and sandwich rectangular plates are investigated for the first time. The equations of motion are obtained via multi-modal energy approach based on Lagrange equations and by using classical and higher-order shear deformation theories with von Kármán type nonlinearities. The numerical 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 performed to obtain the natural frequencies and modes of vibration. Then, the energy functional is discretized by using the natural modes of vibration and a system of nonlinear ordinary differential equations with quadratic and cubic nonlinear terms is obtained. A code based on pseudo arc-length continuation and collocation scheme is used for bifurcation analysis and the near resonance nonlinear vibrations are studied. The frequency-response curves, time histories and phase space diagrams are presented.