Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges

Large amplitude vibrations of completely free rectangular plates are investigated for the first time. 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 studied. Geometric imperfections are taken into account. 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 conducted 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 cubic and quadratic nonlinear terms is obtained. A pseudo arc-length continuation and collocation scheme is used for bifurcation analysis of periodic solutions. Moreover, to validate the numerical results, experimental tests have been conducted for the fundamental vibration mode at several harmonic excitation amplitudes by using a Laser Doppler Vibrometer to characterize the nonlinear response of the plate with imperfections and closed-loop control of the excitation. The effects of geometric imperfections on the trend of nonlinearity and on natural frequencies are fully discussed and the convergence of the numerical solutions has been verified.