Nonlinear Vibrations of Imperfect Thin Rectangular Plates Using the Asymptotic Numerical Method

This paper deals with the nonlinear forced vibration analysis of thin rectangular plates exhibiting initial geometric imperfections, using the Asymptotic Numerical Method (ANM). The geometric nonlinearities are described through the von Kármán strain–displacement relations. By applying Hamilton’s principle in conjunction with the harmonic balance method, the nonlinear governing equations of motion are recast into an operational form suitable for numerical computation. In the first stage, the nonlinear free vibration response is examined to obtain the backbone curves and to assess the influence of imperfections on the dynamic characteristics. Subsequently, the forced vibration behavior of an imperfect plate subjected to a harmonic point load is studied. The ANM expands the unknown quantities displacements and frequencies into power series with respect to a control parameter. Introducing these expansions into the governing equations converts the nonlinear problem into a set of linear subproblems that share a common stiffness matrix. These are efficiently solved using the finite element method. To enhance the convergence and computational efficiency, Padé approximants are incorporated along with a continuation strategy to trace complete nonlinear response curves. The proposed methodology provides a reliable and computationally efficient framework for analyzing the nonlinear dynamic behavior of imperfect thin plates under diverse loading and boundary conditions. Numerical examples for both free and forced vibrations are presented and compared with available references, confirming the accuracy and convergence of the present approach for various imperfection amplitudes.