Nonlinear forced vibration of damped plates coupling asymptotic numerical method and reduction models

This work concerns the computation of the nonlinear solutions of forced vibration of damped plates. In a recent work (Boumediene et al. in Comput Struct 87:1508-1515, 2009), a numerical method coupling an asymptotic numerical method (ANM), harmonic balance method and Finite Element method was proposed to resolve this type of problem. The harmonic balance method transforms the dynamic equations to equivalent static ones which are solved by using a perturbation method (ANM) and the finite element method. The numerical results presented in reference (Boumediene et al. in Comput Struct 87:1508-1515, 2009) show that the ANM is very efficient and permits one to obtain the nonlinear solutions with few matrix triangulation numbers compared to a classical incremental iterative method. However, putting a great number of harmonics (6 or greater) into the load vector leads to tangent matrices with a great size. The computational time necessary for the triangulation of such matrices can then be large. In this paper, reduced order models are proposed to decrease the size of these matrices and consequently the computational time. We consider two reduced bases. In the first one, the reduced basis is obtained by the resolution of a classical eigenvalue problem. The second one is obtained by using the nonlinear solutions computed during the first step of the calculus which is realized with the ANM. Several classical benchmarks of nonlinear damped plates are presented to show the efficiency of the proposed numerical methods.