Abstract
This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.
Highlights
Recent advances in non-intrusive reduced-order finite element modeling of geometrically nonlinear structures offer new perspectives for nonlinear prediction in structural computation
We focus on two main points: (i) a precise validation of the STiffness Evaluation Procedure” (STEP) using reference analytical models of flat structures and (ii) the modal convergence of the reduced-order model (ROM), especially when internal resonance is involved
A study at the coefficient level has confirmed that a large part of the ROM coefficients are equal to zero, which enables to significantly reduce the computational burden of the stiffness evaluation procedure, in terms of number of coefficients to compute and number of nonlinear static problems
Summary
Recent advances in non-intrusive reduced-order finite element modeling of geometrically nonlinear structures offer new perspectives for nonlinear prediction in structural computation. Some authors propose strategies to naturally compute relevant in-plane displacement field with prescribed modal loads, with so-called implicit condensation with expansion approach [35] or dual modes strategies [33] These methods have the drawback that the ROM coef-. Among the huge literature described above, the present article addresses the computation of the nonlinear frequency response of thin structures using a continuation method and finite element modal reduced-order models, obtained by the STEP. Within this subject, we focus on two main points: (i) a precise validation of the STEP using reference analytical models of flat structures (beams and plates) and (ii) the modal convergence of the ROMs, especially when internal resonance is involved. Convergence results are obtained, showing that very reduced ROM, composed of a few transverse oscillators coupled by cubic terms only, is efficient to compute nonlinear modes even in the case of internal resonances and vibration localization
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