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R. Heuer:
"Free large vibrations of buckled laminated plates";
in: "Proceedings of the IUTAM Symposium on Dynamics of Advanced Materials and Smart Structures", K. Watanabe, F. Ziegler (Hrg.); herausgegeben von: IUTAM; Kluwer Academic Publishers, Dordrecht, 2003, ISBN: 1-4020-1061-3, S. 105 - 114.

In the present contribution large amplitude natural flexural vibrations of plates about a buckled reference configuration are investigated. In particular, thermally buckled polygonal plates made of multiple transversely isotropic layers are considered. In case of hard hinged supports of the straight boundary segments of skew or even more generally shaped polygonal plates, a multi-modal approach combined with the Galerkin procedure gives a finite nonlinearly coupled set of ordinary differential equations of the Duffing type. In the following article that formulation is extended to the (unsymmetric) free vibrations about the thermally buckled plate position. The latter is assumed to be associated to a (postcritical) spatial distribution of cross-sectional mean temperature. In the special case of layered plates with physical properties symmetrically disposed about the middle surface, a correspondence to moderately thick homogeneous plates is found.
The analysis starts with the equations of motion according to the dynamic version of the von Kármán plate theory , modified by Mindlin´s kinematic hypothesis in order to include shear deformation. Berger´s approximation is applied which is known to be a reliable simplification if the in-plane displacements are constrained on the boundary. A multi-modal approach, where the eigenfunctions of the corresponding linear plate problem are selected as the space variables, gives a finite nonlinearly coupled set of ordinary differential equations containing both quadratic and cubic nonlinearities. In a single-term approximation, a closed-form solution is found in terms of Jacobian elliptic functions which is independent of the special polygonal plate geometry. For an evaluation of the real-time spectrum of the nonlinear natural fundamental frequency from this unifying similarity solution, only the Dirichlet-Helmholtz-eigenvalue of the corresponding plate must be known. The influence of multiple degrees-of-freedom on the fundamental frequency is studied numerically by means of the computer program MAPLE.

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