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R. Heuer, H. Irschik, F. Ziegler:
"Multi-Modal Approach for Large Natural Flexural Vibrations of Thermally Stressed Plates";
Nonlinear Dynamics, 1 (1990), 449 - 458.

By means of Berger's approximation, suitable for plates with immovable edges, the geometrically nonlinear problem is considerably simplified. Thermally loaded plates with polygonal planform under hard-hinged support conditions are considered, taking into account the effect of shear in transverse isotropy. The class of symmetric vibrations about the flat plate position is represented by a homogeneous and coupled set of Duffing-oscillators as a result of a multi-mode expansion. A unifying non-dimensional closed-form solution for the corresponding nonlin¬ear natural vibration periods is given, which is independent of the special planform. The individual shape of the plate enters the transformation into real time through the linear natural frequencies, or equivalently through the linear eigen-values of an effectively prestressed membrane of the same planform.