Exact solution for free in-plane vibration analysis of an eccentric elliptical plate
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- Hasheminejad, S.M., Ghaheri, A. & Vaezian, S. Acta Mech (2013) 224: 1609. doi:10.1007/s00707-013-0829-y
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A two-dimensional analytical model is formulated for free extensional vibrations of a thin elastic plate of elliptic planform with an arbitrarily located elliptical cutout, based on Navier’s displacement equation of motion for the state of plane stress. The analytic solution is obtained by invoking Helmholtz’s decomposition theorem and utilizing the method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions. The first three panel in-plane natural frequencies are tabulated in a systematic fashion for selected panel/cutout aspect ratios, and cutout location/orientation parameters, under different combinations of classical (clamped/free) edge conditions. Also, selected two-dimensional vibration mode shapes are represented in vivid graphical form. The accuracy of the solutions is ensured through proper convergence studies, and the validity of results is demonstrated with the aid of a commercial finite element package as well as by comparison with the existing data. The herein reported data are believed to be the first rigorous attempt on the in-plane free vibrational characteristics of thin eccentric circular/elliptical plates for a wide range of geometric parameters.