JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 290 (2006) 465–489 Vibration of shear deformable plates with variable thickness — first-order and higher-order analyses I. Shufrin, M. Eisenberger à Faculty of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel Received 9 December 2003; received in revised form 1 April 2005; accepted 4 April 2005 Available online 11 July 2005 Abstract This work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions. The thickness variation in one or two directions of the plate is taken in polynomial form. The first-order shear deformation plate theory of Mindlin and the higher-order shear deformation plate theory of Reddy have been applied to the plate analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the Lagrangian function. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity, which provides for the derivation of the exact dynamic stiffness matrix of varying cross-sections strips. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate theory and with published results. r 2005 Elsevier Ltd. All rights reserved. 1. Introduction Plate elements with varying thickness are used in civil, mechanical, aeronautical and marine structures. The consideration of free vibration of such plates is essential to have an efficient and ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.04.003 à Corresponding author. Tel.: +972 4 8292413; fax: +972 4 8295697. E-mail address: cvrmosh@tx.technion.ac.il (M. Eisenberger).