A novel Bessel function method is proposed to obtain the exact solutions for the free-vibration analysis of rectangular thin plates with three edge conditions: (i) fully simply supported; (ii) fully clamped, and (iii) two opposite edges simply supported and the other two edges clamped. Because Bessel functions satisfy the biharmonic differential equation of solid thin plate, the basic idea of the method is to superpose different Bessel functions to satisfy the edge conditions such that the governing differential equation and the boundary conditions of the thin plate are exactly satisfied. It is shown that the proposed method provides simple, direct, and highly accurate solutions for this family of problems. Examples are demonstrated by calculating the natural frequencies and the vibration modes for a square plate with all edges simply supported and clamped.
Exact Solutions for Free-Vibration Analysis of Rectangular Plates Using Bessel Functions
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Wu, J. H., Liu, A. Q., and Chen, H. L. (April 23, 2005). "Exact Solutions for Free-Vibration Analysis of Rectangular Plates Using Bessel Functions." ASME. J. Appl. Mech. November 2007; 74(6): 1247–1251. https://doi.org/10.1115/1.2744043
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