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This work presents the single finite Fourier sine integral transform method for finding the natural frequencies of flexural vibration of rectangular Kirchhoff plates with two opposite simply supported edges (x = 0, x = a) and two clamped edges (y = 0, y = b), where the origin is at a corner. For free harmonic vibrations, the problem is represented mathematically as a fourth order partial differential equation (PDE) over the domain, and boundary conditions along the edges. Application of the transform with respect to the x coordinate variable converts the Boundary Value Problem (BVP) to an integral equation which satisfies all the Dirichlet boundary conditions along x = 0, and x = a, due to the sinusoidal kernel function of the transform used. The integral equation is further simplified to a system of ordinary differential equations (ODEs), which is solved to obtain the unknown deflection in the transform space. Transforms of the boundary conditions are used in the solution of the ODE to generate a system of homogeneous equations in terms of the unknown integration constants. The condition for nontrivial solution is used to obtain the characteristic frequency equation which is solved by iteration methods to obtain the natural frequencies for any given mode of vibration. The results obtained are identical with results obtained by previous researchers who used Galerkin-Vlasov methods, Levy’s single trigonometric series method, finite element method, and energy methods.

Single Finite Fourier Sine Integral Transform Method for the Determination of Natural Frequencies of Flexural Vibration of Kirchhoff Plates