he principal concerned in this paper is the finite Hankel integral transform method for solving the symmetrical free vibration problem of rectangular thin plate. The considered plate has two opposite simply supported edges and the others are partially simply supported mixed with free. Based on the Levy -plate solution, the mixed boundary conditions between the simple and free supports on the same side of the plate can be written as dual -series equations. However, with the advantages of the

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... ent method, these mentioned equations are analytically reduced to homogeneous integral equation of Fredholm -type. Significantly, the nature of moment singularities where the simple support changes to a free edge has been brought into consideration. They can be isolated and treated analytically with a weight function. From this aspect, the attention is placed specifically on deriving an integral equation of the problem considered here instead of the partial differential equation. As a result, those with mixed boundary conditions and the moment singularities in the order of an inverse -square -root type have been successfully obtained.