In this paper,the problem of a finite ciycular cylinder with a penny-shaped crack is considered, which is reduced to the solution of a Fredholm integral equation of the second kind by using Hankel and finite sine transform.

Then it solves the partial differential equation by using finite Fourier sine transform.

Propose mathematical expression of fractional cosine transforms based on different character values,give the reason of multiplicity and analyze the relationship of fractional cosine,sine transforms and fractional Fourier transform.

By defining the generalized cosine,sine transform and using the pairity of function,the corresponding relations between the generalized cosine,sine transform and the generalized Fourier transform under full time domain are discussed,as well as the symmetry of the generalized cosine and sine transform.

The theoretical solution of eigenfrequencies and vibration models of rectangular thin plate on foundation with completely free boundary was derived by the double finite cosine integral transform method.

Free vibration analysis of a thin plate with completely supported by finite cosine integral transform method;