Free Axisymmetric Vibrations of a Hollow Cylinder of Finite Length Made of a Functionally Graded Material

On the basis of the three-dimensional elasticity theory, we study a problem of free axisymmetric vibrations of inhomogeneous hollow cylinders of finite length made of functionally graded materials under different boundary conditions imposed on their end faces. The elastic properties of the material continuously vary in the radial direction. We propose a numerical-analytic approach for the solution of this problem. The original problem of the elasticity theory containing partial differential equations is reduced to a boundary-value problem for the systems of ordinary differential equations of high order in the radial coordinate by using spline approximations and collocation methods. The obtained one-dimensional problem is solved by using a stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present the results of determination of the frequencies and modes of vibrations of the cylinder made of a functionally graded material obtained as a composition of stainless steel and nickel for different types of boundary conditions imposed at the end faces and different values of temperature. © 2017, Springer Science+Business Media New York.