An analytical method for free vibrations of functionally graded cylindrical shells with arbitrary intermediate ring supports

An analytical method is presented to study free vibrations of functionally graded material (FGM) cylindrical shells with arbitrary intermediate ring supports. Material properties continuously vary in thickness direction in accordance with the four-parameter power-law distributions in terms of volume fractions of constituents. To establish the final governing equation, the shell is firstly divided into several shell segments according to locations of ring supports. Based on the first-order shear deformation theory (FSDT), differential equations about displacements are solved by expanding displacements as exponential functions in axial direction and Fourier series in circumferential direction. Correspondingly, five displacements and five forces at any cross section of segments are further expressed as 10 unknown coefficients for every circumferential mode number. The artificial spring technique is adopted to restrain displacements at boundaries and ring supports, and all segments are assembled to the overall shell through continuity conditions. To test the validity of the developed model, natural frequencies and mode shapes of several homogenous and FGM cylindrical shells with/without ring supports are compared with appropriate ones in the literature or calculated by finite element method (FEM), which demonstrates high accuracy and wide application of the proposed method. Furthermore, effects of ring supports and material parameters are analyzed. The results reveal that restraining radial and/or circumferential displacements at ring supports significantly affects natural frequencies, but mode shapes are mainly affected by radial displacements. In addition, natural frequencies of the FGM shell obviously vary with the power-law exponent, whereas the corresponding mode shapes almost keep unchanged.