Natural dynamic characteristics of a circular cylindrical Timoshenko tube made of three-directional functionally graded material

The natural dynamic characteristics of a circular cylindrical tube made of three-directional (3D) functional graded material (FGM) based on the Timoshenko beam theory are investigated. Hamilton’s principle is utilized to derive the novel motion equations of the tube, considering the interactions among the longitudinal, transverse, and rotation deformations. By dint of the differential quadrature method (DQM), the governing equations are discretized to conduct the analysis of natural dynamic characteristics. The Ritz method, in conjunction with the finite element method (FEM), is introduced to verify the present results. It is found that the asymmetric modes in the tube are controlled by the 3D FGM, which exhibit more complicated shapes compared with the unidirectional (1D) and bi-directional (2D) FGM cases. Numerical examples illustrate the effects of the axial, radial, and circumferential FGM indexes as well as the supported edges on the natural dynamic characteristics in detail. It is notable that the obtained results are beneficial for accurate design of smart structures composed from multi-directional FGM.