Dynamic equations for a functionally graded cylinder
Paper in proceedings, 2016
This work considers the analysis and derivation of dynamical equations of functionally graded (FG) solid cylinders. The proposed method is based on the 3D equations of motion using a power series expansion of the displacement fields in the radial coordinate of the cylinder. This assumption results in sets of equations of motion together with consistent sets of boundary conditions. These derived equations are hyperbolic and asymptotically correct to all studied order. A hierarchy of partial differential equations is thus constructed in a systematic fashion to any order desired. The construction of the equations are systematized by the introduction of recursion relations which relates higher order displacement terms with the lower order terms. Results for longitudinal, torsional and bending motion are obtained where the material distribution vary with radial and circumferential coordinates. Eigenfrequency results and plots on mode shapes and stress distributions are presented for cylinders using different truncations orders. The approach may render benchmark solutions provided higher order truncations are used, and act as engineering cylinder equations using low order truncation.