Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams

In this paper, the motion differential equations of the bi-directional functionally graded Timoshenko beam are established using Hamilton’s principle. The material properties of the beam change exponentially in both axial and thickness directions. Using the variable substitution method, state space differential equations of the structure are established. First, the dynamic stiffness matrix is formed using the traditional method. Then, the author proposes a new method to directly form an exact dynamic stiffness matrix by using state space differential equations, and this method is compared with the traditional dynamic stiffness matrix method. At the same time, the natural frequency of the structure is computed by combining the Wittrick–William algorithm with a non-iterative algorithm. The influence of gradient parameters α,β on the fundamental frequency, mode shape and frequency response function is analysed through the establishment of the dynamic stiffness matrix of the overall structure. Finally, using the Lagrange equation and the method of modal superposition, structural dynamic differential equations under a harmonic moving load are derived. Using the precise integration method, the dynamic response of the displacement is computed and the influence of gradient parameters α,β on the dynamic response is analysed.