Free Vibration of a Functionally Graded Timoshenko Beam using the Dynamic Stiffness Method

Abstract

The dynamic stiffness method is used to investigate the free vibration behaviour of a functionally graded beam (FGB). The material properties of the FGB are assumed to vary in the thickness direction based on a power-law. The kinetic and potential energies of the beam are formulated using the Timoshenko beam theory. The governing differential equations of motion in free vibration for the FGB are derived using Hamilton’s principle. The analytical expressions for axial force, shear force and bending moments at any cross-section of the FGB are obtained as a by-product of the Hamiltonian formulation. The differential equations are solved in closed analytical form for harmonic oscillation. The dynamic stiffness matrix of the FGB is then formulated by relating the amplitudes of forces and displacements at the ends of the beam. The Wittrick-Williams algorithm is used as solution technique to yield natural frequencies and mode shapes of the FGB. A parametric study is carried out by varying significant beam parameters and boundary conditions. The investigation required a substantial amount of validation exercise to confirm the predictable accuracy of the dynamic stiffness method. The results are discussed and some concluding remarks are made.