Keywords: free vibration, functionally graded beams, dynamic stiffness method, Wittrick-Williams algorithm, Timoshenko beam theory.

This paper develops and applies the dynamic stiffness method (DSM) to investigate the free vibration of a functionally graded beam (FGB) using Timoshenko theory. The dynamic analysis of FGB has become an area of intense research activity and the literature on the topic is rapidly increasing. A majority of the investigations use traditional finite element or other approximate methods. Apparently there has not been any attempt to solve the problem using the DSM. This paper is intended to fill this gap. The proposed DSM uses exact member theory based on frequency dependent shape functions obtained from the exact solution of the governing differential equations of motion of the FGB in its free vibration. The method provides exact results for all natural frequencies and mode shapes of the FGB without making any approximation en route. The DSM is more accurate than traditional finite element and other approximate methods.

The research is carried out in following steps. First the material properties of the FGB are given variations through its cross-section according to a power law. Then the kinetic and potential energies are formulated using the Timoshenko beam theory with particular reference to the FGB. Next, the governing differential equations of motion in free vibration are derived using Hamilton's principle and making use of symbolic computation. The expressions for axial and shear forces and bending moment at any cross-section of the beam are obtained as a by-product of the Hamiltonian formulation. For harmonic oscillation, the governing differential equations are solved in closed analytical form for displacements and bending rotation. Axial force, shear force and bending moment are also obtained in explicit analytical form using the solutions of the governing differential equations. The frequency dependent dynamic stiffness matrix is then derived by relating the amplitudes of the forces to those of the displacements. Finally, the eigenvalue problem is solved by using the Wittrick-Williams algorithm to yield natural frequencies and mode shapes of the FGB. The investigation required a substantial amount of validation exercise for which computed results from the present theory are compared with those in the literature. A parametric study is carried out by varying significant beam parameters, such as the length to height ratio and the effect of power law distribution. For different boundary conditions natural frequencies of some illustrative examples computed using the present theory are discussed and compared with published results. Some representative mode shapes are also shown and commented on. The method is computationally efficient and numerically accurate and thus can be used as an aid to validate finite element and other approximate methods.

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