Dynamic stiffness formulation and free vibration analysis of functionally graded beams

The dynamic stiffness method is developed to investigate the free vibration behaviour of functionally graded beams. Material properties are assumed to vary continuously in the beam thickness direction according to a power law distribution. Hamilton's principle is used to derive the governing differential equations of motion and natural boundary conditions in free vibration. For harmonic oscillation the differential equations are solved in closed analytical form. The dynamic stiffness matrix is derived by relating the amplitudes of forces to those of the displacements at the beam ends. The Wittrick-Williams algorithm is used as the solution technique when applying the dynamic stiffness matrix to compute the natural frequencies and mode shapes. A parametric study is carried out to demonstrate the effects of the length to thickness ratio and the variation of the power law index parameter. Numerical results are discussed and compared with the published ones wherever possible. Some conclusions are drawn.