Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

Free vibration of nonuniform axially functionally graded Timoshenko beams subjected to combined axially tensile or compressive loading is studied. An emphasis is placed on the effect of tip and distributed axial loads on the natural frequencies and mode shapes for an inhomogeneous cantilever beam including material inhomogeneity and geometric non-uniform cross section. The initial value method is developed to determine the natural frequencies. The method's effectiveness is verified by comparing our results with previous ones for special cases. Natural frequencies of standing/hanging Timoshenko beams are calculated for four different cross sections. The influences of shear rigidity, taper ratio, gradient index, tip force, and axially distributed loading on the natural frequencies of clamped-free beams are discussed. Material inhomogeneity and geometric non-uniform cross-section strongly affect higher-order vibration frequencies and mode shapes.