Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams

Free vibration of non-uniform functionally graded beams is analyzed via the Timoshenko beam theory. Bending stiffness and distributed mass density are assumed to obey a unified exponential law. For various boundary conditions, exact frequency equations are derived in closed form. These frequency equations can reduce to those for classical Timoshenko beams if the gradient index disappears. Moreover, the frequency equations of exponentially graded Rayleigh, shear, and Euler-Bernoulli beams can be obtained as special cases of the present. The gradient index has a strong influence on the natural frequencies. For Timoshenko beams, there exist two critical frequencies depending on the gradient index. Harmonic vibration cannot be excited for frequencies less than the lower critical frequency. The obtained results can serve as a benchmark for examining the accuracy of numerical frequencies based on other approaches for analyzing transverse vibration of non-uniform axially graded Timoshenko beams. The results also apply to bending vibration of rectangular Timoshenko beams with constant thickness and exponentially decaying/amplifying width.