Exact frequency equations of free vibration of exponentially functionally graded beams

Free vibration of axially inhomogeneous beams is analyzed. For exponentially graded beams with various end conditions, characteristic equations are derived in closed form. These characteristic or frequency equations can analytically reduce to the classical forms of Euler–Bernoulli beams if the gradient index disappears. The gradient has a strong influence on the frequency spectrum, and the natural frequencies noticeably depend on the variation of the gradient parameter and end support conditions. For certain beams with exponential gradients, there exists a critical frequency depending on the gradient parameter. Vibration can be only excited by propagating waves with frequencies in excess of the critical frequency, and otherwise vibration is prohibited for pseudo-frequencies lower than the critical frequency. For some gradient index with small change, the natural frequencies have an abrupt jump when across its critical frequencies. Obtained results can serve as a benchmark for other numerical procedures for analyzing transverse vibration of axially functionally graded beams. The minimal natural frequency can be sought for certain gradient index, and this helps engineers to optimally design vibrating nonhomogeneous beam structures. Obtained results also apply to free vibration of nonuniform beams with constant thickness and exponentially decaying width.