Free vibration analysis of a nonuniform piezoelectric rod

Piezoelectric materials, which have fast response and low energy usage features, are widely used as a sensor and actuator. It is important to know the vibration characteristic of piezoelectric materials due to the active role in their working principle. Free longitudinal vibration of non-uniform piezoelectric (PZT-4) rod is solved by complementary functions method for arbitrary cross-section area under three different boundary conditions. The solution procedure is not confined to any particular choice of cross-sectional area model; it is equally suitable for arbitrary functions defining the variation of cross-sectional area. Complementary functions method (CFM) will be infused into the analysis to convert the problem to an initial value problem, which can be easily solved by, for instance, Runge-Kutta methods with great accuracy. Solutions thus obtained are compared to analytical solutions which solved with Bessel functions to validate the method presented. The solution procedure is well-structured, simple and efficient. Numerical examples are given to show the effect of arbitrary geometry on the dynamic characteristic of the rod.