Numerical Study of Wave Propagation and Reflection in Semi-Infinite Long Piezoelectric Cylinders

A numerical procedure is presented for evaluating approximate solutions of free-end reflections in a semi-infinitely long composite piezoelectric circular cylinder. The governing equations are discretized by a semi-analytical finite element formulation where the discretization occurs through the cylinder’s thickness. Solutions are constructed with modal data from a spectral decomposition of the differential operator governing its natural vibrations. These modal data consist of all propagating modes and edge vibrations and they constitute the basis for a wave function expansion of the reflection of waves arriving at the traction-free end of the cylinder. This traction-free end condition is enforced at the Gaussian integration points over the end cross-section on the combination of incoming and reflected wave fields. Both least-square and virtual work methods are used for evaluation the amplitudes of the reflected wave field. Reflections due to monochromatic incoming axisymmetric (m = 0) and flexural (m = 1) waves are studied and a numerical example is presented for a two-layer PZT-4 cylinder. For an incoming axisymmetric wave, there is a particular frequency that induces an end resonance, which is characterized by high (but finite) amplitudes of end displacements vis-a-vis those of neighbouring (i.e., slightly different) frequencies. This phenomenon is illustrated in the two-layer cylinder example. It is possible to modify the passive reflection event by imposing some voltage distribution over the free end. For an oscillating end voltage that is out-of-phase with the incoming wave, it is possible to extract electrical energy from it. i.e., energy harvesting. An example of such an oscillating voltage with a particular radial distribution is given, that illustrate the amount of extracted energy as a function of the frequency of the incident monochromatic wave.