Eigensolution of piezoelectric energy harvesters with geometric discontinuities: Analytical modeling and validation

Although cantilevered beams are the most prolific design for resonant piezoelectric energy harvesters, other topologies have been studied for their compactness or conformability to their host structures' geometry. These more complex structures have been analyzed using custom analytical models developed from the first principles or finite-element methods to compute their eigensolutions and piezoelectric coupling effects. This article discusses the use of the transfer matrix method to derive analytical solutions to beam structures with pointwise discontinuities, bends, or lumped inertias between members or at the tip. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure between beam segments. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated, and the base excitation problem is considered. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage and angle between members. Finally, these results are compared to finite-element solutions using COMSOL, and the modeling discrepancies are discussed. Based on the favorable comparison between these two methods, the utility and accuracy of the transfer matrix method are proven.