Abstract

This paper proposes an explicit solution for the design of a target-frequency-customized, one-dimensional phononic crystal (PNC) with a defect for piezoelectric energy harvesting under longitudinal waves. Due to the innate narrow bandwidth nature of the defect modes of a PNC at the target frequency, there is a great need to generate an electromechanically coupled defect band of a piezoelectric-defect-introduced PNC. This work considers the transfer matrix method which has been widely used in analytical approaches. The need for defect bands to be included in a bandgap inspires the use of a quarter-wave stack as a unit cell to match the bandgap's central frequency with the target frequency. In band structure analysis, considering that the electromechanically coupled defect band corresponds to a set of real wavenumbers despite being within the bandgap, several possible solutions for the piezoelectric defect's length are derived in an explicit fashion. Since switching from a short- to an open-circuit condition causes defect bands to slightly increase due to piezoelectric effects, an explicit solution that reflects the piezoelectric defect's electrical characteristics is finally proposed. Finite-element-based numerical validation studies are conducted to study two aspects, specifically parametric studies (i.e., the natural numbers in the solution to the piezoelectric defect's length, the supercell sizes, and the defect locations) and supporting studies (i.e., the electrical boundary conditions and unit cell designs). At the target frequency, it is demonstrated that the proposed PNC design actualizes the formation of one defect band and the representation of the peak output voltage.