An Analytical Method for Vibration Modelling of a Piezoelectric Bimorph Beam for Power Harvesting

Abstract

This paper presents the development of a mathematical method for modelling a piezoelectric bimorph beam under two input base-transversal and longitudinal excitations. The piezoelectric bimorph beam model was based on the Euler-Bernoulli beam coupled with polarity-electric field for low power harvesting. The piezoelectric bimorph beam with brass centre shim was also coupled to a simple electrical circuit of resistor component. The existence of input base-longitudinal motion can affect the overall strain, polarity and electric field of the cantilevered piezoelectric bimorph, identified to have predominant bending due to input transverse-base motion. The characteristic physical behaviour of the bimorph model for parallel connection can create mode vector configurations of X-poling due to longitudinal extension form and Y-poling due transverse bending form. Conversely, the effect of series connection of the physical bimorph model can create X-poling due to transverse bending and Y-poling due to longitudinal extension forms. A new method of solving the piezoelectric bimorph under two input base-motions using coupling superposition of the elastic-polarity field has been introduced. The governing dynamics equations can be derived analytically using the weak form of Hamiltonian theorem to obtain the constitutive equations. DuBois-Reymond lemma can be used to separate three constitutive dynamic equations based on independent coefficients of the virtual displacement vectors. Furthermore, the solution forms for the three governing dynamics equations were assumed using the three independent normal modes of displacement functions based on the normal modes in the transversal, longitudinal and electric potential mode forms. To this end, the dynamic equations for frequency response, dynamic displacements, accelerations and electric voltage can be further computed analytically according to the suggested formulations.