Nonlinear Piezoelectric Vibration Energy Harvesting of a Cantilever Beam Using Homotopy Analysis Method

The vibration energy collectors based on piezoelectric resonators are promising elements for energizing remotely located systems. However, differences between the resonant frequency of these traditional harvesters and the vibration frequency can drastically decrease the collected energy and make them ineffective. Appropriate mathematical models, different analyses, and optimization techniques to tune the resonant frequency of piezoelectric collectors have been researched. In this study, the model of an inverted vertical cantilever beam with a piezoelectric patch and a tip mass is used for energy harvesting. The beam is subjected to base excitations that can induce large lateral displacements of the tip, and consequently large deformation for the piezoelectric patch. Applying the homotopy analysis method (HAM) to the coupled electromechanical governing equations of motion, novel analytical solutions of the transverse displacement of the cantilever beam, its amplitude and phase as well as the output voltage obtained from the piezoelectric patch are derived. The analytical solutions are derived for the transversal displacements of the beam, even if it presents a varying cross-sectional area. The analytical solution considers the nonlinear behavior characteristics emphasizing the capabilities of a first-order approximation of HAM to present highly accurate closed-form solutions. The accuracy of this approximation of HAM is confirmed by comparison to numerical integration methods.