Exact energy harvesting analysis of multimodal piezoelectric beams using the dynamic stiffness method

Piezoelectric vibration energy harvesting holds great potential for converting ambient vibrations into electrical energy. Establishing a suitable theoretical model to predict the performance of piezoelectric harvesters under base excitation is essential. This paper proposes a dynamic stiffness (DS) modeling technique to predict the electromechanical coupling responses of piezoelectric beams. The modeling technique is sufficiently general to be applied to a wide range of simple cantilever or generally complex piezoelectric beam structures. For demonstration purposes, this technique is applied to model beams equipped with three typical tip attachments and connected to three representative external circuits, enabling a comprehensive multimodal analysis. The Wittrick–Williams (WW) algorithm is employed to efficiently calculate the eigenvalues of DS matrices with any desired accuracy. This aids in tuning the natural frequency of piezoelectric beams to match the ambient environmental vibration frequency. The concepts of electrically-induced stiffness and electrically-induced damping are introduced to characterize the impact of energy harvesting circuits on natural frequencies and output voltage of piezoelectric harvesters. Theoretical research specifically conducted on piezoelectric beams explores the effects of material parameters, structural dimensions, load resistance, and base excitation on vibration energy harvesting. Finite element simulation confirms that the proposed DS model, based on the exact solution of governing differential equations, can accurately and effectively predict the output performance of piezoelectric harvesters. The proposed method can emerge as a powerful tool for the design and optimization of piezoelectric energy harvesters.