This paper presents a novel formulation and exact solution of the frequency response function (FRF) of vibration energy harvesting beam systems by the distributed transfer function method (TFM). The method is applicable for coupled electromechanical systems with nonproportional damping, intermediate constraints, and nonclassical boundary conditions, for which the system transfer functions are either very difficult or cumbersome to obtain using available methods. Such systems may offer new opportunities for optimized designs of energy harvesters via parameter tuning. The proposed formulation is also systematic and amenable to algorithmic numerical coding, allowing the system response and its derivatives to be computed by only simple modifications of the parameters in the system operators for different boundary conditions and the incorporation of feedback control principles. Examples of piezoelectric energy harvesters with nonclassical boundary conditions and intermediate constraints are presented to demonstrate the efficacy of the proposed method and its use as a design tool for vibration energy harvesters via tuning of system parameters. The results can also be used to provide benchmarks for assessing the accuracies of approximate techniques.

The purpose of this paper is to investigate the effect of boundary flexibility on the performance of piezoelectric vibration energy harvester (PVEH) beam systems, which has not been studied comprehensively in the literature despite its importance. The coupled electromechanical equations of motion of a piezoelectric cantilever beam with a tip mass are established, with the base boundary constrained by translational and rotational springs. An exact closed-form solution of the frequency response function (FRF) of the PVEH is obtained by the distributed transfer function method (DTFM). The DTFM is a systematic powerful tool for the dynamic analysis of distributed parameter continua with non-classical boundary conditions, intermediate constraints, coupled fields, and non-proportional damping without adding much complexity to the solution formulation. Moreover, the DTFM computes the derivatives of the response, that is, the strains, which are required in the electromechanical coupling formulation, simultaneously without any differentiation. Numerical results showing the effects of boundary flexibility on energy harvesting efficiency are presented. A first-order rational function relating the boundary stiffness parameters and the harvesting efficiency is determined by nonlinear curve fitting of the calculated data. Physical insights and applicability of this analytical function for end-of-line quality check of the boundary of PVEH are discussed.