Low frequency, broad bandwidth and adjustable frequency have always been adopted as crucial indexes to measure the performance of piezoelectric energy harvesters (PEHs). The axial preload is deemed as one of the effective ways to improve these indexes. To explore the effect of preload on the energy harvesting characteristics, the electromechanical coupling governing equations were derived based on the Euler-Bernoulli beam theory and Gauss Theorem. The asymptotically analytic solutions of the displacement, the voltage and the average output po-wer were obtained with Galerkin discretization and the multi-scale approach. Then, through theoretical analysis, this paper obtained the expressions of the short-circuit and open-circuit resonance frequencies, the open-circuit voltage amplitude, the optimal output power and the optimal load resistance. For the cantilever beam model, the validity of the theory was verified by numerical simulation. Finally, the influence of preload on open-circuit voltage and optimal output power was analyzed. The results reveal that the axial preload can improve the energy harvesting efficiency of piezoelectric cantilever beam. Compared with the case without preload, the resonant frequency decreases by 31.6%, the amplitude of open-circuit voltage increases by 120.8%, and the optimal output power increases by 40.0%, when the preload is 20 N.

At the micro-nano scale, the flexoelectric effect exhibits a higher electromechanical coupling conversion efficiency in comparison with the traditional piezoelectric effect. Therefore, it holds potential applications in the fields of sensing, actuation, and energy harvesting. However, most commercially available finite element analysis software packages lack the flexoelectric constitutive model, rendering it impossible to perform a precise numerical simulation of flexoelectric structures. Therefore, this study incorporated the flexoelectric effect into the calculation model based on the development of Abaqus user-defined element (UEL) subroutine. Additionally, we derived the electromechanical coupling equations of the flexoelectric structure and developed a flexoelectric element in Abaqus, which provides numerical simulation technology for analyzing the structural flexoelectric response. Compared with the traditional mixed finite element method (MFEM), this proposed method has the advantages of easier modeling, high efficiency and low computational cost. And its deflection and electric field strength metrics are closer to the analytical solution than either of the other methods. Then, using this numerical simulation method, this study established a flexoelectric structural response analysis numerical model, conducted force-electromechanical coupling response calculations, and analyzed straight and curved beams under different boundary conditions. The analysis results show the mechanism of the geometric parameters of the beam affects the structural strain gradient, and show that the output voltage can be controlled by controlling the deformation gradient when designing a flexoelectric beam using the same material. It is shown that changing the degree of bending of the beam is effective in increasing the open-circuit voltage of the beam output and reducing the deflection of the beam. The open-circuit voltage of a curved cantilever beam bent downward is greater than the open-circuit voltage of a curved cantilever beam bent upward for the same degree of bending. When the circular arc angle of the upward bending curved beam is 38°, the left edge is a sliding bearing, and the right edge is a hinge bearing, the open-circuit voltage is the maximum, up to 214.07 mV, which is five times more than that of the cantilevered rectangular beam. In addition, by considering the piezoelectric effect alone, the open-circuit voltage will decrease by 89.3% in the same model.