Abstract The steady-state response of a nonlinear piezoelectric energy harvester subjected to external and parametric excitation is investigated based on the Mathieu-Duffing nonlinear oscillator model. The parametric excitation is introduced to amplify the external harmonic excitation and extend the capabilities of the nonlinear piezoelectric energy harvester device. To obtain the approximated solution of the nonlinear periodic responses for displacement and electrical voltage of the energy harvester, the incremental harmonic balance method in combination with the path-following technique is adopted. It is assumed that the proposed nonlinear model consists of cubic and quadratic nonlinearity, where parametric amplification appears in the form of a trigonometric function. The frequency is tuned as one-to-one and the one-to-two ratio between external and parametric excitation. The effects of quadratic and cubic nonlinearity as well as parametric amplification are studied in detail, and their incredible properties to extend harvester application performance is illustrated. It is explicitly demonstrated that for some particular combination of the system parameters, vibration amplitudes and harvested power can be amplified up to three or five times in comparison to the classical broadband nonlinear energy harvester based on the forced Duffing oscillator. This extraordinary amplification shown to be a key motivation to realize the proposed concept in practice. The presence of combined quadratic and cubic nonlinearities resulted in both hardening and softening spring behavior and leading to the appearance of coexisting periodic solutions in the amplitude-frequency responses. Periodic orbits obtained by the proposed methodology are verified with the results from direct numerical integration and fine agreement is demonstrated. Moreover, a significant influence of the parametric amplification on the instantaneous power is revealed in time response diagrams, thus showing better performance of the proposed energy harvester system.

中文翻译：

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参数放大 Mathieu-Duffing 非线性能量采集器

摘要 基于Mathieu-Duffing 非线性振荡器模型研究了非线性压电能量采集器在外部和参数激励下的稳态响应。引入参数激励以放大外部谐波激励并扩展非线性压电能量收集器设备的能力。为了得到能量采集器位移和电压非线性周期响应的近似解，采用增量谐波平衡法结合路径跟踪技术。假设所提出的非线性模型由三次和二次非线性组成，其中参数放大以三角函数的形式出现。频率被调谐为一对一和外部激励和参数激励之间的一对二比率。详细研究了二次和三次非线性以及参数放大的影响，并说明了它们在扩展收割机应用性能方面令人难以置信的特性。明确证明，对于系统参数的某些特定组合，与基于强制达芬振荡器的经典宽带非线性能量采集器相比，振动幅度和采集功率可以放大三到五倍。这种非凡的放大被证明是在实践中实现所提出的概念的关键动机。二次和三次非线性组合的存在导致弹簧硬化和软化，并导致在幅频响应中出现共存的周期解。通过直接数值积分的结果验证了通过所提出的方法获得的周期轨道，并证明了良好的一致性。此外，时间响应图中显示了参数放大对瞬时功率的显着影响，从而显示了所提出的能量收集器系统的更好性能。