This paper investigates the dynamical response of a nonlinear piezoelectric energy harvester under a hard harmonic excitation and assesses its output power. The system is composed of a unimorph cantilever beam with a tip mass and exposed to an harmonic tip excitation with a hard forcing amplitude. First, the governing dimensionless nonlinear electromechanical ordinary differential equations (ODEs) are obtained. Next, the multiple scales method (MSM) is exploited to provide an approximate-analytical solution for the ODEs in hard and soft forcing scenarios. It is observed that, the hard force results in sub- and super-harmonic resonances. The MSM-based solutions are then validated by a numerical integration method and a good agreement is observed between the approximate-analytical and numerical results. Furthermore, utilizing the MSM-based solutions for the subharmonic, superharmonic, and soft primary resonances cases, the associated frequency and force response curves are constructed. It is revealed that the hard excitation leads to a remarkable voltage generation in the secondary resonances; this leads to a broadband energy harvesting. In addition, the time-domain electrical responses of the secondary resonances are also obtained and compared with each other. Finally, the three-dimensional graphs of the electrical power versus detuning parameter and time constant ratio in the cases of the secondary resonances are plotted. The results show that the optimum output power of the superharmonic resonance is considerably larger than the maximum power of the subharmonic resonance case.

中文翻译：

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硬谐波激励下非线性压电梁二次共振的能量收集

本文研究了非线性压电能量收集器在硬谐波激励下的动态响应并评估其输出功率。该系统由具有尖端质量的单压电晶片悬臂梁组成，并暴露于具有强强迫振幅的谐波尖端激励。首先，获得控制无量纲非线性机电常微分方程（ODE）。接下来，利用多尺度方法 (MSM) 为硬和软强迫场景中的 ODE 提供近似解析解。据观察，硬力导致次谐波和超谐波共振。然后通过数值积分方法验证基于 MSM 的解决方案，并在近似解析结果和数值结果之间观察到良好的一致性。此外，利用针对次谐波、超谐波和软初级谐振情况的基于 MSM 的解决方案，构建相关的频率和力响应曲线。结果表明，硬激发导致二次共振产生显着的电压；这导致宽带能量收集。此外，还获得了二次谐振的时域电响应并相互比较。最后，绘制了二次谐振情况下电功率与失谐参数和时间常数比的三维图。结果表明，超谐波谐振的最佳输出功率远大于次谐波谐振情况的最大功率。构建了相关的频率和力响应曲线。结果表明，硬激发导致二次共振产生显着的电压；这导致宽带能量收集。此外，还获得了二次谐振的时域电响应并相互比较。最后，绘制了二次谐振情况下电功率与失谐参数和时间常数比的三维图。结果表明，超谐波谐振的最佳输出功率远大于次谐波谐振情况的最大功率。构建了相关的频率和力响应曲线。结果表明，硬激发导致二次共振产生显着的电压；这导致宽带能量收集。此外，还获得了二次谐振的时域电响应并相互比较。最后，绘制了二次谐振情况下电功率与失谐参数和时间常数比的三维图。结果表明，超谐波谐振的最佳输出功率远大于次谐波谐振情况的最大功率。这导致宽带能量收集。此外，还获得了二次谐振的时域电响应并相互比较。最后，绘制了二次谐振情况下电功率与失谐参数和时间常数比的三维图。结果表明，超谐波谐振的最佳输出功率远大于次谐波谐振情况的最大功率。这导致宽带能量收集。此外，还获得了二次谐振的时域电响应并相互比较。最后，绘制了二次谐振情况下电功率与失谐参数和时间常数比的三维图。结果表明，超谐波谐振的最佳输出功率远大于次谐波谐振情况的最大功率。