计及边界非线性和几何非线性俘能器的振动分析与实验

The Vibration Energy Scavenging Project is an important project for the purpose of running hand-held low-powered electronics. The area has captivated both academics and industrialists..Nonlinear piezoelectric energy harvester, one of the most popular vibrational energy harvesters, often involves a cantilever subjected to magnetic interaction. In such harvesting prototype, a magnet is attached to the free end of the piezoelectric cantilever, the tip magnet suffers a nonlinear magnetic force, and the cantilever is clamped on an excited base so as to harvest vibrational energy. In the past investigations, Galerkin method was always used to truncate the distributed-parameter model. In this project, the continuous model will be directly analyzed without any truncation for three cantilevered piezoelectric energy harvesters. .The single cantilever /double cantilevers-based piezoelectric energy harvester will be considered here. The fundamental beam theory will be used to establish the equation of motion, a nonlinear integro-partial-differential equation. The effect of geometric nonlinearity will be considered. The magnetic force serves as nonlinear boundary condition dependent on time. The Kelvin viscoelastic damping will be used to account for the energy dissipation. The perturbation method, such as the multi-scale method, will be applied to solve the steady-state response of the continuous model. The analytical solution to the governing equation will be pursued approximately. .The effects of the system parameters on the output power will be analyzed by using bifurcation theory. The result will help to design nonlinear energy harvesters. To avoid the charge cancellation, the length of the piezoelectric patch will be designed as well. The analytical result can be devoted to theoretically solve it for all vibration modes. Interesting to us, through adjusting the system parameters, the frequency-response curve might vary among the linear, the softening, the hardening and the double-jumping behaviors. These theoretical results will be verified numerically and experimentally.

俘能器引入非线性磁力后，可以克服工作频带窄的缺陷，从而将环境中废弃振动有效转为电能，应用于微电子领域。制约压电悬臂式俘能器产业化的关键问题是难以全面解析结构参数对俘能效果的影响。针对该问题，拟发展摄动法分析足够精确的连续体模型，探讨多种激励、多种共振形式下的动力学和电学特性，并通过数值和实验验证。首先，针对三种最具代表性悬臂模型，考虑几何非线性建模得积分-偏微分方程组且含动态非线性边界，近似解析获取可解性条件，判断稳定性。然后，从解析结果分析软硬转换机制，明确系统参数对俘能器工作带宽的影响，根据分岔理论设计悬臂结构参数和电流回路。接着，发展含动态非线性边界受激系统的数值仿真算法，从简谐激励和随机激励两方面验证。最后，实验设计和测试多种装置，对比不同方案俘能效率。综上，借助本项目研究，将提供一套处理复杂边界下电-磁耦合悬臂系统非线性振动的连续体建模与分析方法，为研发振动型俘能器奠定理论基础。

振动源广泛存在于地震波、潮汐能、流动的风、运转中的机械仪器、交通工具和人类活动等，因此振动型能量采集技术的潜在应用前景广阔。又因该技术不需要电池、绿色环保且可持续使用，所以特别适合生态文化发展的需要。而悬臂式压电俘能器利用压电效应原理转换机械应变，具有结构简单、不发热、无电磁干扰、易于微型化和集成化等优点而颇受关注。.本项目围绕几种悬臂式压电俘能器开展研究，建立了相应的连续体模型。控制方程都是积分-偏微分方程组且含动态非线性边界。针对单臂梁的主共振与双臂梁的内共振，分别发展了直接多尺度方法近似解析获取可解性条件，并判断其稳定性。也发展了伽辽金方法来处理这类含动态非线性边界的连续体模型，将其离散化后使用多尺度方法研究稳态响应及其稳定性。两种方法所得结果吻合度较高。研究发现：.(1)压电片的布局问题对俘能器工作带宽的影响巨大。将压电片合理布置在相应模态的应变节点之间可以获取较大的输出功率，否则会引起电荷中和，从而减小能量输出，甚至采集电量为零。.(2)压电式俘能器中负载电阻对采集功率的影响较大。由于存在内部电容，机电耦合系数较大时可能出现两个最优值，机电耦合系数较小时则只有一个最优电阻。该理论结果通过解析分析得到，需要使用对数坐标方能发现，实验中确实不易找到。.(3)可以利用内共振、组合共振和亚谐波共振采集能量。由于1:2内共振、和式组合共振以及二阶亚谐波共振都可以引起振动失稳，因此，除了主共振之外，也可以合理设计系统参数利用次共振和内共振俘能。.综上，借助本项目研究，提供了一套处理复杂边界下电-磁耦合悬臂系统非线性振动的连续体建模与分析方法，为研发振动型俘能器奠定理论基础。

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