This paper reports a predictive model for vibration-to-electrical energy harvesters (e-VEHs) based on an in-plane, gap-closing variable capacitor with frequency up-conversion triggered by the impact between the electrodes. Since the output power is proportional to the output frequency, rectifying low-frequency ambient vibrations (1-50 Hz) to high-frequency electrical signals (200-600 Hz) increases the power output. While such device has been previously reported experimentally, this is the first time to describe a model able to predict the experimental data. The model is based on lumped approximation. The central area supporting the mobile electrodes, or the shuttle mass, is represented by a point mass suspended by springs and has its own equation of motion. The motion of the electrodes attached to the shuttle is described by a set of two equations, each associated with a distinct dynamic mode. In these equations, the electrodes are represented by an equivalent mass and spring constant. The first equation describes the separate motion of mobile and fixed electrodes. In this mode, the electrodes experience damped free vibration due to the electrostatic and air damping between them. The second equation describes the combined motion when the two sets of electrodes move together. In this mode, the air damping forces between the collided electrodes are eliminated and the electrostatic force is kept constant. The motion equations are solved simultaneously with Kirchhoff's law to compute the voltage drop across a resistor in series with the variable capacitor and a DC bias voltage source. Predictions are shown to be consistent with the experimental results, and frequency up-conversion effects are observed with exponentially decaying voltage amplitude as seen in experiments. A parametric study is also carried out to identify main parameters that affect the up-conversion, laying the foundation for future design optimization to maximize the power output.

中文翻译：

---

基于冲击的频率上变频静电能量收集器的预测模型

本文报告了一种基于平面内间隙闭合可变电容器的振动到电能收集器 (e-VEH) 的预测模型，该电容器具有由电极之间的冲击触发的频率上变频。由于输出功率与输出频率成正比，因此将低频环境振动 (1-50 Hz) 整流为高频电信号 (200-600 Hz) 会增加功率输出。虽然之前已经通过实验报道过这种设备，但这是第一次描述能够预测实验数据的模型。该模型基于集总近似。支撑移动电极的中心区域或穿梭质量由弹簧悬挂的点质量表示，并具有自己的运动方程。连接到穿梭机的电极的运动由一组两个方程描述，每个都与不同的动态模式相关联。在这些方程中，电极由等效质量和弹簧常数表示。第一个方程描述了移动电极和固定电极的单独运动。在这种模式下，由于它们之间的静电和空气阻尼，电极会经历阻尼自由振动。第二个方程描述了当两组电极一起移动时的组合运动。在这种模式下，碰撞电极之间的空气阻尼力被消除，静电力保持恒定。运动方程与基尔霍夫定律同时求解，以计算与可变电容器和直流偏置电压源串联的电阻器上的电压降。预测结果与实验结果一致，如实验中所见，随着电压幅度呈指数衰减，观察到频率上变频效应。还进行了参数研究以确定影响上变频的主要参数，为未来设计优化以最大化功率输出奠定基础。