# A Novel Ropes-Driven Wideband Piezoelectric Vibration Energy Harvester
## 2 Working Principle  
### Frequency Upconversion and Multimodal Mechanisms  
As shown in Figure 1a, the proposed wideband PVEH system is composed of one high-frequency beam attached with piezoelectric material as the generating beam (GB) and multiple low-frequency beams as driving beams (DBs) having different frequencies arranged face-to-face with a predetermined space, and multiple DBs are connected mechanically with the GB using ropes. For the structure of an individual DB driving the GB, the architecture and schematic model are shown in Figure 1b,c. The equivalent schematic model contains (1) an excitation oscillator as the DB with stiffness \(k_{0}\), damping coefficient \(c_{0}\), and proof mass \(m_{0}\); (2) another oscillator as the GB, which acts as a generator oscillator, is placed at a distance of \(x_{0}\) above the excitation oscillator, with higher stiffness \(k_{1}\) than \(k_{0}\), damping coefficient \(c_{1}\), and proof mass \(m_{1}\); and (3) the rope with stiffness \(k_{2}\), which is used to connect the excitation oscillator and generator oscillator with a length of \(x_{1}\). The operation mechanism is depicted in Figure 1d. When the LFDB is excited with sufficiently large amplitude that exceeds the rope margin \(\Delta x\) (defined as length of rope \(x_{1}\) minus \(x_{0}\)), the LFDB pulls the HFGB by the rope and vibrates together with the HFGB for a short period. After this period, the HFGB gets mechanical energy from LFDB by the rope and oscillates with exponentially attenuating amplitude at its higher resonant frequency, and the cyclic deformation of piezoelectric layer on HFGB will be transformed into electricity due to the piezoelectric effect. The operating principle of LFDB driving HFGB to output energy by HFGB can be treated as a frequency upconversion mechanism, which can well-realize wideband energy-harvesting and improve output power efficiency for low-frequency applications [12; 13; 14; 15; 16; 17; 18]. Based on the equivalent schematic model and operating mechanism of the proposed VEH system, as shown in Figure 1c,d, the differential equations of the motion can be written as  
\[\left\{ \begin{array}{l}(m_{0}+m_{1})\ddot{z}+(c_{0}+c_{1})\dot{z}+(k_{0}+k _{1}+k_{2})z-(k_{1}+k_{2})\;\;\Delta x=-(m_{0}+m_{1})\ddot{y}\quad(z\geq\Delta x )\\ m_{0}\ddot{z}+c_{0}\dot{z}+k_{0}z=-m_{0}\ddot{y}\quad(z<\Delta x)\end{array} \right. \tag{1}\]  
The operating mechanism and equivalent schematic model of individual rope-driven frequency-upconverted EHs are similar to the impact-driven frequency-upconverted EHs using a stopper, which has been analyzed and demonstrated in the literatures [13; 14; 15; 16; 17; 18; 32]. Hence, similar derivations can be used in this novel EH model. Here, assuming that the stiffness \(k_{2}\) of rope is far smaller than the stiffness \(k_{0}\) and \(k_{1}\), and the proof mass of \(m_{0}\) is far bigger than the mass of \(m_{1}\), Equation (1) can be rearranged as follows:  
\[\left\{ \begin{array}{l}m\ddot{z}+(2\xi_{0}\omega_{0}+2\xi_{1}\omega_{1})\dot{ z}+(\omega_{0}^{2}+\omega_{1}^{2})z-\omega_{1}^{2}\Delta x=-m\ddot{y}\quad(z \geq\Delta x)\\ m\ddot{z}+2\xi_{0}\omega_{0}\dot{z}+\omega_{0}^{2}z=-m\ddot{y}\quad(z<\Delta x )\end{array} \right. \tag{2}\]  
where \(y(t)=Ysin(\omega t)\) is a harmonic base excitation, \(Y\) is the amplitude of the base excitation, \(z(t)\) is the displacement of LFDB relative to the base, \(\xi_{0}\) and \(\omega_{0}\) are the LFDB damping and frequency characteristics, and \(\xi_{1}\) and \(\omega_{1}\) are the HFGB damping and frequency characteristics, which are defined as \(2\xi_{0}\omega_{0}=c_{0}/m_{0}\), \(2\xi_{1}\omega_{1}=c_{1}/m_{1}\), \({\omega_{0}}^{2}=k_{0}/m_{0}\), and \({\omega_{1}}^{2}=k_{1}/m_{1}\). Thus, the amplitude of the LFDB displacement against excitation frequency \(\omega\) can be obtained numerically based on Equations (3) and (4).