# A Novel Ropes-Driven Wideband Piezoelectric Vibration Energy Harvester
## 2 Working Principle
### Frequency Upconversion and Multimodal Mechanisms
\[Z_{1}=-2a\xi_{0}\pi\frac{\omega}{\omega_{0}}-a\xi_{1}(\pi-2\varphi-sin2\varphi )\frac{\omega\omega_{1}}{\omega_{0}^{2}} \tag{3}\]  
\[Z_{2}=\pi a(1-\frac{\omega^{2}}{\omega_{0}^{2}})-\left[\frac{1}{2}a\frac{ \omega_{1}^{2}}{\omega_{0}^{2}}\left(2\varphi-sin2\varphi-\pi\right)+2\frac{ \omega_{1}^{2}\Delta x}{\omega_{0}^{2}Y}cos\varphi\right] \tag{4}\]  
where \(\varphi=sin^{-1}(\Delta x/\alpha Y)\) is the phase angle when the LFDB pulls the HFGB, and \(a\) is the dimensionless amplitude with respect to frequency \(\omega/\omega_{0}\). Detailed derivations can be found in the literatures [17,32]. The instantaneous output voltage of HFGB with optimal impedance as a function of time \(t\) can be expressed by [14,15]  
\[V(t)=\left\{ \begin{array}{l}\frac{V_{OC}}{\omega_{c}}sin(\omega_{c}t)e^{- \xi_{0}\omega_{c}t},2n\pi/\omega<t<3(n+1)\pi/2\omega_{c},\textit{phase1}\\ \frac{V_{OC}^{2}}{V_{OC}^{2}}sin(\omega_{1}(t-3(n+1)\pi/2\omega_{c})+\frac{3 \pi}{2})e^{-\xi_{1}\omega_{1}t},3(n+1)\pi/2\omega_{c}\leq t\leq(n+1)\pi/2 \omega,\textit{phase2}\end{array} \right. \tag{5}\]  
where phase 1 is the coupled vibration of LFDB and HFGB, phase 2 is the separation vibration of HFGB, \(n=\) 0, 1, 2, 3,...... is the number of the harvester's cycle, \(\xi_{t0}\) and \(\xi_{t1}\) are the total damping ratio of the coupled vibration and the HFGB respectively, \(\omega_{c}\) is the frequency of the coupled vibration (which can be calculated based on the assumption that the LFDB and the HFGB connected with rope never separate after pulling, and then given by \(\omega_{c}=\sqrt{(k_{0}+k_{1})/m}\), and \(V_{oc}\) is the maximum open voltage generated by HFGB. It needs to be pointed out that, for the proposed VEH system, when an individual DB pulls the GB to realize wideband energy harvesting, the working principle is similar to the impact-driven frequency upconverting wideband VEH using a stopper [13, 14, 15, 16, 17, 32]. However, unlike the bandwidth of impact-driven VEHs that get wider but are still limited [17,18,32], the novel VEH system uses the rope to replace the direct mechanical collision and can easily realize wider or even unlimited bandwidth just by increasing the number of LFDBs, using the ropes to transfer the energy from multiple LFDBs to HFGB. Therefore, appropriately setting the number and resonant frequencies of LFDBs can achieve unlimited bandwidth to match a variable environment, which is named \"multimodal harvesting mechanism\". Based on the structure of HFGB driven by multiple LFDBs together, the proposed VEH is based on frequency upconversion and multimodal mechanisms to achieve a superiorly wide bandwidth. Obviously, the novel ropes-driven wideband energy harvester offers two advantages. Firstly, it has the potential to achieve wider or even unlimited bandwidth by adding LFDBs connected with ropes. Secondly, the output power will not be weakened with the increasing number of LFDBs, unlike arrays of beams vibration energy harvester (AB-VEH) for wider bandwidth [19, 20, 21, 22, 23], and the reason in shown below.