## 2 Conceptual Model
Figure 2: Dimensional definitions for piezoelectric bimorph as used in forcing approximation. This represents a very simplistic approximation of impact behavior between two compliant structures, neglecting factors such as modal vibration effects, adhesion or other nonlinear interfacial forces, and the sensitivity of CoR to angle and location of impact. Nonetheless, the model in (1)-(6) will be seen to predict certain characteristic behaviors later observed in experimental testing. Motion output is measured at the far tip of the passive beam, a distance \(R\) from the compliant spring connection, with vertical output displacement, \(x_{out}\) approximated for small angles by  
\[x_{out}\approx x_{L}+R\theta_{L} \tag{7}\]  
To illustrate the behavior of this class of the interacting components, the model in (1)-(7) was simulated numerically under square wave inputs with varying frequency, using parameters listed in Table 1. Output behavior is characterized from time domain results and dominant frequency components of the resulting motion after achieving quasi-steady-state oscillation, as shown in Figure 3.  
Notably, the simulated output oscillations do not occur at the input frequency, even when the PZT beder is driven near resonance. Rather, output oscillations are produced in the frequency range of amplification mechanism dynamics, but with tunability between dominant frequencies by input. Figure 4 shows sample simulation trajectories at 3.7 Hz, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Quantity** & **Variable** & **Value** & **Units** \\ \hline Equiv. PZT mass & \(m_{p}\) & 0.12 & g \\ Equiv. PZT stiffness & \(k_{p}\) & 230 & N/m \\ PZT damping & \(c_{p}\) & 0.008 & N/m/s \\ Piezoelectric coeff. & \(d_{31}\) & 350 & pC/N \\ PZT thickness & \(t_{p}\) & 0.19 & mm \\ PZT width & \(w_{p}\) & 3 & mm \\ PZT length & \(L_{p}\) & 29 & mm \\ Output equiv. mass & \(m_{L}\) & 0.37 & g \\ Output equiv. inertia & \(J_{L}\) & 25 & g-mm\({}^{2}\) \\ Output stiffness & \(k_{xx}\) & 0.77 & N/m \\ Matrix terms & \(k_{\phi\theta}\) & 1 & N-mm/rad \\  & \(k_{x\theta}\) & \(-7\) & mN/rad \\  & \(k_{\theta x}\) & \(-7\) & mN-m/m \\ CoR & \(\alpha\) & 0.5 & mm \\ Impact offset & \(r\) & 2.5 & mm \\ Tip length & \(R\) & 15 & mm \\ Gap & \(d\) & \(-0.04\) & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 3: Magnitude of largest output frequency component versus square wave input frequency. 75 Hz, and 225 Hz (bender resonance), with gradually increasing frequency and amplitude of quasi-periodic output oscillations. Trends are shown more comprehensively across input frequencies in an output displacement spectrum, shown in Figure 5, with input excitation frequencies generating substantial output amplitudes at various frequencies between 1 and 250 Hz, and corresponding output frequencies rising from about 2.5 to 6.5 Hz. These oscillations occur between the natural frequencies of the compliant system in (3) alone (2.3 Hz and 10.5 Hz), with both rotational and translational deformation present. Highest amplitudes are achieved when the differing response times of those axes complement each other as measured at the mechanism output. However, while the frequencies at which this occurs are dependent on input frequency, in simulation, the uniformity of motion tends to become worse as input frequency is increased. Overall, the basic arrangement of a high-frequency oscillator impacting a compliant structure produces extremely large amplification ratios between piezoelectric bender displacement and output displacement. These can be as high as 60:1 when the bender is driven substantially below resonance. In addition, a variety of input-output frequency ratios are possible, either for tuning the output frequency or minimizing energy consumption. In the next section, we examine whether these types of behavior can be achieved in a physical setting, with characterization of an experimental prototype with dimensions near that of the simulated design.