# Nonlinear Dynamics of a Resonant-Impact Dielectric Elastomer Actuator
## 3 Numerical and Experimental Study Results  
### Frequency Response Overview  
To provide a general concept of the effects of the constraint on the frequency responses of the DEA, two sets of frequency responses curves are generated by using the nondimensional numerical model. These following values are adopted for the nondimensional variables and parameters in the simulations: \(\Gamma=0.0045\), \(\mu_{A}/\mu_{B}=1\), \(\alpha_{A}=3\), \(\alpha_{B}=2\), \(\delta_{A}=0.3\), \(\delta_{B}=60\), \(l=0.8\), \(k=2.5\times 10^{3}\), and \(\delta_{i}=2\), \(\phi_{p-p}=0.4\). Unless specified otherwise, the same values are utilized in the rest of this paper. The simulated frequency response results of the DEA with no constraint (i.e., \(g=+\infty\)) and with \(g=0.1\) are shown in Figure 6a,b, respectively. It can be noted from Figure 6a that the oscillation amplitude of the DEA with no constraint first increases with the increasing excitation frequency \(\omega_{e}\), then peaks at its resonant frequency before dropping to a low value as the excitation frequency increases further, which is demonstrated in the experiments in Figure 6b as well. In addition, note that the forward and backward frequency response curves overlap completely. As a comparison, it can be observed from Figure 6b that the constraint results in strong distortions in the frequency response curves of the resonant-impact DEA system. Although the oscillation amplitudes increase with the increasing excitation frequency before resonance, the upper dead-end of the DEA can only increase to close to \(g\), while the bottom dead-end continues to increase until its resonance. It is worth noting that the resonant frequency in the forward frequency step simulation in Figure 6b is much higher than that in the backward step due to the strong nonlinearity introduced by the constraint. The amplitude of the impact force \(\left\lvert f\right\rvert\) (i.e., the largest force experienced by the constraint during an impact in a steady state response) against the excitation frequency is plotted in Figure 6c. It can be noted that the amplitude of the impact force \(\left\lvert f\right\rvert\) follows the same trend as the oscillation amplitude results in Figure 6b, which indicates that higher oscillation amplitudes lead to a larger impact force due to the higher energy level in the system. In the potential applications of such a resonant-impact DEA system (i.e.,  
Figure 6: Frequency sweep results of the DEA (**a**) without constraint and (**b**) with constraint, \(\phi_{p+p}=0.4\), \(g=0.1\); here, \(X\) is the maximum/minimum displacement at steady state. (**c**) Impact force amplitude \(\left\lvert f\right\rvert\) of the resonant-impact DEA system against the excitation frequency \(\omega_{e}\). (**d**) Demonstration of the resonant-impact DEA system losing its stability when experiencing a disturbance in its excitation voltage signals. DEA-driven crawling robots) a larger impact force is desirable, as it typically means faster response/locomotion velocity or better performance in general. However, it is worth noting from Figure 6d that higher oscillation amplitudes (which lead to a larger impact force) can affect the stability of the system, as a small perturbation from the environment or the input can cause the high amplitude oscillation to lose its stability and transit into low amplitude oscillation. This is due to the multiple stable solutions in the resonant-impact DEA system near its resonance (i.e., the forward and backward frequency response curves do not overlap near the resonance in Figure 6b). This stability issue can significantly limit the controllability and robustness of the resonant-impact DEA system in practical applications where perturbations are unavoidable. To address this limitation, in this work we mainly focus on the impact frequency band (i.e., the overlapping region of the forward and backward curves), where a single stable solution exists.