# Nonlinear Dynamics of a Resonant-Impact Dielectric Elastomer Actuator
## 2 Methods
### Experimental Methods
#### 2.3.2 Experimental Setups and Model Validations  
_Quasi-static force-displacement characterization_. The experimental setup is illustrated in Figure 4a. In this test, a single DEA unit was fixed to the testing rig and was deformed out-of-plane by a linear rail (X-LSQ150B-E01, ZABER, Vancouver, BC, Canada) from 0 to 4 mm, then returned at 0.05 mm/s. A load cell (S/N 835827, FUTEK, Irvine, CA, USA) measured the reaction force of the membrane and a laser displacement sensor (LK-G152 and LKGD500, Keyence, Osaka, Japan) measured the deformation. A DC voltage of 3.3 kV was applied to the DEA unit using a high voltage amplifier (10/40A-HS, TREK, Fort Collins, CA, USA) to characterize its electromechanical coupling effects. The measured and modelled force-displacement curves are plotted in Figure 4b, where the identified Ogden  
Figure 3: Illustration of the fabrication process of a single unit. (**a**) The initial state of the DE membrane. (**b**) Pre-stretching of the membrane. (**c**) Bonding to acrylic frames. (**d**) Hand-brushing custom carbon grease.  
parameters in the model are \(\mu_{A}=3.15\times 10^{5}\) Pa and \(\alpha_{A}=2.6\). It can be noted that the model adopted in this work is able to characterize the force-displacement relationship of the DEA unit with good accuracy.  
Frequency response characterizationThe experimental setup is plotted in Figure 5a and is described as follows. The DEA and the constraint were horizontally mounted to the testing rig. The two units of the DEA were actuated by two anti-phase sinusoidal frequency sweep (0 to 150 Hz in 600 s) voltage signals with amplitudes of \(\Phi_{p,p}=1.7\), 2.5 and 3.3 kV from two high-voltage amplifiers. The constraint gap \(G\) was adjusted manually using a high precision two-axis slider. Two laser displacement sensors were used to measure the displacements of the DEA and the center of the constraint, respectively, at a sampling rate of 40 kHz. Figure 5b,c plots the experimental and modelled forward frequency sweep results of the DEA, respectively, with \(G>>\) the resonant amplitude (i.e., the constraint has no effects on the dynamics). The identified modelling parameters are \(\mu_{B}=1.4\times 10^{5}\) Pa, \(\alpha_{B}=2\), \(\eta_{B}=1.4\times 10^{4}\), and \(\eta_{A}=220\). Note that the proposed model is able to characterize the frequency responses of the DEA accurately in all three voltage cases. The modelled frequency sweep results with constraint gap \(G=0.6\) mm are compared with the experimental results. It can be noted from Figure 5d,e that the impact model developed this work characterizes the resonant-impact responses with good accuracy.  
Figure 4: (**a**) Experimental setup of the quasi-static force-displacement characterization and (**b**) comparison of the experimental and modelling force-displacement results.  
Figure 5: (**a**) Experimental setup of the frequency response characterization. (**b**) Measured forward frequency sweep results of the DEA with G >> resonant amplitude and at voltage amplitudes of 1.7, 2.5, and 3.3 kV. (**c**) Modelling results of the DEA at the same voltage cases. (**d**) Measured frequency sweep results of the resonant-impact DEA system with G = 0.6 mm. (**e**) Modelling results of the resonant-impact DEA system.