# Nonlinear Dynamics of a Resonant-Impact Dielectric Elastomer Actuator
## 2 Methods
### Numerical Model  
#### 2.2.1 Equation of Motion Development  
To investigate the nonlinear dynamics of the resonant-impact DEA system, it is necessary to first develop a dynamic model of it. A sketch of the system is illustrated in Figure 2a;  
Figure 1: **(a)** Schematic diagram of the resonant-impact DEA system. **(b)** Motion of the resonant-impact DEA system in one typical impact oscillation cycle. **(c)** Displacement of the moving mass and the reaction force on the supporting rig. Red curves: proposed resonant-impact DEA system; blue curves: identical DEA without the impact constraint. it can be noted that the system model mainly includes the nonlinear dynamic model of the DEA unit and its interaction with the elastic constraint. To avoid overcomplication of the system model, the following simplifying assumptions are made here: (i) the DEA unit is single degree-of-freedom, i.e., only its translation along the cylindrical axis is analyzed; (ii) the deformations of the DEA membranes are truncated conical deformations; (iii) the strain distributions on the DEA membranes are homogeneous; (iv) the circumferential deformations of the DEA membranes remain constant; (v) the electrical response time of the DEA unit is negligible; (vi) the constraint is massless, i.e., the system is single degree-of-freedom. The equation of motion of the moving mass is  
\[\vec{m}\vec{d}+F_{contact}+F_{DE\_I}-F_{DE\_II}=0 \tag{1}\]  
where \(m\) is the total moving mass, \(d\) is the out-of-plane deformation of \(DE\) unit \(I\) (close to the constraint), \(F_{contact}\) is the force exerted by the constraint, and \(F_{DE\_I}\) and \(F_{DE\_II}\) are the force exerted by \(DE\) units \(I\) and \(II\), respectively. The contact force of the constraint is provided by the following equation:  
\[F_{contact}=\left\{ \begin{aligned} 0,\ d-L/2\ <\ G\\ -K\left(d-L/2-G\right)-c\dot{d},\ d-L/2\geq\ G\end{aligned} \right. \tag{2}\]  
where \(G\) is the gap distance, \(L\) is the rod length, \(K\) is the stiffness of the constraint, and \(c\) is its damping coefficient. The force exerted by the membrane along the cylindrical axis \(F_{DE\_i}\) can be written as  
\[F_{DE\_i}=2\pi a\frac{T_{0}}{\lambda_{1\_i}\lambda_{2\_i}}\sigma_{DE\_i}sin \theta\_i \tag{3}\]  
where \(i=I\) and \(II\) for \(DE\) units \(I\) and \(II\), respectively, \(a\) is the radius of the central disk, \(T_{0}\) is the initial thickness of the membrane, \(\lambda_{1}\) and \(\lambda_{2}\) are the radial and circumferential stretch of the membrane, respectively, \(\sigma_{DE}\) is the radial stress of the membrane (and is developed further below), and \(\theta\) is the out-of-plane deformation angle of the membrane. To characterize the strain-stress relationship of the dielectric elastomer, the hyperelasticity, viscoelasticity, and electromechanical coupling are taken into account, as illustrated in Figure 2b. A first-order Ogden model is adopted to describe the hyperelastic characteristic [41], and a rheological model with parallel Kelvin-Voigt and Maxwell models is utilized to account for viscoelasticity [42]. The total radial stress of the membrane when subjected to a stretch and a voltage can be written as  
\[\sigma_{DE}=\sigma_{h}+\sigma_{v}+\sigma_{e} \tag{4}\]  
Figure 2: (**a**) Sketch of the resonant-impact DEA system model and (**b**) illustration of the material model. where \(\sigma_{h}\) is the hyperelastic stress, \(\sigma_{v}\) is the viscoelastic stress, and \(\sigma_{e}\) is the electrostatic stress. The hyperelastic stress is described by a first-order Ogden model [41]:  
\[\sigma_{h}=\mu_{A}\left(\lambda_{1}^{\alpha A}-\lambda_{1}^{-\alpha A}\lambda_{2 }^{-\alpha A}\right)\] (5a) where \[\mu_{A}\] is the shear modulus of the spring in the first branch of this material model and \[\alpha_{A}\] is a power constant.