Incremental harmonic balance method for periodic forced oscillation of a dielectric elastomer balloon

Dielectric elastomer (DE) is suitable in soft transducers for broad applications, among which many are subjected to dynamic loadings, either mechanical or electrical or both. The tuning behaviors of these DE devices call for an efficient and reliable method to analyze the dynamic response of DE. This remains to be a challenge since the resultant vibration equation of DE, for example, the vibration of a DE balloon considered here is highly nonlinear with higher-order power terms and time-dependent coefficients. Previous efforts toward this goal use largely the numerical integration method with the simple harmonic balance method as a supplement. The numerical integration and the simple harmonic balance method are inefficient for large parametric analysis or with difficulty in improving the solution accuracy. To overcome the weakness of these two methods, we describe formulations of the incremental harmonic balance (IHB) method for periodic forced solutions of such a unique system. Combined with an arc-length continuation technique, the proposed strategy can capture the whole solution branches, both stable and unstable, automatically with any desired accuracy.