In-plane nonlinear vibration of circular dielectric elastomer membranes with extreme stretchability

Abstract

This work investigates large in-plane stretches in thin circular membranes pre-stretched mechanically and dynamically excited by fluctuating voltages applied through their thicknesses. The nonlinear equation of motion is derived using Hamilton’s principle. Nonlinearity arises from the elastomer’s material behavior and electromechanical coupling from applied voltages. Primary and secondary resonances exhibit typical softening nonlinearity characterized by a range of frequencies where two coexisting vibrations, referred to as the upper branch and lower branch response, are possible. Our focus is the upper branch response, where extremely large dynamic stretches in the hundreds of percent are possible. These large stretches have not been fully investigated by prior research. The membrane’s upper branch vibrations can be found by decreasing frequency sweeps through resonance, and may not be captured if only increasing frequency sweeps are analyzed. Large-amplitude upper branch stretches occur when the membrane vibrates about both small-stretch and large-stretch equilibria. In addition to larger amplitudes, the upper branch dynamic stretches near secondary resonances have dominant frequencies different from the excitation frequency. Whether the dielectric elastomer membrane vibrates on the lower or upper branch depends on its initial conditions, and the initial conditions that result in each case are quantified. A technique is demonstrated to switch the membrane’s vibration from the lower to upper branch using impulse-like voltages superposed on the nominal sinusoidal component. The large-amplitude nonlinear dynamic stretches highlighted in this work, and the strategies we propose for accessing and controlling them, could be leveraged for improved performance in existing soft transducer technologies.