Non-Linear Vibration of Thick Dielectric Membrane Disks With Radial Loads

This study analyzes the large-amplitude, non-linear vibration of dielectric elastomer membrane disks with applied voltages through their thickness and mechanical loads applied radially around their outer circumferential surface. The material is modeled as an isotropic ideal dielectric, with the large-stretch mechanical stiffening captured using the Gent hyperelastic constitutive model. The fully non-linear equation of motion for the coupled electromechanical system is derived using Hamilton’s principle. The disk comes to a steady equilibrium where the compressive stresses due to the applied voltage balance the tensile stresses from the applied radial loads. The equilibria are calculated numerically for a wide range of radial loads, applied voltages, and limiting stretches. It is possible for the disk to have two stable steady equilibria at given radial load and applied voltage, which gives rise to an instability where extreme stretches occur for infinitesimal changes in applied voltage. The equation of motion is determined for small vibrations of the system about equilibrium. Unlike for thin membrane disks, the vibrating mass of thick membrane disks depends on the steady equilibrium stretch. The natural frequency for membrane disks meaningfully decreases with increasing thickness due to the inertia associated with dynamic changes in the membrane thickness. The amount of axial inertia depends on the ratio of the nominal disk thickness to its radius and the steady equilibrium stretch. Large amplitude vibrations are numerically investigated for a wide range of system parameters. The frequency response characteristics of circular membranes due to sinusoidal voltage fluctuations are analyzed about small and large equilibrium stretches. Whereas axial inertia meaningfully alters the frequency response about small equilibrium stretches, it has negligible effects on the frequency response about large equilibrium stretches.