Analytical Study on Nonlinear Dynamic Response for Electromechanical Behavior of Dielectric Elastomer Membranes

Abstract

Dielectric elastomer is a prosperous material commonly used in electromechanical systems because it can effectively transform electrical energy to mechanical work. Applications exploiting the dynamic behavior of dielectric elastomers include loudspeakers, active noise control, frequency tuning, etc. Although there already exists literature that focuses on dynamic response of dielectric elastomers, a thorough understanding of its dynamic behavior is still far from enough.

This thesis focuses on the analytical study on the dynamic response of dielectric elastomer membranes subject to voltage and other kinds of loading such as pressure and/or external forces. The dynamic response analysis is mainly conducted through an excellent quantitative analytical approximate method, the Newton–harmonic balance (NHB) method. The NHB method enables the governing equation be linearized prior to applying the harmonic balance (HB) method. It results in simple algebraic equations and avoids solving complex nonlinear algebraic equations. Besides, unlike the traditional HB method the higher-order approximations can be readily derived. The NHB method is valid for both small and large amplitudes of oscillation, and it is not only restricted to systems with small parameters. The work completed in this dissertation is summarized as follows.

Firstly, the large amplitude nonlinear free vibration of a spherical dielectric balloon subjected to static pressure and voltage is studied analytically. Base on the thermodynamic theory, the neo-Hookean material model is adopted to illustrate the hyper-elasticity of the dielectric elastomer. The deformation of the spherical balloon is modeled as an autonomous nonlinear differential equation with general and negatively powered nonlinearities. The period and periodic solution are derived through the NHB method, as the initial stretch ratio is known. Even for such a nonlinear system with negatively powered variable and non-classical non-odd nonlinearity, the NHB method is capable of deriving highly accurate approximation numerical integration solutions is also presented and excellent agreement has been observed. Several practical examples with different initial stretch ratios are solved to illustrate the dynamic inflation of elastomeric spherical balloons. When the initial amplitude is sufficiently large, the system will lose its stability. Comparison with Runge-Kutta numerical integration solutions is also presented and excellent agreement has been observed.

Secondly, the effects of initial stretch ratio, pressure and voltage on nonlinear free vibration of a spherical dielectric elastomer balloon are investigated qualitatively and quantitatively. Through a linear stability analysis of the equilibrium states, the safe regime of initial stretch ratio for the deformation of dielectric elastomer balloon is confined. Under specific static driving pressure and voltage, the system oscillates about the stable equilibrium and there is no oscillation in the neighborhood of the unstable equilibrium. Besides, the critical pressure and voltage are determined by the static bifurcation theory. Beyond the critical values, there is no periodic oscillation. Along with the stability analysis, complex dynamical behavior such as drastic change of output regime, sporadic instability and sudden bifurcations can be predicted. Applying the Newton Harmonic Balance method for quantitative analysis, it is found that the nonlinear free vibration frequency decreases with increasing initial stretch ratio and control parameters (pressure and voltage).

Thirdly, the nonlinear free vibration for a bi-stable circular dielectric elastomer membrane is studied. The Gent material model is adopted to illustrate the strain-stiffening effect of polymer chains. Subject to static voltage and equal biaxial force, the deformation of circular membrane is modeled as an autonomous nonlinear differential equation. The governing equation is more complex than the neo-Hookean dielectric spherical balloon because, except for the general and highly-powered nonlinearities in the restoring function, a fraction with both the denominator and numerator expressed with highly negatively-powered terms exists. Solving such a system itself is a challenge and also an effective validation for the superiority of the NHB method. The governing equation will be single-stable or bi-stable under specific voltage region. Before the free vibration of the dielectric elastomer membrane undergoing large deformation is studied through the NHB method, the voltage regions for the single-stable and bi-stable are confined respectively. For the single-stable case, the periodic oscillation occurs around the stable equilibrium. For the bi-stable case, the periodic oscillation will be around one of the stable equilibria or surround all the equilibria. Comparisons with the exact period and exact periodic solution present excellent agreement. It is found that even for large amplitude oscillation, the second-order approximation keeps its accuracy. Based on the angular frequency obtained by NHB, it is found that for the single-stability case, the angular frequency will firstly decrease and then increase with the augment of the initial stretch ratio due to the strain-stiffening effect of the dielectric elastomers. For the bi-stability case, as only the oscillation around one of the stable equilibria is studied, the augment on the initial stretch ratio will lower the angular frequency.

Date of Award	11 Apr 2019
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	C W LIM (Supervisor) & Ling Hong (External Supervisor)