Nonlinear Response Characteristics of Electrostrictive Material Sensors

In this study, the nonlinear response characteristics of electrostrictive material sensors subjected to stochastic excitation were investigated. Van der Pol nonlinear differential items were developed to interpret the hysteretic phenomena of electrostrictive materials. The hysteretic constitutive models of electrostrictive materials were proposed, and the nonlinear dynamic model of electrostrictive material sensors subjected to stochastic excitation was developed. The expression of the dynamic response of the system was obtained, and the bifurcation conditions and chaos characteristics of the system were determined. Finally, the effects of system parameters on the dynamic characteristics of the system were analyzed. The simulation results show that there were bifurcation phenomena in the system, which can be suppressed by adjusting the parameters. The stochastic noise intensity had an important effect on the system dynamical response, and the stochastic resonance phenomenon occured with the variation in stochastic noise intensity. The results of this study were helpful for the optimal design and improvement of electrostrictive material sensors.


Introduction
Electrostrictive materials are kinds of smart materials that can convert electrical and mechanical energies into each other. They have the advantages of high conversion efficiency, high energy density, large electrodeformation, small viscoelastic lag, and high response speed. (1)(2)(3) They have great potential applications in microactuators, artificial muscles, and robots. A highly sensitive sensor is an important component in modern electronic products. An electrostrictive material sensor is a new kind of sensor. It has high accuracy and self-generating characteristics, and is used widely where the power supply is very difficult to be obtained.
Many researchers have studied the characteristics of electrostrictive materials. Kuang performed a thorough study of electroelastic and magnetoelastic solids by using the variational principle. (4) The electrostrictive constitutive equation, electric field volume force, and mechanical boundary conditions were discussed in detail. (4) The axisymmetric stresses in hollow electrostrictive cylinders were discussed by Jiang and Gao. (5) In 2006, Beom et al. studied the gradual formation of the insulation crack tip in electrostrictive materials. (6) Sunder and Newnham determined the electrostrictive constants of low-dielectric-constant materials. (7) Cheng et al. studied the electrostrictive properties of copolymers. (8) Della Schiava et al. discussed the enhanced figures of merit for a high-performing actuator in electrostrictive materials. (9) The effects of interface diffusion on the strain and stress stability of particulate reinforced electrostrictive materials were determined by Li and Wang. (10) Although many achievements in electrostrictive materials were reported in previous years, theoretical results on the dynamic characteristics of electrostrictive material sensors are few. The nonlinear response characteristics of electrostrictive material sensors subjected to stochastic excitation are examined in this study. Van der Pol nonlinear differential items are developed to interpret the hysteretic phenomena of electrostrictive materials. The hysteretic constitutive models of electrostrictive materials are proposed, and the nonlinear dynamic model of electrostrictive material sensors subjected to stochastic excitation is developed.

Hysteretic Nonlinear Model of Electrostrictive Material Sensors
The strain-stress curve of electrostrictive materials is shown in Fig. 1. Obviously, hysteretic phenomena are observed in the curve. A new differential item is introduced to describe the hysteretic phenomena as In Eq. (1), only the hysteretic characteristic of the strain-stress curve of electrostrictive materials is considered. In this study, a new nonlinear differential item, which is developed from a Van der Pol differential item, is introduced to describe the hysteretic nonlinear characteristics of the strain-stress curve of electrostrictive materials in a differentiable function so as to obtain more theoretical results. The partial least-squares regression software SIMCA-P is used to determine the relationships among strain and stress. The partial least-squares regression method reduces the sum of squared errors to find the best match of data and is usually used for curve fitting.
The model developed to describe the strain-stress curve of electrostrictive materials indicates that the final relationship between stress and strain is The result of the forecast test to Eq. (1) is shown in Fig. 2, where the red line represents the real data and the black line represents the forecast value. Equation (1) can describe the real curve well.

Dynamic Characteristics of the System
Considering the complex characteristics of the materials, we introduce Hamilton's principle to the dynamic modeling of the system. The Hamilton function can be presented as where T is the kinetic energy, . Moreover, ρ is the density, V is the volume, E is the elastic modulus, and A is the cross-sectional area. According to Hamilton's principle, Thus, the nonlinear dynamic model of the system can be shown as Then, the dynamic equation of system response can be solved from Eq. (5) by the Galerkin's method as (2 ) ( ) q c q c q c q c q q eq t η ς where 2 c m η = , , e is the intensity of stochastic excitation, and ς(t) is the Guass white noise, whose intensity is 2D.
We can obtain the Ito equation corresponding to Eq. (6) using the stochastic average method as where B(t) is the standard Wiener process and m(H) and ( ) H σ are the drift and diffusion coefficients, respectively.    The vibration amplitude of the system is shown as the output voltage. As shown by the figures, the stochastic noise intensity has an important effect on the system dynamical response. When the intensity of stochastic excitation is low, it is slightly effective for vibrating; with the increase in the intensity of stochastic excitation, the system output increases. However, with the further increase in the intensity of stochastic excitation, the system vibration decreases again. The system vibration increases significantly when the intensity of stochastic excitation increases. This phenomenon is called stochastic resonance.

Conclusions
The nonlinear dynamic response of electrostrictive material sensors in stochastic excitation is described in this paper. Nonlinear differential items are introduced to interpret the hysteretic phenomena of electrostrictive materials, and the nonlinear dynamic model of electrostrictive material sensors is developed. The dynamic response of the system is obtained. The results of numerical experiments show that the stochastic noise intensity has an important effect on the system dynamical response. When the stochastic noise intensity reaches a certain value, stochastic resonance occurs. The system safety will be strengthened by a control strategy. The results of this paper are helpful for the application of electrostrictive material sensors in engineering fields.