Nonlinear Dynamic Characteristics of PZT/Polymer Gripper

Abstract

A kind of PZT/polymer gripper is proposed in this paper and its nonlinear dynamic response in bounded noise is described. In this paper, a polymer piezoelectric material is applied in gripper to substitute the traditional PZT to improve the response range. Nonlinear differential items are introduced to interpret the hysteretic phenomena of the PZT/polymer piezoelectric composites material, and the nonlinear dynamic model of the PZT/polymer gripper in bounded noise is developed. The dynamic response of the system is obtained, and the bifurcation characteristics of the system are analyzed. The results of numerical simulation and experiments show that the stochastic noise intensity has important influence on the system's dynamical response, and the stochastic resonance phenomenon occurs with the stochastic noise intensity variety.

1. Introduction

Gripper is a basic kind of mechanical structure and used widely in industries of electronics, information technology, optics, medicine, and bio-technology. The specifications to realize such a gripper is to obtain quasi-static motion to have high accuracy in positioning, a large-stroke actuator to grasp the maximum types of object, and the use of electrostatic forces in order to avoid high power consumption.

Piezoelectric ceramics is a kind of smart material. It can be used to convert mechanical energy or electrical energy into another. But the hard and brittle characteristics make it difficult for forming process. It has bad shock resistance which restricts its application. Polymers as matrix materials offer several advantages, such as good electric insulation, high dielectric breakdown strength, low mechanical brittleness, and flexible manufacturability into complex shapes^1–2. Composite PZT and polymer materials, can overcome the shortcomings of the two, obtain low brittleness, low density and low dielectric coefficient, simple manufacturing process and low cost.

Many scholars have studied PZT/polymer piezoelectric composites materials. A kind of high-sensitivity passive magnetic transducer based on PZT plates and a Fe-Ni fork substrate was designed ³ . The connectivity of the ferroelectric phase in a composite microstructure plays a critical role in composite design and optimization ⁴ . PZT ceramics and their compositionally modified variants with soft and hard dopants, all having an MPB composition, have been exploited in many sensor and transducer applications ⁵ . Owing to its tailorable properties, a lot of research work have been carried out recently to investigate the effect of various parameters, namely ceramic/matrix type ⁶ and ceramic-polymer interface^7–8, in order to enhance the dielectric and piezoelectric properties of composites. This growing interest have opened up doors for a wide range of potential applications of piezoelectric composites^9–11.

Although much progress has been obtained, the theoretical results of the dynamic characteristics of PZT/polymer gripper are limited. Duo to the hysteretic nonlinear characteristics of PZT/polymer piezoelectric composites materials, PZT/polymer gripper has complex nonlinear dynamic characteristics, which cause the difficulty to control. In this current paper we aim to provide a kind of method to solve the nonlinear dynamical characteristics of PZT/polymer gripper.

2. Hysteresis Nonlinear Model of Pzt/Polymer Gripper

The strain-stress curves of PZT/polymer piezoelectric composites materials are shown in Figure 1 . Obviously, there is hysteretic nonlinearity. Most of the models are based on thermodynamics theory and micromechanics theory, where the percentage content of martensite is taken as main variable of stress-strain equation. As results, those models are mostly shown as equations with subsection function or double integral function, and hard to be analyzed in theory. Usually, research results to those models can only be obtained by numerical method or experiment method. In this paper, improved Van der Pol hysteretic model was introduced to describe the hysteretic nonlinear characteristics of PZT/polymer piezoelectric composites materials.

Figure 1

Strain-stress curves of PZT/polymer piezoelectric composites materials

In this paper, the Van der Pol hysteretic model used to describe the strain-stress curves is developed as follows:

$$\begin{array}{c}\sigma =a_1\epsilon +a_2{\epsilon }^2+a_3{\epsilon }^3(a_4\epsilon +a_5{\epsilon }^2+a_6{\epsilon }^3+a_7{\epsilon }^4\\ +a_8{\epsilon }^5+a_9{\epsilon }^6+a_{10}{\epsilon }^7+a_{11}{\epsilon }^9+a_{12}{\epsilon }^9+a_{13}{\epsilon }^{10})\dot{\epsilon }\end{array}$$

(1)

where σ is stress, ε is strain, a _i (i = 1 ∼ 12) are coefficients.

Partial least-square regression method is used to test the fitting effect of Eq. (1). The analyze result of principal component based on experimental data is shown in Figure 2 , and the values of the coefficients are shown in Figure 3 . We can see that all of the items are remarkable.

Figure 2

Analysis result of principal component

Figure 3

Values of the coefficients

The result of forecast test to Eq. (1) is shown in Figure 4 , where red line is real data and black line is forecast value. We can see that Eq. (1) can describe the real curve well.

Figure 4

Results of forecast test

3. Nonlinear Dynamic Characteristics of Pzt/Polymer Gripper

The structure of PZT/polymer gripper is shown in Figure 5 . The gripper is made up of PZT/polymer piezoelectric composites thin film, Si₃N₄ layer, and Si substrate. Gripping jaws are fabricated in Si substrate by erosion, Si₃N₄ layer is fabricated in Si substrate by nitriding, and PZT/polymer piezoelectric composites thin film is fabricated in Si₃N₄ layer by magnetron sputtering.

Figure 5

The structure of PZT/polymer gripper

According to Hamilton's principle, we obtained:

$${\displaystyle {\int }_{t_1}^{t_2}[\delta \left(T-U+W_e\right)+\delta W]dt}=0$$

(2)

where T is kinetic energy of structure, U is potential energy of structure, W _e is electric energy, W is work done by external force, $T=\frac{1}{2}{{\displaystyle \int }}^{\text{}}\rho \stackrel{.}{{\dot{u}}^{2}dV,U=\frac{1}{2}{{\displaystyle \int }}^{\text{}}\sigma \epsilon dV,W_e=\frac{1}{2}{{\displaystyle \int }}^{\text{}}EDdV,\delta W=\underset{0}{\overset{L}{{\displaystyle \int }}}\delta uFdx+\delta u(I,t)F+\delta \phi q.}$ ρ is density, V is volume, σ is stress, ε is strain, u = u(x, t) is deflection of composite cantilever beam, E is electric field intensity, D is electrostrictive displacement, F = e $\zeta$ (t) is axial stochastic excitation, $\phi$ is electric potential, q is electric charge.

The constitutive model of PZT/polymer piezoelectric composites materials can be shown as follows:

$$\{\begin{array}{l}\sigma =c_{11}^E\epsilon -e_{31}{E}_3-{\gamma }_{113}\epsilon E_3-\frac{1}{2}{\beta }_{133}{E}_3^2+c_{111}^E{\epsilon }^2\\ D_3={\epsilon }_{33}^s{E}_3+e_{31}\epsilon +\frac{1}{2}{\text{γ}}_{113}{\epsilon }^2+{\beta }_{133}\epsilon E_3+v_{333}{E}_3^2\end{array}$$

(3)

where C₁₁ ^E is stiffness coefficient, e₃₁ is piezoelectric coefficient, ε ^S ₃₃ is dielectric constant, γ₁₁₃ is electroelastic coefficient, β₁₁₃ is electrostrictive coefficient, C₁₁₁ ^E is nonlinear stiffness coefficient, v₃₃₃ is nonlinear dielectric constant.

Let u(x, t) = ψ(x) y(t), the dynamic equation of piezoelectric cantilever beam subjected to axial stochastic excitation can be solved from Eq. (2) as follows:

$$\begin{array}{c}\stackrel{\mathrm{..}}{y}+b_{1}y+b_3{y}^3+b_5{y}^5+\left(2\eta +{\displaystyle \sum _{i=6}^{13}{b}_i{y}^{i-5}}\right)\stackrel{.}{y}\\ =e\sin \left(\Omega t+\text{χ+}\sigma \text{B}\left(t\right)\right)\end{array}$$

(4)

where y is displacement, U is voltage, η is damping, b _i (i = 1 ∼ 13) are coefficients, e is intensity of stochastic excitation, ω is the center frequency, χ is the uniformly distributed random phase between (0, 2π), σ is the intensity of B(t), B(t) is the standard Wiener process. A noise with narrow band e sin (Ωt + χ + σB(t)) presents in Eq. (4). The excitation force in engineering is mostly harmonic excitation, but the disturbance of random factors can not be ignored. These two are coupled together to form a narrowband excitation. This form of excitation force is more in line with the actual situation.

If we let p = y A cos( $\omega \Theta$ ) and $q=\dot{y}=-A\omega \sin (\omega \Theta )$ , then we can obtain the Ito equation (4) using the stochastic average method as follows:

$$\{\begin{array}{l}dA=m_1(A,{\Delta }^{\prime })dt\\ d{\Delta }^{\prime }=m_2(A,{\Delta }^{\prime })dt+\sigma dB\left(t\right)\end{array}$$

(5)

where A, Δ’ are two-dimensional diffusion processes.

$$\begin{array}{c}{\Delta }^{\prime }=\Omega t+\sigma B\left(t\right)+\chi -\Theta \\ m_1\left(A,{\Delta }^{\prime }\right)=\pi (b_5{A}^2+\frac{1}{4}{b}_7{A}^4+\frac{1}{8}{b}_9{A}^6+\frac{5}{64}{b}_{11}{A}^8\\ +\frac{7}{128}{b}_{13}{A}^{10})-\frac{eA}{2\eta }\cos {\Delta }^{\prime }\\ m_2\left(A,{\Delta }^{\prime }\right)=2\pi \Omega -\frac{1}{\eta }\left(\frac{3}{4}\pi b_3{A}^3+\frac{5}{8}\pi b_5{A}^5-\frac{e}{2}\sin {\Delta }^{\prime }\right)\end{array}$$

The averaged Fokker-Planck-Kolmogolov (FPK) equation on the probability density f = f (A, Δ’, t) is as follows:

$$\frac{\partial f}{\partial t}=-\frac{\partial }{\partial A}\left(m_{1}f\right)-\frac{\partial }{\partial {\Delta }^{\prime }}\left(m_{2}f\right)+\frac{{\sigma }^2}{2}\frac{{\partial }^{2}f}{\partial {\Delta }^{r2}}$$

(6)

The initial condition:

$$f\left(A,{\Delta }^{\prime },t\right)=f\left(A_0,{\Delta }_0{}^{\prime }\right)$$

(7)

The boundary condition:

$$f=finite,A=0$$

(8)

$$f,\partial f/\partial A\to 0,A\to \infty$$

(9)

4. Numerical and Experimental Results

The numerical results of the system response are presented in Figures 6–8, where η = 0.6, b₃ = 0.3, b₅ = 0.1, b₇ = 0.15, b₉ = 0.05, b₁₁ = 0.005 and b₁₃ = 0.01.

Figure 6

The probability density when σ = 0.1

Figure 7

The probability density when σ = 0.4

Figure 8

The probability density when σ = 0.8

From Figure 6 , we can see that a high peak and a low cycle occur in the probability density, which means the system will stable at the origin point or move at the cycle. Obviously, the probability of staying at the origin point is bigger.

From Figure 7 , we can see that a cycle takes place of the peak and the outer cycle breaks in the probability density comparing with Figure 6 , which means the system will move at a certain cycle. Compare with Figure 6 , the peak is turning into a cycle and the probability value is reducing. This phenomenon means the system may be unstable and the system's instability is increasing with the increasing of the stochastic noise intensity.

From Figure 8 , we can see that the outer part is recycling again in the probability density, which means the system's stability may be enhance a little again.

Above all, we can obtain a conclusion as follows: the system will be stable when the stochastic noise intensity is weak; with the stochastic noise intensity increasing, the stable state of the system breaks and the instability is increasing; with the stochastic noise intensity further increasing, the stable state of the system enhances a little and the instability decreases. This phenomenon is called stochastic resonance phenomenon.

The experimental results of the gripper are shown in Figures 9–11, where the frequency ω = 50Hz. The vibration amplitude of system is shown as output voltage of sensor. The system's motion is periodic in the case of weak excitation, and becomes unstable with the increase of the excitation, finally becomes periodic again with the further increase of the excitation. The experimental results also indicate that stochastic resonance phenomenon occur when the stochastic noise intensity varies.

Figure 9

Response of the gripper when σ = 0.1

Figure 10

Response of the gripper when σ = 0.4

Figure 11

Response of the gripper when σ = 0.8

5. Conclusions

A kind of PZT/polymer gripper is proposed in this paper and its nonlinear dynamic response in bounded noise magnetic fields is described. The results of numerical simulation and experiments show that the stochastic noise intensity has important influence on the system's dynamical response, and the stochastic resonance phenomenon occurs with the stochastic noise intensity variety. This phenomenon can increase the amplitude of system flutter and reduce the movement accuracy of the gripper.