Nonlinear dynamic behaviors and PID control of viscoelastic dielectric elastomer balloons

Abstract

As a type of intelligent electroactive polymer, dielectric elastomer (DE) exhibits viscoelastic properties. It’s worth pointing out that the relaxation time has great significance for studying the mechanical behavior of viscoelastic polymer. In this paper, a generalized Maxwell model is used to describe the viscoelastic property of dielectric elastomer balloon. Meanwhile, a theoretical model with multiple relaxation times is used and the natural frequency of small amplitude oscillation is derived. Subsequently, the model is validated by comparing with experimental results. The model with double relaxation times can describe the deformation of the dielectric elastomer balloon effectively. Then the effect of relaxation time and shear modulus on the dynamic response of DE balloon is studied. Furthermore, the dielectric elastomer balloons in practical application exhibit the strong nonlinearity and the viscoelastic dissipation. Therefore, it is important to precisely control the dynamic response. The proportional-integral-differential (PID) controller in the form of nonlinear combination is adopted to control the above nonlinear dynamic systems actively. The results indicate that it is feasible to achieve desired control effect.

Keywords

Dielectric elastomer balloon viscoelastic relaxation nonlinear dynamic response active control

1. Introduction

As a kind of intelligent material called artificial muscle, dielectric elastomers (DE) can generate electrically induced large deformation, which have gradually attracted the attention of researchers (Pelrine et al., 2000a, 2000b; Rus and Tolley, 2015; Smela et al., 1995). The dielectric elastomers are able to transform electrical energy into mechanical energy. In addition, they also have characteristics of small density, large deformation, and fast response. The dielectric elastomers can be widely applied in some fields such as energy converters, flexible robot, and sensors (Christianson et al., 2018; Gu et al., 2017b; Heydt et al., 2000). As an unique structure actuator, dielectric elastomer balloons have several practical applications, such as high-frequency pumps (Cao et al., 2019), loudspeakers (Hosoya et al., 2019), and vibrotactile display (Lee et al., 2014).

In recent decades, a great deal of experiments and theoretical works on the mechanical behavior of dielectric elastomers have been carried out. Pelrine et al. (1998) revealed the mechanical driving mechanism of the dielectric elastomers through experiments. Toupin (1956) firstly proposed the nonlinear theory of elastic dielectric. Dorfmann and Ogden (2005) derived the governing equations by using the nonlinear electroelastic theory. Then, Suo’s group (Suo, 2010; Suo et al., 2008; Zhao and Suo, 2008; Zhao et al., 2007) proposed energy approach to establish the constitutive relationship of dielectric elastomers, which describes the deformable dielectric by means of continuum mechanics and thermodynamics. Considering the viscoelasticity of the dielectric elastomer, Zhao et al. (2011) established a dissipation model of dielectric elastomers with Neo-Hookean model based on the theory of non-equilibrium thermodynamics. Hong (2011) proposed a method for studying the electro-viscoelastic behavior of the deformable dielectric based on non-equilibrium thermodynamics. Zhou et al. (2018) developed a micro–macro constitutive model for finite-deformation viscoelasticity of elastomers considering nonlinear viscosity.

The stability has always been an important factor limiting the application of the dielectric elastomers. Zhao and Suo (2007) analyzed electromechanical stability of dielectric elastomers by Hessian matrix of the free energy function. Li et al. (2011) and Wang et al. (2020b) investigated the effect of pre-stretch on the pull-in instability of dielectric elastomer membrane. Kollosche et al. (2015) studied the effects of pre-stretch and loading rates on the wrinkles instability of dielectric elastomers by experiments. Yong et al. (2012) and He et al. (2011) analyzed the electromechanical stability in anisotropic dielectric elastomers and a dielectric elastomer spherical shell. Godaba et al. (2019) investigated the instabilities induced by the voltage of dielectric elastomer membrane, such as wrinkling, buckling, and crumpling.

The dynamic behavior of dielectric elastomers is also an important research topic. Fox and Goulbourne (2008) used experimental method to describe the dynamic behavior of dielectric elastomer drivers. Zhu et al. (2010a, 2010b) studied the nonlinear vibration behavior of dielectric elastomer balloons under periodic excitation and the resonance behavior of dielectric elastomer by perturbation method. Xu et al. (2011) proposed an analytical model for the dynamic analysis of dielectric elastomer actuator by the Euler-Lagrange equation. Yong et al. (2011) described the dynamic behavior of the thick-walled dielectric elastomer spherical shell. Chiang Foo et al. (2012) developed a viscoelastic model to predict the dynamic response of the dielectric elastomer by the Gent model and compared it with experiments. Gu et al. (2017a) studied the creep and hysteresis of dielectric elastomer subjected to cyclic voltage by experiment and simulation. Liu et al. (2018) utilized the shooting method and arc length continuation method to capture the bifurcation behavior and jump phenomenon of the nonlinear oscillation of dielectric elastomers. Zhang et al. (2017a) studied viscoelastic creep and relaxation behavior of dielectric elastomers using the Kelvin-Voigt-Maxwell model. Zhang et al. (2017b) analyzed the effect of the geometrical size of viscoelastic dielectric elastomer membrane on the dynamic response by generalized Maxwell model, which is proposed by Khan et al. (2013). In addition, Lv et al. (2018) investigated on the dynamic performance of dielectric elastomer balloons incorporating stiffening and damping effect by modeling the viscoelastic effect as damping force. Li et al. (2019) studied the dynamic performance of viscoelastic dielectric elastomers considering nonlinear material viscosity. Jin and Huang (2017) investigated the random oscillation around the stable state of equilibrium of an idealized DE balloon subjected to random perturbation of voltage or pressure. In addition, many researchers also paid attention to the viscoelastic and dynamic modeling in the dielectric elastomers (Kashyap et al., 2020; Khurana et al., 2021; Patra and Sahu, 2015; Wang et al., 2013; Yarali et al., 2020).

Due to the advantages of large deformation, high flexibility, and short response time, dielectric elastomers are gradually used in the control components of the actuators (Godaba et al., 2016; Gupta et al., 2019; Lau et al., 2017; Rizzello et al., 2016a; Zhang et al., 2018). Moreover, because of the influence of nonlinear dynamic response and viscoelastic dissipation of dielectric elastomers in engineering applications, it is important to accurately control the dynamic response of materials (Chen et al., 2020; Jones and Sarban, 2016; O’Halloran et al., 2008; Tian et al., 2018; Wang et al., 2020a). The soft wall-climbing robot which is controlled by a combination of dielectric elastomer artificial muscles and electrostatic feet was developed by Gu et al. (2018). Papaspiridis and Antoniadis (2008) used the active control strategy to control the dynamic system composed of dielectric elastomers to obtain the desired output. Rizzello et al. (2015) designed a nonlinear PID controller by using a direct loop-shaping robust control design approach and a Linear Parameter Varying (LPV) Controller in combination with a Linear Matrix Inequality (LMI) algorithm (Rizzello et al., 2016b) for the control of the bias viscoelastic DE actuators. Poulin and Rosset (2019) proposed an open-loop method to increase the response speed and suppress the viscoelastic creep of dielectric elastomer actuators based on a quasi-linear viscoelastic model. Li et al. (2018) adopted a linear proportion integral differential (PID) closed-loop controller to eliminate the nonlinear response of the dielectric elastomer membrane.

It has been found experimentally (Plante and Dubowsky, 2007) that the DEs are highly viscoelastic, and the viscoelastic elastomer possess multiple relaxation times. This behavior can be incorporated in the model by introducing more parallel units of springs and dashpots (Chiang Foo et al., 2012). In this paper we use the generalized Maxwell model with double relaxation times to show the evolution process of the beating phenomena and stability. In order to further understand the underlying mechanisms of dynamic relaxation in dielectric elastomer balloons and describe the dynamic behavior of DE materials more accurately, the effects of equilibrium modulus and relaxation times on the time response of DE balloons are studied. The results show that the multiple relaxation elements in dielectric elastomers will interact with each other. In addition, the Gent model is used to describe the strain hardening effect (Boyce and Arruda, 2000; Wang et al., 2016; Zhao and Suo, 2010). Furthermore, based on the approach by Li et al. (2018), we adopt the PID controller of nonlinear combination form to actively control the dynamic behavior of dielectric elastomer balloons. The results show that the PID controller of nonlinear combination form can achieve desired control effect.

2. Governing equations

Figure 1 shows the deformation of a balloon made of dielectric elastomer film, where the mass density is $ρ$ . In the reference state, the radius is $R$ , and the thickness is $H$ . When it is subjected to internal pressure $ρ$ and external voltage φ, the radius and thickness are changed to $r$ and $h$ , respectively.

The deformation gradient $F$ of the dielectric elastomer balloon is expressed as

F = (\begin{matrix} λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix})

(1)

where $λ_{1}$ , $λ_{2}$ , $λ_{3}$ are the principal stretches of DE balloons. For the incompressible and ideal dielectric elastomer balloons, the hoop stretch of the balloon film along the radius is

λ = \frac{r}{R}

(2)

The volume of the balloon can be approximated as

V = 4 π R^{2} H

(3)

Since the volume of DE balloon is unchanged, namely, $4 π R^{2} H = 4 π r^{2} h$ , we have

h = \frac{H}{λ^{2}}

(4)

Thus, the deformation gradient $F$ satisfies $λ_{1} = λ_{2} = λ$ , $λ_{3} = λ_{1}^{- 1} λ_{2}^{- 1} = λ^{- 2}$ in equation (1).

According to the viscoelasticity theory of finite deformation and non-equilibrium thermodynamics (Chiang Foo et al., 2012; Hong, 2011; Lee, 1969), the deformation gradient between the reference state and the intermediate state is donated as $F^{i}$ , and that of the current state with respect to the intermediate state is donated as $F^{e}$ . Therefore, the actual deformation gradient $F$ can be multiplicatively decomposed into

F = F^{e} F^{i}

(5)

in which the deformation gradient of the elastic part $F^{e}$ is

F^{e} = (\begin{matrix} λ_{1}^{e} & 0 & 0 \\ 0 & λ_{2}^{e} & 0 \\ 0 & 0 & λ_{3}^{e} \end{matrix})

(6)

and the deformation gradient of the inelastic part $F^{i}$ is

F^{i} = (\begin{matrix} λ_{1}^{i} & 0 & 0 \\ 0 & λ_{2}^{i} & 0 \\ 0 & 0 & λ_{3}^{i} \end{matrix})

(7)

Note that the superscript “ $e$ ,”“ $i$ ” represent the elastic and inelastic parts. Similarly, deformation gradient $F^{e}$ and $F^{i}$ of the elastic part and the inelastic part satisfy $λ_{1}^{e} = λ_{2}^{e} = λ^{e}$ , and $λ_{1}^{i} = λ_{2}^{i} = λ^{i}$ respectively.

It is assumed that the deformation of dielectric elastomer balloon is in isothermal condition, and thus the balloon is a thermodynamic system which can be described by the Helmholtz free energy density. The total Helmholtz free energy can be described as (Hong, 2011)

W = W^{EQ} (λ, D) + W^{NEQ} (λ^{e})

(8)

where the equilibrium Helmholtz free energy density $W^{EQ}$ is determined by the applied voltage and the total deformation, and the non-equilibrium Helmholtz free energy density $W^{NEQ}$ is just a function of elastic deformation. For an ideal dielectric elastomer, the relationship between the electric displacement and the electric field is $D = ε E$ in current state (Suo, 2010). Therefore, the equilibrium Helmholtz free energy is written as (Suo, 2010):

W^{EQ} = W_{s} + \frac{D^{2}}{2 ε}

(9)

where the $D$ represents the electrical displacement of the membrane in the current state, that is,

D = \frac{Q}{4 π r^{2}}

(10)

$W_{s}$ is the strain energy density function which is related to the total deformation, and $ε$ is the permittivity. According to the works of Zhao et al. (2011) and Hong (2011), $W_{s}$ and $W^{NEQ}$ have the similar strain energy function form.

To describe the viscoelastic behavior of dielectric elastomer materials, Hong (2011) and Chiang Foo et al. (2012) chose the standard linear solid model which consists of two parallel elements, that is, one element consists of a spring and a dashpot, and the other element includes one spring. By utilizing this rheological model, viscoelastic relaxation could be achieved. However, this model contains only one relaxation element, while the elastomers have multiple relaxation times (Chiang Foo et al., 2012). Thus, more parallel elements need to be used to describe the viscoelastic properties of dielectric elastomer. A generalized Maxwell model with multiple relaxation elements is used to describe the viscoelastic dielectric elastomer balloons, as shown in Figure 2. In this model, the deformation is characterized by $λ$ for the spring $α$ . For the nonlinear spring $β_{m}$ (Chiang Foo et al., 2012; Hong, 2011; Li et al., 2018), the deformation is characterized by a different stretch $λ_{m}^{e}$ .

By taking into account strain hardening, Gent model (Gent, 1996) is adopted to describe the strain energy density of the dielectric elastomer balloon.

W_{s} = - \frac{μ^{EQ} J^{EQ}}{2} \ln (1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}})

(11)

W^{NEQ} = - \sum_{m = 1}^{n} \frac{μ_{m}^{NEQ} J_{m}^{NEQ}}{2} \ln (1 - \frac{2 {(λ_{m}^{e})}^{2} + {(λ_{m}^{e})}^{- 4} - 3}{J_{m}^{NEQ}})

(12)

When a small disturbance is given in the current state of dielectric elastomer membrane, the change of the free energy of the membrane is equal to the sum of the work done by the pressure, inertial forces, and the electric field force (Zhu et al., 2010a), that is,

4 π R^{2} H δ W = ϕ δ Q + 4 π r^{2} p δ r - 4 π R^{2} H ρ \frac{d^{2} r}{d t^{2}} δ r

(13)

By substituting equations (2), (4), (8), (9), and (10) into (13), we can obtain:

\begin{matrix} \frac{\partial W}{\partial λ} = \frac{2 ϕ D λ}{H} + \frac{pR}{H} λ^{2} - R^{2} ρ \frac{d^{2} λ}{d t^{2}} = \frac{2 μ^{EQ} (λ - λ^{- 5})}{1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}}} \\ + \sum_{m = 1}^{n} \frac{2 μ_{m}^{NEQ} (λ {(λ_{m}^{i})}^{- 2} - λ^{- 5} {(λ_{m}^{i})}^{4})}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}} \end{matrix}

(14)

\frac{\partial W}{\partial D} = \frac{ϕ}{H} λ^{2} = \frac{D}{ε} = E

(15)

Substituting equation (15) into (14) to obtain

\begin{matrix} \frac{2 ε ϕ^{2} λ^{3}}{H^{2}} + \frac{pR}{H} λ^{2} - R^{2} ρ \frac{d^{2} λ}{d t^{2}} = \frac{2 μ^{EQ} (λ - λ^{- 5})}{1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}}} \\ + \sum_{m = 1}^{n} \frac{2 μ_{m}^{NEQ} (λ {(λ_{m}^{i})}^{- 2} - λ^{- 5} {(λ_{m}^{i})}^{4})}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}} \end{matrix}

(16)

When the last term in equation (16) is removed, the motion equation of the hyper-elastic DE balloon without damping is obtained, which is consistent with the previous works (Liu and Zhou, 2018; Zhu et al., 2010a). Meanwhile, the general trend of the time response of DE balloons is also in agreement with the work (Liu and Zhou, 2018). The inelastic elongation in equation (14) must satisfy the thermodynamic evolution law (Li et al., 2020; Zhou et al., 2015), which is expressed as

- \frac{1}{2} F \frac{d [{(C^{i})}^{- 1}]}{dt} F^{T} \cdot (b^{e})^{- 1} = γ^{- 1} : σ^{NEQ}

(17)

where $C^{i} = (F^{i})^{T} F^{i}$ , $b^{e} = F^{e} (F^{e})^{T}$ , $σ^{NEQ} = 2 F^{e} \frac{\partial W^{NEQ}}{\partial C^{e}} (F^{e})^{T}$ , $C^{e} = (F^{e})^{T} F^{e}$ . $γ^{- 1}$ is an isotropic fourth-order modified tensor, which can be written as

γ^{- 1} = \frac{1}{2 η_{v}} (I^{4} - \frac{1}{3} I \otimes I)

(18)

where $η_{ν}$ is the shear viscosity, $I$ is the second-order identity tensor, and $I^{4}$ is the fourth-order symmetric identity tensor.

Equation (17) is expanded as,

\frac{d λ_{m}^{i}}{dt} \cdot \frac{1}{λ_{m}^{i}} = \frac{μ_{m}^{NEQ}}{3 η_{vm}} \frac{λ^{2} {(λ_{m}^{i})}^{- 2} - λ^{- 4} {(λ_{m}^{i})}^{4}}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{- 4} - 3}{J_{m}^{NEQ}}}

(19)

From the above equations, the governing equations (16) and (19) can be used to describe the dynamic characteristics of the viscoelastic dielectric elastomer balloons considering damping effect. According to the previous works (Zhang et al., 2014), when the inertia term of the equation (16) vanishes, the constitutive relation of the viscoelastic DE with pre-stretch $λ_{p}$ is expressed as

\begin{matrix} \frac{μ^{EQ} (λ^{2} - λ^{- 4})}{1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}}} + \sum_{m = 1}^{n} \frac{μ_{m}^{NEQ} (λ^{2} {(λ_{m}^{i})}^{- 2} - λ^{- 4} {(λ_{m}^{i})}^{4})}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}} \\ - \frac{ε ϕ^{2} λ^{4}}{H^{2}} \\ - \frac{μ^{EQ} (λ_{p} - λ_{p}^{- 5})}{1 - \frac{2 λ_{p}^{2} + λ_{p}^{- 4} - 3}{J^{EQ}}} λ = 0 \end{matrix}

(20)

Then we introduce dimensionless parameters as $T = t / R \sqrt{ρ / μ}$ with $μ = μ^{EQ} + \sum_{m = 1}^{n} μ_{m}^{NEQ}$ , and ${\tilde{τ}}_{m} = τ_{m} / R \sqrt{ρ / μ}$ is the dimensionless relaxation time, where $τ_{m} = η_{vm} / μ_{m}^{NEQ}$ is the viscoelastic relaxation time. In the following analysis, the units of the axes in figures are 1 unless otherwise specified. Thus, the dimensionless governing equation can be expressed as:

\begin{matrix} \frac{d^{2} λ}{d T^{2}} = \frac{2 ε ϕ^{2} λ^{3}}{μ H^{2}} + \frac{pR}{μ H} λ^{2} - \frac{μ^{EQ}}{μ} \frac{2 (λ - λ^{- 5})}{1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}}} \\ - \sum_{m = 1}^{n} \frac{μ_{m}^{NEQ}}{μ} \frac{2 (λ {(λ_{m}^{i})}^{- 2} - λ^{- 5} {(λ_{m}^{i})}^{4})}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}} \end{matrix}

(21)

\frac{d λ_{m}^{i}}{dT} = \frac{λ_{m}^{i}}{3 {\tilde{τ}}_{m}} \frac{{(λ)}^{2} {(λ_{m}^{i})}^{- 2} - λ^{- 4} {(λ_{m}^{i})}^{4}}{1 - \frac{2 {(λ)}^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}}

(22)

3. Equilibrium state and natural frequency

The generalized Maxwell model contains n Maxwell elements, and we only consider two Maxwell elements. When the dielectric elastomer balloon is only affected by internal pressure or static voltage, the balloon will reach static equilibrium after the viscoelastic relaxation. Then, we rewrite equation (21) as

\frac{d^{2} λ}{d T^{2}} + f (λ, λ_{m}^{i}, p, ϕ) = 0

(23)

\begin{matrix} f (λ, λ_{m}^{i}, p, ϕ) = - \frac{2 ε ϕ^{2} λ^{3}}{μ H^{2}} - \frac{pR}{μ H} λ^{2} + \frac{μ^{EQ}}{μ} \frac{2 (λ - λ^{- 5})}{1 - \frac{2 λ^{2} + λ^{- 4} - 3}{J^{EQ}}} \\ + \sum_{m = 1}^{n} \frac{μ_{m}^{NEQ}}{μ} \frac{2 (λ {(λ_{m}^{i})}^{- 2} - λ^{- 5} {(λ_{m}^{i})}^{4})}{1 - \frac{2 λ^{2} {(λ_{m}^{i})}^{- 2} + λ^{- 4} {(λ_{m}^{i})}^{4} - 3}{J_{m}^{NEQ}}} = 0 \end{matrix}

(24)

By solving equations (22) and (24), the stretch of the dielectric elastomer balloon at equilibrium state is obtained. Moreover, when the balloon is in equilibrium, a slight disturbance is applied at time $\tilde{T}$ , which can be expressed as:

λ (\tilde{T}) = λ_{eq} (\tilde{T}) + Δ (\tilde{T})

(25)

where $λ_{eq}$ also satisfies the equation (23), that is

\frac{d^{2} Δ}{d T^{2}} + \frac{\partial f}{\partial λ} (λ, λ_{m}^{i}, p, ϕ) | λ = λ_{eq} Δ = 0

(26)

According to the work of Zhu et al. (2010a), the dimensionless natural frequency of small amplitude oscillation of dielectric elastomer balloon with multiple relaxation times is

{\tilde{ω}}_{n} = \sqrt{\frac{\partial f}{\partial λ} (λ_{eq}, λ_{m}^{i}, p, ϕ)}

(27)

4. Nonlinear dynamic analysis

4.1. Model validation

Firstly, we compare the simulations results with the experiment to verify our model. Combined with the equations (19) and (20), the simulation results and the experimental results (Zhang et al., 2017a) are compared and the fitting parameters are obtained. In order to better characterize the experimental results, Zhang et al. (2017a) used the Kelvin-Voigt-Maxwell (KVM) model, where the inelastic terms are described by two dashpots. However, we use the generalized Maxwell model with double relaxation times, and the parameters of single relaxation time and double relaxation times obtained by fitting experiment are given in Table 1. As shown in Table 1, for the model with single relaxation time, the parameters are consistent with the materials parameters reported in literature for VHB 4910 (Zhang et al., 2014, 2017a). For the model with double relaxation times, the parameters are also within a reasonable range.

Table 1.

Fitting parameters obtained by the experimental results (the parameters of single relaxation time are given by Zhang et al. (2017a)).

	$μ^{EQ}$ (kPa)	$μ_{1}^{NEQ}$ (kPa)	$μ_{2}^{NEQ}$ (kPa)	$J^{EQ}$ (1)	$J_{1}^{NEQ}$ (1)	$J_{2}^{NEQ}$ (1)	$τ_{1}$ (s)	$τ_{2}$ (s)
Single	16	45	—	115	70	—	200	—
Double	20	120	60	152	152	152	15	10

As shown in Figure 3(a), compared with the single relaxation time, the simulation results with double relaxation times fit the experimental data better, particularly in the initial jumping process. In addition, except for the initial moment, the absolute value of relative error for double relaxation times in Figure 3(b) is below 2%. The slight difference between the theoretical modeling and the experiment may be caused by the inhomogeneous deformation at boundary. The comparison also shows the validity of the model with double relaxation times.

4.2. Effect of shear modulus on the time response of DE balloons

In this section, the effects of the shear modulus on dynamic responses of the dielectric elastomer balloon are studied. For the dielectric elastomer balloon, the fixed internal pressure is $pR / μ H$ , and a DC voltage and a sinusoidal AC voltage $ϕ (T) = ϕ_{DC} + ϕ_{AC} \sin (\tilde{ω} T)$ are applied to the dielectric elastomer balloon, where $ϕ_{dc}$ is the DC voltage, and $ϕ_{ac}$ is the AC voltage. All the numerical examples can be solved by the Runge-Kutta method.

The parameters obtained from the fitting experimental data are set as: $μ^{EQ} / μ = 0.1$ , $μ_{1}^{NEQ} / μ = 0.6$ , $μ_{2}^{NEQ} / μ = 0.3$ , $J^{EQ} = 152$ , $J^{NEQ} = 152$ . The fixed internal pressure $pR / μ H = 0.02$ , voltage $ε ϕ_{dc}^{2} / μ H^{2} = 0.02$ , $ϕ_{ac} / ϕ_{dc} = 0.5$ , the excitation frequency $\tilde{ω} = ω_{n} = 3.1778$ and initial conditions $λ (0) = λ^{i} (0) = 1.071$ , $d λ (0) / dT = 0$ are applied. When we set $R = 0.3$ $m$ , $ρ = 1200$ $kg / m^{3}$ and the relaxation times are $τ_{1} = 15$ s, $τ_{2} = 10$ s, we can get the dimensionless relaxation times ${\tilde{τ}}_{1} = 645, {\tilde{τ}}_{2} = 430$ . The time responses of the DE balloons with double relaxation times are shown in Figure 4(a). It can be found that time response of DE balloon takes a long beating process. In order to show the periodicity and stability of the time response more clearly, only the phase diagrams and Poincare’ maps of the stable time interval are shown in Figure 4(b). As shown in Figure 4(b), phase diagram forms a closed curve, and the Poincare’ maps can aggregate into a point. It reveals that the DE balloon experiences a stable periodic vibration.

In order to investigate the influence of shear modulus on the time-dependent response of the DE balloons, we change the equilibrium shear modulus into $μ^{EQ} / μ = 1 / 3$ and $μ_{1}^{NEQ} / μ = μ_{2}^{NEQ} / μ = 1 / 3$ . The other parameters remain the same as in the Figure 4. Figure 5(a) shows the time response of DE balloons with double relaxation times when the equilibrium shear modulus is increased. It is evident that the amplitude and duration of the beating phenomenon increase with the increase of the equilibrium shear modulus. As shown in Figure 5(b), after a period of beating, the time response of DE balloons becomes stable, which is consistent with Figure 4(b).

4.3. Effect of relaxation time on the time response of DE balloons

In this section, the effect of relaxation time on the dynamic evolution process of DE balloon is considered. Here, in order to show the whole evolution process of dynamic response, another set of parameters is selected. First, the single relaxation time is studied, and the relaxation time is set as $τ = 0.63$ s. The dimensionless relaxation time is $\tilde{τ} = τ / R \sqrt{ρ / μ} = 15$ in Figure 6. We choose the same parameters: $μ^{EQ} / μ = 0.5$ , $J^{EQ} = J^{NEQ} = 100$ as in Zhang et al. (2017a). We set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 0.5$ , the excitation frequency $\tilde{ω} = ω_{n} = 3.152$ and initial conditions $λ (0) = λ^{i} (0) = 1.041$ , $d λ (0) / dT = 0$ .

By comparing with the previous work (Liu and Zhou, 2018) about undamped case, the beating phenomenon will gradually disappear with time in Figure 6(a). Phase diagram (blue line) gradually forms a closed curve, and the Poincare’ maps (red points) aggregate into a point (green point), as shown in Figure 6(b). It indicates that the dynamic process will evolve from non-periodic vibration to periodic vibration.

To be more accurately describe the viscoelastic dissipation behavior of the DE balloon, the dynamic response of DE balloon with double relaxation times is studied. We choose the same parameters used by the reference (Zhang et al., 2017b): $μ^{EQ} / μ = μ_{m}^{NEQ} / μ = 1 / 3$ , $J^{EQ} = J_{m}^{NEQ} = 100$ $(m = 1, 2)$ . We set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 0.5$ , excitation frequency $\tilde{ω} = ω_{n} = 3.053$ , and initial conditions: $λ (0) = 1.07$ , $λ_{1}^{i} (0) = λ_{2}^{i} (0) = 1.07$ , $d λ (0) / dT = 0$ . The dynamics response of DE balloons with double relaxation times is given in Figure 7. By comparing with Figure 6(a), the dynamics responses of DE balloons show obvious difference when the second relaxation time is considered. The amplitude in beating phenomenon will increase with the second relaxation time increase. In addition, it can be found that with the increase of the second relaxation time, the duration of beating phenomenon lasts longer when the first relaxation time is fixed. However, when the second relaxation time reaches a higher value ${\tilde{τ}}_{2} = 100$ ( $τ_{2} = 2.4$ s), the duration of the beating no longer increases as shown in Figure 7(b) and (c). Then, we increase the first relaxation time, and the duration of beating continues to increase as shown in Figure 7(d). Hence it is reasonable to note that the multiple relaxation elements in dielectric elastomers will interact with each other.

In the Figure 8, it can be seen that the phase diagram (blue line) gradually forms a closed curve, and the Poincare’ maps (red points) converge to a point. The results indicate that the vibration will change from non-periodic vibration to periodic vibration. In addition, it also can be found that the similar evolution in Figure 8(b) and (c). The results are consistent with the time responses of Figure 7(b) and (c). Furthermore, there is a longer path to converge to a point with the longer relaxation time in the Poincare’ maps.

Then the amplitude frequency curves by taking the difference (Amp) between the maximum stretch and the minimum stretch of the steady state region as a function of excitation frequency ( $\tilde{ω}$ ) are shown in Figure 9. In order to show the effect of relaxation time on the frequency response clearly, different double relaxation times magnitudes with Figure 7 are used. In addition, for better comparison with the amplitude-frequency response of single relaxation time, we kept one of the double relaxation times consistent with the single relaxation time.

As illustrated in Figure 9(a) and (b), the primary resonance occurs near the natural frequency and the peak of the primary resonance increases with the increase of relaxation time. In addition, when the excitation frequency is very small, the amplitude-frequency curve will jump at a small excitation frequency. However, the peak of the jump will be eliminated by the larger relaxation time. In addition, sub-harmonic resonance occurs around the twice natural frequency and there are also obvious second and third order super-harmonic resonance phenomena at about the half and a third of the natural frequency. As shown in Figure 9(c), the amplitude-frequency curve shifts to the left when we add the second relaxation time and the amplitudes of the harmonic oscillations (at about $\tilde{ω} = 3$ ) decrease as the second relaxation time is added. However, the amplitudes of the super-harmonic oscillations (at about $\tilde{ω} = 1.5$ ) increase as the second relaxation time increases.

Afterward, we study the dynamic response of DE balloon by taking super-harmonic frequency, harmonic frequency, and sub-harmonic frequency from Figure 9(b) as the excitation frequency. As shown in Figure 10, when the double relaxation times are considered, the DE balloon shows the super harmonic response, harmonic response, and sub harmonic response, respectively. The $Φ$ is dimensionless voltage.

5. Nonlinear PID active control of dielectric elastomer balloons vibration

In practical applications, the mechanical response of the dielectric elastomer balloons is complicated due to their nonlinearity and viscoelastic dissipation. Thus, the active control is usually used to get the desired results (Li et al., 2018). As the first developed control strategy, the PID controller is widely used in process control and motion control owing to its simple practical algorithm and high reliability. The traditional PID controller takes the difference between the given value and the actual output value as the control deviation. Then, the present (P), past (I), and future (D) of the control deviation are substituted into the control law as expressed in equation (28) to control the controlled object (Liu, 2016).

u (t) = K_{p} [e (t) + \frac{1}{T_{I}} \int_{0}^{t} e (t) dt + \frac{T_{D} de (t)}{dt}]

(28)

where $K_{P}$ is the proportionality factor, $T_{I}$ is the integration time constant, and $T_{D}$ is the derivative time constant. The $e$ is control deviation. However, the reference input may not be differentiable or discontinuous and the actual output may be disturbed by noise for the specific problems. It will cause the contradiction between overshoot and the faster response of the PID control in the closed-loop system (Han, 1994; Valluru and Singh, 2017).

This paper adopts the nonlinear PID controller by introducing a nonlinear PID module (Su et al., 2005), which is shown as:

f (e, α, δ) = {\begin{matrix} {| e |}^{α} sign (e) | e | \geq δ \\ \frac{e}{δ^{1 - α}} | e | < δ \end{matrix}

(29)

where the $sign (e) = {\begin{matrix} 1 & e \geq 0 \\ - 1 & e < 0 \end{matrix}$ , and the function $f (e, α, δ)$ represents the rate of feedback of the errors. The $δ$ represents the linear range of the function.

As shown in Figure 11, it is the principle block diagram of the control law of the nonlinear PID controller (Valluru and Singh, 2017). The control equation is given as

\begin{matrix} u (t) = K_{P} f_{p} (e_{p}, α_{p}, δ_{p}) e (t) + K_{I} f_{i} (e_{i}, α_{i}, δ_{i}) \int edt \\ + K_{D} f_{d} (e_{d}, α_{d}, δ_{d}) \frac{de}{dt} \end{matrix}

(30)

where $K_{P}$ , $K_{I}$ , $K_{D}$ are obtained by the manual tuning, and $α_{p} \in [0, 1]$ , $α_{i} \in [- 1, 0]$ , $α_{d} > 1$ . The $u (t)$ is the output of the nonlinear PID controller and the input of the DE balloons. By combining the principle block diagram with the control law shown in Figure 11, the dynamic control equations of nonlinear PID control for DE balloons are obtained as

\overset{\cdot\cdot}{λ} + f (λ, λ_{m}^{i}, p, ϕ (λ, \overset{\cdot}{λ}, K_{P}, K_{I}, K_{D})) = 0

(31)

where $ϕ$ is the input voltage. It can be seen that we can actually adjust the input voltage to control output by applying the appropriate gain coefficient to get the desired results.

In the following part, the nonlinear PID control is used for the vibration of DE balloon. As a consequence of the special structure of the sphere, the vibration of the DE membrane exhibits the unique nonlinear characteristics. Here different scenarios are considered: super-harmonic response, beating phenomena, phase shift, and step response.

Figure 12(a) shows the super-harmonic response vibration of a DE balloon with single relaxation time, which is a typical nonlinear dynamic response. This phenomenon will distort the sound of a loudspeaker designed by dielectric elastomers (Zhu et al., 2010b). In this case, we set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 1$ , $\tilde{ω} = 1.46$ , and $\tilde{τ} = 5$ . With the nonlinear PID controller, the parameters $α_{p} = 0.2$ , $α_{i} = - 0.01$ , $α_{d} = 1.3$ , $K_{p} = 200$ , $K_{i} = 75$ , $K_{d} = 65$ , $δ = 0.003$ are employed to obtain the expected results as shown in Figure 12(b). The expected result is $λ = 1.1 + 0.15 \cos (3 T)$ . It can be seen that the errors between the expected results and the controlled outputs are very small in Figure 12(c), which indicates that the nonlinear PID controller is effective.

Then we investigate the phase shift through nonlinear PID controller. By controlling the dynamic response shown in Figure 13(a), the controlled results are obtained as shown in Figure 13(b). In this case, we set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 1$ , $\tilde{ω} = 3.152$ , and $\tilde{τ} = 5$ . The control parameters are $α_{p} = 0.2$ , $α_{i} = - 0.02$ , $α_{d} = 1.4$ , $K_{p} = 280$ , $K_{i} = 65$ , $K_{d} = 90$ . The expected result is $y = 1.1 + 0.2 \cos (3 T + 3)$ . The dotted line in Figure 13(b) is the results of eliminating beating phenomena with controller, and the solid red line is the results with phase shift based on the dotted line. It is seen that the desired results are obtained.

Figure 14(a) presents super-harmonic response in the DE balloon with double relaxation times. The expected result is $y = 1.25 + 0.15 \cos (3 T)$ . We set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 1$ , $ω = 1.25$ , and $\tilde{τ} = 5, 15$ . By applying PID controller (the control parameters are $α_{p} = 0.1$ , $α_{i} = - 0.01$ , $α_{d} = 1.5$ , $K_{p} = 650$ , $K_{i} = 300$ , $K_{d} = 200$ ), the controlled results are obtained as shown in Figure 14(b). Figure 14(c) shows the errors between the controlled results and the expected results. It reveals that the errors are around 0 except at the initial moment by the nonlinear PID controller. This also shows that the nonlinear PID controller has a good performance on the control of dielectric elastomer actuators.

Figure 15 shows the implementation of phase shift in DE balloons with double relaxation times by nonlinear PID controller, and the control parameters of phase shift are $α_{p} = 0.2$ , $α_{i} = - 0.04$ , $α_{d} = 1.2$ , $K_{p} = 400$ , $K_{i} = 60$ , $K_{d} = 90$ . We set $pR / μ H = 0.1$ , $ε ϕ_{dc}^{2} / μ H^{2} = 0.05$ , $ϕ_{ac} / ϕ_{dc} = 1$ , $\tilde{ω} = 3.053$ , and $\tilde{τ} = 5, 15$ . It can be seen that the beating phenomenon is eliminated, and the phase shift is achieved.

As can be seen from Figure 16, when we apply the step voltage shown in Figure 16(a) and fixed internal pressure $pR / μ H = 0.1$ on the DE balloons, there is a period of oscillation before each step response is stable. The step response with nonlinear PID controller is shown in Figure 16(c). The material parameters are set as: $μ^{EQ} / μ = μ_{m}^{NEQ} / μ = 1 / 3$ , $J^{EQ} = J_{m}^{NEQ} = 100$ $(m = 1, 2)$ , $\tilde{τ} = 5, 15$ . The control parameters are $α_{p} = 0.4$ , $α_{i} = - 0.02$ , $α_{d} = 1.3$ , $δ = 0.003$ , $K_{p} = 1000$ , $K_{i} = 180$ , and $K_{d} = 80$ . The initial conditions are set as: $λ (0) = 1.028$ , $λ_{1}^{i} (0) = λ_{2}^{i} (0) = 1.028$ , and $d λ (0) / dT = 0$ . It can be seen that the initial oscillation process in each step is almost eliminated. This also indicates that the nonlinear PID controller can deal with the step signal.

Through the above analysis, as long as the appropriate control parameters are applied in nonlinear PID active controller, the nonlinear dynamic response of DE balloon is precisely controlled, and the desired results can be obtained.

6. Conclusions

In this paper, to accurately describe the nonlinear dynamic behavior of the DE balloons, a series of governing equations with multiple relaxation times are obtained by using the generalized Maxwell model. The simulation results with double relaxation times can match with the experimental results compared with single relaxation time. This also shows the necessity of multiple relaxation times for the study of the dynamic behavior of DE polymer materials. Then the nonlinear dynamic performance of viscoelastic dielectric elastomer balloons with considering multiple relaxation times are analyzed. The results show that the viscoelastic dielectric elastomer balloons experience a period of beating process when it is subjected to a sinusoidal voltage and internal pressure. However, shear modulus and the interaction of multiple relaxation times can influence the amplitude and duration of the beating process. From the phase diagram and the Poincare’ maps, the dielectric elastomer balloons undergo a process from non-periodic vibration to periodic vibration. The modeling approach in this work could be extended to other viscoelastic models. A nonlinear PID controller is used in this paper to eliminate several nonlinear responses of dielectric elastomer balloons with multiple relaxation times and desired control effect is achieved. It provides a method for precise control of dielectric elastomer in engineering application. In addition, bifurcation behavior of viscoelastic dielectric elastomers and the electromechanical stability with multiple relaxation times will be studied in the future work.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (nos. 11872195 and 11472120) and the 111 Project (no. B14044).

ORCID iD

Huadong Yong

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