Dynamic model and analysis of a dielectric elastomer vibration isolator

Abstract

Dielectric elastomers are a kind of electroactive polymer and have a great potential for application to soft actuators. Dielectric elastomer materials also feature nonlinear response owing to the electromechanical coupling. In this work, a theoretical dynamic model was developed to characterize the nonlinear performance of a dielectric elastomer actuator used as a vibration isolator. The vibration reduction mechanism of dielectric elastomer structures is theoretically described. An experiment was also performed to validate the model. The experimental results tend to match well with the numerical calculation.

Keywords

Dielectric elastomer dynamic model vibration isolator energy method

1. Introduction

Dielectric elastomers (DEs) as soft electroactive polymers (Carpi et al., 2010) are being studied to manufacture flexible sensors, robots, actuators, energy harvesters, and biological devices (Jung et al., 2008; Liu et al., 2009; Maas and Graf, 2012; Shian et al., 2015; Zhao and Niu, 2015). Because the material shear moduli of DEs are small, DEs are highly flexible and capable of undergoing large deformation (2200% strain) (An et al., 2015). Moreover, besides being lightweight and relatively inexpensive, DEs also possess high energy density (3400 J/kg) (O’Halloran et al., 2008). A dielectric elastomer actuator (DEA) is generally made from a DE film (Kornbluh et al., 2002), both sides of which are coated with compliant electrodes. When the DE is subjected to a voltage, the positive and negative charges spread out on their respective plate electrodes. Then, the Maxwell stresses compress the DE, causing a reduction in its thickness and an increase in the area of the plane of the film (Madden, 2008).

The main advantage of DE-based devices lies in their large deformation capacity. However, the actuated deformation and dynamic performance of the DE structures are also significantly influenced by the intrinsic material viscoelasticity, which is still not well understood (Kofod et al., 2003; Plante and Dubowsky, 2006). Viscoelasticity limits the application of DEs to a certain extent. DE structures also exhibit strong nonlinearity, energy dissipation, and creep characteristics (Yang et al., 2005). Conversely, viscoelasticity can become an advantage for DE materials in the field of vibration reduction (Plante and Dubowsky, 2007). Moreover, the natural frequency of DE structures can be varied with changes in applied voltage (Pelrine and Kornbluh, 2008). Because of these characteristics, DE structures are considered to have potential applications in active vibration control. At present, few studies on the application of DE for vibration control have been reported. Most of these works focused on DE actuator designs and control strategies for vibration reduction. On the basis of the Gent model, Zhang et al. (2015b) modeled a planar DE resonator using the principle of virtual work and studied the passive vibration characteristics with different parameters. Sarban and Jones (2012) used a DE tubular actuator for adaptive vibration isolation, where an electromechanical model of the actuator was utilized as a part of the feedforward controller estimation scheme. Kaal et al. (2017) undertook the design, simulation, and experimental investigation of a stack DE for vibration isolation. Zou et al. (2017) proposed a feedforward control approach for creep and vibration compensation of a cone DE actuator. A vibration compensator based on a zero-vibration input shaping (ZVIS) technique was developed to suppress the vibrational dynamics of the creep-compensated DEA.

Other works regarding DE vibration are limited in their treatment of the influence of membrane viscoelasticity. Few works focused on the dynamic performance of DE-based vibrational devices. To investigate the vibration properties of the DEs, Zhang et al. (2017) investigated the influence of viscous damping with the Kelvin–Voigt model on the vibration of DE membranes. Zhang et al. (2015c) showed that the resonant behavior and vibration of DE membranes under an electric field and different mechanical loading states are highly time-dependent because of the material viscoelasticity. Using the perturbation method, Zhu et al. (2010) and Li et al. (2014) applied nonlinear vibration analysis to viscoelastic DE oscillators.

In this work, we propose a modeling method for a vibration isolator based on the energy method and theoretically describe the vibration reduction mechanism of DE structures. This work can also provide guidelines for better predicting the dynamic performance and optimal design of DE vibrational devices. Furthermore, an experiment was performed to verify the model of the DE vibration isolator.

This article is organized as follows: the next section describes the design and dynamic model of the DE vibration isolator; the parameter analysis of the model is also presented. The subsequent section describes the use of a perturbation method to obtain the equilibrium state and natural frequency of the DE actuator; the results of the experiment are also discussed. This is followed by concluding remarks in the final section.

2. Model development and analysis

2.1. Model development

The diagram of the DEA considered in this work is shown in Figure 1(b). A similar actuator shape was also considered by York et al. (2010) and Rizzello et al. (2015). York et al. (2010) designed an experiment conducted with a particular focus on the hysteretic and rate-dependent material behavior. Rizzello et al. (2015) developed a dynamic model composed of a set of nonlinear, time-invariant differential equations describing the dynamic relationship between the input voltage and the output actuator displacement. Although the existing model accurately describes the behavior of the device, it is difficult to describe the vibration property of the DEA. In the following section, we describe a dynamic model with a focus on the vibration analyses of the DEA vibration isolator.

This section presents a model of the DEA vibration isolator based on the modeling method presented by Hodgins et al. (2013). Using the energy method, the model established in this work reveals the dynamic performance of DE-based vibrational devices. The DE membrane is equiaxially pre-stretched before bonding to the polymethyl methacrylate (PMMA) framework. A sandwich structure is formed by coating compliant carbon grease and flexible electrodes on both sides of the DE membrane. Then, by adding a weight and bias spring to the framework, the cone-shaped DE vibration isolator is established. The shape of the cross-section of the deformed DE film is approximately that of a truncated cone, as shown in Figure 1(b).

For the circular geometry, the state of deformation of the membrane is described by the radial, circumferential, and thickness stretches, denoted as $λ_{l} = l / l_{0}$ , $λ_{c} = 1$ , and $λ_{t} = t / t_{0}$ , respectively (Suo, 2010). A pre-stretch also exists in the DE membrane; we define $λ_{pre} = x_{0} / X_{0} = y_{0} / Y_{0}$ and $λ_{pret} = t_{0} / T_{0}$ as pre-stretch of the DE. The symbols $t_{0}, x_{0}, and y_{0}$ represent the thickness, length in the x direction, and length of y direction before pre-stretch. The symbols $T_{0}, X_{0}, and Y_{0}$ represent the thickness, length in the x direction, and length in the y direction after pre-stretch. The process of the pre-stretch of the DE membrane is described in Figure 1(a). The following quantities are defined (Zhou et al., 2016)

λ_{l, tot} = λ_{l} λ_{pre}

(1)

λ_{c, tot} = λ_{c} λ_{pre}

(2)

λ_{t, tot} = λ_{t} λ_{pret}

(3)

The symbols $λ_{l, tot}, λ_{c, tot}, and λ_{t, tot}$ represent the total stretch in the radial, circumferential, and thickness directions. Because the material is incompressible, the following equation applies

λ_{l} λ_{c} λ_{t} = λ_{l} λ_{t} = \frac{tl}{t_{0} l_{0}}

(4)

When a voltage is applied to the DE, the charge Q distributed on the electrode of the vibration isolator can be expressed as

Q = DS = \frac{D π l_{0} t_{0} (l_{0} + 2 r)}{t} = \frac{D π l_{0} t_{0} (l_{0} + 2 r)}{T_{0}} (λ_{l} {λ_{pre}}^{2})

(5)

where S represents the area of the electrode on the DE membrane and D represents the electrical displacement. By defining the voltage applied on the DEA as $φ$ , the electric field E can be represented as

E = \frac{φ}{t} = \frac{φ}{T_{0}} (λ_{l} {λ_{pre}}^{2})

(6)

The electrical displacement of the DE can be represented as $D = E ε$ , where $ε$ is the permittivity of the DE. We can obtain the relationship between charge Q and voltage $φ$

Q = ε \frac{φ {(λ_{l} {λ_{pre}}^{2})}^{2}}{{T_{0}}^{2}} π l_{0} t_{0} (l_{0} + 2 r)

(7)

Let $λ = λ_{l}$ for the convenience of expression; the partial differential of equation (7) can be expressed as

\begin{matrix} δ Q = & ε \frac{{({λ_{pre}}^{2} λ)}^{2}}{{T_{0}}^{2}} π l_{0} t_{0} (l_{0} + 2 r) δ φ \\ + 2 ε φ \frac{{λ_{pre}}^{4} λ}{{T_{0}}^{2}} π l_{0} t_{0} (l_{0} + 2 r) δ λ \end{matrix}

(8)

When a voltage is applied, the work done by the voltage is calculated as $φ δ Q$ . We ignore the inertia of the DE membrane because the weight of the DE is relatively small. To describe the influence of the DE viscoelastic, the damping force of the DE is represented as $f_{c} = cld λ / dt$ , where c is the damping coefficient. By its displacement integrated along the radial direction, it is not difficult to determine the work done by the damping force (Sheng et al., 2014; Xu et al., 2012). It can be expressed as

W_{c} = c \frac{d λ}{dt} δ λ \int_{0}^{l_{0}} ldl = \frac{1}{2} {cl}_{0}^{2} δ λ \frac{d λ}{dt}

(9)

Thermodynamics dictates that, for arbitrary variation of the system, the variation of the free energy of the membrane should equal the work done by the voltage, the stress, and the damping force

V_{pre} δ W = φ δ Q + P l_{0} δ λ - \frac{1}{2} {cl}_{0}^{2} δ λ \frac{d λ}{dt}

(10)

$V_{pre} = π l_{0} t_{0} (l_{0} + 2 r)$ is the volume of the DE membrane. Substituting equation (8) into equation (10) yields

\begin{matrix} δ W = & φ (ε \frac{{({λ_{pre}}^{2} λ)}^{2}}{{T_{0}}^{2}} δ φ + 2 ε φ \frac{{λ_{pre}}^{4} λ}{{T_{0}}^{2}} δ λ) \\ + \frac{P l_{0} δ λ - \frac{1}{2} {cl}_{0}^{2} δ λ \frac{d λ}{dt}}{π l_{0} t_{0} (l_{0} + 2 r)} \end{matrix}

(11)

The vertical force equilibrium on the biasing mass can be expressed as

P \cos θ - mg - k l_{0} (λ - 1) \cos θ + m l_{0} \frac{d^{2} λ}{d t^{2}} \cos θ = 0

(12)

where m is the mass of the weight, k is the stiffness, and $θ$ is the deflection angle of the DE membrane. We can solve P as

P = - m l_{0} \frac{d^{2} λ}{d t^{2}} + k l_{0} (λ - 1) + \frac{mg}{\cos θ}

(13)

Substituting equation (13) into equation (11) and differentiating by $λ$ , we obtain

\begin{matrix} \frac{\partial W}{\partial λ} = & - m l_{0} \frac{d^{2} λ}{d t^{2}} = φ (ε \frac{{({λ_{pre}}^{2} λ)}^{2}}{{T_{0}}^{2}} δ φ + 2 ε φ \frac{{λ_{pre}}^{4} λ}{{T_{0}}^{2}} δ λ) \\ + \frac{P l_{0} δ λ - \frac{1}{2} {cl}_{0}^{2} δ λ \frac{d λ}{dt}}{π l_{0} t_{0} (l_{0} + 2 r)} \end{matrix}

(14)

To characterize the hyperelastic behavior of the DE, the free-energy density function W is used, and is given in the Gent (1969)form

W = - \frac{μ J_{m}}{2} \ln (1 - \frac{{λ_{t, tot}}^{2} + {λ_{l, tot}}^{2} + {λ_{c, tot}}^{2}}{J_{m}})

(15)

where $μ$ is the shear modulus, and $J_{m}$ is a material constant related to the limiting stretch.

Combining equations (1) to (3), (7), and (15) yields

\begin{matrix} W = & - \frac{μ J_{m}}{2} \ln (1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}}) \\ + \frac{ε}{2} {λ_{pre}}^{4} {(\frac{φ}{T_{0}})}^{2} λ^{2} \end{matrix}

(16)

Substituting equation (16) into equation (14) yields

\begin{matrix} \frac{m l_{0}}{μ π t_{0} (l_{0} + 2 r)} \frac{d^{2} λ}{d t^{2}} = - \frac{1}{2} \frac{c l_{0}}{μ π t_{0} (l_{0} + 2 r)} \frac{d λ}{dt} \\ + \frac{ε {λ_{pre}}^{4} λ}{μ} {(\frac{φ}{T_{0}})}^{2} + \frac{k l_{0} \cos θ (λ - 1)}{μ π t_{0} (l_{0} + 2 r)} \\ + \frac{mg}{μ \cos θ π t_{0} (l_{0} + 2 r)} - \frac{- {λ_{pre}}^{- 4} λ^{- 3} + {λ_{pre}}^{2} λ}{1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}}} \end{matrix}

(17)

To be more general, the variables are simplified to be nondimensional. Consequently, equation (17) can be re-expressed in the following form (Li et al., 2014)

\begin{matrix} \frac{d^{2} λ}{d T^{2}} = & - \tilde{c} \frac{d λ}{dt} + \frac{M}{\cos θ} + K \cos θ (λ - 1) \\ - \frac{- {λ_{pre}}^{- 4} λ^{- 3} + {λ_{pre}}^{2} λ}{1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}}} + {λ_{pre}}^{4} \emptyset^{2} λ \end{matrix}

(18)

where $T = t / \sqrt{m l_{0} / (μ π t_{0} (l_{0} + 2 r))}$ is dimensionless time, $\tilde{c} = (1 / 2) c \sqrt{l_{0} / (m π t_{0} (l_{0} + 2 r))}$ is dimensionless damping, $K = k l_{0} / (μ π t_{0} (l_{0} + 2 r))$ is dimensionless stiffness, $M = mg / (μ π t_{0} (l_{0} + 2 r))$ is the dimensionless mass, $\emptyset = (φ / T_{0}) \sqrt{ε / μ}$ is the dimensionless voltage, and $\cos θ^{- 1} = l_{0} λ / \sqrt{{(l_{0} λ)}^{2} - {(R - r)}^{2}}$ .

2.2. Parameter analysis of the DE vibration isolator

In order to describe the vibration parameters of DE-based vibrational devices, we studied the influence of pre-stretch, applied voltage, dimensionless stiffness, and dimensionless mass. We set the applied voltage equal to zero in the study of the free vibration; the results show the ability of DE to reduce vibration. Equation (18) can be expressed as

\begin{matrix} \frac{d^{2} λ}{d T^{2}} = & - \tilde{c} \frac{d λ}{dt} + \frac{M}{\cos θ} + K \cos θ (λ - 1) \\ - \frac{- {λ_{pre}}^{- 4} λ^{- 3} + {λ_{pre}}^{2} λ}{1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}}} \end{matrix}

(19)

The parameters used in this study were J_m = 100, $μ = 28 kPa$ , and $ε = 3.98 \times 10^{- 11} F m^{- 1}$ (Zhang et al., 2015a). The DE pre-stretch $λ_{pre} = 2$ . By the internal viscoelastic damping of DE, the vibration of the DEA also can be damped, which is termed as passive vibration isolation. We assumed the undeformed state has the initial conditions for calculation, that is, $λ (0) = 1.1$ and $λ (\overset{\cdot}{0}) = 0$ . Figure 2 describes the results of the performance of DE vibration isolator by comparing a different dimensionless damping coefficients ∼c equal to 0.05, 0.10, and 0.15. The parameters $λ_{pre} = 2$ , K = 1.5, and M = 4.5. It can be easily found that a larger coefficient $\tilde{c}$ can induce better vibration attenuation performance. Figure 3 plots the influence of vibration parameters by changing the dimensionless stiffness K to 1.0, 1.5, and 2.0. The parameters $λ_{pre} = 2$ , $\tilde{c} = 0.1$ , and M = 4.5. With the increasing K, the spring force increases and the DE device performs a strong vibration and a weakened vibration attenuation. Furthermore, a larger dimensionless spring stiffness K reduces the DEA frequency. Figure 4 shows that the performance of vibration isolation declines with increasing load mass, because the load mass increases the inertia of the DE system. The mass values we chose were 3.5, 4.5, and 5.5. The parameters $λ_{pre} = 2$ , $\tilde{c} = 0.1$ , and K = 1.5. Figure 5 shows the effect of the pre-stretch of the DE vibration isolator. The pre-stretch was set to 1.5, 2.0, and 2.5. The parameters M = 4.5, $\tilde{c} = 0.1$ , and K = 1.5. It is obvious that the larger pre-stretch causes greater stiffness of the system and better performance of the DE vibration attenuation.

3. Dynamic characteristics analysis

3.1. Equilibrium state and natural frequency for different voltages

In order to study the effect of the DE vibration isolation with different applied voltages, we re-present equation (18) as

\frac{d^{2} λ}{d T^{2}} + \tilde{c} \frac{d λ}{dt} + g (λ, \emptyset, M, K) = 0

(20)

\begin{matrix} g (λ, \emptyset, M, K) = - \frac{M}{\cos θ} - K \cos θ (λ - 1) \\ + \frac{- {λ_{pre}}^{- 4} λ^{- 3} + {λ_{pre}}^{2} λ}{1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}}} - {λ_{pre}}^{4} \emptyset^{2} λ \end{matrix}

(21)

When the DEA is in a static balance state, equation (20) can be reduced to the equilibrium equation, that is, equation (21) should equal zero. We can obtain the relationship between applied voltage $\emptyset$ and stretch at balance position $λ_{eq}$ . The parameters used in the simulation were $M = 4.5$ and $K = 1$ . Figure 6 shows the relation between the voltage applied on the DE isolator and the stretching ratio of the equilibrium position under different pre-stretching conditions. When the pre-stretching of the DE is 2.5, 3.0, and 3.5, the equilibrium position of the DEA is 1.04, 1.08, and 1.19. It can be observed that with the increased pre-stretching, the stretch at the equilibrium position increases gradually. Furthermore, the stretch at the equilibrium position also increases with greater applied voltage.

In order to study the vibration of the DE with alternating voltage load, we treated the vibration of the DE as perturbed at the equilibrium position. The time-dependent stretch of the DE membrane can be expressed as (Zhu et al., 2010)

λ (T) = λ_{eq} + Δ (T)

(22)

where $Δ (T)$ is the amplitude of the small perturbation. Substituting equation (22) into equation (20) and expanding the function $g (λ, \emptyset, M, K)$ as a power series around the equilibrium stretch $λ_{eq}$ to the first order, we obtain

\frac{d^{2} Δ}{d T^{2}} + Δ \frac{\partial g (λ, \emptyset, M, K)}{\partial λ} + \tilde{c} \frac{d Δ}{dT} = 0

(23)

From equation (23), we obtain the dimensionless natural frequency of small-amplitude oscillation around the equilibrium state

\begin{matrix} ω^{2} = {\frac{\partial g}{\partial λ} |}_{λ = λ_{eq}} \\ = \frac{({λ_{pre}}^{2} + 3 {λ_{pre}}^{- 4} λ^{- 4}) (1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}})}{{(1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}})}^{2}} \\ + \frac{\frac{2}{J_{m}} {({λ_{pre}}^{2} λ - {λ_{pre}}^{- 4} λ^{- 3})}^{2}}{{(1 - \frac{{({λ_{pre}}^{2} λ)}^{- 2} + {(λ_{pre} λ)}^{2} + {(λ_{pre})}^{2} - 3}{J_{m}})}^{2}} \\ - {λ_{pre}}^{4} \emptyset^{2} - M \frac{\sqrt{λ^{2} - 1} - λ^{2} {(λ^{2} - 1)}^{- \frac{1}{2}}}{λ^{2} - 1} \\ - K \frac{λ (λ - 1) {(λ^{2} - 1)}^{- \frac{1}{2}} - \sqrt{λ^{2} - 1} (λ - 2)}{λ^{2}} \end{matrix}

(24)

When we chose different pre-stretch values of 2.5, 3.0, and 3.5, we obtained the relationship between the dimensionless voltage squared applied on the DE system and the natural frequency squared of the dimensionless dimension. From Figure 7, we can conclude that the natural frequency decreases first and then increases with increasing voltage. When the applied voltage is low, the deformation of the system is small and the material is soft (Dubois et al., 2008), so the natural frequency is relatively small. With increasing voltage, the deformation becomes large and the DE material becomes hard, and the natural frequency is also increased.

According to this analysis, the DEA has the characteristic of moving frequency with changing voltage. The natural frequency of the system can be changed to avoid the resonance region, and it can be used as a vibration isolator.

3.2. Experimental validation and results

In this study, a cone-shaped DE vibration isolator was designed and fabricated for modeling. The structure and dimensions of the circular DEA are shown in Figure 8. A high-bond membrane (3M VHB 4910 Acrylic) was equiaxially pre-stretched before bonding to the PMMA framework. Then, the membrane was coated with compliant carbon grease on both sides. A spring supplied pre-bias for the DEA. The stiffness of the spring was 0.2 N/m, and the mass was 2 g. When driven by a high voltage, the DEA moved vertically.

To investigate the influence of high applied voltage on the natural frequency of the DEA, a series of tests was carried out to measure the natural frequency of the DEA. In this study, we obtained the responses of the DEA under both a sweep frequency excitation signal and a pulse signal. Figure 9 shows the DEA under sweep frequency excitation in the time domain and in the frequency domain. The frequency range was 0.1–60 Hz, and the sweep time was 240 s. In the amplitude-frequency curve shown in Figure 9(b), we observe that the amplitude of the output increases first and then decreases. The natural frequency of the DEA is at the peak of the curve. The natural frequency obtained by the sweep frequency excitation test was 20.01 Hz. Similarly, the natural frequency obtained from the pulse test was 19.49 Hz. The displacement curve and the frequency curve are shown in Figure 10. The DEA in both of these tests was under 4 kV applied voltage.

Figures 10 to 14 illustrate the impulse response of the DEA under different applied voltages. The applied voltage varies from 0 to 4 kV with the interval 1 kV, and the natural frequency of the DEA is 15.11, 14.67, 15.5, 17.92, and 19.49 Hz. Figure 15 shows the change in natural frequency of the DEA with varying voltage for both the experimental and simulation data. The parameters used in the simulation were $M = 4.5$ , $K = 1$ , and $λ_{pre} = 4$ . The tendency of the natural frequency matches well with the model data. Therefore, the dynamic model developed in this work is to represent the dynamic characteristics of the DEA.

4. Conclusion

In this study, a nonlinear dynamic model based on the energy method of the DEA was established. Through the dynamic characteristics’ analysis, we theoretically describe the vibration reduction mechanism of DE structures. Dynamic response of the DEA can be actively changed by tuning the natural frequency with applied voltage. The experimental results match well with the numerical calculation. This model can also provide guidelines to better predict the dynamic performance and design control strategies of DE vibrational devices.

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 was supported by the National Natural Science Foundation of China (grant no. 11372179), the National Key Technologies R&D Program of China (grant no.2015BAF07B03), and Innovation Project of Shanghai (grant no. 15JC1402600).

ORCID iD

Lili Meng

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