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Dynamic Electromechanical Modeling of Dielectric Elastomer Actuators With Metallic Electrodes Rahimullah Sarban, Benny Lassen, and Morten Willatzen

Abstract—In this paper, a nonlinear electromechanical model for a PolyPower dielectric elastomer actuator is proposed based on an electric circuit model coupled with a viscoelastic mechanical model. The parameters of the model are found by fitting to an electrical step impulse for the mechanical part and by standard methods for the electric circuit. The resulting model is compared with experiments for a range of sinusoidal stimuli. The comparison shows good agreement between experiments and model results. Index Terms—Dielectric elastomer actuator, frequency response, spring–dashpot networks, system modeling.

I. INTRODUCTION DIELECTRIC elastomer (DE) is an emerging smart material that has gained increasing attention from scientists and engineers over the last decade [1]–[3]. DEs, when utilized as actuators, have the potential to be an effective replacement for many conventional actuators due to their light weight, noiseless operation, large strain, modest force density, and response time. DE actuators are constructed from thin sheets of DE material sandwiched between compliant electrodes. The actuation principle of DE actuators is based on the electrostatic pressure being induced between oppositely charged electrodes. The electrostatic pressure causes a compression of the DE sheets in the thickness direction. The thickness compression results in an extension of the material in the plane perpendicular to the thickness direction which is used for actuation [4], [5]. Although the physical actuation principle of DE actuators can be easily understood, predicting the actuation for a given electrical stimulus is not as straightforward. Extensive research has been dedicated to develop models that can predict the electromechanical behavior of DE [4]–[6]. Focus has been on developing quasistatic electromechanical models where the mechanical output of the material for a given electrical stimulus is assumed time invariant. Polymers are viscoelastic materials whose mechanical responses are time dependent even for a step input, i.e., a DE’s

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Manuscript received February 14, 2010; revised April 18, 2011; accepted April 23, 2011. Recommended by Technical Editor A. Menciassi. The work of R. Sarban was supported in part by a sponsorship from the Institute of Sound and Vibration Research, Southampton, U.K., and in part by the Danish Ministry of Science. R. Sarban is with Danfoss PolyPower A/S, 6430 Nordborg, Denmark (e-mail: [email protected]). B. Lassen and M. Willatzen are with the Mads Clausen Institute, University of Southern Denmark, 6400 Sonderborg, Denmark (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2011.2150239

mechanical output is time varying for a step electrical stimulus (creep) [11]. Furthermore, many applications involving, e.g., pumps, dynamic positioning, sound generation, energy scavenging, and active vibration control require dynamic actuation [7]. If DE actuators are to be used in such applications, the electrical stimulus will be time varying. Since a DE sheet can be regarded as a capacitor, i.e., an insulating material sandwiched between conducting electrodes, applying a time-varying stimulus will introduce a time- and frequency-dependent mechanical response. Hence, a dynamic electromechanical model is necessary to account for the capacitive and viscoelastic properties of DE. Extensive research has been carried out on the viscohyperelastic characterization and modeling of DE [12]–[15]. Lochmatter et al. have carried out a simulation study on DE actuators stability, efficiency, and energy density, using a model of the electromechanical characteristics for an acrylic DE [16]. Son and Goulbourne [17] have developed and validated a numerical model for the dynamic response of tubular DE transducers for the first time. The mechanical characteristics are modeled by a Mooney–Rivlin hyperelastic model and the dynamic response of the model is validated at two distinctive frequencies. Kaal and Herold [21] have developed a dynamic model of a tubular DE actuator for dynamic applications. This paper is focused on the development of a comprehensive and at the same time tractable dynamic electromechanical model for silicone-based tubular DE actuators undergoing relatively small deformations. The dynamic model is intended to be used for model-based active vibration control purposes, and hence, the frequency range of the model is adapted to this particular application. However, the model can be expanded to accommodate a dynamics at lower frequencies. First, in Section II, a description of the PolyPower DE tubular actuators used in this study is given together with a description of the complete system and the experimental setup used. A dynamic electromechanical model of the DE actuator is developed in Section III. Parameter identification and estimation of other component phenomena in the system such as amplifier and sensor are carried out in Section IV. And finally, in Section V, model results are compared with experimental data. Conclusion is presented in Section VI. II. DE ACTUATOR AND SYSTEM A. PolyPower DE PolyPower DE is a silicone DE manufactured by Danfoss PolyPower A/S, Denmark. PolyPower DEs, used for the construction of the actuators under consideration, are produced in thin sheets of 40-μm thickness. A corrugated microstructure

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Fig. 2. Overall dynamic of a DE system subdivided into dynamic of each component. H represents the frequency-dependent transfer function of each device.

Fig. 3. Fig. 1. (a) Actuation direction of a PolyPower DE with corrugated silver electrodes. (b) Schematic of rolling DE sheets into tubular form. (c) Photograph of the PolyPower tubular DE actuators.

is imprinted on one side of the thin sheet with amplitude and period of the corrugations given by 5 and 10 μm, respectively. A 100 nm silver electrode is then sputtered on the corrugated surface using a physical vapor deposition process [8]. The metallic electrodes increase the dynamic range of the PolyPower DE; however, due to high stiffness and no prestraining, the tubular actuators have smaller strain than conventional tubular DE actuators. For actuator fabrication, a laminate of two sheets placed back to back is used. The resulting laminate has a thickness of 80 μm with corrugated electrodes on both the upper and lower surfaces. Charging the corrugated electrodes reduces the thickness of the laminate causing elongation in the compliant direction as illustrated in Fig. 1(a). The DE laminate is then, in a semiautomated process, rolled to form a tubular actuator. Fig. 1(b) shows a schematic of the rolling of the laminate to form the tubular actuator. Note that the actuators have an active length L0 , which is the area being actuated and passive areas at either side to avoid electrical short circuiting. Fig. 1(c) is a photograph of the assembled actuator with electrical connections and end caps. The advantages of corrugated metallic electrodes are as follows. 1) Low electrical resistance reduces ohmic losses in the electrodes. Conventionally used compliant electrodes exhibit a relatively high resistance that reduces the electromechanical efficiency of the actuator, generates heat, and increases the phase lag between the electrical stimuli and mechanical response. 2) The electrode corrugation ensures that the expansion of the DE material is unidirectional which results in an increase of the force and stroke of unidirectional actuators. 3) Industrial processes for applying thin films of metal to surfaces are well established allowing for a high volume production of this type of DE material being compara-

Experimental setup diagram for DE system dynamic measurements.

tively straightforward compared to alternative DE material technologies. The resulting actuators shown in Fig. 1(c) are core free, selfsupporting DE actuators without prestrain. This is the type of actuators used in experiments for model validation in this study. B. DE System In order to induce a pronounced elongation of the tubular actuator, an applied voltage over the two electrodes in the kilovolt range is required. High-voltage amplifiers and power supplies are normally used to supply this voltage. For the actuators under consideration in this paper, the operating voltage is kept below 2800 V to keep the electric field below 35 MV/m in order to avoid electrical breakdown. Displacement or acceleration sensors are used for measuring the mechanical response of the actuators for a given stimulus. Digital controllers are typically used to generate a desired input to the amplifier and acquire the signals from the sensors. The overall dynamic response of a DE system will, therefore, consist of the dynamics of the DE itself as well as the dynamics of the amplifier, the sensors, and the controller (see Fig. 2). The main emphasis will be on the dynamic characteristic of DE actuators, although the time delays introduced from other components are also taken into account in the modeling. C. Experimental Setup The experimental setup for measuring the dynamic response of the DE system is shown in Fig. 3. A National Instrument digital controller model NI-USB 6212 is used for signal generation and measurements; a Matsusada high-voltage amplifier model AMP-10B40 is used to amplify the signal from the controller to the high voltage fed to the DE actuator. A linear variable differential transducer (LVDT) is used to measure the stroke of the DE actuator. The signal from the LVDT is fed to the controller and converted into physical values based on the sensitivity of the sensor. LabView software is used to control NI-USB 6212.

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The output from the controller VCON , the voltage and current outputs from the amplifier (VAM P and IAM P , respectively), and the LVDT signal VLVDT are measured simultaneously by the controller. These measurements are used for parameter identification in the model validation. III. MODELING Our purpose is to develop and verify a dynamic electromechanical model of a DE actuator. The model is a combination of two submodels describing the mechanical and electrical parts. These models are subsequently coupled via electrostatic forces. In addition, the following assumptions are made. 1) No hyperelastic behavior is assumed as the DE actuator under consideration is without prestrain and subject to an actuation strain of less than 5%. 2) The operating temperature is assumed to be above the glass transition temperature of the DE material; thus, temperature effects are ignored. 3) Plasticity is assumed to be negligible as the material is a cross-linked polymer. 4) The magnetic field, induced in the actuator, during charging and discharging, is assumed to be negligible (hence, we describe the dynamics as being quasi-electrostatic). 5) Static hysteresis has been neglected in accordance with our experiments (albeit not shown here). 6) Electrostatic pressure difference between the inner and outer walls of the cylinderical actuators is assumed to be negligible due to their small thickness/radius (80 μm/ 8 mm) ratio. Similarly, the nonuniform charge concentration on the corrugated surface is assumed to have negligible effects on the electrostatic field, based on COMSOL simulations, albeit not included here. A. Mechanical Model Polymers are viscoelastic materials, i.e., they exhibit both elastic solid and viscous fluid characteristics. The mechanical output of a viscoelastic material is time varying even for a step input. The viscoelastic behavior of the polymers, assuming negligible hyperelasticity, can be modeled as linear elastic spring and dashpot networks. Several models exist [9] to predict different viscoelastic behaviors. A Kelvin model, a parallel spring and dashpot, can model the viscoelastic characteristics of thermoset polymers with a single time constant. We emphasize that the stresses in this model for a step-strain input contain a Dirac delta function—a phenomenon not observed in real polymers. This defect is rectified by using the three-parameter Voigt–Kelvin model [9], [16], where a spring is added in series with the Kelvin model. However, a three-parameter Voigt–Kelvin model has one time constant, i.e., the model can predict the viscoelastic characteristics of a thermoset polymer in a small interval of time only. Experiments indicate that multiple time constants are needed in order to capture the response of these polymers at larger time scale. In this study, a Voigt–Kelvin five-parameter model is chosen to predict the viscoelastic behavior of the DE actuators under consideration (see Fig. 4). This model is chosen so as to have a model predicting an initial immediate response

Fig. 4.

Voigt–Kelvin’s five-parameter viscoelastic model.

Fig. 5.

Voigt–Kelvin’s five-parameter viscoelastic model with inertial force.

to a step-stress input followed by a period with creep (timedependent increase in the strain slowly leveling out). It can be shown that the strain response S(t) to a step-stress input σ0 , i.e., the creep compliance D(t), of the five-parameter Voigt–Kelvin model is    2  1 1 S (t) t = D (t) = + 1 − exp − (1) σ0 E0 n =1 En τn where τn = ηn /En are the retardation time constants [9]. Equation (1) shows that the first spring E0 expresses the immediate response of the material, and the second and third (E1 and E2 ) control the magnitude of the creep relative to the retardation times τ1 and τ2 . The five unknown parameters of the model in Fig. 4 can be found by fitting its creep compliance to a measured electrical step response of a DE actuator and will be discussed in Section IV. When the DE actuators are operated at higher frequencies, the model in Fig. 4 is inadequate as inertial forces (i.e., forces stemming from the mass of the actuator and load on the actuator) are absent. In order to describe DE actuator characteristics in a broader range of operation, a mass is added to the model via the inertial parameter M, see Fig. 5 (the unit of M is kg/m and it takes into account the cross-sectional area and the length L0 of the actuator, see Section IV). We note that the location of the load is unique, in practice, as it is always placed on top of the actuator. The stress–strain relationship in Fig. 5 can be represented by a fourth-order ordinary differential equation as follows [9]: 4  n =0

dn S (t)  dk σ (t) = ak dtn dtk 2

bn

(2)

k =0

where the superscripts n and k represent the order of differentiation, and the coefficients of (2) are given as a0 = E0 (E1 + E2 ) + E1 E2 a1 = E0 (E1 τ1 + E2 τ2 ) + E1 E2 (τ1 + τ2 ) a2 = E1 E2 τ1 τ2 b0 = E0 E1 E2 , b1 = b0 (τ1 + τ2 ) b2 = M a0 + b0 τ1 τ2 , b3 = M a1 , b4 = M a2 .

(3)

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Fig. 6.

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Circuit model of a DE actuator. Fig. 7.

Equation (2) is the dynamic mechanical equation of the DE actuator used in the numerical implementation. The stiffness and viscous damping parameters of the model are acquired by fitting (1) to the measured creep response of the actuator. B. Electrical Model A DE sheet is a dielectric material sandwiched between compliant electrodes; hence, a DE sheet can be modeled as a straindependent variable plate capacitor. However, dielectric materials (including DEs) are nonideal as they also exhibit a certain conducting current. Therefore, a more realistic model of a DE actuator is a variable capacitor in parallel with a resistor. A DE’s electrode resistance and resistances in wiring and connections also influence the electric circuit dynamics of the system and must be included in the electrical model. Fig. 6 shows a circuit representation of the electrical model for the DE system employed in this paper. In the figure, RS is the resistance of the electrodes and the wiring, RL is the resistance of the DE material, C is the capacitance of the DE sheet, VAM P is the voltage over the power source, and VC is the voltage over the DE sheet. The quasi-electrostatic pressure at the surface of the DE sheet is induced by VC which is given by (see Fig. 6) [10] dvC (t) 1 1 + · vAM P (t) · vC (t) = dt τe RS C

Electromechanical model diagram of DE actuator.

σ. The electrostatic pressure is the input to the mechanical part resulting in a mechanical strain S. If the actuation strain S is large, it changes the capacitance C, thickness d, and the stiffness of the material (hyperelasticity). These characteristics of DE are shown by a dashed feedback line in Fig. 7; however, they are not included in the model due to small actuation strain of the DE actuators under consideration here. For detailed information about strain-dependent capacitance and hyperelasticity of the DE material, under consideration, refer to [20]. D. DE Response of the Known Stimuli The DE dynamic response, for a given stimulus vAM P (t), can be acquired by first solving (4) to determine the voltage across DE vC (t). The result of (4) can then be inserted into (6) to estimate the electrostatic pressure σ(t) and, finally, inserting the solution of (6) into the right-hand side of (2) and solving for S(t) the strain of DE can be estimated. In this paper, a single-frequency harmonic stimulus with a constant bias term for vAM P (t) is considered, see (7).The bias is commonly used for DE’s harmonic actuation as it enables bidirectional movement of the actuator

(4) vAM P (t) = (A sin (ω0 t) + B) h(t)

where τe is the electrical circuit’s time constant and is given as τe =

RL RS C . RL + RS

(5)

C. Electromechanical Model A DE actuator is an electromechanical device that changes its mechanical shape due to the induced electrostatic pressure. The electrostatic pressure in a planar DE sheet is given by  2 vC (t) (6) σ (t) = ε0 εr d where ε0 , εr , and d are the permittivity of vacuum, the relative permittivity of the DE material, and the distance between the electrodes, respectively. Although the actuator under consideration is not a planar structure, (6) is still a good approximation of the electrostatic pressure as the ratio of thickness to the cylinder radius is less than 1%, see [4] and [6] for detailed derivation. Furthermore, as the amplitude of the corrugation is substantially smaller than the thickness of the elastomer sheets, the influence of the corrugation is negligible (for a detailed discussion refer to [18]). Fig. 7 shows a diagrammatic representation of the electromechanical model of DE. Fig. 7 shows that the input voltage vAM P generates a voltage across the capacitor vC which induces an electrostatic pressure

(7)

where h(t) is the Heaviside function. Inserting (7) into (4) and solving for vC (t) gives vC (t) = A1 sin (ω0 t) − A2 cos (ω0 t) + A3 exp (−τe t) + A4 (8) where the coefficients Ai are given as ARL (RL + RS ) ARL2 RS Cω0 , A2 = φ φ ARL RS Cω0 (RL + RS ) − RL Bφ (9) A3 = (RL + RS ) φ RL B , φ = (RL RS Cω0 )2 + (RL + RS )2 . A4 = RL + RS A1 =

Inserting (8) into (6), the electrostatic pressure can be written as σ (t) = α0 +

2 

(αn cos (nω0 t) + βn sin (nω0 t))

n =1

+

2 

(χn cos (ω0 t) + δn sin (ω0 t)+γn ) exp (−nτC t).

n =1

(10)

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The coefficients of (10) are given as  K 2 A1 + A22 + 2A24 2  K 2 A2 − A21 α1 = −2KA2 A4 , α2 = 2 β1 = 2KA1 A4 , β2 = −KA1 A2 α0 =

χ1 = −2KA2 A3 , γ1 = 2KA3 A4 ,

δ1 = 2KA1 A3 , γ2 = KA23 ,

χ2 = δ2 = 0 ε0 εr K= 2 . (11) d

Equation (10) shows that the electrostatic pressure, for a biased harmonic stimulus, has a constant term α0 , oscillatory terms with frequencies ω0 and 2ω0 , and an exponentially decaying oscillatory term. The latter takes the transient characteristic of the electrostatic pressure into account. In steady state, the electrostatic pressure of (10) can be reduced to σss (t) = α0 +

2 

(αn cos (nω0 t) + βn sin (nω0 t)).

Fig. 8.

Voigt–Kelvin model fit to experimental creep data. TABLE I VOIGT–KELVIN MODEL PARAMETERS

(12)

n =1

Equation (12) is a Fourier series of second order which is a linear representation of the nonlinear electrostatic pressure. Inserting (12) or, for a more general prediction, (10) into (2) and solving for S(t) gives the dynamic electromechanical relation between the stimulus voltage vAM P (t) and output strain S(t) for biased harmonic stimuli as in (7). The DE sheet used in this paper is not planar due to the corrugated surfaces; however, it has been shown that as long as effective stiffness’s are used, the effect of the corrugations on both the electric field and the strain fields can be disregarded [18]. Furthermore, the tubular geometry also affects the strain/stress characteristics; however, this can again be described by effective stiffness. For a more detailed discussion of the effect of the geometry, refer to [19].

time constant of DE is small enough; 2) a powerful high-voltage power amplifier is used; and 3) the material is constrained in one direction due to corrugated surface and metallic electrodes. The operating frequency values of interest in this study are above 2 Hz; hence, the model is fitted to the measured strain S(t) for a period of 0.5 s. In Fig. 8, it is seen that the strain of the actuator increases with time which indicates the presence of creep in the material.

IV. PARAMETER IDENTIFICATION A. Voigt–Kelvin Model Parameters The five parameters of the Voigt–Kelvin model are estimated by fitting (1) to experimental creep data using MATLAB’s lsqcurvefit function (see Fig. 8). The fitted parameter values of (1) are given in Table I. The experimental data are acquired, with 10 kHz sampling frequency, using the experimental setup shown in Fig. 3. A step voltage is applied and it is assumed that this corresponds to a constant pressure σ0 . Viscoelastic parameters are, typically, determined with mechanical creep experiments. The DE actuators under consideration are multilayer cylindrical structures with both ends sealed by the manufacturer. A passive stretch–stress test will exert more pressure on the outer layers of the actuator compared to the inner layers, hence reducing the quality of parameter identification. Ideally, in creep tests, all layers of the structure must be subject to uniform stress while the strain is measured. For the actuators studied in this paper, a uniform stress can be achieved by applying a step-voltage input. This approach is used and shown to be valid due to 1) the electrical

B. Inertial Parameter M The inertial parameter M of the mechanical model is given as M=

(mDE + mLVDT + mLOAD ) × L0 A0

(13)

where mDE is the mass of the moving part of the actuator, mLVDT is the mass of the sensor, mLOAD is the mass of the load, A0 is the cross-sectional area of the actuator being proportional to the length and thickness of the DE laminate, and L0 is the initial length of the actuator. The total mass of the DE actuator is measured to be 100.5 g. The moving mass of the actuator mDE is the mass of the active and upper passive areas of the actuator. The lower passive area of the actuator is not moving during actuation (see Fig. 3). The mass of the active area is calculated from its known volume and mass density. The mass of the passive end is assumed to be half the difference between the total mass and the active area’s mass. The mass of the sensor mLVDT is acquired from its datasheet. The mass of the load mLOAD is chosen to be 100, 300, and 500 g in three different

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TABLE II INERTIAL PARAMETER DATA

the hysteretic response of the overall system; hence, it must be taken into account even for low operating frequencies. The high-voltage amplifier model AMP10B40 produced by Matsusada Inc., CA, has a maximum current and voltage range of ±120 mA and ±10 kV, respectively. If the current limit of the amplifier is reached, saturation will occur and thus the electromechanical response of the DE system will be affected. Knowing the resistance RL which is several orders of magnitude larger than the resistance RS (refer to Table III), the required current IR in a DE system, for a sinusoidal stimulus, can be approximated as IR = 

TABLE III ELECTRICAL PARAMETERS OF THE DE SYSTEM

experiments carried out in this paper. Table II shows the different parameters appearing in (13). C. DE Circuit parameters C, RL , RS The capacitance C of the DE sheet is measured directly with a multimeter. The DE material resistance RL is determined experimentally by measuring the steady-state leakage current of the actuator for a step voltage of 2 kV after 200 s. The serial resistance RS is determined experimentally from the phase relation between the amplifier current IAM P and the voltage VAM P in Fig. 3. In the experiment, a sinusoidal control signal VCON (t) = 1.25 sin (20π t) + 1.25

(14)

is generated and the amplifier voltage and current are measured simultaneously. Assuming a material resistance RL much larger than serial resistance RS , the phase lag between the amplifier’s current and voltage can be approximated as   1 IAM P  ≈ arctan (15) VAM P RS Cω where ω is the angular frequency. Based on the results of the experiments, the left-hand side of (15) is found to be 1.56 rad. Inserting the measured phase lag into (15), the serial resistance RS can be calculated. Table III shows the values of the electrical parameters of the DE system. D. Sensor and Amplifier The displacement sensor used in the experiments (LVDT model DC15) has a time delay of 0.4 ms which is acquired from its datasheet. It is assumed that the sensor’s delay is constant in the range of frequencies in this paper. A constant sensor delay acts as a mechanical backlash in the system amplifying

ωm ax C 1 + (ωm ax RS C)2

Vm ax

(16)

where ωm ax and Vm ax are the maximum operating voltage’s angular frequency and amplitude, respectively. The frequency and amplitude of all experiments are chosen within the limits of the amplifier in order to avoid amplifier saturation. The frequency response of the amplifier input and output voltages (VCON and VAM P ), for the frequency range of interest, is determined experimentally. The maximum deviation between the input and output voltages is measured to be 0.1 dB, and therefore, the amplifier dynamics is assumed to be negligible and, hence, disregarded in the model. Parameter identification, for other configuration of DE actuators, can be divided in three parts: 1) electrical parameters can be determined using the purposed approach as electrical parameters are less sensitive to geometrical configuration; 2) geometrical parameters are normally provided by the manufacturer or known; and 3) for PolyPower DE, the mechanical parameter identification approach will be the same as it is assumed that the electrostatic pressure in thickness direction is transferred along the compliant direction only due to the corrugation and metallic electrodes. V. IMPLEMENTATION AND RESULTS A. Implementation The model developed in Section III is implemented in MATLAB. Equation (2) is solved, for the electrostatic pressure of (10), using the MATLAB’s ode45 function that numerically solves ordinary differential equations with the Runge–Kutta method. The output of the model was delayed by 0.4 ms to take into account the delay of the sensor. B. Results Experiments are performed using the expression for the control voltage VCON given by (14) which is then amplified 1000 times by the high-voltage amplifier. In Fig. 9, the measured strain of the actuator is compared with model’s steady-state predictions using the identified model parameters in Section IV. Fig. 9 shows that the simulated-strain results are in close agreement with experimental data. The measured response of the actuator has the following three important characteristics. 1) Stimulus and strain are not linearly proportional. This is due to the quadratic term of equation (6). 2) The strain exhibits

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Fig. 9.

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Simulated and measured stroke for a sinusoidal stimulus of 10 Hz.

hysteretic behavior. This is partly due to the phase lag introduced in the DE’s mechanical and electrical characteristics and partly due to sensor delay. 3) The mechanical output (strain) is biased. This effect occurs primarily due to the viscoelastic characteristics of DE materials (creep). However, in some cases, the saturation of the amplifier’s outputs could also contribute to such effects, and hence, care must be taken to distinguish between viscoelasticity and amplifier’s saturation. Model simulation and experiments, albeit not included here, have shown that amplitudes of the terms associated with 2ω0 in (10) are reduced substantially when the biased term B of (7) is much larger than the amplitude A. Thus, the output strain of DE, when BA, can be assumed to be dominated by the fundamental frequency term ω0 . This condition enables determining an approximated frequency response function (FRF) of the DE by plotting the magnitude of the ω0 term. The model is simulated for a frequency range of 1–100 Hz with 0.5 Hz resolution and A = 500 V and B = 4 A. Fig. 10(a) shows the simulated and experimental magnitudes of %Strain/V for three different loadings at steady state. Fig. 10(b) shows the corresponding phase lag between input voltage and output strain. Fig. 10 shows that the simulated and measured FRFs of the actuator are in close agreement. The magnitude at the resonance point with 500 g load is, however, predicted to be larger than measured. This is due to the slight buckling of the actuator at 500 g load that reduces the amplitude of vertical movement. The model, however, does not take the buckling into account hence predicting higher vertical movement. Table IV shows the error between predicted and measured resonance frequencies E(fn) and amplitudes E(An) for the three loading conditions. The developed model has shown to predict the dynamic electromechanical response in the frequency range of 2–100 Hz. This frequency range is chosen specifically for active vibration control purposes where the model will be implemented in a model-based control design. The model accuracy, however, will degrade if the operation frequency is lower than 1 Hz. The suggested viscoelastic parameter identification method can be used for PolyPower DE due to its constrain in one direction. Moreover, the model cannot predict any buckling of the DE actuators if they are loaded with a force close to or beyond their buckling forces.

Fig. 10. Simulated and measured frequency response functions of the DE actuator. (a) Magnitude. (b) Phase. TABLE IV MODEL ERROR IN RESONANCE FREQUENCY AND AMPLITUDE

VI. CONCLUSION A dynamic electromechanical model for a tubular DE actuator has been proposed. The model was derived by combining an electrical circuit model for DE sheets with a mechanical viscoelastic model where the coupling is given by the quasielectrostatic pressure. The model was implemented and simulated in MATLAB. It is shown that the model is in close agreement with the measured response of the DE actuator. A dynamic electromechanical model of a DE actuator can be used to predict its bandwidth and hysteretic behavior for control purposes and possibly lead to improvements of the controller’s performance and stability. The model can also be used to design actuators for use in applications with certain load and bandwidth requirements. Future work will consist of expanding the model for large actuation strains where some of the model parameters are strain dependent.

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Rahimullah Sarban received the B.Sc. degree in engineering and the M.Sc. degree in mechatronics (with the research project being carried out in conjunction with Danfoss PolyPower A/S, Nordborg, Denmark) both from the University of Southern Denmark, Sonderborg, Denmark in 2006 and 2008, respectively. He currently holds an industrial Ph.D. Scholarship and is involved in a project on active vibration control using DEAP Actuators. The project partners are the University of Southern Denmark, The University of Southampton, and the newly formed DEAP manufacturing company, Danfoss PolyPower A/S. The project is expected to complete by September 2011.

Benny Lassen received both the M.Sc. degree in mathematics and the Ph.D. degree in mathematical modeling from the University of Southern Denmark, Sonderborg, Denmark, in 2002 and 2005, respectively. After receiving the Ph.D. degree, he was at Lund University where he was financed by a Villum Kann Rasmussen Postdoctoral stipend for one year. He is currently an Assistant Professor at the University of Southern Denmark. His main research interests include modeling of smart material systems, especially dielectric elastomer and piezoelectric systems, nanoscale semiconductor heterostructures, specifically, electronic bandstructures, transport, strain, and piezoelectric effects.

Morten Willatzen received the M.Sc. degree in mathematical physics (quantum-field theory) from the University of Aarhus, Aarhus, Denmark, in 1989, and the Ph.D. degree in theoretical semiconductor physics with emphasis to applications of quantumconfined structures from the Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark, in 1993. He was a Postdoctoral Fellow at TeleDanmark Research from 1993 to 1994, and at the Max-PlanckInstitute for Solid State Physics, Stuttgart, Germany, from 1994 to 1995, and he was employed at Danfoss A/S conducting research on thermofluid systems and ultrasonic flowmeter applications from 1995 to 2000. From May 2000 to May 2004, he was employed as an Associate Professor and from June 2004, as a Full Professor and Group Leader (Mads Clausen Institute for Product Innovation). Since 2007, he has been the Head of Mathematical Modeling in the Faculty of Engineering, University of Southern Denmark, Sonderborg, Denmark. From 2001 to 2006, he served as Vice Dean of the Faculty of Science and Engineering, SDU. His research interests include solid-state physics and electronic bandstructures of semiconductors, flow acoustics, piezoelectric transducer modeling, and modeling of thermo-fluid systems.