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Sensors and Actuators A 170 (2011) 121– 130

Contents lists available at ScienceDirect

Sensors and Actuators A: Physical

jo u rn al hom epage: www.elsev ier .com/ locate /sna

numerical model for electro-active polymer actuatorsith experimental validation

ark Potter ∗, Kevin Gouder, Jonathan F. Morrisonepartment of Aeronautics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

r t i c l e i n f o

rticle history:eceived 17 November 2009eceived in revised form 23 August 2010ccepted 23 August 2010vailable online 10 December 2010

a b s t r a c t

This paper presents a theoretical model for predicting the behavior of electro-active polymer actuators.This model takes the form of an array of masses with the interconnecting forces derived using Hooke’slaw coupled with an electrostatic force, and is solved using time-marching integration. Forming themodel in this way allows the response of actuators of various sizes and pre-strains to be simulated witha range of electrode designs and mechanically stiffened sections. To validate the response of the model a

eywords:lectro-active polymerinite-difference modelingre-strainxperimental validation

series of Nusil MED4905 polydimethylsiloxane actuators were built and tested with differing electrodedesigns and with additional mechanical stiffening of the membrane. Through a comparison of maximumelectrode displacement, with applied electrode voltage, tracking of a grid on the membrane surface anddynamic testing, the model was found to show excellent agreement with the experimental results.

© 2010 Elsevier B.V. All rights reserved.

ynamic

. Introduction

In recent years an increasing need for new small-scale actuatorsas led to significant interest in the field of electro-active polymersEAPs). These materials have the potential to form compliant, ver-atile, low cost and low density devices [1]. A particular class ofAPs known as dielectric elastomer actuators (DEAs) are based onhe field-induced deformation of elastomeric polymers with com-liant electrodes and can produce a large strain combined with aast response time and high electromechanical efficiency [6]. Givenhe potential benefits inherent with actuators of this form a vastange of possible applications have been considered. These includeightweight artificial muscles for robots, diaphragm actuators forumps and speakers and flow control devices such as dimples oribrating surfaces for aerodynamic applications.

.1. Principle of operation: the dielectric elastomer actuatorechanism

The basic principle of operation for a dielectric elastomer actu-tor is shown in Fig. 1. An elastomeric film is sandwiched betweenwo electrically conductive, compliant electrodes [5]. This consti-

utes a compliant capacitor, made from an incompressible andighly deformable elastomeric material with electrodes which areompliant, meaning they do not mechanically stiffen the dielec-

∗ Corresponding author. Tel.: +44 207594 5141.E-mail address: m.potter@imperial.ac.uk (M. Potter).

924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2010.08.027

tric medium [3]. By applying a voltage difference between theelectrodes, positive charges appear on one electrode and negativecharges on the other [3], giving rise to Coulomb electrostatic forceswhich cause a contraction of the actuator along the direction of theelectric field and an expansion of it in the two orthogonal directions[5]. The electrostatic loading on the membrane, as a result of thisapplied voltage, is modeled as an effective pressure acting acrossthe membrane thickness [4]. By considering the change in elec-trostatic energy per unit displacement of the film in the thicknessdirection, Pelrine et al. [4] demonstrate that this effective pressureis given by

p = εε0E2 = εε0V2

h2(1)

where E is the electric field, V is the applied voltage and h is themembrane thickness. This effective pressure is exactly twice thepressure in a charged parallel plate capacitor. This difference isexplained by the change in membrane area as the actuator contractsowing to the incompressibility of the membrane material.

To exploit this mechanism, in order to form an actuator, a mem-brane of electro-active polymer is pre-strained and an electrodeis applied to only one section of the membrane. This effectivelydivides the actuator into an active region that contains the elec-trode, and a passive region that does not. When a voltage is appliedto the actuator, the resulting compressive load in the electrode

area relaxes the in-built pre-strain within that area. The resultingimbalance in the stresses in the membrane causes an expansionof the electrode and a contraction of the passive region until astress balance is restored. When the voltage is removed the mem-

122 M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130

of a d

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Fig. 1. Principle of operation

rane returns to the non-actuated position. By tuning the levelsf pre-strain independently in each of the two in-plane axes ofhe membrane and controlling the location and area of the activelectrode region, the direction and magnitude of actuation can beptimized for a specific application. This form of actuator is referredo as an in-plane EAP actuator and modeling the effects of activeegion area and location, to allow design optimization withoutxtensive prototyping, is the focus of this paper.

.2. Modeling a dielectric actuator

A search of the literature reveals relatively few attempts toodel the behavior of electro-active polymer actuators. An earlyodel proposed by Pelrine et al. [4], was based on the assumption

f conservation of membrane volume owing to a Poisson’s ratio of ≈ 0.5. By considering a case of equal strain in the orthogonal in-lane directions, the strain in the out-of-plane direction is relatedo the in-plane strain. To determine the strain in the membranehickness, a uniaxial loading case is examined, relating the stressue to the Maxwell loading across the membrane to the strain viahe Young’s Modulus. Solving the resulting equation gives a predic-ion of displacement against applied voltage for a given actuatorize. While this model gives an approximate fit to experimentalork carried out in validation, the assumption of equal in-plane

trains and a linear stress–strain relationship between the electro-tatic loading and out of plane strain seriously limits the cases forhich this model is valid. An improvement to this basic modeling

pproach was made by Trujillo et al. [7]. While a case of symmetri-al in-plane strains is again assumed, Hooke’s law is used to relatehe in-plane strains to the out-of-plane strain and Maxwell load-ng due to the applied voltage. Experiments carried out on a rolledctuator setup are found to be in good agreement with the proposedodel, however this model is again limited by requiring symmetri-

al in-plane strains and by having an electrode that covers the totalembrane area. Carpi et al. [1] considered a similar model, based

n Hooke’s law coupled with an electrostatic loading, in order toodel the behavior of a pre-strained flat, in-plane actuator with

otal electrode coverage. The resulting model equations are solvedtatically to predict the displacement of a simple actuator withespect to applied voltage. This work shows excellent agreementith experimental work carried out. However, due to the sim-le construction of the model, the response of different electrodeesigns and therefore the effect of having active and passive mem-rane regions cannot be generated. Wissler and Mazza [8] movedway from Hooke’s law based model and instead used three hyper-

lastic models with three strain energy formulations (Yeoh, Ogdennd Mooney-Rivlin) to describe the behavior of a small element oflectro-active membrane. A finite element (FE) model is then con-tructed based on a fine mesh of eight-noded linear elements. Using

ielectric elastomer actuator.

this construction the 3D response of a pre-strained circular actua-tor was investigated and compared against experimental devices.By comparing the three models based on the different strain energypotentials, it is demonstrated that the Yeoh form, based on theleft Cauchy–Green deformation tensor, shows excellent agreementwith the response of an experimental actuator. The use of a FEmodel formulation adds greatly to the flexibility of this model tosimulate membranes divided into active sections containing elec-trodes, and passive sections, without electrodes. Furthermore thiswork highlights the danger of using uniaxial test data to determinematerial properties as this can lead to significant errors in the mod-eled deformation. The model presented in this paper is designed toallow rapid prototyping of actuator designs as an aid to developingand optimizing new actuators in terms of both displacement anddynamic response. The goal is to allow different electrode designsand different levels of in-plane pre-strain in each of the orthogo-nal in-plane axis to be investigated, along with stiffened sectionsbonded to the membrane. For these reasons the model is based ona FE type approach whereby the membrane is divided into a seriesof lumped masses or nodes. Based on the loading applied to eachnode, the electrode coverage can be controlled specifically allowingfor different regions of passive and active membrane sections. Toavoid an overly complicated model with long computation times,the membrane is modeled in 2D by a single sheet of point masses.While the change in thickness is accounted for in determining theMaxwell loading, this simplified construction greatly reduced thecomputational complexity of the model. As an aid to model simpli-fication and therefore computational speed, the inter-nodal forcesare derived using Hooke’s law. Given that the actuator designs beingconsidered by this work generally display deformation strains of<5%, the use of Hooke’s law, rather than a hyperelasticity model, isconsidered justified.

2. Development of the lumped parameter model

To model the behavior of the actuator, the membrane is modeledas a series of discrete masses in a regular rectangular lattice. Usingthe three-dimensional Hooke equations, the forces between adja-cent nodes are calculated and the total resultant force on each nodeis used to calculate the acceleration of that node. A time-marchingapproach is then used to resolve the movement of the membranein response to an applied electrode design and voltage. The follow-ing is a brief description of the model formulation and the methodused to determine the resulting membrane displacements.

2.1. Model formulation

According to Hooke’s law for an isotropic material, the threeprincipal strains within a material, ε11, ε22 and ε33 can be expressed

M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130 123

i

ε

ε

ε

wrkcr�

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ε

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s

2.2. Implementation

Fig. 2. One node and its relationship with its four neighbors.

n terms of the principal stresses as

11 = 1E

(�11 − �(�22 + �33)) (2)

22 = 1E

(�22 − �(�11 + �33)) (3)

33 = 1E

(�33 − �(�11 + �22)) (4)

here E is the Young’s modulus and � is Poisson’s ratio of the mate-ial. By modeling the membrane as a series of discrete nodes, withnown displacements, the in-plane strains, ε11 and ε22, can be cal-ulated directly from the relative separation of adjacent nodes. Byearranging Eqs. (2) and (3), the principal in-plane stresses �11 and22 can be shown to relate to the in-plane strains by

11 = E

1 − �2(ε11 + �ε22) + �2

1 − �2�33 (5)

22 = E

1 − �2(ε22 + �ε11) + �2

1 − �2�33 (6)

here �33 is the stress applied to the membrane in the out-of-planeirection. For an electro-active polymer actuator, �33 is equal tohe pressure applied on the material by the applied voltage at thelectrodes. This stress is given by Eq. (1). To determine the strainsithin the membrane, the deformed shape of the node grid is used.

ig. 2 shows the arrangement of five nodes in the undeformed case,n white, and in a deformed case in black. If we consider the strainf the left hand node in relation to the center node, the strain in theirection extension is given by

11 ≈ L1 − L0

L0(7)

here L1 is the length of separation between the deformed nodes,nd L0 is the original length separating the undeformed nodes. Toetermine the strain in the left-hand node, in relation to the centerode, in the direction perpendicular to the direction of extensionn average of the strains in the adjacent nodes is used.

22 ≈ L2 + L4 − 2L0

2L0(8)

The length between the central node and node 1, on the left handide, is determined based on the nodal displacement in the in-plane

Fig. 3. A discrete node with dimensions and principal stresses labeled.

directions, x and y. Based on trigonometry the nodal spacing can beexpressed as

L1 =√

(X(i − 1, j) − X(i, j))2 + (Y(i − 1, j) − Y(i, j))2 (9)

where X and Y are the locations of each node in the plane of themembrane. Having converted the nodal displacements into theprincipal stresses, �11 and �22, the nodal forces can be calculatedbased on the dimensions of the nodes. By considering Fig. 3 show-ing the central node from Fig. 2 and by combining Eqs. (5), (7)–(9)the force acting between the central node and node 1 on the lefthand side of Fig. 2 is found to be

f1 = L0yh0E

1 − �2

(L1 − L0x

L0x+ �

L2 + L4 − 2L0y

2L0y

)+ L0yh0�2

1 − �2�33 (10)

where L0x and L0y are the lengths between the nodes before stretch-ing and h0 is the thickness of the membrane before stretching. Byusing this formulation the membrane can be given any level of pre-strain in either the x and y directions, as required for many actuatordesigns. To determine the total force acting on the central node theforces f1–4 are each calculated and then resolved in the two orthog-onal in-plane directions. For the first node considered above, thistakes the form

f1x = f1

(X(i − 1, j) − X(i, j)

L1

)(11)

f1y = f1

(Y(i − 1, j) − Y(i, j)

L1

)(12)

At each node, a force matrix for the x and y directions is formedwith Fx and Fy defined by summing Eqs. (11) and (12) respectively.From the resultant force matrix the acceleration of each node iscalculated using Newton’s law. Each node within the array has anoriginal length L0x by L0y and original thickness h0. The mass of eachelement is given by

m = �L0xL0yh0. (13)

From this, the acceleration matrices are therefore given by

X = Fx − cX

�h0L0xL0y(14)

Y = Fy − cY

�h0L0xL0y(15)

where c is the damping coefficient of the membrane and is foundlater in the paper.

The model that has been laid out defines a series of dis-crete nodes and calculates the inter-nodal forces and accelerationswithin the array of these nodes. To convert these accelerations

1 Actua

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24 M. Potter et al. / Sensors and

nto estimations of actuator displacement, the accelerations of theembrane are numerically integrated. To perform this integration,

numerical time-marching approach is adopted. First the models initialized, and then a loop is used to advance one time step at aime. Firstly the membrane is initialized by defining the dimensionsf the actuator and the material properties of the electro-activeolymer. The required levels of pre-strain are then used to deter-ine the lengths L0y and L0x. At this point an iterative loop is

nitialized:

. At each node the four forces f1, f2, f3 and f4 are calculated.

. For stiffened or actuated nodes additional nodal forces are added.

. The nodal forces are resolved into the x and y directions andsummed to form the total force matrices Fx and Fy.

. The acceleration matrices for each of the in-plane directions xand y, are calculated.

. Using the acceleration matrices the system is advanced one timestep.

This procedure has advanced the model by one time step. Toesolve the dynamic response of the model, this process is repeatedntil the desired time has been reached, or in the case of a static test,ntil the membrane displacement has converged to a final value.

t is this aspect of the solver that allows the dynamic response ofhe model to be evaluated. Given that a time marching approachs adopted to solve the equations of the lumped parameter model,he resulting simulated behavior will be time step dependent. Tonsure that the choice of time step does not influence the resultsf the simulation the size of time step was reduced until the out-ut response converged. This lead to a time step of dt = 10−5 beinghosen.

. Experimental apparatus and procedure

In order to validate the lumped parameter model, two actuatoresigns were chosen for experimental testing and model validation.

ig. 4. The two electrode configurations selected for modeling. Dimensions of actuator u0% pre-strain.

tors A 170 (2011) 121– 130

An actuator consisting of a rectangular membrane with a biaxialin-plane pre-strain was selected with dimensions of 54 mm in thedirection of actuation and 255 mm in the passive direction. Themembrane was pre-strained by 70% in the active direction andby 250% in the passive direction. This high level of pre-strain inthe passive direction was selected to prevent membrane bucklingand to stiffen the membrane in the passive direction, thereforepromoting greater actuation in the active direction. Two differentelectrode designs were chosen for testing to determine the viabilityof using the lumped parameter model as a tool for the optimiza-tion of electrode design. The two electrode designs selected formodeling and experiment were (i) a central electrode in the mid-dle of the pre-strained membrane. This electrode is designed toexpand symmetrically about its center-line upon actuation (ii) aside electrode on one side of the pre-strained membrane designedto expand towards the passive side upon actuation. The two elec-trode designs are shown in Fig. 4. The following table shows theamounts of pre-strain values chosen for investigation.

Electrode location SideElectrode dimensions 175 mm × 27 mmPre-strain values 250% in y and 70% in x

Electrode location CenterElectrode dimensions 175 mm × 20 mmPre-strain values 250% in y and 70% in x

Smooth-edged and round-cornered frames were fabricatedfrom glass-fibre composite sheeting, onto which the pre-strainedactuator membrane could be mounted. A bespoke pre-strain jigwas fabricated out of Perspex and consists of two clamps whichcan be moved away from each other on two stainless steel rails andlocked in position. Once the necessary pre-strain was applied inboth directions (see Fig. 5), the glass-fibre frame coated in double-

sided tape was lowered onto the pre-strained membrane and aline of silicone glue applied along the perimeter of the frame. Theassembly was then left to dry before electrodes were applied usinga brush-on method. A highly conductive expanded graphite powder

nder consideration: length: 255 mm with a 250% pre-strain, width: 54 mm with a

M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130 125

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3

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ohtumtwouahom

ftmaatrm

3

V

repeated at a higher voltage. Fig. 7 shows the resulting experimen-tal actuator responses. To ensure that the central electrode actuatessymmetrically, both sides were tracked and are presented.

Fig. 5. The membrane in the pre-strain jig.

rom SGL Carbon Group of SGL Technologies GmbH, ConductographFG5, was applied using a sponge applicator. A mask made fromeak-glue tape ensured that the electrodes were of the requiredimension and crisp, facilitating strain measurement with a cam-ra. Leads made from aluminum foil were applied to the assemblyn such a way that the connection between the lead and electrodelways occurred on the stiff frame rather than on the membraneroviding a robust and safe connection.

.1. Material properties

Several experiments were carried out to obtain the materialroperties of the membrane for use with the lumped parameterodel. The final thickness of the membrane after pre-straining was

nitially estimated using an assumption of conservation of volume.he thickness was measured using a screw gauge to an accuracy of0 �m before the membrane was stretched to the required levelsf pre-strain and again after. Given the accuracy to which the finalhickness was predicted using a conservation of volume assump-ion, a Poisson’s ratio of � = 0.5 was adopted for the model.

Biaxial tensile and relaxation tests were carried out in order tobtain the Young’s Modulus of the silicone. Silicone is known toave highly non-linear tensile material properties, the modulus ofhe membrane was therefore determined at the levels of pre-strainsed in the actuator. A frame rig was used to strain a sample of theembrane to 250% in the passive direction, before loading into the

ensile testing machine. Both the tensile and the relaxation testsere then performed between strains of 60% and 80%. The results

f the relaxation test are shown in Fig. 6. It is clear from the fig-re that the material is highly viscoelestic with a stress relaxationfter the initial rise. This behavior is addressed later in the paper,owever analysis of the tensile testing led to a Young’s Modulusf E = 0.55 MPa being adopted for the steady state response of theodel.To determine the damping coefficient, c, of the membrane, a

ree vibration test was performed using the biaxial tensile rig. Theensile load cell was removed from the setup and replaced with a

ass sufficient to maintain a strain of 70%, which was tracked using laser triangulation sensor. The mass was then displaced verticallynd released while the amplitude of the resulting oscillations wasracked to determine the decay rate of the response envelope. Theesults of this test lead to a value of c = 0.13 being adopted in theodel.

.2. Actuator testing procedure

To test the actuators each unit was connected to a TREK High-oltage amplifier (model 609E-6). The low-voltage signal to the

Fig. 6. Relaxation test on Nusil polydimethylsiloxane MED4905.

TREK HV amplifier was provided by the analogue output port of aNI DAQ card which was controlled via a LabView interface designedto slowly ramp up the voltage to the required level. This precau-tion was taken to avoid subjecting the actuator to a step changein voltage and the high current transients that this would induce.To record the displacement of the actuators, a CCD camera wasmounted directly over the electrode edge with a scale in view. Eachimage recorded by the camera was analyzed using a fully auto-mated procedure within Matlab. By using the contrast change in theimage, when moving from the electrode edge to the membrane sur-face, the location of the electrode was found to within an accuracyof 2 pixels, corresponding to an error of 60 �m.

4. Steady state response

To examine the behavior of the two actuator designs, both weresubjected to a range of voltages from 1 kV up to 3.5 kV. The volt-age applied to the actuator was ramped up slowly, to avoid a largecurrent loading, then held constant until the displacement of themembrane had reached a maximum steady state value. The appliedvoltage was then ramped down and the whole procedure was

Fig. 7. Maximum displacement against applied voltage for an experimental actua-tor.

126 M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130

4

ets

4

tietdacgdtcmatana

tfmbceesc

4

pcbTid

However, rather than considering only the displacement of theelectrode, the displacement of a grid drawn onto a region of themembrane was tracked using the same camera setup as before.By recording the initial and final locations of the grid intersec-

Fig. 8. Nodal displacement of electrode edge for a range of node numbers.

.1. Experimental results

Fig. 7 shows that the displacement of each side of the centrallectrode is approximately half that of the side electrode and thathe displacement of the membrane in both cases scales with thequare of the applied voltage.

.2. Steady state model response

The use of a discrete model to simulate the behavior of a con-inuous system can lead to significant quantization errors if annsufficiently fine mesh is selected for the model. To determine theffect that node density has on modeled membrane displacement,he two electrode design cases were simulated for a range of nodeistributions. Fig. 8 shows the maximum membrane displacementgainst the nodal distribution, in nodes/mm, for each electrodease at an actuation voltage of 3.5 kV. With a nodal distribution ofreater than 3 nodes/mm, the model has converged and no furtherisplacement is seen with an increased density. With a nodal dis-ribution of 1 node/mm, the side and middle electrode cases haveonverged to within 1.1% and 3.3% respectively of the displace-ent obtained at 3 nodes/mm. Given that computation time scales

pproximately with nodal density it may prove time efficient toest early designs with a coarse grid, and then move to a finer grids the actuator converges towards a final design. To ensure thatodal distribution does not play a factor in the following analysis,ll models are run with a nodal distribution of 3 nodes/mm.

To form a comparison with the experimental results, each of thewo modeled actuators designs was subjected to a range of voltagerom 0 V to 3.5 kV. Each model was run until the displacement of the

embrane had reached steady state before the maximum mem-rane displacement was recorded. Fig. 9 shows the voltage and theorresponding displacement as determined by the lumped param-ter model. The model accurately predicts that the displacement ofach side of the central electrode is approximately half that of theide electrode and that the displacement of the membrane in bothases scales with the square of the applied voltage.

.3. Static displacement comparison

Fig. 10 shows a comparison of the responses for the two lumpedarameter model simulated actuators and the experimental dataollected from the test actuators. There is clearly a good agreement

etween the response of the model and the experimental work.he quality of this fit suggests that the lumped parameter models indeed able to predict the effects that pre-strain and electrodeesign have on the steady state displacement of the actuator and

Fig. 9. Maximum displacement against applied voltage for the finite-differencemodel.

therefore offers a useful prototyping tool for developing actuatorsof this type.

5. Stiffened sections

New actuator designs have included stiffened reinforced sec-tions that are bonded to the membrane to maximize the area ofdisplacement. By bonding the stiffened section against the elec-trode, the displacement experienced by the electrode is transferredto a potentially far larger, stiffened section, which displaces uni-formly as a single unit. In this way large regions of a surface canbe driven by a far smaller electrode at the edge. To account forthis development within the lumped parameter model, additionalforces are applied to nodes within the selected stiffened area. Theadditional nodal forces for the stiffened sections are calculatedusing Eqs. (7) and (10), with an increased Young’s modulus E andzero pre-strain.

5.1. Experimental setup

To test the validity of modeling a stiffened section in this way,an actuator with a side electrode and a stiffened section, as shownin Fig. 12, was fabricated. The same frame geometry and pre-strain that were used for the previous experiments were selected.

Fig. 10. Comparison of displacements for an experimental actuator and the finite-difference model.

M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130 127

Ft

trwdc

5

sbmbstmatdtncttat

5

appsstsnt

F

ig. 11. Actuator with stiffened section with captured region highlighted (insidehe dotted-line rectangle).

ions within the camera frame, the displacement of the membrane,ather than just the electrode, was found. The tracked region alongith the grid is shown in Fig. 11 before actuation. The higher gridensity around the stiffened section was employed to capture theurvature in the deflection in this region.

.2. Experimental results

Fig. 13 (top) shows the locations of the grid intersections, ashown in Fig. 12, before actuation in gray and after actuator inlack. A voltage of 3.9 kV was chosen to maximize the grid displace-ent. To make the figure clearer, the horizontal grid lines have

een included. As expected the displacements at the right-handide edge of the figure and at the top edge of the figure are closeo zero given that these are solid frame boundaries. The displace-

ent lines in the lower portion of the left-hand edge experiencepproximately similar displacement as would be expected due tohe presence of the stiffened section. Fig. 13 (bottom) shows theisplacement of each grid reference node, obtained by subtractinghe position of the node before actuation from the position of theode after actuation. Again horizontally adjacent nodes have beenonnected for clarity. The uppermost line in the figure correspondso the electrode as it bulges out between the stiffened section andhe frame. The horizontal lines connected to the stiffened sectionre seen to converge as they all have zero displacement with respecto each other at the stiffened section.

.3. Stiffened section model response

To form a comparison with the experimental results, the samectuator with stiffened section was simulated with the lumpedarameter model. Fig. 14 (top) shows the modeled grid referenceoints before and after actuation at 3.9 kV while Fig. 14 (bottom)hows the displacements of each grid reference node, obtained byubtracting the position of the node before actuation from the posi-

ion of the node after actuation. The grid lines in contact with thetiffened section are seen to converge to a single point as there iso relative movement between the nodes and the electrode is seeno bulge out between the frame and the stiffened section.

ig. 12. Stiffened experimental actuator with grid for displacement capture.

Fig. 13. Top: Experimental grid reference points before actuation (gray) and afteractuation at 3.9 kV (black). Bottom: Experimental displacement of grid referencepoints after actuation at 3.9 kV.

5.4. Stiffened section comparison

In Fig. 15 the displacements of the grid reference points forboth the experimental data and the lumped parameter model, forthe mechanically stiffened actuator are presented. There is clearlyexcellent agreement between the displacements predicted by thelumped parameter model and the displacements measured exper-imentally.

6. Dynamic behavior

The method proposed in Section 2.2, to implement the lumpedparameter model, is a time marching algorithm based on the accel-eration of each node due to the applied load on that node. Thismodel can therefore be used to determine the dynamic response ofthe model to a time varying input voltage. With this in mind a seriesof dynamic tests were performed on a side electrode actuator. Thesame frame geometry and pre-strain that were used for the previ-ous experiments were selected. However the actuator was drivenby a rectified sine wave of 3.4 kV at a range of frequency between1 and 20 Hz.

6.1. Experimental setup and results

To perform the dynamic tests on the membrane, the same exper-

imental procedure used to determine the steady state response wasused. An overhead camera captured images of the electrode edge ata rate of 180 frames/s. These images were then analyzed to deter-mine the displacement of the electrode as a function of time. The

128 M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130

Faa

ariutp

wta

Fa

presented in Fig. 6, shows that the actuator material is viscoelastic.This behavior is not represented in the initial lumped parameter

ig. 14. Top: Modeled grid reference points before actuation (gray) and after actu-tion at 3.5 kV (black). Bottom: Modeled displacement of grid reference points afterctuation at 3.5 kV.

ctuator was then subjected to a series of rectified sine wave inputsanging in frequency from 1 Hz up to 20 Hz. To avoid a sudden load-ng of the membrane the amplitude of the sine wave was rampedp linearly over 3 s, held for 4 s and then ramped down over a fur-her 3 s. The response to the 1 Hz and the 20 Hz input signals isresented in Fig. 16.

Simulations performed using the numerical model suggest that

hen driving the membrane at frequencies up 20 Hz, the actua-

or should produce a response that matches the static response inmplitude and returns to zero displacement as the voltage drops to

ig. 15. Comparison of the displacement of grid reference points for an experimentalctuator and the finite-difference model after actuation at 3.5 kV.

Fig. 16. Actuator response to a 1 Hz driving signal (top) and a 20 Hz driving signal(bottom).

zero. Fig. 16 shows that the amplitude of the maximum displace-ment of the electrode at 20 Hz and even 1 Hz is lower than the staticresponse and that the displacement does not return to zero whenthe voltage drops to zero, instead the amplitude of oscillation isreduced and develops an offset.

Fig. 17 summarizes the results of the dynamic tests, plotting themaximum achieved deflection against actuation frequency alongwith the minimum return deflection against frequency. The staticresponse for the same actuator is included at 0 Hz for completeness.

It is clear from the figure that as the actuation frequencyincreases there is a marked drop in the amplitude of actuation andthat the electrode develops an offset as the material is unable toreturn to zero before the next actuation occurs. This behavior sug-gests that the properties of the silicone membrane are frequencydependent as expected from the results of the relaxation test.

6.2. Modeling viscoelastic effects

The relaxation test performed on the silicone membrane, and

model and therefore an addition is required. To model the vis-coelastic effects a generalized Maxwell model was employed [2].

Fig. 17. Minimum and maximum actuator electrode displacement against drivingfrequency.

M. Potter et al. / Sensors and Actuators A 170 (2011) 121– 130 129

Tmeet

t

�

w

�

e

�

�

t

ε

ε

cncwte

Et

ε

ε

pfpr

ε

ε

that the experimental trend for a reduction in actuation displace-ment and a failure to return to a zero displacement state have beenfaithfully captured. The addition of viscous terms into the model

Fig. 18. Generalized Maxwell model.

his model is shown in Fig. 18, consisting of a purely elastic ele-ent in parallel with a series of damped elastic elements. For the

lectro-active polymer actuator application two damped elasticlements were found to be sufficient to simulate the response ofhe relaxation test.

By considering the model in Fig. 18, the total stress exerted byhe element is given by the equation

T = E(

ε + E2

Eε2 + E3

Eε3

)(16)

here the force due to the damping elements is given by

D = �εD (17)

Considering the force balance in each arm of the viscoelasticlements leads to

2 = E2ε2 = �2ε2D = �(ε − ε2) (18)

3 = E3ε3 = �3ε3D = �(ε − ε3) (19)

Allowing the strain in each viscous element to be related to theotal strain by

˙ 2 + E2

�2ε2 = ε (20)

˙ 3 + E3

�3ε3 = ε (21)

To determine the Young’s moduli E2 and E3 and the dampingonstants �2 and �3 the relaxation test was used. By applying aear instantaneous step change in strain and holding that strainonstant, Eq. (20) states that �2 and �3 will decay exponentiallyith a time constant given by the ratio of the damping coefficient to

he Young’s modulus. Using a least square fit based on two decayingxponentials, the four coefficients were determined from Fig. 6.

To form a discreet model for these pseudo-strains a forwarduler is applied to the pseudo-strains and a backward Euler to theotal strain, leading to

2(t+1) =(

1 − E2

�2dt

)ε2(t) + ε1(t) − ε1(t−1) (22)

3(t+1) =(

1 − E3

�3dt

)ε3(t) + ε1(t) − ε1(t−1) (23)

Having formed a numerical model for the behavior of theseudo-strains within the membrane, the additional inter-nodal

orces resulting from the viscoelastic behavior must be incor-orated into the lumped parameter model. This is done byepresenting the strains in the in-plane directions as

′11 = ε11 + ε11(2)

E2

E+ ε11(3)

E3

E(24)

′22 = ε22 + ε22(2)

E2

E+ ε22(3)

E3

E(25)

Fig. 19. Model response to a 1 Hz driving signal (top) and a 20 Hz driving signal(bottom).

These strains are then used to determine the forces betweennodes as before using Eq. (5). It should be noted that in the limit asε2 and ε3 tend to zero the original steady state lumped parametermodel is fully recovered.

6.3. Dynamic model response

Having modified the lumped parameter model to incorporatethe viscoelastic effects of the silicone membrane, the experimen-tal dynamic tests performed on the side electrode actuator weresimulated in full. The simulated cases of 1 Hz and 20 Hz actuationfrequency are shown in Fig. 19, with an overlay of the experimentaldata shown in Figs. 20 and 21.

6.4. Dynamic displacement comparison

A summary of the simulated dynamic tests is shown in Fig. 22,overlaid with the experimental response. We see from the figure

Fig. 20. Comparison of model and actuator electrode displacement for a 1 Hz drivingsignal.

130 M. Potter et al. / Sensors and Actua

Fig. 21. Comparison of model and actuator electrode displacement for a 20 Hz driv-ing signal.

Fd

ct

7

upBloptttiamvmmt

[

[[

[

[

[

[

[

quent visits to the Department of Mechanical and Aerospace Engineering, Princeton

ig. 22. Comparison of model and electrode minimum and maximum electrodeisplacement against driving frequency.

learly allows the frequency dependency of the actuator, withinhe tested frequency range, to be captured.

. Conclusions

In this paper a novel lumped parameter type model for sim-lating the behavior of electro-active polymer actuators, for theurpose of developing new actuator designs, has been presented.ased on a discrete array of masses with the internal forces calcu-

ated from Hooke’s law, this model is able to simulate the effectf varying a range of design parameters. The levels of in-planere-strain can be selected independently in the two in-plane direc-ions and any conceivable electrode design can be imposed onhe actuator and additional stiffening sections can be bonded tohe membrane. In addition a viscoelastic based model has beenncorporated into the model to allow the dynamic response of thectuator to a time varying driving voltage to be examined. Experi-ents conducted on three prototype actuators have been used to

alidate this model with good agreement being found between theodeled and recorded responses. The quality of the fit between theodel and experimental work for both steady state and dynamic

esting suggests that this work offers a highly useful tool for the pur-

tors A 170 (2011) 121– 130

pose of prototype actuator and electrode designs without needingto physically build an actuator, and while the experimental investi-gation has been confined to a rectangular pre-strained actuator, themodel is not restricted to simulating devices of this form, but offersa fully versatile model for testing a range of EAP actuator designs.

References

1] F. Carpi, P. Chiarelli, A. Mazzoldi, D. De Rossi, Electromechanical characterisa-tion of dielectric elastomer planar actuators: comparative evaluation of differentelectrode materials and different counterloads, Sensors and Actuators A 107(2003) 85–95.

2] R.M. Christensen, Theory of Viscoelasticity, Academic Press, 1982.3] G. Kofod, Dielectric elastomer actuators. Ph.D. Thesis, The Technical University

of Denmark, 2001.4] R.E. Pelrine, R.D. Kornbluh, J.P. Joseph, Electrostriction of polymer dielectrics

with compliant electrodes as a means of actuation, Sensors and Actuators A:Physical 64 (1998) 77–85.

5] R.E. Pelrine, R.D. Kornbluh, J.P. Joseph, S. Chiba, Electrostriction of polymer filmsfor microactuators, IEEE 0-7803-3744-1 (1997) 238–243.

6] R.E. Pelrine, P. Sommer-Larsen, R. Kornbluh, R. Heydt, G. Kofod, Q. Pei, P.Gravesen, Applications of dielectric elastomer actuators, in: Y. Bar-Cohen (Ed.),Proceedings of SPIE: Smart Structures and Materials 2005: Electroactive PolymerActuators and Devices (EAPAD), vol. 4329, 2001, pp. 335–349.

7] R. Trujillo, J. Mou, P.E. Phelan, D.S. Chau, Investigation of electrostrictive poly-mers as actuators for mesoscale devices, The International Journal of AdvancedManufacturing Technology 23 (2004) 176–182.

8] M. Wissler, E. Mazza, Modeling of a pre-strained circular actuator made of dielec-tric elastomers, Sensors and Actuators A 120 (2005) 184–192.

Biographies

Dr. Mark Potter holds a post-doctoral research position in the Flow Control groupof the Department of Aeronautics at Imperial College London. After completing aDPhil at Oxford University, investigating the stability of non-return valves in cabinbleed systems and developing hardware-in-the-loop control testing methods forthese valves, he moved to Imperial College to work on control of turbulent flows.The increasing use of electro-active polymers (EAPs), as actuators for the control offluids, leads to work on developing improved actuator designs. To facilitate this workhe has developed a series of numerical models to predict actuator displacement andtherefore optimize actuator designs. More recently his work has moved on to lookat the control of flow over a backward facing step using synthetic jet actuators andcontrol based state observers.

Mr. Kevin Gouder holds a pre-doctoral position in the BioMedical Flow Group ofthe Department of Aeronautics at Imperial College London. Mr. Gouder will defendhis PhD thesis entitled “Turbulent Friction Drag Reduction Using Electroactive Poly-mer Surfaces” in September 2010. In this dissertation, work was carried out on thereduction of friction-drag by means of spanwise wall oscillations. Two such actua-tors were developed, namely, a flat plate driven by an electromagnetic linear motorand an electroactive polymer (EAP) version as a first step towards the develop-ment of a travelling wave actuator. The effects of the control were measured withhot-wire anemometry, PIV and directly using a bespoke drag balance. Through hisexperience in EAP actuator fabrication, he helped in the experimental validation ofthe finite difference code for EAP development. More recently, his work has movedto human, in-vivo measurements of air-flow in the human air-ways using hot-wireanemometry in a multi-disciplinary effort to better understand the dynamics ofhuman breathing and, aerosol deposition studies in the human upper airways andtrachea in the early stages of the development of inhaled (i.e. non-invasive) methodsof drug-administration.

Prof. Jonathan Morrison holds the chair of Experimental Fluid Mechanics in theDepartment of Aeronautics at Imperial College London. Much of his work has beeninvolved with wall turbulence including coherent structures and in particular, theirrelation to changes in imposed length-scale, to energy transfer and to pressure fluc-tuations. Recent work has concentrated on control of turbulent flows, and the useof micro-electro-mechanical (MEMS) devices for active control of instabilities andturbulence. His present approach focuses on novel approaches to the control of fluidflow using electro-active polymers (EAPs) in the form of dimples, time-dependentsurface depressions for the generation of on-demand vortices, and development ofEAP for surface sensing in the development of a ‘smart’ surface. In the first call of the‘Active Aircraft’ programme, work began on the use of ink-jet printing to fabricate,with EAP, monolithic smart flow-control surfaces. With further support from Air-bus and QinetiQ, surfaces capable of generating travelling surface waves are beingdeveloped. Novel algorithms are being formulated for real-time feedback controlusing wall-based sensing and actuation. Ferrari S.p.A. supports his studies of pulsedjets for the control of separated flows. As Visiting Fellow, he has recently made fre-

University, for collaboration on the internationally recognised Superpipe project. Hehas recently investigated the effects of large roughness on turbulent channel flow.He chaired the Scientific Committee for the IUTAM Symposium, Flow Control andMEMS, held at Imperial College (2006), the proceedings of which have recently beenpublished by Springer.