O

UD

a

ARR2AA

KMDVRM

1

wteF2btafitoiidmaEb(eta2

0h

Mechanics Research Communications 55 (2014) 72– 76

Contents lists available at ScienceDirect

Mechanics Research Communications

jo ur nal ho me pag e: www.elsev ier .com/ locate /mechrescom

n the resonance frequencies of a membrane of a dielectric elastomer

ttam Kumar Chakravarty ∗

epartment of Mechanical Engineering, University of New Orleans, 2000 Lakeshore Drive, New Orleans, LA 70148, USA

r t i c l e i n f o

rticle history:eceived 21 September 2012eceived in revised form6 September 2013ccepted 12 October 2013vailable online 24 October 2013

a b s t r a c t

The resonance frequencies of a pre-stretched circular membrane of a dielectric elastomer are investigated.The resonance frequencies increase with mode and thickness of the membrane, but they decrease in airfrom those in vacuum due to the added mass of air. The damping of air is low and has negligible effecton the frequencies; however, it helps to reduce the amplitude of vibration, comparing with that in thevacuum. The frequencies decrease with an increase of the applied voltage, the mass of the electrodes,and the radius of the circular membrane. The effect of applied pressure on the resonance frequencies of

eywords:embraneielectric elastomeribration

the membrane is not significant.© 2013 Elsevier Ltd. All rights reserved.

esonance frequenciesode shapes

. Introduction

Dielectric elastomers are smart and electroactive materialshich deform due to the applied voltage. The membrane of dielec-

ric elastomers has many applications (Heydt et al., 2000; Duboist al., 2008; Koh et al., 2009; McKay et al., 2010; Zhu et al., 2010;ox and Goulbourne, 2008, 2009; Chakravarty and Albertani, 2011a,012) that include in robotics, cardiac membrane pump, an artificialicep for orthotic and prosthetic technology, programmable hap-ic surfaces, loud speakers, micro air vehicles, energy harvesting,ctive noise control, sensing, and RF filtering where the resonancerequencies may need to be very precisely controlled. It is verymportant to investigate the resonance frequencies of the dielec-ric elastomer’s membrane for these types of device design andptimization. Recently, many people are investigating the mechan-cal properties of the membrane of dielectric elastomers due tots diverse applications, ease of fabrication, low cost, and higheformability. Quasi-static experiments were conducted for esti-ating the hyperelastic material properties of the membrane (Fox

nd Goulbourne, 2008, 2009; Chakravarty and Albertani, 2011b).xperimental vibration analysis of the membrane was presentedy Fox and Goulbourne (2008, 2009), Chakravarty and Albertani2011a, 2012), and Jenkins and Korde (2006). It is found from thexperimental results that the resonance frequencies and ampli-

ude of vibration of a membrane change due to the added massnd damping of surrounding air (Chakravarty and Albertani, 2011a,012).

∗ Tel.: +1 504 280 6191; fax: +1 504 280 5539.E-mail address: uchakrav@uno.edu

093-6413/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.mechrescom.2013.10.006

The nonlinear vibration characteristics of the membrane wereinvestigated by a few researchers (Zhu et al., 2010; Fox andGoulbourne, 2008, 2009; Chakravarty and Albertani, 2011a, 2012;Gonc alves et al., 2009; Jenkins and Leonard, 1991; Mockensturmand Goulbourne, 2006; Tuzel and Erbay, 2004; Jiang et al.,1992), where hyperelastic material models were considered.There are several hyperelastic material models available, suchas Mooney–Rivlin, neo-Hookean, Ogden, Yeoh, and Arruda–Boycehyperelastic material models (Chakravarty and Albertani, 2011b;Mooney, 1940; Rivlin, 1948; Treloar, 1944; Boyce and Arruda, 2000;Yeoh, 1993). Gonc alves et al. (2009) developed the analytical andfinite element (FE) models for examining the dynamic behavior ofa radially pre-stretched circular membrane without the effect ofvoltage and the added mass of surrounding fluid. Zhu et al. (2010)investigated the resonant behavior of a pre-stretched membrane ofa dielectric elastomer without the effect of added mass and damp-ing of surrounding air.

This paper presents the analytical and FE models for investigat-ing the vibration characteristics of a radially pre-stretched circularmembrane of a dielectric elastomer. The effect of voltage, addedmass, damping, and pressure on the resonance frequencies of themembrane is examined. The variations of resonance frequencieswith the mass of two electrodes, thickness and radius of the mem-brane are also investigated.

2. Analytical model

For the analytical solution, a flat circular membrane specimen ofinitial radius of Ri and initial thickness of hi is radially pre-stretchedand clamped at the boundary with a rigid ring of radius of Rf (stretch

earch Communications 55 (2014) 72– 76 73

raofpavc

f{tpop[m(c

w

wtfJ(e

ω

woeoadm

2c

3

eataTbTogtfp

ca

U.K. Chakravarty / Mechanics Res

atio � = Rf/Ri). Thin carbon grease electrodes are attached to the topnd bottom surfaces of the membrane for applying the DC voltagef E. The Mooney–Rivlin hyperelastic material model is consideredor the membrane and the Mooney–Rivlin hyperelastic materialarameters are C1 and C2. The out-of-plane deformation of a pointt an arbitrary location X (r, �) on the membrane is w(r, �, t) due toibration. The equation of motion can be expressed in cylindricaloordinate system as (Chakravarty, 2013).

∂2w

∂t2= cs

2

(∂2w

∂r2+ 1

r

∂w

∂r+ 1

r2

∂2w

∂�2

)(1)

or 0 ≤ r < Rf, 0 ≤ � ≤ 2� and w(Rf, �, t) = 0, where cs2 =

[2(C1 + C2) + 2C2(�2 − 1)](1 − 1/�6) − ε[E/(hf �)2} /�, ε ishe permittivity of the membrane, hf is the thickness of there-stretched membrane (hf = hi/� 2), and � is the densityf the membrane-electrode material system. The princi-al in-plane pre-stresses of the membrane are equal to2(C1 + C2) + 2C2(� 2 − 1)](� 2 − 1/� 4) − εE2. The free vibration

odes wmn(r, �, t) are found by solving the above equationChakravarty, 2013), based on the required boundary andontinuity conditions, i.e.

mn(r, �, t) = AmnJm

(�mn

r

Ri

)cos(m�) cos(ωmnt) (2)

here Amn is an arbitrary constant and known as the modal ampli-ude of vibration; Jm is the m-th order (m = 0, 1, 2, . . ., ∞) Besselunction of the first kind; �mn are the zeros of the Bessel function,m (n = 1, 2, 3, . . ., ∞); ωmn is the circular resonance frequency of them, n) mode of vibration and can be calculated from the followingquation.

mn = �mm

√[2(C1 + C2) + 2C2(�2 − 1)](1 − 1/�6) − ε[E/(hf �)]2

�Ri2

(3)

The surrounding air exerts force and opposes the movementhen the membrane vibrates in air. The added mass is the mass

f the surrounding air that is required to accelerate for the accel-ration of the membrane. As a result, the resonance frequenciesf the membrane decrease due to the effect of added mass. Thedded mass depends on the geometry of the membrane and theensity of air. The added mass ma of the circular membrane isa = (8/3)R3

f�a when it vibrates in air (Chakravarty and Albertani,

011a, 2012; Azuma, 2006), where Rf and �a are the radius of theircular membrane and density of surrounding air, respectively.

. Results and discussion

For this paper, a circular membrane specimen of the dielectriclastomer VHB 4910 is radially pre-stretched at stretch ratio of 3nd attached to a circular rigid ring of radius of 50 mm. The ini-ial thickness, density, and dielectric constant of the membranere 1.0 mm, 960 kg/m3 and 4.55, respectively (Zhu et al., 2010).he Mooney–Rivlin hyperelastic material parameters of the mem-rane are C1 = 16 kPa and C2 = 7.3 kPa (Fox and Goulbourne, 2008).he mass of the electrodes is considered 4 times higher than thatf the membrane specimen (Zhu et al., 2010). The stretch ratio,eometry (radius and thickness) of the membrane, and the mass ofhe electrodes remain same as above for computing the resonancerequencies, excluding where they are mentioned in the following

aragraphs.

An FE model is also developed for investigating the vibrationharacteristics of the pre-stretched circular membrane, clampedt the boundary, using the FE analysis software, Abaqus 6.10®

Fig. 1. Resonance frequencies vs. voltage plots for the membrane specimen in air.

(SIMULIA, Providence, Rhode Island 02909, 2010). The FE modelis developed considering the following three steps.

(1) Calculate the principal in-plane pre-stresses of the membraneincluding the effect of the electrostatic force.

(2) Define the pre-stresses of the membrane as input data and runthe linear and nonlinear static analysis.

(3) Run the resonance frequency analysis of the membrane includ-ing the effect of damping, added mass of air, and mass of theelectrodes.

M3D6 (6-node quadratic triangular membrane) type of ele-ments are selected for the membrane. The effect of added massof surrounding air is included in the FE model. The added mass ofsurrounding air is added with the actual mass of the membranein the FE model. Rayleigh damping is considered and the damp-ing is provided in the FE model as Rayleigh damping parameters(Chakravarty and Albertani, 2011a, 2012; Cook et al., 1989). Theconvergence of the resonance frequencies of the membrane is stud-ied and it is found that the frequencies converge on the order of1000 degrees of freedom. It is also found that the FE model corre-lates well with the experimental data, reported by Chakravarty andAlbertani (2012).

Resonance frequencies of the pre-stretched circular membranespecimen at different voltages are computed in vacuum and inair (at atmospheric pressure) by using both the analytical and FEmodels and the first three mode resonance frequencies in air areshown in Fig. 1. A good correlation is found among the resonancefrequencies computed by the analytical and FE models (vary lessthan 0.05%). Fig. 1 depicts that resonance frequencies increase withmode. It is well known that the first (0, 1) mode frequency meansthe fundamental (lowest) frequency of vibration. So the second (1,1) mode frequency is higher than that of the first mode and so on.Resonance frequencies of the membrane specimen decrease 4.54%in air from those in vacuum due to the added mass of surroundingair. The frequencies decrease because the mass of the membraneincreases due to the added mass of the surrounding air, although thestiffness of the membrane remains constant. Fig. 1 also shows thatresonance frequencies decrease with voltage. The applied voltagehelps to reduce the internal stress of the pre-stretched mem-brane (Zhu et al., 2010). As a result, the stiffness of the membranedecreases which leads to the decrease of the resonance frequenciesdue to an increase of applied voltage.

The variation of the first (0, 1) mode resonance frequency withthe ratio of the masses of the electrode and the membrane speci-men at three different voltages (0, 5, and 10 kV) in air is shown inFig. 2. The ratio is calculated by dividing the mass of the electrodes

74 U.K. Chakravarty / Mechanics Research Communications 55 (2014) 72– 76

Fa

b5mmt

ebFwctmTpatoba

tvi

paring with that in the vacuum (Chakravarty and Albertani, 2011a,

Fv

ig. 2. First (0, 1) mode resonance frequency vs. ratio of the masses of the electrodesnd the membrane specimen plots in air.

y the mass of the circular deformed membrane of radius of0 mm. The frequency decreases with an increase of the ratio of theasses of the electrode and the membrane because the mass of theembrane-electrode specimen increases, although the stiffness of

he membrane remains constant.The combined effect of added mass of air and mass of the

lectrodes on the first (0, 1) mode resonance frequency of the mem-rane of the dielectric elastomer is also investigated and shown inig. 3. The variation of the first (0, 1) mode resonance frequencyith the ratio of the mass of the electrodes and the mass of the

ircular deformed membrane of radius of 50 mm, and the ratio ofhe added mass of air and the mass of the same membrane speci-

en at three different voltages (0, 5, and 10 kV) is shown in Fig. 3.he density of air can change due to the change of the atmosphericressure and as a result, the added mass of the surrounding air canlso vary. The frequency decreases with an increase of the ratio ofhe masses of the electrode and the membrane, and also the ratiof the added mass of air and the mass of the membrane specimenecause the mass of the membrane-electrode specimen increases,lthough the stiffness of the membrane remains constant.

The variation of the first (0, 1) mode resonance frequency with

he initial thickness hi of the membrane specimen at three differentoltages (0, 5, and 10 kV) in air is shown in Fig. 4. The frequenciesncrease with an increase of the thickness of the membrane due

ig. 3. Plots of the variation of the first (0, 1) mode resonance frequency of the membroltages.

Fig. 4. First (0, 1) mode resonance frequency vs. initial thickness of the membranespecimen plots in air.

to the combined effect of the added mass of surrounding air andthe applied voltage (if applicable). The frequency increases quiterapidly initially and then increases slowly with the thickness of themembrane. The effect of thickness on the frequency is negligiblewhen the initial thickness of the membrane is greater than 10 mm.

The variation of the first (0, 1) mode resonance frequency withthe radius of the deformed circular membrane specimen (i.e., radiusof the ring Rf) with stretch ratio of 3 at three different voltages(0, 5, and 10 kV) in air is shown in Fig. 5. (The corresponding ini-tial/undeformed radius Ri of the membrane can be calculated bydiving the deformed radius with the stretch ratio.) The frequencydecreases quite rapidly initially and then decreases slowly with theradius of the deformed circular membrane specimen.

It is found from the previous experimental investigation thatthe damping ratios are about 0.5% when the membrane specimenis vibrated in air (Chakravarty and Albertani, 2011a, 2012) andresonance frequencies of the membrane specimen in air decrease0.001% when compared with those in vacuum due to the damp-ing. The small damping does not have influence on the resonancefrequencies, but it helps to reduce the amplitude of vibration, com-

2012).FE model is also developed for investigating the effect of pres-

sure on the resonance frequencies of the pre-stretched membrane

ane specimen with the mass of the electrodes and the added mass of air at three

U.K. Chakravarty / Mechanics Research Communications 55 (2014) 72– 76 75

Fig. 5. First (0, 1) mode resonance frequency vs. radius of the deformed circularmembrane specimen plots in air.

Fp

sdu(

Fig. 7. Change of the volume enclosed by the deformed membrane specimen withpressure plots in air.

Fp

ig. 6. Static out-of-plane deformation of the membrane specimen in air at appliedressure of 500 Pa and voltage of 5 kV.

pecimen. The membrane specimen is deformed initially into a

ome shape due to the applied pressure, shown in Fig. 6. The vol-me between the initial and deformed shapes of the membranei.e., the volume of the dome) is estimated. The volume enclosed

ig. 9. (a) 1st (0, 1) mode shape at 124.23 Hz; (b) 2nd (1, 1) mode shape at 198.11 Hz; andressure of 500 Pa and voltage of 5 kV.

Fig. 8. First (0, 1) mode resonance frequency of the membrane specimen vs. appliedpressure plots in air.

by the deformed membrane (i.e., static out-of-plane deformation)increases with pressure, depicted in Fig. 7. After applying the pres-sure, the frequencies of the statically deformed membrane of adome shape are computed from the FE model and the analyticalmodel is not valid in this situation. The first (0, 1) mode resonance

frequency of the membrane specimen increases less than 0.15 Hzdue to an increase of pressure from 0 to 1000 Pa, shown in Fig. 8.The effect of pressure on the frequency of the membrane specimen

(c) 3rd (2, 1) mode shape at 265.38 Hz of the membrane specimen in air at applied

7 earch

issat

4

imuodsmodeipei

A

CL

R

A

B

C

C

6 U.K. Chakravarty / Mechanics Res

s minimal because the stiffness of the membrane does not changeignificantly due to an increase of pressure. The first three modehapes of the membrane specimen from the FE model in air atpplied pressure of 500 Pa and voltage of 5 kV are shown in Fig. 9,o get an idea about the mode shapes of the membrane.

. Conclusions

This paper presents the analytical and FE models for investigat-ng the vibration characteristics of a radially pre-stretched circular

embrane specimen of a dielectric elastomer in air and in vac-um. Resonance frequencies increase with mode and the thicknessf the membrane, but they decrease in air from those in vacuumue to the effect of added mass of surrounding air. Damping of air ismall and has minor influence on the resonance frequencies of theembrane, although it helps to reduce the out-of-plane amplitude

f vibration, comparing with that in the vacuum. The frequenciesecrease with an increase of the applied voltage, the mass of thelectrodes, and the radius of the circular membrane. The membranes deformed initially due to the applied pressure. The static out-of-lane deformation of the membrane increases with pressure. Theffect of pressure on the resonance frequencies of the membranes minimal.

cknowledgment

The author appreciates support through a National Researchouncil Research Associateship Award at the U.S. Air Force Researchaboratory.

eferences

zuma, A., 2006. The Biokinetics of Flying and Swimming. American Institute ofAeronautics & Astronautics, Inc., Reston, VA.

oyce, M.C., Arruda, E.M., 2000. Constitutive models for rubber elasticity: a review.Rubber Chemistry and Technology 73, 504–523.

hakravarty, U.K., 2013. Analytical and finite element modal analysis of a hyper-elastic membrane for micro air vehicle wings. ASME Journal of Vibration andAcoustics 135 (5), 051004:1–6.

hakravarty, U.K., Albertani, R., 2011a. Modal analysis of a flexible membrane wingof micro air vehicles. Journal of Aircraft 48 (6), 1960–1967.

Communications 55 (2014) 72– 76

Chakravarty, U.K., Albertani, R., 2011b. Energy absorption behavior of a hypere-lastic membrane for micro air vehicle wings: experimental and finite elementapproaches. International Journal of Micro Air Vehicles 3 (1), 13–23.

Chakravarty, U.K., Albertani, R., 2012. Experimental and finite element modal analy-sis of a pliant elastic membrane for micro air vehicles applications. ASME Journalof Applied Mechanics 79 (2), 021004:1–6.

Cook, R.D., Malkus, D.S., Plesha, M.E., 1989. Concepts and Applications of FiniteElement Analysis. John Wiley & Sons, Inc., New York.

Dubois, P., Rosset, S., Niklaus, M., Dadras, M., Shea, H., 2008. Voltage control of theresonance frequency of dielectric electroactive polymer (DEAP) membranes.Journal of Microelectromechanical Systems 17, 1072–1081.

Fox, J.W., Goulbourne, N.C., 2008. On the dynamic electromechanical loading ofdielectric elastomer membranes. Journal of the Mechanics and Physics of Solids56, 2669–2686.

Fox, J.W., Goulbourne, N.C., 2009. Electric field-induced surface transformationsand experimental dynamic characteristics of dielectric elastomer membranes.Journal of the Mechanics and Physics of Solids 57, 1417–1435.

Goncalves, P.B., Soares, R.M., Pamplona, D., 2009. Nonlinear vibrations of a radiallystretched circular hyperelastic membrane. Journal of Sound and Vibration 327,231–248.

Heydt, R., Pelrine, R., Joseph, J., Eckerle, J., Kornbluh, R., 2000. Acoustical perfor-mance of an electrostrictive polymer film loudspeakers. Journal of the AcousticalSociety of America 107, 833–839.

Jenkins, C., Korde, U., 2006. Membrane vibration experiments: an historical reviewand recent results. Journal of Sound and Vibration 295, 602–613.

Jenkins, C.H., Leonard, J.W., 1991. Nonlinear dynamic response of membranes: stateof the art. Applied Mechanics Reviews 44, 319–328.

Jiang, L., Haddow, J.B., Wegner, J.L., 1992. The dynamic response of a stretched cir-cular hyperelastic membrane subjected to normal impact. Wave Motion 16,137–150.

Koh, S.J.A., Zhao, X., Suo, Z., 2009. Maximal energy that can be converted by a dielec-tric elastomer generator. Applied Physics Letters 94 (262902), 1–3.

McKay, T., O’Brien, B., Calius, E., Anderson, I., 2010. An integrated, self-primingdielectric elastomer generator. Applied Physics Letters 97 (062911), 1–3.

Mockensturm, E.M., Goulbourne, N., 2006. Dynamic response of dielectric elas-tomers. International Journal of Non-Linear Mechanics 41, 388–395.

Mooney, M., 1940. A theory of large elastic deformation. Journal of Applied Physics11 (9), 582–592.

Rivlin, R.S., 1948. Large elastic deformations of isotropic materials. IV. Further devel-opments of the general theory. Philosophical Transactions of the Royal SocietyA 241 (835), 379–397.

Treloar, L.R.G., 1944. Strains in an inflated rubber sheet and the mechanism of burst-ing. Transactions of the Institution of the Rubber Industry 19, 201–212.

Tuzel, V.H., Erbay, H.A., 2004. The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension. InternationalJournal of Non-Linear Mechanics 39, 515–537.

Yeoh, O.H., 1993. Some forms of the strain energy function for rubber. Rubber Chem-istry and Technology 66, 754–771.

Zhu, J., Cai, S., Suo, Z., 2010. Resonant behavior of a membrane of a dielec-tric elastomer. International Journal of Solids and Structures 47 (24),3254–3262.