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Modal Analysis of a Flexible Membrane Wing of Micro Air Vehicles

Uttam Kumar Chakravarty

U.S. Air Force Research Laboratory, Eglin Air Force Base, Florida 32579

and

Roberto AlbertaniOregon State University, Corvallis, Oregon 97331

DOI:10.2514/1.C031393

A biologically inspired exible wing for micro air vehicles applications is constructed with a prestrained latex

membrane, attached to a thin aluminum ring with elliptical planform shape. The wing is placed inside a low-speed

wind tunnel and the strains of the membrane due to the aerodynamic loads are measured using the visual image

correlation technique at different anglesof attackand freestream velocities. Finiteelement models aredeveloped for

investigating the modal characteristics of the wing including the effect of added mass, damping, and aerodynamic

loads.For validatingthe nite element models, experimentalmodal analysis of a prestrainedmembrane is conducted

inside a vacuum chamber at different ambient pressures. Natural frequencies of the wing increase with mode and

strainlevelof themembrane, butdecreasein airfromthosein vacuum, dueto theaddedmassof air. Damping ofair is

low and has minimal effect on the natural frequencies of the wing, but assists to reduce the out-of-plane modal

amplitude of vibration.

Nomenclature

a,b = major and minor radii of an ellipse, respectivelyC1,C2 = hyperelastic MooneyRivlin material parametersc = viscous damping coefcientF = forcef = frequencyh = thicknessk = stiffnessm = massR = radius of a circleu,v,w = Cartesian coordinate displacementsV

1 = freestream velocity

x,y,z = Cartesian coordinate directionsm,k = Rayleigh damping parameters = damping ratio = density! = circular frequency (2f)

I. Introduction

M ICRO air vehicles (MAVs) are small (wing span on the orderof 100 mm) and can y at low ight speeds (up to15 m=s) inareas, where operate large aircraft are not feasible or expensive tooperate, especially for surveillance and measurement purposes. Thedesignand operation of MAVs of similar proportions to natural yersemphasize the intricate but vital aeroelastic features mastered by

biological systems. A particular form of these enhanced yingabilities benets from the use ofexible lifting surfaces: eitherxedor apping. Birds and bats twist and bend their wings whilemaneuvering for optimal aerodynamics. Locusts use specializeddome-shaped sensory organs (campaniform sensillae) within thestructure of their wings [1]. These feedback sensors respondspecically to wing deformation in order to trigger thewing structure

to operate at resonancefrequency. Furthermore, bats cancontrol theirwing characteristics by changing the level of prestrain in their wingsmembrane, thus effectively changing the wing camber and thepassive aeroelastic dynamic feedback of the membrane to the aero-dynamic loading [2]. Insects are by far the oldest, most numerous,and smallestying machines. Wootton [35] andEnnosand Wootton[6] investigated various insects, morphology of insect wings, ightmechanics, and control behavior. Combes and Daniel [7] and Danieland Combes [8] examined the effect of aerodynamic and inertial-elastic forces on the bending deformation and modal characteristicsof insect wings. Wings of the insects are twisted for controlling theight. Clearly, wing stiffness distribution and exibility are essential

aspects when considering natural

yers. The well-known class ofmaterials which function in supportive systems through deformationhas been classied by biologists as pliant materials and includeproteins, soft connective tissues, and cartilage.

Biologically inspired MAV wings (Fig. 1) are constructed byattaching prestrained hyperelastic membrane to metallic/compositereinforced structures and the modal characteristics (natural frequen-cies and mode shapes) of the wings are controlled mainly by theprestrained membrane due to signicantly lower stiffnesses thanthose of metallic/composite reinforced structures. The modal char-acteristics of thewings canbe tuned by changing theprestrainlevel ofthe membrane and also depend on the added mass, damping, andaerodynamic pressure of surrounding air.

Recently, there has been signicant progress in the understandingof the aerodynamics of low Reynolds number articial and natural

ight [9], though the structural dynamics is still under earlynumerical modeling efforts. The aerodynamic models and ightcontrol design ofxed [9] and apping wings [10] must include thewing exibility and structural dynamics, an area where very littleexperimental data is available. Experimental modal analysis of themembrane was presented by Chakravarty and Albertani [11], Sewallet al. [12], Gaspar et al. [13], and Jenkins and Korde [14]. Graveset al. [15] conducted the dynamic deformation measurements of aMAVRICI semispan wing in a wind tunnel. Shape and strainmeasurements using visual image correlation (VIC) and aerodynam-ic coefcients evaluations for different congurations of membraneMAV wings in the wind tunnel were performed in steady conditions[16,17]. Critical experimental work showed the dependency ofmodal characteristics of microstructures with ambient pressure [18].

Analytical and nite element (FE) models to explain the structuralbehavior of natural and articial xed and apping wings are at anearly stage of development. The large amplitude of wing s deform-ation, the nonlinear interaction with the ow, and the lack of

Received 2 February 2011; revision received 16 May 2011; accepted forpublication 17 May 2011. This material is declared a work of the U.S.Government and is not subject to copyright protection in the United States.Copies of this paper may be made for personal or internal use, on conditionthat thecopier paythe $10.00per-copy feeto theCopyright Clearance Center,Inc.,222 RosewoodDrive,Danvers, MA 01923; include the code 0021-8669/11 and $10.00 in correspondence with the CCC.

National Research Council Postdoctoral Research Associate, MunitionsDirectorate. Senior Member AIAA.Research Associate Professor, School of Mechanical, Industrial, and

Manufacturing Engineering. Member AIAA.

JOURNAL OF AIRCRAFTVol. 48, No. 6, NovemberDecember 2011

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quantitative experimental results for validation limit the numericalmodels applications. The nonlinear vibration of hyperelasticmembrane wasinvestigatedby a fewresearchers [1925]. Gonalveset al. [19] developed analytical and FE models for examining thedynamic behavior of a radially stretched circular hyperelasticmembrane without theeffectof added mass of surroundinguid. Zhuet al. [20] investigatedthe resonant behavior of a prestrained circular

membrane of a dielectric elastomer without the effect of damping ofsurroundinguid.This paper presents the FE models to evaluate the modal char-

acteristics of a circular membrane at different prestrain levels and aexible membrane wing of MAVs. The FE modal analysis of theprestrained circular membrane is validated by the experimentalresults. The prestrained membrane is attached to a rigid circular steelring, mounted on the shaker, andplacedinside a vacuumchamber forinvestigating the effect of added mass and damping of air on themodal characteristics at different ambient pressures. The exiblemembrane wing of MAVs is constructed by attaching the prestrainedmembrane to a thin elliptical aluminum ring. The wing is placed in alow-speed wind tunnel, where the strain levels of the membrane dueto aerodynamic loads are measured at different angles of attack andfreestream velocities. FE model is developed for investigating the

effect of added mass, damping, and aerodynamic loads on the modalcharacteristics (natural frequencies and mode shapes) of the wing.

II. Experimental Setup and Procedure

A. Test Specimen

A rubber latex membrane at different prestrain levelsis attached toa circular steel ring of inner andouter diameters 102.5 and113.2 mm,respectively, as shown in Fig. 2. The steelring is considered rigid dueto signicantly high stiffnesses comparing with those of latexmembrane and the steel-ring-membrane specimen provides only the

membrane characteristics. The thickness and density of the latexmembrane are estimated 0:1016 0:0508 mm and 980 kg=m3,respectively, by the manufacturer (TAN Thera-Band band, HygeniCorporation). The average thickness of the membrane specimens ismeasured by the authors and 0:15 0:01 mm is a reasonableassumption for the FE models. MooneyRivlin material model[26,27] is considered for this hyperelastic latex membrane. The

MooneyRivlin material parameters are calculated based on theuniaxial tension experimental data and the parameters are C1 18:088E4 Paand C2 18:088E3 Pa[28].

A exible membrane wing (Fig. 3) is constructed for investigatingthe effect of added mass, damping, and aerodynamic loads on themodal characteristics. The prestrained latex membrane (thickness of0:15 0:01 mm) is attached to an elliptical aluminum ring(thickness of 1.5 mm) of major and minor inner radii, 96 mm and46 mm; and outer radii, 100 mmand 50 mm; respectively. The ring isattached (symmetric to the minor axis) to a thin rectangular plate ofdimensions,100 3 10 mmwith a round of radius, 10 mm at acorner. The rectangular plate is attached to thexture inside the windtunnel and the plate is rounded at a corner (which faces the airowdirection) for reducing the drag. Aluminum Al 6061 T6 (density of2700 kg=m3, modulus of elasticity of 68.9 GPa, and Poissons ratio

of 0.33) is selected as the material of the ring and the plate.

B. Membrane Deformation Measurement

It is important to measure the applied strains at different directionson the membrane because the modal characteristics depend on thestrain level of the membrane. The strain levels of the membrane aremeasured using a noncontact method, VIC technique [29]. Imagesare captured with two high-speed Phantom version 7 CMOScameras, capable of storing 2,900 frames in an in-camera ash-memory buffer. Typical data results obtained from the VIC systemconsist of geometry of the surface in Cartesian coordinates (x, y,and z) and the corresponding displacements (u, v, and w). Apostprocessing option involves calculating the in-plane strains, "xx,

"yy, and"xy.

C. Experimental Setup

The membrane specimen with different prestrain levels is attachedto a circular steel ring, mounted on the shaker (shown in Fig. 2), andplaced inside the vacuum chamber (shown in Fig.4), where ambientpressure can be controlled for investigating the effect of added massof air on the modal characteristics of the membrane. The complexburst chirp signal is considered as the base excitation for all prestrainlevels of the membrane and the frequency sweeps range from 2 to1000 Hz [30]. The time varying out-of-plane deformation (w) of themembrane specimen is recorded by scanning laser Doppler vibro-meter, Polytec PSV-400. MEscope is used toprocess the data by fastFourier transform and identify the modes (modal frequencies anddamping ratios). The eigenvectors (mode shapes), eigenvalues(natural frequencies), and damping ratios of the membrane aremeasured at different ambient pressures inside the vacuum chamber.

Fig. 1 Two MAVs from the MAV Laboratory at the University of Florida, Gainesville, Florida.

Fig. 2 Prestrained latex membrane attached to the circular steel ringmounted on the shaker.

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The elliptical wing is also placed inside a low-speed wind tunnel atdifferent angles of attack and freestream velocities, where the strainlevel of the wings membrane due to the aerodynamic loads ismeasured using the VIC system.

III. Finite Element Models

The nite element (FE) models are developed for investigating themodal characteristics of a circular membrane and an elliptical wingusing the FE analysis software, Abaqus 6.9 [31]. The FE models aredeveloped using M3D6 (six-node quadratic triangular membrane)and C3D10 (10-node quadratic tetrahedron element) type ofelements for the membrane and aluminum ring attached to the plate,respectively. Elastic material model, where stress is directly propor-tional to strain (Hookes law), can not be used for the membrane dueto large nonlinear deformation behavior at quasi-static loading.

Mooney [26], Rivlin [32,33], Rivlin and Saunders [34], and Treloar[35] are the pioneers of developing the hyperelastic material models.There are several nonlinear hyperelastic material models available,such as MooneyRivlin, Ogden, neo-Hookean, Yeoh, and ArrudaBoyce [27,3645]. It is found that the MooneyRivlin and Ogdenhyperelastic material models are considered to be the most accuratefor predicting the deformation using biaxial and uniaxial stretchingtests [46]. FE models are developed based on the hyperelasticMooneyRivlin material model of themembrane. The added mass ofsurrounding air is added in the FE models. The damping is providedin the FE models as Rayleigh damping parameters. The convergenceof thenatural frequencies of thecircular membrane is studied andit isfound that thefrequencies converge even at lower degreesof freedom(on the order of 1000). The strain level of the membrane of theelliptical wing due to the aerodynamic loads is calculated from thewind-tunnel test data at different angles of attack and freestream

Fig. 3 Undeformed and mode shapes of the ring, membrane, and wing for spatial average membrane prestrains,"xx 0:0464 and"yy 0:0334 (noaerodynamic strain).

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velocities. These aerodynamic strains are added with theprestrains ofthe membrane and provided as input data for the FE models. The

in-plane shear strain"xyis neglected for the FE models because "xyissmall, compared with"xxand "yy.

A. Added Mass

For simplicity, a single-degree-of-freedom spring-mass-dashpotsystem is considered for explaining the concept of added mass of thesurrounding air. The equation of motion for free vibration of thesystem in vacuum is

m x c _x kx 0 (1)

wherem, c, k, and x are mass, viscous damping coefcient, stiffness,and absolute displacement of the mass, respectively.

The surrounding air exerts force, Fa when it vibrates in air. The

added mass maof the surrounding air opposes the movement duringvibration. In this situation, the equation of motion is

m x c _x kx Fa (2)

) me x c _x kx 0 (3)

whereFa ma x andthe effective massme mma. Thenaturalfrequencyfof the system is

f 12

k

me

s (4)

As a result, thenatural frequenciesdecrease dueto theeffectof added

mass. The damping ratio can be calculated from the followingequation:

ckme

p (5)

So the added mass is the mass of the air that is required to acceleratefor the acceleration of the body. The added mass depends on thegeometry of the body and the density of air. The added mass macanbe calculated for a circular membrane when it vibrates in air from thefollowing equation [47]:

ma 83R3f (6)

where R and fare the radius of the circular membrane and thedensity of surrounding air, respectively. The density of the circularmembrane is increased due to the added mass of air and can becalculated by dividing the added mass with the volume of themembrane specimen from the following equation:

a added mass

volume of the membrane specimen 8

3

R

h

f (7)

whereh is the thickness of the membrane. As a result, the effectivedensity of the membrane specimen is, e a.

The added mass ma can be calculated for an elliptical thinspecimen when it vibrates in air and a 2b from the followingequation [47]:

ma 0:414

3a

2

bf (8)

where a, b, and fare major and minor radii of the elliptical thinspecimen and density of surrounding air, respectively. The effect ofadded mass is consideredfor the membrane andring, butis neglectedfor the attached plate of the wing for the FE models.

B. Damping

The surrounding air also adds damping. Rayleigh damping isconsidered for the FE models. In Rayleigh damping, the globaldamping matrix [c] is proportional to both the mass matrix [m] andstiffness matrix [k] by the constants mand kand can be expressedas

c

m

m

k

k

(9)

where

m 2!1!21!2 2!1

!22 !21k

22!2 1!1!22 !21

and!nand narethe nth-mode naturalfrequencyin vacuumandnth-mode damping ratio, respectively. The natural frequencies of aspecimen are calculated using the FE model in vacuum.

IV. Results and Discussion

A. Validation of the Finite Element Models

The FE models are validated by experimental modal analysis of a

prestrained latex membrane attached to a circular steel ring. Therstthree natural frequencies of the membrane specimens are computedat different ambient pressures inside a vacuum chamber by bothlaboratory experiments and FE modeling and shown in Figs.5and6for two different prestrain levels. The frequencies at zero ambientpressure indicate that the frequencies are computed in vacuum by theFE model. It is not possible to perform experiments at ambientpressure less than 9.325 kPa with the available vacuum chambersetup. So the modal characteristics of themembrane can notbe foundthrough experiments in vacuum (without the effect of added mass). Itis noted from Figs. 5 and6 that natural frequencies increase withprestrain level of the membrane and mode, but reduce with ambient

Fig. 4 Membrane specimen mounted on the shaker and positionedinside the vacuum chamber. Partial VIC system is also shown in the

gure.

Fig. 5 Natural frequenciesvs ambient pressure plots forthe membrane

specimen with spatial average"xx 0:0524 and"yy 0:0579 prestrainlevel.

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pressure. Natural frequencies decrease due to increase in ambientpressure because the added mass of air increases with ambientpressure and reduces the naturalfrequencies. Natural frequencies areexpected to vary between those computed by experimental and FEtechniques within an acceptable limit due to the nonuniform thick-ness nature of the membrane. Additionally, nonuniform prestraindistributions are a source of error, as well as the orthotropic nature ofthe membrane (from the rolling process). Damping ratios arecomputed from the experimental data and provided as input

parameters to the FE models. The variations of damping ratios anderror bars at 90% condence intervals of the membrane with ambientpressure for two different spatial average prestrain levels are shownin Figs. 7 and 8. Itis found fromFigs. 7 and 8 that the damping ratiosgenerally increase with ambient pressure, but decrease with mode.Damping ratios are small (n 0:5%approximately), can be noisybecause of experimental error, and have minor inuence on thenatural frequencies. For example, natural frequencies fn of themembrane reduce 0.00125% for damping ratios n 0:5% whencompared with those in vacuum due to

fn in fluid

fn in vacuum

1 2n

p

But damping ratios reduce the amplitude of vibration. The rst threemode shapes of the membrane specimen with spatial average"xx 0:0524and "yy 0:0579prestrain level from the FE model in air atatmospheric pressure are shown in Fig. 9.

B. Finite Element Model of the Elliptical Wing

The wing is constructed by attaching the latex membrane ofspatial average prestrains "xx 0:0464 6:0816E 05 and "yy0:0334 3:4599E 05 to a thin elliptical aluminum ring. Thevariations of the in-plane spatial average strains "xxand "yywith90%condence intervals (error bars) of the membrane of the wing due tothe aerodynamic loads with angle of attack at three differentfreestream velocitiesfrom the low-speed wind-tunnel testsare shownin Fig. 10. Some sorts of variations of the in-plane strains of the

membrane are expected at higher freestream velocities due to thevortex shedding. The in-plane strains of the membrane of the wingdueto theaerodynamicloadsdo notchange with theangleof attack atlower freestream velocities, 6 and 7:8 m=s. The in-plane strains ofthemembraneof thewingdue to theaerodynamic loads increase withangle of attack up to a certain limit and then decrease because of thestall of the wing at relatively higher freestream velocity,V1 13 m=s. Damping ratios are assumed 0.5% for the FE modelof the wing based on the experimental modal analysis of the circularmembrane, shown in Figs.7and8.

Natural frequencies of the wing increase with mode and strainlevel of the membrane. Mode shapesof thewing also changewith themode and strain level of the membrane. The rst three mode shapes

Fig. 7 Damping ratios vs ambient pressure plots for the membranespecimen with spatial average"xx 0:0524 and"yy 0:0579 prestrain

level.

Fig. 8 Damping ratios vs ambient pressure plots for the membrane


Fig. 9 Mode shapes for the membrane specimen with spatial average "xx 0:0524 and "yy 0:0579 prestrain level from the FE model in air atatmospheric pressure.

Fig. 6 Natural frequencies vs ambient pressure plots forthe membrane



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of the ring, membrane, and wing from the FE model in air atatmospheric pressure are shown in Fig. 3, where the effect ofaerodynamic loads is not considered. The natural frequencies of thering are higher than those of the membrane. As a result, the naturalfrequencies and mode shapes of the wing are dominated by themembrane characteristics.

The rst three mode shapes of the ring, membrane, and wing fromthe FE model in air at atmospheric pressure are shown in Fig. 11,where the effect of strains of the membrane due to the aerodynamic

loads at freestream velocity of13 m=sand angle of attack of 20 isconsidered. The natural frequencies of the membrane increase due tothe increase of the strain level from the previous case (Fig.3), wherethe effect of aerodynamic loads is not considered. The inuence ofthering on themodalcharacteristics of thewingincreaseswith mode.The rst natural frequency of the ring is higher than that of themembrane. So therstnatural frequency andmode shape of thewingare similar to the membrane characteristics. The second and thirdnatural frequencies of the wing are less than those of the membranedue to the deformation of the ring. The second mode shape of thewing shows a slightdeformation of thering, butthe third mode shapeof the wing clearly depicts the deformation of the ring as well as themembrane.

The variation of therst (fundamental) natural frequency of thewing with angle of attack at three different freestream velocities is

shown in Fig. 12. Natural frequencies of the wing remain unchangedwith angle of attack at lower freestream velocities, 6 and 7:8 m=s,butincrease (due to increase of strain level of the membrane) with angleof attack at relatively higher freestream velocity, 13 m=s, up to acertain limit and then decrease due to the stall of the wing.

Fig. 10 Variation of the in-plane spatial average strains: a)"xx andb)"yy of the membrane of the wing due to the aerodynamic loads with

angles of attack at three different freestream velocities.

Fig. 11 Mode shapes of the ring, membrane, and wing for spatial average membrane strains,"xx 0:0947 and"yy 0:0781 (prestrains"xx 0:0464and"yy 0:0334 and aerodynamic strains"xx 0:0483 and"yy 0:0447), freestream velocity of 13:0 m=s, and angle of attack of 20

.


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V. Conclusions

This paper presents the modal analysis of a circular latexmembrane and an elliptical wing for potential MAVapplications. TheFE model can predict the natural frequencies of the circular latex

membrane reasonably well comparing to those estimated byexperiments at different ambient pressures. A reasonable amount ofuncertainty is expected among the frequencies predicted by the FEmodel and experiments due to the variation of the thickness of themembrane, nonuniform prestrain distributions, and the orthotropicnature of the membrane (from the rolling process). It is found fromthe FE and experimental results that natural frequencies of themembrane decrease with increase in ambient pressure of the vacuumchamber by increasing the added mass effect of air. Naturalfrequencies of the membrane increase with mode and prestrain level,as expected.The damping ratiosare low andhavevery minimal effecton the natural frequencies, but contribute to the decrease of theamplitude of vibration of the membrane.

The FE model is developed for investigating the modal char-

acteristics of a

exible membrane wing of MAVs. The wing isconstructed by attachingthe prestrained membrane to a thin ellipticalaluminum ring. The aerodynamic strains and the prestrain of themembrane, added mass, and damping of air are considered for themodal characteristics of thewing. It is found that aerodynamic strainsofthe membrane increase upto the stall ofthewingandthendecreasewith angle of attack at relatively higher freestream velocity. Theinuence of angle of attack on aerodynamic strains of the membraneof the wing is minimal for lower freestream velocity. Naturalfrequencies of the wing increase with mode and strain level of themembrane therefore the modal frequencies are not signicantlyinuenced by the dynamic pressure loads for low and mediumvelocities. The inuence of the ring on the natural frequencies andmode shapes of the wing is increased with the strain level of themembrane by increasing the aerodynamic loads (i.e., by increasing

the angle of attack and freestreamvelocity upto the stall of the wing).The modal characteristics of the wing are dominated by themembrane at lower strain level of the membrane, but the ring alsoplays a signicant role at relatively higher strain level of themembrane.

Acknowledgments

This research was performed while therst author held a NationalResearch Council Research Associateship Award at the U.S. AirForce Research Laboratory. The authors would like to thank thesupport from the U.S. Air Force Ofce of Scientic Research undercontract FA9550-09-1-0072, with Victor Giurgiutiu (initiator) andDavid Stargel as project monitors. The continuing support forresearch activities from the U.S. Air Force Research Laboratories atEglin Air Force Base and Wright-Patterson Air Force Base is alsogreatly appreciated. The authors would also like to thank the experi-

mental support from Joshua Martin. The authors also appreciate thesupport from Mark Costello of Georgia Institute of Technology.

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Fig. 12 First mode frequency of the wing vs angle-of-attack plots atthree different freestream velocities.

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