Root, Courtland et alia 1999

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Proceedingsof the11th International Symposium on Unmanned Untethered Submersible Technology, 1999, Pp. 378-388.

SWIMMINGFISHANDFISH-LIKEMODELS:THEHARMONICSTRUCTUREOFUNDULATORYWAVESSUGGESTSTHAT

FISHACTIVELYTUNETHEIRBODIESRobertG.Root1,Hayden-WilliamCourtland2,CharlesA.Pell3,BrettHobson3,

EamonJ.

Twohig

2

,Robert

B.

Suter

2,William

R.

Shepherd

III

2,Nicholas

C.

Boetticher

2

andJohnH.Long,Jr.21

Dept.ofMathematics ,LafayetteCollege,Easton,PA,18042,(610)250-5280voice,[email protected]

2Dept.ofBiology,VassarCollege,Poughkeepsie ,NY,12604,

(914) 437-7305 voice, (914)437-7315 fax,[email protected] ,[email protected] ,[email protected] ,[email protected]

3NektonTechnologies ,Inc.,710WestMainStreet,Durham,NC 27701,

(919)682-3993 voice, (919)682-8779 fax,[email protected], [email protected]

Toevaluate the versimilitude of theoretical and mechanical models of fish and fish-likepropulsivesystems,wecompared quantitatively the motion of undulatory body wavesusing aFourier-basedmethod. Theundulatorymotionsofafreely-swimmin glampreywerecomparedtoaseriesof theoretical andphysical models. For each swimming trial, Fourier series of up tofive frequencieswerefit to thex(axial)andy (lateral)motionof31pointsreconstructedalongthe midlineof each fish or model. Using a multiple regression technique, only frequenciesthat significantly (p 0.05) described the axial and lateral displacements were included asexplanatory variables. We found that models of all types, and backward-swimmin g lamprey,undulate with harmonicsof higher-order than the fundamental frequency and 1st harmonic inlateral and axial motion, respectively. In contrast, the forward-swimmin g lamprey has farfewer higher-order harmonics . Sincehigher-orderharmonicsproducepoint trajectories that arekinematicallycomplexandenergeticallycostly,forward-swimminglampreyhavesufficientmechanicalmotivationtoswiminamannerconsistentwiththehypothesisthatfishareabletoactivelytunetheirbodies inorder toeithersuppressorremovehigherharmonicsfromtheirundulatorywavemotion.

Introduction

Using the body as a propeller, most fish swim using undulations, flexural body waves thattravel fromhead to tail. Thispropulsivewaveofmotion,oftenover-simplified as a sinusoidalwave of constant or linearly-increasing lateral amplitude, is geometrically complex, withmidline flexion accelerating fasterthanthelateralmotion(Cheng etal. 1998; Katz& Shadwick,1998; Jayne & Lauder,1995)andaxialmotions producing figures-of-eight trajectories at thetail (Bainbridge,1963). Consequently, thekinematiccomplexityofundulatoryswimming hasnotbeen accuratelyand quantitatively described. Only with the proper analytic tool can we measure andthus comparetheperformanceofswimmingfishwith the theoreticalandmechanicalmodelsoffish- likeswimmerscurrentlybeingproduced(Carlingetal.1998;Ekeberg,1993;Librizzietal,Inpress; McHenry et al. 1995; Root & Long, 1997). To develop this tool, and to test itsaccuracyand usefulnessonfishandfish-likemodels,isthegoalofthisstudy.

Giventhat the wavemotions of a steadily-swimmin g fish are periodic (Long et al. 1997),a Fourier-basedanalysis, inwhichperiodicmotion isdecomposed into its harmonic constituents ,was deemed appropriate. While Fourier-based methods have been employed previously(Videler & Hess,1984),wesoughttoinclude,inadditiontolateralmotion,(1)axialmotion and(2)astatistical basisfortheinclusionorexclusionofharmonics.
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Studiesofwhole-bodykinematicsandmuscleactivationserveas thebasisformanytheoreticalmodels. In an attempt to mechanically model undulatory swimmers, researchers have employedtwo generalapproaches: (1)neurobiological models basedoncoupledneuraloscillationsdirectedbythecentralpatterngeneratorand(2) dynamicbendingmodelsbased on fundamentalmechanicalprinciplesandhydrodynami cconsiderations . Someof themost importantneurobiologicalstudieswere

performed

using

the

marine

lamprey,

and

for

that

reason

we

use

lamprey

in

this

study.

Bydeveloping interaction schemes for different locomotor neurons, computer simulations

have demonstrated signal patterns which closelymimic the lampreys neural activity (Grillneretal.,1993;Grillneretal.1995). Ekeberg(1993) extended thesemodelsby simulatinga mechanicalenvironment inwhichanattachedneuralnetworkcould function. The results of this studyhave suggested that the neural control circuits charted by Grillner and others are sufficient indescribingthemotionofintactlamprey. Beginninginsteadwiththemechanicalprinciplesgoverningmotion,recent development s have modeled undulatory swimmers as continuous, dynamically-bending beams (Cheng et al. 1994; Cheng et al. 1998). In addition, the importance ofwhole-body accelerations, viscous forces, and rotational forces has been presented within thecontextofatwo- dimensionalundulatorymodel(Carlingetal.1998).

A central difficulty is faced by all models of undulatory swimming, whether they areneurobiological ,mechanicalortheoreticalhowdoestheinvestigatorknowif,and towhatdegree,the model mimics the undulatory motion of the biological system? Studies of whole-bodykinematicsandother invivoparameters givedetaileddescriptions which establish apparentlygeneralsimilaritiesanddifferencesamongspecies,but comparisons based on neuromechanicalprinciples or mathematical representation s are not easily derived from the resulting data.Descriptivemodels,ontheotherhand,areversatileintheirpredictivecapabilities,butareseverelylimited by the assumptions made and by the often unnatural physical restraints placed on themodel. Forexample,negligibleaxialmotions, simple sinusoidalmotion in the lateral dimension,and phase constraints have been employed throughout the literature (Videler & Hess, 1984;Grillneretal.1995;Gillis,1998). Thesesimplificationsmaskmanyof theunderlyingsubtletiesthatarevitaltoourunderstandingofundulatoryswimmingmaneuvers.

InthispaperwefitFourierseriesofstatistically-determine dharmonics toundulatorymotions,therebymathematicallydescribingthemotions thatcharacterizesteadyswimminginfishandfish-likemodels. Inaddition,weanalyzeandcompare themotionsofrealfishto those of theoreticalandmechanicalmodelsandfind,tooursurprise,thatforward-swimminglamprey lackmanyof thehigher-orderharmonicsseeninmodelsandbackward-swimmin g lamprey. Thissimplerharmonicstructure suggests that forward-swimmin g lamprey are actively (i.e., muscularly) tuningtheir undulatorymotioninordertosuppressorremovehigher-orderharmonics.

Wechosespecificmodelsandbehaviorsinordertoaddressthefollowingquestions(relevantmodelsorbehaviorsaregivenparentheticallyanddiscussedindetailintheMethodssection):

1. HowaccuratelydoestheFourier-basedmethodreconstructundulatorymotion?(theoreticalkinematicmodels)

2.Whatistheharmonicstructureoftheundulatorymotiongeneratedbyatravelingwaveofsinusoidallateraldisplacementorsinusoidalmidlineflexion?

(theoreticalkinematicmodels)3. Dotheoreticalandmechanicalmodelsdrivenbyinternalforcesinafluidproduceundulatorymotionwith

harmonicstructuredifferentfromthatproducedbysinusoidalwaves?(theoreticalplatemodelandmechanicaloscillatoryimpellers)

4. Doswimmingfishproduceanundulatorymotionwithharmonicstructuredifferentfromthatproducedbytheoreticalandmechanicalmodels?

(forwardandbackward-swimminglamprey)5. Doswimmingfishmodulatetheharmonicstructureoftheirundulatorymotioninordertoproducedifferent


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ModelsandAnimals MaterialsandMethods

Theoretica lmodels. Wepresentfourdifferent theoreticalmodels:(1)acontinuousconstant-lengthmidlinewithasinusoidalwaveofconstantlateralamplitude(10%oftotalbody length,L)travelingatconstantvelocity,(2)amidlinewith30segmentswithawaveof sinusoidally varyingjointflexion(increasingexponentiallyfromhead to tail;values from Gillis,1998) travelingatconstantvelocity,(3)amidlinewith30segmentswithawaveofsinusoidallyvaryingjointflexion(increasingexponentiallyfromheadtotail)travelingwithacceleratingvelocity(valuefromKatz&Shadwick,1998),and(4)acontinuouselasticplatewithinternalforcegeneration("muscles")andsimpleviscosityexternally(Librizzietal.Inpress). Pleasenotethatthefirstthreeof thesemodelsarekinematiconly;i.e.,motionismodeledwithoutreferencetointernalorexternalforces. Thesekinematicmodels sharedseveralfeaturestheheadoscillated laterallybutnotaxiallyand bodylength,L,washeldconstant.

Theelasticplatemodelisanumericalimplementationofaninitial-boundaryvalueproblem fora fourth-orderhyperbolicpartialdifferentialequationdescribed in detailelsewhere (Librizzi,etal.,In Press;Root&Long,1997). Themodeledfishwas20cminlengthwithamaximumthicknessof1cmandamaximumdepthof5cm. TheYoungsmodulus,E,ofthebody was 0.18 MPa, basedon bendingtestsofsunfish(Longetal.1994)andgar(Longetal.1996); for themodel, theheadwas assumed tobe five timesasstiffas thebody. The muscle stresses never exceeded 15%of themaximumof1.5x107 Nm-2 perfiberasmeasuredbyAltringhametal. (1993). The muscleswere activated ina sinusoidalpatternwithawavelengthof8cmanda frequency of 3.7 Hz.

Thismodelstwomostimportantlimitations aretheomissionofinertialfluidforcesactingonthefish(onlyviscousdampingisused),andthemodelingofthefish'sbodyasperfectlyelastic.Mechanicalmodels. Twothree-dimensiona lNektors(oscillatoryimpellers)werecreated atNektonTechnologies,Inc.usingapolyurethanecastingmolded intoa14cmlongfish-likebody.Themodels,ofidenticalshape,variedinmaterial stiffness, withthestiffNektorhavinganEof15.5MPaandtheflexibleNektorhaving an E of 5.5 MPa; modulus was measured usinga Shore A hardness durometer (model 306L, Pacific Transducer Corp.). Analogous to ourprevious method(McHenryetal.1995), themodelswereactuated toswimataconstantvelocity(netthrust=netdrag)inawatertunnelbywayofasteeldriveshaftcastwithinthemodelatapproximately0.2L driven by an oscillating couple from computer-drive n servomotor (model 630, brushlessDC, Compumotor). During swimming, the shaft was permitted to move laterally and axially.Nektors wer e drivenat a frequency of12Hzand a couple amplitude of 20, conditionsthat generated traveling waves of flexures and, hence, undulatory locomotion. Nektorswimming was captured ventrally at 250 images per second (ips) using high-speed video(model 1000, Red Lake) with images captured from amirror angled45 to thebottom of thewatertunnel.Lamprey. Theparasiticphasesealamprey,Petromyzonmarinus(L=14.1cm)used in thisstudywasoneofseveralobtainedfromtheLakeHuronBiologicalStation inMillersburg,Michiganandwashousedat20Cinan80LaquariumatVassarCollege. Forward and backwardswimmingtrials were recorded using high-speed video (Kodak model Ektapro 1000; 500 or 1000ips). Sequences of volitional swimming were captured in a still-water aquarium and ranged inswimming speeds from 0.164 to 2.844 body L/s-1. During filming,lamprey were backlit by a 500watt halogen light diffused through a sheet of 0.5 cm whiteacrylic inorder to capture clear, high contrast images. The ventral surface of each fish wastapedby reflecting the lightpathoff of a mirror locatedatanangleof45tothetanksbottom.Backward swimming in lamprey was initiated by gently approaching or stimulating the rostralendof the fish with a rubber tipped rod. Forward swimming was initiated by applying asimilar stimulus to the caudal end. Only swimming sequences where the fish swam ina


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straightlineatnearlyconstantvelocitywith the trailingedge


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passing through a minimum of two opposite lateral extremes were used for kinematicmeasurements.

DigitizingandmidlinereconstructionVideosequencesforlampreyandNektortrialsweredigitizedusingavideodeck(model

SVO-9500MD,Sony

Corp.)

linked

to

acomputer

(model

Macintosh

IIfx,

Apple

Corp.).

A

10

x

10cmsquaregridwasplacedinthevideofieldforcalibrationof thedigitizedimage. Usingavideo

genlock(modelTeleveyesPro,DigitalVisionInc.),thedigitallyfrozenimagewasoverlaidonthecomputerscreen,whichcontainedanx,ycoordinategrid(Imagesoftwareversion1.51,NIH). Fortheslenderlamprey,aseriesof20points wasmanuallyplottedon the fishs midlinefor 15-30framespertailbeatcycle(seeLongetal.1997);themidlinewasestimatedonscreenas thepointatagivenpositionequidistantfromtheleftandrightsidesofthefish. SincethebodyofaNektortaperssubstantially,wecouldnot,usingthemethoddescribedforlamprey,reliably digitize themidline;insteadweselected20pointsalongeachsideof theNektor. Acustomvideodigitizingprogram(Jayne&Lauder,1993)wasusedtoreconstructamidlineofnearlyequalsizedsegmentsfromthedigitizeddata. Foreachdigitizedimage, theprogramemployedcustomspline fitting toreconstructamidlinewith30bodysegments(31x,ybodypoints).Preliminary

Fourier

transforms

MidlinesfromfishandmodelswerereadintoacustomMathematicaprogram(FishFourier)foranalysis. Tobegin,a seriesof fastFourier transforms (FFT) foreachof the 31 body pointsidentified frequencycomponentsconstituting thecontinuouswaveform ofundulatory locomotion.These transforms, and all subsequent analyses, were conducted separately for x and ycoordinate data inorder toelucidateandverify theiruniquecontributions to theoverallmotion.Transforms were windowed, selecting for frequencies smaller than twice an estimated tailbeatfrequency (visual inspection) to prevent aliasing. The linear portion of the data, i.e., constantvelocitymotion,wasfitby least-squares at each body point and extracted from the FFT, leaving a filteredFourier transform (or fFt), which represents the lampreys periodic motion and acceleration.Velocities were then computed vectorially from every points fFt and averaged to give thecompositevelocity vector,s,ofeachlampreytrial. Withineachtrialsanalysis, themagnitudeanddirection of s never varied by more than 1% and 2, respectively. The frequency andamplitude of the maximum- amplitudefrequency(fundamentalfrequency, 1) in the windowwerefoundforeverybodypointineachtrial'sfFt. Frequencieswerethenaveragedusing theamplitudesasweightsand thetailbeatperiodwascalculatedto thenearestmultipleoftheinterframetime. Followingthe removalofconstantvelocitymotionandthedeterminationof the tailbeatperiod,a coordinate transformationwas performed on the x, y position data to yield a coordinate system in the fishs frameof reference; that is, the positive x-axis points in the direction of the fish's progression.

Thistransformationappliedarotationmatrixtoeachx,ydatapairyieldingtherespectivelytransformedcoordinatespairsuandv.Fourierfitting

Havingestablishedatailbeatperiodandaveragevelocityfor the fish, thedisplacementsweremodeled with Fourier series. In order to determine the smallest set of Fourier harmonicsthat accuratelypredict the separatebehavior of u and v data, we performed a forward selectionmultiple regression (Sokal&Rohlf,1981),modifiedforourparticularsituation. In the standardforward selection procedure, the order in which the independent, or predictor, variables areintroducedintothe model depends on either the strength of their partial correlation or ability tosignificantly increase thecoefficientofdetermination(r2 value). Because our predictorvariables are the individual Fourierfrequencyfunctionswehad an a priori reason to believethat contribution of each frequency to the coefficient of determination would decrease with


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increasingfrequency; i.e.,


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theeffectsofthefundamentalfrequency,whichcorresponds to thetailbeatfrequency inthelateraldirection, are expected to dominate. Thus, harmonics were added in order of increasingfrequency. Notice that this procedure thus provides a conservative test of the hypothesis thatfrequenciesother thanthefundamenta lareimportantinpredictingundulatorymotion.

Tobeginthefitting,thefirsttailbeatcycleofeachtrialwasfitwithaseriesofu,vregressions todetermine

statistically-significan t

components

of

the

undulatory

motion.

The

first

regression

gavethebestfitlinearfunctionforboththeu andv coordinates. For eachcoordinate independently,

it usestheF-statistic for the model to determinewhetherthe fit is statisticallysignificant,usinga specified thresholdp-value (0.05 for thisstudy). If the fit is significant , then t is one of thefitting functions used;ifnot,thentheanalysisbeginswithonlyconstantfunctionsused in the fit.The analysis proceeds through all the harmonics of the tailbeat cycle, beginning with thefundamental, 1,andendingwiththefifth, 5. Foreachharmonic,thecosineandsineareregressedagainstthefunctionsalreadybeingusedinthefittodeterminetheresiduals. Theseconstitutethecomponentsofthefunctionsorthogonaltothefittingfunctionsalreadybeingused. The residualsof thedata(thatisthecomponent notalreadyexplainedby thecurrentfittingfunctions) are then regressedagainst theresidualsof thecosineandsine. Using theF-statisticof thismodeland the samethresholdpvalueasabove,thesignificanceofthefitisassessed. Ifitisstatisticallysignificantthattheresidualsofthedataarefitby thecomponentsof thenewharmonicindependentof thecurrentfittingfunctions,thentheharmonicisaddedtothefittingfunctions. Aswiththelinearregression,thefittingofthe u andv coordinatesofapointaredoneindependentlyofoneanothertodescribethekinematics ofindividualbodypoints. Onceallthespecifiedharmonicshavebeen tested, thedataisregressedagainstthefitting functions to givethe finalresult. The fits for a givenbodypointsaxialandlateralmotionarerepresentedasfollows:

(1) up ~Cp0+Cp1t+pi(cos( pit+ pi)) axial,ui

(2) vp~Dp0+Dp1t+ pi(cos( pit+ pi)) lateral,vi

In these equations, Cp0 andDp0are the initial average values for the pth points u and vcomponents,respectively . TheseFourier expansionsarenot centeredon a horizontalline,as isusual(Haberman,1998);instead,theaveragepositionmovesalongalinewithslope Cp1 orDp1 fortheu orv component ofthepthpoint,respectively. Other than this lineardisplacementof theaverageposition,themotion is assumed periodic,and for thepurposes of this paper theperiodisthetailbeatperiod,T,whichisthetimerequiredforonefulloscillationof thetrailingedgeofthefish. Byharmonicanalysiswemeanthedescriptionof this periodicmotionas a sum ofoscillationswhosefrequenciesareintegermultiplesofthefundamentalfrequency 1= 2

T.

Thehigherfrequency, i= i 1,whereiisanintegergreaterthan1,iscalledthe(i1)stharmonic.Itisawell-knowntheoremthatperiodicmotioncanbedescribedasaninfinite sumof sines andcosinesofthesehigherharmonics(Haberman,1998). Ratherthantheusualpresentationof theithfrequency component of the motion as Aicos( i t)+Bisin( i t), we prefer to express theexpansionintermsofcosinefunctionswithamplitudeandphase. Thisisequivalenttotheusual

presentationusingthefollowingrelation. Fortheu componentstherelationis:i = Ai +Bis the

2 2

Bi

amplitudeof the motion,and tan i= Ai

. For v components, corresponds to ,and

correspondsto ;alltheseparametersareindexedonboth the frequency(i ) and thepoint (p )whosemotionisdescribed.


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The above equations treat each of the thirty-one midline points as discrete entities.Consequently ,theinclusionorexclusionofharmonicsisbasedsolelyonthemotionofthebody


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0 0

i

pointbeinganalyzed. Thisapproachenablesapointbypoin tcomparison of periodic motionalongthe lengthof thefishappropriateforelucidatingtheroleof localizedvariationingeneratingwhole-bodyswimmingmotions.

In our analysis, we havenot treated the infinitesequence of higher harmonic frequenciesrequiredbythemathematicaltheory;wetruncatedthehigherharmonicsatthefourthfivetimesthe

fundamental

frequency.

While

our

choice

of

the

fourth

harmonic

was

heuristic,

it

is

clear

fromenergeticconsiderations thatthemotionof the fish cannot havelargeamplitudemotionsathigh

harmonics. To see this, note that the energy required over a time interval is the integral of thepower expended over the interval, and the power is the product of the force exerted and thevelocity of the segment. That is, for a small segment of the fish undergoing periodic rigidmotion, theenergy requiredformotionatthe(i1)stharmonicisasfollows:

t1 2 3 t1

(3) Ei =

tmav dt= m

i i t

cos( it+ i)sin( it+ i)dt

wheremisthemass of thesegment. Overan entirecycleat this harmonic,that is, if the timeintervalt1t0haslengthT/i,theenergyiszero,since thesegmenthasperformedaclosed loop;noworkisperformed. However,oversmallerintervalswecanseethattheenergyrequired increaseswiththesquareofmultipleofthefundamental,i 2, and with the square of the amplitude 2.Considertheintervalbetweent = T and t0

2i 1 2

T asanexampleofanintervalwiththe4i

maximum possible energy required. Using standard Calculus, one can compute the energy 2 22

requiredoverthisintervaltobe Ei = 2m(T) i

i

. Thereare2isuchintervalsinatailbeatperiod,T.Sincethereareaninfinitenumberofharmonics,theenergyassociatedwiththe(i1)stharmonicmusteventuallygrowsmallwithincreasingi,andsimply to maintaina constant magnitudeofenergy the amplitude I must grow smaller on the order of 1/i. In practice, the fourthharmonic appears infrequent in themotionofactual fish,andalwayswithsmallamplitude. Bytruncating the series at this frequency, we lost no appreciabl e amount of the harmonicanalysiss ability to reconstructthemotion.Results

Steadilyswimmingfish-likemodels,oscillatoryimpellers,and lamprey all generate flexuralwaveswhosemotionisprimarily the sum of the two-dimensiona lperiodicmotionsat individualpoints(Fig.1). Thepresenceofharmonics,andtheirrelativeamplitude,variesasafunctionof thekind of swimmer (features of model or behavior of fish), the dimension (axial or lateral),and position

on

the

body

(Fig.

2).

In

response

to

the

questions

posed

in

the

Introduction,specific resultsarepresentedbelow.

1. The Fourier-based technique is highly accurate. As measured by the coefficientof determination(r2 value), the Fourier-based fitting of the harmonic structure of the axial andlateral motionofbodypointsonundulatorymodelsrecoversnearly100% of the totalvariance inmotion (Fig.2,thefirstthreeoffourtheoreticalmodels).2. Theharmonicstructureoftheundulatorymotiongeneratedatravelingwaveofsinusoidallateraldisplacemen t or sinusoidal midline flexion is dominated, in amplitude, by thefundamental frequency in v and the first harmonic in u (Fig. 2, first three of four models).


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Together,thesetwo frequenciesproduceafigure-of-eighttrajectoryatanypoint(Fig.1). Whilethisfigureofeightis


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the predominant trajectory, the motion also contains higher-order harmonics in bothdimensions (Fig.2). Thus,apurelysinusoidallateralor flexural input produces an undulatorywavehaving complexharmonicstructure.3. The theoretical model (Librizzi et al. In Press) and the mechanical impellers (Nektors),which share the properties of having internal forces generating body waves that interact withexternal

fluid forces,

have

wave

motions

with

harmonic

structure

that

differs

from

that

seen

inthe theoretical kinematic models(Fig.2). Whilethefundamental and1st harmonic in u and v,

respectively,are presentinbothkindsofmodels,themodelswithinternalandexternalforceshavehigherharmonics present inapattern different than that of the kinematicmodels. Thesedifferent patterns are correlatedwithtrajectories(Fig.1)showingasymmetryandellipticalpaths,inadditiontofigure-of- eightmotionsnearthetail.4. Lamprey swimming backward produceundulatorymotionwithharmonicstructure thatbearssomeresemblancetothatofthestiffNektor,withthefundamentalfrequencypresent inuandv,the1st harmonicpresent inv,andhigherorder harmonics in v (Fig. 2). Lamprey swimmingforward,however,produceundulatorymotionwithharmonicstructurethathasonly13higher-orderharmonics(where higher-orderrefersto

2, 3, 4,and 5 invand 3, 4,and 5 inu)ineither

dimension. Themodelwiththenext-lowest numberof higherharmonics is the theoreticalforcemodel(Librizzietal.InPress),whichhas20.5. Whenthelampreyswimsbackward, compared to swimmingforward,itproduces undulatorymotionwithalmostfourtimesthenumberofhigherharmonics(68v.20,seeFig.2).Discussion

Higher-orderharmonics ,bywhichwemeanthoseabovethe lateralfundamentalandthe axial1

stharmonicrequiredtocreateafigure-of-eighttrajectory,arepresentatmostpointsofthebody in

the undulatory motionofallmodels and inbackward-swimming lamprey (Fig. 2). Given thebroad rangeofconditionsrepresentedby thesemodels (Fig. 1) simple travelingwavesoflateral displacement or midline flexion; muscularly driven undulations; polyurethane impellersgenerating thrust in water it is striking that forward-swimmin g lamprey completely lackhigher-order harmonics in body points 4 to 25 (almost two-thirds of the body length). Whileonlyone trial is presentedhere,thisgeneralresultholdsforasamplesizeof20 trialsfromsevenlamprey(Rootetal.Inreview).

Whatare themechanicalconsequence sofhavingfewerhigher-orderharmonics? Since theenergy to move a segment periodically is proportional to the cube of the frequencyunder consideration (Eq. 3), higher-order harmonics require that the model or lampreyexpend more energy to undulate. Compared to swimming backward, a forward swimminglamprey also has lower lateralamplitudesatmostbodypoints,aresultsimilar to that found inforward and backward swimming eels, Anguilla anguilla (D'Aout & Aerts, 1999). Asmentioned by D'Aout & Aerts (1999),higher amplitudesareenergetically costly. In fact, theenergycostofmotionisproportionalto the square of the amplitude (Eq. 3). Hence, backward swimming, for both lamprey andeels, appears to be a far less energetically efficient mode of transport, especially when oneconsiders the slowspeedatwhichlampreymovebackward(Fig.1).

Thepresenceofhigherharmonicsinbackward-swimming lampreyand in the theoreticalandmechanicalundulatorsleadsus topredictthefollowing: forward-swimming lampreyareactivelytuning theirbodiesinorder tosuppressorremovehigher-orderharmonics. Thishypothesisisnovel, and suggests to the engineer that biomimetic swimmers and oscillatory impellers willrequire mechanical filters and/or driver signal modulation in order to approach the energy


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efficiencyand harmonicsimplicityofforward-swimmin glamprey.


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Howmightlampreyactivelytunetheirbodies? American eels,Anguilla rostrata,have theabilitytotriple the flexuralstiffness of theirbodies using their lateralbody musculature(Long,1998). By modulatingbody stiffness using muscles,fish couldvary theirnaturalfrequency toreducethemomentneededtodriveandpropagatethetravelingwaveofbending(Long& Nipper,1996). Inaddition,eelsalsopossessthecapacitytoactivelyvarythebody'sdampingcoefficientbyan

order

of

magnitude

(Long,

1998).

Since

damping

moments

absorb

the

kinetic

energy

of

thetravelingwave,dampingcoefficientscouldbeactivelytuned toattenuatethehigher-frequency

vibrationscausedbytheharmonicsofundulatorymotion.Itisalsopossiblethatlampreysuppresshigher-orderharmonicsatthesourceof theundulatory

signal. Since higher-order harmonics are generated from sinusoidal signals in thetheoretical models (Fig. 2, first three of four theoretical models), source suppression wouldrequire non- sinusoidal signalgeneration. Non-sinusoidal periodicstrainhasbeenmeasured inthe locomotor muscleoflivemilkfish,Chanoschanos ,using implantedsonomicrometers (Katzet al. 1999). Since, under optimal conditions, muscles generate bending moments roughly inproportion to their strain rate or shortening velocity, it is possible that active tuning may beaccomplishedby signal suppressionatthemuscles.

Inconclusion,wehavepresentedaFourier-basedmethodthataccuratelydescribestheharmonicstructure of the point-by-poin t periodic motions of undulating bodies. The isolatedperiodic motions are variants of a figure of eight (Fig. 1), a trajectory created by thepresence of a fundamental frequency and 1st harmonic in the lateral and axial directions,respectively. In addition, most undulatory motion is characterized by the presence ofharmonics higher than those just mentioned. That these higher-order harmonics appear insimple kinematic models (Fig. 2), would appear to require a new null hypothesis forundulatory motion that undulatory motion is harmonicall y complex. Thus, when thosehigher-harmonic s are missing when undulatory motion is harmonically simple mechanisms , heretofore unrecognized , must be at work to suppressor remove them. Themechanismwepropose is that fishuse theirmuscles toadaptively tune their bodies, changingflexural stiffness and damping to mechanically modulate the propagationof the travelingpropulsivewaveofbending.Acknowledgements

ThisworkwasfundedbyONRgrants toRGR& JHL (ONR #N-00014-97-1-0292) and toCAP(ONR#N-00014-97-C-0462 ,Award:STTRPhaseII). WewishtothankJeffCynxforhiscriticaldiscussionsoftheanalysisofharmonics.References

Bainbridge,R.(1963). Caudalfinandbodymovementinthepropulsionofsomefish. J.exp.Biol.40,23-56.Carling,J.,Williams,T.L.,&Bowtell, G. (1998).Self-propelledanguilliformswimming: simultaneous solutionofthetwo-dimensionalNavier-StokesequationsandNewtonslawsofmotion.J.Exp.Biol.201,3143-3166.Cheng,J-Y.&Blickhan,R.(1994).Bendingmomentdistributionalongswimmingfish.J.Theor.Biol. 168, 337-348.

Cheng, J.-Y., Pedley, T.J. & Altringham, J.D. (1998). A continuous dynamicbeam model for swimming fish.Phil.Trans.R.Soc.Lond.B353, 981-997.D'Aout,K.& Aerts,P. (1999). A kinematiccomparisonof forwardandbackwardswimming in theeelAnguillaanguilla. J.exp.Biol.202, 1511-1521.


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Root, Courtland et alia 1999