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Comparing the mathematical models of Lighthill to the performance of a biomimetic fish

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2008 Bioinspir. Biomim. 3 016002

(http://iopscience.iop.org/1748-3190/3/1/016002)

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IOP PUBLISHING BIOINSPIRATION & BIOMIMETICS

Bioinsp. Biomim. 3 (2008) 016002 (8pp) doi:10.1088/1748-3182/3/1/016002

Comparing the mathematical models ofLighthill to the performance of abiomimetic fishRobert L McMasters1, Casey P Grey1, John M Sollock1,Ranjan Mukherjee2, Andre Benard2 and Alejandro R Diaz2

1 Department of Mechanical Engineering, Virginia Military Institute, Lexington,VA 24450, USA2 Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824,USA

Received 14 July 2007Accepted for publication 19 December 2007Published 4 February 2008Online at stacks.iop.org/BB/3/016002

AbstractThe mathematical models for the performance of aquatic animals developed byM Lighthill are compared with the experimental performance of a biomimetic fish. Theequations developed by Lighthill are evaluated at steady-state conditions. Equilibrium velocityand mechanical efficiency are calculated using Lighthill’s mathematical model and comparedwith experimental results. In both cases, a pattern is found wherein an optimum combinationof tail frequency and amplitude maximizes equilibrium velocity. Differences between thetheoretical and experimental results are attributed to mechanical limitations in the drivetrain.

(Some figures in this article are in colour only in the electronic version)

Nomenclature

A(l) area of a circle with diameter equal to the overall heightof the tail (ft2)

B amplitude of fish oscillations (ft)D drag force (lb)η mechanical efficiency of swimmingh displacement of the body centerline from the

longitudinal axis (ft)l overall length of the body (ft)P power expended (W)ρ fluid density (slug ft–3)ω angular frequency (rad s–1)R radius of curvature (ft)S cross-sectional area (ft2)t time (s)T average thrust developed (lb)T∗ dimensionless thrust

(T

/[12ρU 2

])θ angle of curvature of the tail (rad)U overall fish velocity (ft s–1)V local velocity near the surface of the fish (ft s–1)x longitudinal dimension

Introduction

Many studies have been performed regarding self-propulsionof biological organisms. These include studies of all differentsizes and types of creatures in both air and water media. Thepresent research focuses only on neutrally buoyant organismsin an incompressible fluid. Considering only this portion ofthis spectrum of biomechanical self-propulsion, there are threemajor groups of cases which can be studied, as discussedby Fauci and Peskin [1]. The first group is where viscouseffects dominate; the second includes cases where momentumeffects dominate, and finally, in the third group both effectsare on par with one another. A case in the first group is thelocomotion of single-celled organisms, which corresponds toa Reynolds number of 10−3. The second group corresponds tothe swimming of most fish, with a Reynolds number between103 and 105. Examples in the third group, which correspondsto very small macroscopic creatures such as nematodes, existin the realm of Reynolds numbers near unity.

The present research focuses on the second regime,involving large Reynolds numbers. Within this regime,several methods have been used in analyzing biomechanical

1748-3182/08/016002+08$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK

Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

Figure 1. Top view of a swimming fish, showing the coordinate system used. The origin is at the head of the fish and the y dimension isvertical (out of the page).

propulsion. One of the first of these was set forth byLighthill [2], which was a ‘small displacement’ model. A‘finite displacement’ model is described in Lighthill [3] andis expanded by Childress [4]. Several numerical modelshave also been used, including the analysis of a sinusoidallyoscillating sheet by Fauci and Peskin [1] and the study ofmoving immersed filaments by Fauci [5]. Further workby Lighthill [6] examined aspects of aquatic self-propulsionwhich optimized efficiency. More recent studies by Walkerand Westneat [7] examined multiple modes of self-propulsion,comparing the merits and efficiency of each. More workon self-propelled bodies was done by Triantafyllou et al,[8] specifically focusing on large Reynolds number cases.Although the influence of vorticity was essentially neglectedby Lighthill, the influence of vorticity can be significant andwas examined by Zhu et al [9] with direct application to theswimming motion of fish.

As numerical modeling capabilities have developed,the advent of computational fluid dynamics (CFD) offerssignificant promise. By contrast to the earlier models offeredin [1], where the fluid is actually allowed to flow through thebody, the CFD models offer much more rigorous simulation.One such study by Eldredge [10] examines a three-piece self-propelled device. The study of self-propulsion is still a verycomplex undertaking, even for modern CFD programs, sincethe boundary is continuously moving. The commerciallyavailable CFD programs are typically designed for stationaryboundary problems. Therefore, the shapes for the mechanismsbeing modeled must be fairly simple. Indeed, the body studiedin [10] consists of three elongated ovals in close proximity toone another, but do not touch. The shape of a swimmingfish can be simulated to some degree by coordinating thesethree members together. More sophisticated CFD modelinghas been done by Mittal and Iaccarino [11] where threedimensional solid bodies are meshed and examined in a flowfield. Many other CFD studies have been undertaken asapplied to biomimetics. A summary of this work is given byMittal, [12] including a discussion of some of the challengesfacing the field of biomimetic applications of CFD. Theseinclude the large range of flow regimes encountered as well asthe force interactions between the moving boundary and thefluid.

The field of biomimetics has also brought about manymechanical models which simulate swimming organisms.Many of these studies concentrate on the mechanical controlsand the logic used in making the models move in a waywhich closely approximates biological organisms. One suchstudy is given by Yu et al [13]. Related to these researchefforts, the present research attempts to develop practical

solutions to the mathematical models developed in [2] andcompare experimental results from a biomimetic device tothese calculations. Some of the experimental aspects of thiswork are recorded in [14].

The remainder of the paper is organized as follows. Abrief discussion of the Lighthill small displacement modelfor a biomimetic fish is introduced. Next, the application ofthis model to the present research is discussed, extending thedevelopment of the small displacement model. The parametersassociated with the particular biomimetic fish used in thisresearch are then presented, followed by a discussion of thedata collected from the experiments. Plots are shown forthe purpose of comparing the mathematical models and theexperimental results, followed by conclusions of the research.

The small displacement model

The mathematical models used in this research are analyticaland utilize fundamental equations for fluid motion, requiringvarious simplifying assumptions. Specifically, the swimmingbody is considered as a long cylinder, with lateral movementswhich are very small in comparison to the overall body length.The geometry of this body is depicted in figure 1. The localvelocity near the surface of the body in this model is expressedas

V (x, t) = ∂h

∂t+ U

∂h

∂x(1)

where U is the overall average velocity of the body (i.e. the freestream velocity) and h is the displacement of the centerline ofthe body from the longitudinal axis, as shown in figure 1. Asderived in [2], the equation for the average thrust, T, developedby the fish is

T = ρA(l)

2

[(∂h

∂t

)2

− U 2

(∂h

∂x

)2]

x=l

(2)

where ρ is the density of the fluid, A(l) is the area of a circlecomputed by using the overall dimension of the tail as adiameter, and the squares of the derivative values shown areaverages over a typical cycle. These derivatives are evaluatedat x = l, where l represents the overall length of the fish. Aswith average thrust, the total average power expended by thefish in order to move with a displacement h(x, t) as developedby Lighthill is

P = ρUA(l)

[∂h

∂t

(∂h

∂t+ U

∂h

∂x

)]x=l

. (3)

Some restrictions are placed on this solution in [2], whichare required in order to preserve the assumptions used in the

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Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

development of the small displacement model. The criticalassumption to apply to this work is

0.05U 2 < (∂h/∂t)2. (4)

The above equations, developed in [2], are used as part ofthis research in generating practical solutions, which can becompared to experimental results from a biomimetic device.

Extension and application of the small displacementmodel

As noted in the introduction, the application of the smalldisplacement model for this research is based on steady-state conditions. No attempt is made in this work to studythe transient motion of the biomemetic device. Under theseconditions, with the device at equilibrium velocity, the thrustis exactly balanced with the overall drag force on the body.The drag force is given by

D = 12ρU 2CDS (5)

where D is the drag force, CD is the drag coefficient, ρ is thedensity of the fluid and S is a characteristic cross-sectionalarea of the fish. Using equations (2) and (5) and setting thrustand drag equal to each other at this equilibrium condition, thesteady-state velocity can be solved for as a function of theother parameters:

U 2 =(

∂h∂t

)2

(∂h∂x

)2+ CDS

A(l)

(6)

Once the nature of the motion of the fish is known, and thederivatives in equation (6) can be found quantitatively, theequilibrium velocity of the device can be found. In addition tothe nature of the motion of the fish, other physical parametersmust be known, such as the drag coefficient CD, the cross-sectional area of the body, S, and the overall dimension of thetail, which generates the term A(l). Of these parameters, thedrag coefficient is the one which would be the most difficultto determine without a physical model that could be tested.The only alternative to experimental determination of the dragcoefficient would be to assume a simple shape from whichcoefficients have already been recorded in the literature, orto perform a complete CFD analysis on the proposed shape.Therefore, without an actual physical model, the Lighthillequations cannot be readily used to generate practical results.

Once these parameters are in hand, however, otherperformance factors for the swimming body can be computed.The most significant of these is the swimming efficiency. Thiscan be defined as the ratio of the power required for propulsionto the total amount of power expended by the fish. The amountof power required to push the body through the water is

P required = T U. (7)

Using equations (2), (3) and (6) the required power overexpended power may be expressed as

η = T U

P=

[(∂h∂t

)2 − U 2(

∂h∂x

)2]x=l

2[

∂h∂t

(∂h∂t

+ U ∂h∂x

)]x=l

. (8)

Figure 2. Design for the body of the biomimetic fish used in thisresearch (top view).

Examining the implications of the restriction placed on theserelationships by equation (4), this restriction can actually beseen as a limitation on (∂h/∂x)2 by utilizing equation (6)developed above. Since from equation (6) the relationshipbetween U 2 and (∂h/∂t)2 can be written as

U 2

[(∂h

∂x

)2

+CDS

A(l)

]=

(∂h

∂t

)2

. (9)

It follows from condition (4) that(∂h

∂x

)2

+CDS

A(l)> 0.05 (10)

Although experimental data must be used for the determinationof the dimensionless group CDS/A(l) in this inequality, thederivative term (∂h/∂t)2 can be found as long as the basicgeometry of the fish is known.

Further mathematical development using thespecific device parameters

The geometry of the biomimetic fish used in this research isshown in figure 2. In contrast with the model of the biologicalfish shown in figure 1, the biomimetic fish model in figure 2has only one degree of freedom. The front two-thirds of thebiomimetic fish is rigid and the rear third moves in a simple arcshape. There are no other modes or shapes generated by thetail and the radius of curvature of the tail is uniform throughoutits range of motion. This shape is created by a series of gearsfrom which the tail is made. The displacement h(t) at the endof the tail, as a function of the radius of curvature of the tail, is

h(t) = R(1 − cos θ) (11)

where R is the radius of curvature of the tail and θ is the arc ofthe tail in radians. As the tail sweeps out the path of its motion,the angle θ becomes larger and the radius R becomes smalleras the displacement increases, as given in equation (11). Theangle θ is used as the driving term as a function of time, sinceit is directly proportional to the motor rotation, as dictated bythe gear assembly. A photograph of these gears is shown infigure 4.

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Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

Since one-third of the length of the fish is made up of thetail, the relationship between the tail length and body length is

l

3= Rθ. (12)

Combining equations (11) and (12) and recognizing the angleθ as a function of time, the tail displacement becomes

h(t) = l(1 − cos θ(t))

3θ(t). (13)

If θ (t) is driven sinusoidally, i.e., if θ (t) is of the form

θ(t) = θm sin(ωt), (14)

the tail displacement becomes

h(t) = l [1 − cos(θm sin(ω t))]

3θm sin(ω t), (15)

where θm is the amplitude and ω is the frequency of thesinusoidal motion. Now taking the time derivative of thisexpression gives(

∂h

∂t

)x=l

= lω cos(ω t){θm(sin ωt) sin(θm sin ωt)+cos(θm sin(ω t))−1}3θm sin2(ω t)

.

(16)

Since the spatial derivative for the equations developed in [2]are based on the location x = l, the spatial derivative simply(

∂h

∂x

)x=l

= tan θ = tan(θm sin(ω t)). (17)

The time-averaged values must next be found for the specificgeometry of the arc-tail biomimetic device as given byequations (16) and (17). These are required in order to find thetime averaged values of the spatial and temporal derivativesused in equations (2) and (3), from which the subsequentequations in this research are derived. These time-averagedvalues are simply obtained by integrating each function overone cycle of the body motion and dividing by the period of thebody motion. Specifically,(

∂h

∂x

)2

x=l

= ω

2π

∫ t=2π/ω

t=0

[(∂h

∂x

)x=l

]2

dt (18)

(∂h

∂t

)2

x=l

= ω

2π

∫ t=2π/ω

t=0

[(∂h

∂t

)x=l

]2

dt (19)

(∂h

∂x

) (∂h

∂t

)x=l

= ω

2π

∫ t=2π/ω

t=0

[(∂h

∂x

)x=l

] [(∂h

∂t

)x=l

]dt

(20)

The time-averaged values for these derivatives were foundnumerically for various amplitudes and the results are shownin figure 3. As would be expected, with larger amplitudevalues, the time-averaged slope and velocity for the tail arelarger. Moreover, this relationship would not be expected tobe linear, since the slope becomes extremely large as the angleθ approaches π/2 radians. Note that no plot for the productof slope and velocity appear in this plot, i.e., the results fromequation (20). This is because of the configuration of the arctail used in the design of the biomimetic device. Since both

Figure 3. The integrated square of the slope and the velocity of thetail of the fish over one complete cycle.

Figure 4. The skeleton of the biomimetic fish.

sides of the tail swing are symmetrical, the integration of thetail swing to the right exactly cancels the integration of the tailswing to the left. Therefore, the results for equation (20) willalways be zero for this tail design.

In order to go any further with calculations related toequilibrium velocity, which is found through equation (6), orequilibrium efficiency, which is found through equation (8),a drag coefficient is needed. Therefore, the experimentalbiomimetic device is introduced at this point.

Experimental biomimetic device description

Figure 4 shows the skeleton of the biomimetic device,including the motor, the stainless steel plate frame, and theinterlocking gears, which form the arc tail. The Faulhabermotor is provided with a gear reduction of 133:1 as part ofa factory installed packaged unit, along with the encoder.Further reduction is provided through the 2:1 angle gears justto the rear of the motor, which transition the shaft from ahorizontal orientation to a vertical orientation. This verticalshaft then drives the lead gear in the tail, which, in turn, driveseach of the following gears in a single-degree-of-freedom arc.As shown in figure 2, this arc is tangent to the centerline

4

Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

Figure 5. The device shown in the water with the rigid fiberglassbody and flexible silicone tail.

of the body. The motor is driven by a sinusoidal signal, theamplitude and frequency of which are controlled by a computerprogram. A photograph with the rigid fiberglass body and theflexible skin, placed over the skeleton mechanism, is shown infigure 5. This photograph shows the device in the water witha slight positive net buoyancy. The flanged nature of the rigidfiberglass body allows access to the internal mechanism of thedevice, in addition to providing lateral resistance in the water,minimizing lateral motion. Attempts were made to ballastthe device to a neutrally buoyant state before each test byadding or removing lead weights. Additionally, the forwardand aft trim on the device required periodic adjustment to keepit horizontal in the water. These adjustments were necessarydue to the somewhat variable size of the flexible tail, due toexpansion and contraction of the air inside the body, whichimpacted the buoyancy of the tail.

With the physical model constructed, measurements weremade to determine the parameters CD, S, and A(l). so thatcalculations could be generated from equations (6) and (8).The cross-sectional area, S, was determined to be 0.158 ft2.In order to determine the drag coefficient CD, test were runto determine the force needed to pull the device through thewater at various speeds. The measurements were then plottedas drag force versus velocity squared, as shown in figure 6.The equation on this plot shows the slope between the dragforce and velocity was 0.0951 lb s2 ft–2. Obtaining the densityof water at 70 ◦F from [15], and combining these parameterswith the cross-sectional area S, a drag coefficient of 0.621was calculated. Finally, the tail area parameter, A(l), wasdetermined by calculating the area of a circle whose diameter isdefined as the height of the tail. This parameter was determinedto be 0.196 ft2.

Comparison of mathematical and experimentalresults

With these parameters in hand, equations (6) and (8) couldbe used to generate theoretical plots of equilibrium velocityand efficiency as functions of tail amplitude for various tailfrequencies. The results of these calculations are shown

Figure 6. Results of drag test showing the linear relationshipbetween drag force and velocity squared.

Figure 7. Theoretical curves generated from equation (6) ofequilibrium velocity as a function of tail amplitude for various tailfrequencies ω (T = 2π/ω).

in figures 7–9. Figure 7 shows the theoretical equilibriumvelocity as a function of tail amplitude θm as measured inradians. It provides plots at three different tail frequencies ω,which are more conveniently labeled in terms of tail periodT = 2π/ω at T equal to 1 s, 0.75 s and 0.5 s. The dottedlines in figure 7 indicate the range in which equation (4)is not satisfied. That is, the slope on the end of the tailis too large for the equilibrium velocity, which exceeds thelimitations of the ‘small displacement’ model. As can be seen,however, the general trend is that the higher tail frequenciestheoretically generate the higher equilibrium velocities forthis tail configuration. Additionally, the equilibrium velocityincreases as a function of tail amplitude, which is largelyintuitive. However, there is a point beyond which increasingthe tail amplitude brings about a reduction in equilibriumspeed. For the most part, this regime is beyond the theoreticallimitations of the small displacement model. Still apparent,however, is that large velocities accompanying large tailamplitudes would be offset by increases in drag at the highervelocities. It is interesting to note that the equilibriumvelocities shown in figure 7 can be non-dimensionalized byusing l/T as a scaling factor, where l is the length of the fishand T is the period of tail movement. When this is done,

5

Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

Figure 8. Theoretical curves generated from equation (6) ofdimensionless equilibrium velocity as a function of tail amplitudefor various tail frequencies ω (T = 2π/ω).

Figure 9. Theoretical curves generated from equation (8) ofequilibrium efficiency as a function of tail amplitude for various tailfrequencies ω (T = 2π/ω).

the curves of figure 7 collapse into one curve, as shown infigure 8.

Figure 9 depicts a similar outcome to that of figure 7 in thatthe highest efficiencies and the highest equilibrium velocitiesboth correspond to the highest tail frequencies. Likewise, asthe amplitude increases, a point is reached beyond which theincrease in drag offsets the gains otherwise made at the highervelocities. These portions of the curve are designated withdotted lines. Some of the efficiencies in these areas are verylow, especially for the curve corresponding to a period of 1 s.An additional item of note is that, as the displacement of thetail becomes smaller, the tail velocity terms in equation (8)begin to dominate and the efficiency tends toward 50%. Thiscan be seen graphically in figure 9.

Figure 10 shows the experimental counterpart to figure 7,dealing with equilibrium velocity. As in the previous figures,the regions of angular displacement identified as being outsidethe bounds of the small amplitude model are designated usingdotted lines. Although the experimental velocities shown infigure 10 are lower than the theoretical velocities calculated infigure 7, some of the same trends are exhibited. Specifically,

Figure 10. Experimental equilibrium velocity as a function of tailamplitude for various tail frequencies ω (T = 2π/ω).

for each set of experiments conducted at a particular frequency,the equilibrium velocity tends to increase until some criticalpoint and then begins to decrease as the amplitude becomesextremely large. As in the previous plots, this is logically dueto the fact that drag begins to increase as the velocity increases.One contrast between figures 7 and 10 is that the experimentalvelocities are all of the same relative size, almost independentof frequency. It appears that the equilibrium velocity is muchmore dependent on tail amplitude than on tail frequency. Theauthors believe that this is most likely due to the limitations ofthe mechanical device in terms of available motor torque andthat this factor is exacerbated by gear backlash. Specifically,in order to obtain a large amplitude at a high frequency, alarge amount of torque is required from the motor due to therequired higher velocity of the tail. This would be the operatingregime where the motor would be taxed to the highest degreeand, as the experiments demonstrated, large amplitudes werenot generated in this operating area. Gear backlash is a largerfactor at the higher frequencies, due to the short amount of timeavailable for the tail to move in each direction. At these higherfrequencies, a greater fraction of the time is spent overcominggear backlash, since the time needed to switch tail directionsis basically independent of tail frequency. In this respect,a disparity is inevitably created between the theoretical andexperimental performance of the device.

An additional factor, which may contribute to thediscrepancy between theoretical and experimental velocityplots, is that the theoretical equations are based on a ‘slender’body shape for the fish. Although the researchers made thebody as narrow as possible, while still accommodating thewidth of the motor, the biomimetic device may not be slenderenough to meet the assumptions of the mathematical model.The term ‘slender’ is somewhat vague in the realm of fluiddynamics and no specific requirements are given in [2] formeasures of slenderness, such as the ratio of cross-sectionalarea and overall length. As such, the degree to which thegeometry of the biomimetic fish affects the conformance of thetheoretical and experimental models is somewhat unknown.

The determination of the experimental efficiency of thebiomimetic fish is inherently more difficult than determining

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Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

Figure 11. Experimental measurement of a typical half-cycle of tailmotion at a period of 0.75 s.

the equilibrium velocity. This is because of the unknownfactors regarding motor and drive train efficiency duringtransient conditions. The transient load encountered in thewater cannot accurately be replicated during bench-top testingof the motor, making the transient motor efficiency effectivelyunknown. Additionally, internal inefficiencies in the drivetrain are not accounted for in the theoretical model and aredependent on position and velocity of the motor. So acceptingthat the calculation of efficiency is somewhat of a coarseundertaking, the instantaneous power input to the biomimeticfish was measured using an oscilloscope. These measurementswere taken using a step input to the motor in place of thecomputer controls, due to an electrical incompatibility betweenthe oscilloscope and the computer-driven motor controller.The measurements are shown in figure 11. As depicted in thisfigure, there is a sudden inrush of current when the motor startsat t = 0, along with a corresponding drop in voltage. This isa typical characteristic of motors starting from rest, with alarge current due to the low winding resistance of the motor.The large current is accompanied by a drop in voltage due toline losses and battery resistance. The load on the motor thendrops as the tail swings through mid-travel, since there is notmuch mechanical torque required when the tail is aligned withthe water flow. Finally, current rises again as the tail swingsback into the free stream velocity, requiring more mechanicaltorque. With the mechanical switching scheme used, therewas approximately a 135 ms delay between current reversalsto the motor, which accounts for the additional time involvedbetween cycles beyond the 240 ms shown on the time axis.

Integrating the power curve with respect to time infigure 11, the average input power to the drive motor of thebiomimetic fish is 0.459 W. This power level was measuredat a tail period of 0.75 s, and an amplitude of 1 rad s–1. Theequilibrium fish velocity for this combination was recordedat 0.72 ft s–1. Using this velocity, the corresponding dragforce can be calculated using figure 6, which is 0.0894 lb.The required power to overcome this drag is simply velocitymultiplied by drag force, which is 0.0868 W. Dividing the0.0894 W of power needed to overcome the drag force by the0.459 W power input to the driving motor gives an efficiency

Figure 12. Experimental measurement of efficiency with tailmotion at a period of 0.75 s.

of 18.9% at this particular operating condition. When theefficiency is plotted for all of the tail amplitudes tested at atail frequency of 0.75 s, the results can be seen in figure 12.Unlike the theoretical efficiency plots in figure 9, the maximumefficiency does not occur at the lowest amplitude. Instead,the experimental efficiency tends to follow the same trend asboth the theoretical and experimental velocity. It is believedthat this discrepancy is due to the same factors as thosementioned above for the differences between the experimentaland theoretical velocity as a function of amplitude. Themechanical limitations on motor torque and gear backlash arequite likely the main reason for this difference. Additionally,as noted in the comparisons of the curves for equilibriumvelocity, it may be that the body of the biomimetic fish is notas ‘slender’ as the body on which the theoretical assumptionsare based. However, the mechanical limitations on the drivesystem seem to be a more likely reason for the discrepancy.

Conclusions

The theoretical equations derived by M Lighthill for a slenderswimming body were applied and compared with experimentalmeasurements from tests performed on a biomimetic fish.The behavior of the equilibrium velocity as a function of tailamplitude followed the same trend in both the theoretical andexperimental cases. These results showed that there exists anoptimum tail amplitude for each tail frequency. Efficiencywas much more difficult to measure than velocity and thetrend between the experimental and theoretical results did notappear to match as well as the velocity results. This mismatchoccurred at low tail amplitudes. The discrepancy is believed tobe due to the mechanical limitations on the biomimetic device,such as motor toque and gear backlash. Additionally, someof the discrepancy may have been rooted the possibility thatthe biomimetic fish was not quite slender enough to match theassumptions used in the theoretical development.

Acknowledgments

The authors would like to acknowledge the contributionsof Darren Wellner and Chris Petree who did much of the

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Bioinsp. Biomim. 3 (2008) 016002 R L McMasters et al

construction work on the body of the fish and in arrangingthe control circuits. The contributions of Ron Chandlerin the Virginia Military Institute machine shop and RoyBailiff in the Michigan State University machine shop arealso acknowledged. Additionally, this research would nothave been possible without the financial contributions of theMichigan State University research development fund and theVirginia Military Institute Undergraduate Research Initiative.

References

[1] Fauci L and Peskin C 1988 A computational modelof aquatic animal locomotion J. Comput. Phys.77 85–108

[2] Lighthill M J 1960 Note on the swimming of slender fishJ. Fluid Mech. 9 305–17

[3] Lighthill M J 1975 Mathematical Biofluidynamics(Philadelphia, PA: Society for Industrial and AppliedMathematics)

[4] Childress S 1977 Mechanics of Swimming and Flying (NewYork: Courant Institute of Mathematical Sciences)

[5] Fauci L 1990 Interaction of oscillating filaments: acomputational study J. Comput. Phys. 86 294–313

[6] Lighthill M J 1970 Aquatic animal propulsion of highhydromechanical efficiency J. Fluid Mech. 44 265–301

[7] Walker J and Westneat M 2000 Mechanical performance ofaquatic rowing and flying Proc. Biological Sciencesvol 267, pp 1875–81

[8] Triantafyllou M, Triantafyllou G and Yue D 2000Hydrodynamics of fishlike swimming Ann. Rev. FluidMech. 32 33–53

[9] Zhu Q, Wolfgang M J, Yue D K P and Triantafyllou M S 2002Three-dimensional flow structures and vorticity control infish-like swimming J. Fluid Mech. 468 1–28

[10] Eldredge J 2006 Numerical simulations of undulatoryswimming at moderate Reynolds number Bioinsp.Biomim. 1 S19–24

[11] Mittal R and Iaccarino G 2005 Immersed boundary methodsAnn. Rev. Fluid Mech. 37 239–61

[12] Mittal R 2004 Computational modeling in biohydrodynamics:trends, challenges, and recent advances IEEE J. Ocean Eng.29 595–604

[13] Yu J, Tan M, Wang S and Chen E 2004 Development of abiomimetic robotic fish and its control algorithm IEEETrans. Syst. Man Cybern. B 34 1798–810

[14] Grey C and Sollock J 2007 Measuring the performance of aself-propelled underwater device New Horizons I 17–32

[15] Eshbach O 1963 Handbook of Engineering Fundamentals(New York: Wiley)

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