Mathematical Biology

J. Math. Biol. (2006) 53:843–886 DOI 10.1007/s00285-006-0036-8

Mathematical Biology

An elastic rod model for anguilliform swimming T. McMillen · P. Holmes

Received: 27 October 2005 / Revised: 23 June 2006 / Published online: 14 September 2006 © Springer-Verlag 2006

Abstract We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We employ a geometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor’s assumption of prescribed shape to fail, leading to timeperiodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158–183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation

T. McMillen (B) · P. Holmes Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] P. Holmes Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA e-mail: [email protected] Present Address: T. Mc Millen Department of Mathematics, California State University of Fullerton, Fullerton, CA 92384, USA e-mail: [email protected]

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T. McMillen, P. Holmes

yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.

1 Introduction We develop a continuum mechanical model for anguilliform swimming, as exhibited by lampreys, eels and certain aquatic snakes. These animals propel themselves by propagating traveling waves along their bodies, generally from head to tail, and while their fins may be important in attitude control and complex maneuvers, they are not thought to be critical to the propulsive mechanism. See [2] for an overview of animal locomotion, and [15] for vertebrate swimming in particular. The major motivation for our work comes from experiments on lampreys (Ichthyomyzon unicuspis or Petromyzon marinus) and modeling conducted by the groups of Grillner [21, 22, 26, 27], Williams [5, 6, 9, 10], and Cohen [14, 16, 35]. It has long been known that lampreys possess a central pattern generator (CPG) in the form of a distributed network of bursting interneurons and motoneurons in their spinal cords; indeed, isolated, deafferented regions having as few as 3–4 segments can display periodic bursting. More significantly, entire deafferented cords, with brain removed, can produce waves of activation in antiphase contralaterally and with ipsilateral phase lags from segment to segment characteristic of the bending waves observed in intact swimming animals [16]. However, observations of apparent standing wave patterns in animals moving out of water on a low friction surface [5] suggest that the neuronal activation wave does not simply prescribe the body motion, but that the latter arises from an interaction among activation, passive muscular and body elasticity, and fluid reaction forces. While relatively detailed models of CPG oscillators in lamprey have been developed [7, 32, 53], and simulations of muscle-body models both with and without hydrodynamic loading have been done [5, 6, 9, 10, 21, 22], tractable neuromechanical models are lacking. Our continuum model, which includes neuromuscular activation, passive elasticity and fluid reaction forces, offers a compromise between direct simulations of the Navier–Stokes equations coupled to an actuated elastic body (such as those of Fauci et al. [19, 23], using the immersed boundary method of Peskin [42]), and the analytical studies of Taylor [44] and Lighthill [39]. In this paper we use it to examine the effects of passive elastic and resistive fluid forces, and of material and geometric properties including passive stiffness and taper. In future work we plan to include more realistic muscle models, and to compare our results with immersed boundary method simulations. We focus on lamprey, but our model should be more generally useful for anguilliform swimmers including eels and aquatic snakes. Related studies for non-anguilliform swimmers involving muscle mechanics and hydrodynamic forces include [11, 12, 41, 50]. See the excellent review by Lighthill [40] for a discussion of anguilliform and non-anguilliform swimmers.

An elastic rod model for anguilliform swimming

845

Following the notable early work of Taylor [44], we model the body as an actuated elastic rod subject to resistive (drag) forces due to its motion through the fluid. Taylor’s theory assumes that the hydrodynamic reaction force at a point on the body is equal to that generated by a cylinder tangent to the body axis placed in a uniform flow whose velocity matches the relative velocity between body and fluid at that point. The resulting quasisteady drag force neglects both added mass effects due to acceleration of fluid by the body, and unsteady effects such as vortex shedding. Added or virtual mass and moments of inertia for a rigid body moving in an irrotational, inviscid fluid was addressed by Kirchhoff in the nineteenth century [38]; for a recent example involving propulsion of multiple linked bodies, see [36]. Lighthill [39] developed an inviscid, ‘reactive’ theory for slender body swimming that includes added mass, and Wu [57, 58] also considered the effects of vortex shedding. Recent experimental work, such as that of Tytell [46– 48] on eels, emphasise the importance of unsteady flow, and demonstrate that neither the resistive theory of Taylor, nor Lighthill’s reactive theory, can alone account for impulse production or wake structure, and of course, neither theory can predict the complex vortex patterns and wakes left by swimming bodies. We are primarily interested in probing the interaction among CPG, muscles, body elasticity, fluid reaction forces, and, ultimately, sensory feedback. Taylor’s resistive hydrodynamic theory allows us to do this in the relatively simple context of an actuated elastic rod, and in spite of its shortcomings, we believe that it provides a useful basis for integrative modelling of the neuromechanics of swimming. Here we take a geometrically nonlinear rod model and, rather than assuming a traveling wave of specific (sinusiodal) form as in [44], we impose activation in the form of a time-dependent preferred curvature and seek traveling wave solutions that emerge from the coupled elastic-fluid system. We consider only the ‘feedforward’ problem: body motions in response to stereotypical actuation. The balance of this paper is as follows. The rod model is described in Sect. 2, where we also review the hydrodynamic body forces and other relevant parts of [44], and analyze periodic traveling waves in uniform rods. In Sect. 3 we derive a discretized model for use in numerical simulations and compare it with the rigid link and point mass models of [5, 9, 21], showing how, with appropriate parameters, neural activation and muscle model assumptions, it compares to those models and to Taylor’s assumption of a traveling sine wave in body shape [44]. We find that muscle actions can modulate bending stiffness in the discretized system and we briefly discuss this, although our simulations employ enough elements that it is a small effect. More significantly, we show that laterally-differential neural activations provide a natural turning mechanism. Section 4 contains the main simulation results. After validating our code against analytical results and Taylor’s analysis, we show that stable straight swimming and turning can be achieved, but that at steady state the mass center oscillates about a constant velocity, due to small departures from the intrinsic (sinusiodal) shape corresponding to the preferred curvature caused by the passive elastic response. We also investigate the effects of hydrodynamic loading, activation strength, passive elasticity and tapered geometry on asymptotic speed and acceleration from rest. Section 5 summarizes and concludes the paper.

846


2 The activated elastic rod We consider an isotropic, inextensible, unshearable, viscoelastic rod that obeys a linear constitutive relation. We generally assume that material properties such as density and bending stiffness are constant in time, but we allow them to vary along the rod, and, crucially, we endow the rod with a time dependent preferred curvature in the form of a traveling wave, representing the waves of muscular activation that pass along the bodies of anguilliform swimmers. We adopt the conventions of Coleman, Dill et al. [17, 18], and use an elliptical cross section to compute hydrodynamic reaction forces, although we restrict to planar motions, since lampreys and eels in ‘normal’ steady swimming flex their bodies primarily in the horizontal plane [49, 54] (as leeches do in the vertical plane, cf. [19]). 2.1 Equations of motion of the rod A configuration of a rod is given at each time t by a space curve s → r(s, t) = (x(s, t), y(s, t)) representing the centerline of the rod in the inertial (x, y)-plane, where s ∈ [0, l] denotes arc-length. Derivatives with respect to s and t will be denoted by subscripts. The inextensibility condition |rs | = 1, can be written in terms of the angle ϕ between the tangent to the curve t = rs and the inertial x-axis: xs = cos(ϕ),

ys = sin(ϕ)

(1)

(see Fig. 1.) The normal to r is then given by n = (− sin ϕ, cos ϕ). Each element of the rod is subject to contact forces f = (f , g), a contact moment M, and body forces W = (Wx , Wy ). The contact forces and moment are those exerted on the region (s, s + ds) by [0, s), and the body forces arise from interactions with the external environment. Balancing linear and angular momentum, we obtain the equations of motion (cf. [3, 18]): y

t(s,t) j (s,t)

x

r(s,t) a b

Fig. 1 An elastic rod with elliptical cross-sections and variable semi-axes undergoes bending motions in the (x, y) inertial coordinate plane; s denotes arclength along the body centerline


847

ρA xtt = Wx + fs ,

(2)

ρA ytt = Wy + gs , ρI ϕtt = Ms + g cos ϕ − f sin ϕ.

(3) (4)

Here ρ is the volumetric material density and A and I the cross-sectional area and moment of inertia of the rod. For an elliptical cross-section with semi-axes a and b, as in Fig. 1, A = π ab and the moment of inertia for motions in the (x, y)plane is I = π4 ab3 . We assume that ρ is constant, but allow A = A(s), I = I(s) to vary (both remaining strictly positive). Most of our analysis is for a uniform circular rod, but we also consider a tapered elliptical cross section based on lamprey body geometry. We do not include added mass effects as in the Kirchhoff or Lighthill theories [38, 39], but we remark on them when we consider the tapered elliptical rod in Sect. 4.3.3. The intrinsic shape of the rod is determined by a prescribed function κ = κ(s, t), representing its preferred curvature. When ϕs = κ the rod is in equilibrium, and we adopt the usual linear constitutive relation [3] so that the moment is proportional to the departure from preferred curvature: M = EI (ϕs − κ) + δϕst .

(5)

Here E > 0 and δ ≥ 0 are Young’s modulus and viscoelastic damping coefficient and the flexural rigidity EI, with SI units N m2 , determines the overall stiffness. In the absence of body forces, and with κ independent of time and δ = 0, the system conserves energy [18], leading to oscillations about the equilibrium equilibcurvature ϕs = κ. If δ > 0, oscillations decay and the rod approaches s rium as t → ∞. More generally, solving for φ(s, t) = φ0 = 0 κ(σ , t) dσ and integrating Eqs. (1), we can obtain instantaneous intrinsic equilibrium shapes (x0 (s, t), y0 (s, t)). For moderate curvatures and if the body is oriented approximately along the x-axis, one can express this as a graph in the form y0 (x, t) (see Figs. 6, 7). The equations of motion (2)–(4), the constraints (1) and the constitutive relation (5), along with specified body forces and suitable boundary and initial conditions, form a closed system of evolution equations. The most natural boundary conditions for free swimming are that contact forces and moments vanish at the head and tail: M = f = g = 0 at s = 0, l, but we shall also consider conditions appropriate for periodic waves. One could also consider (idealized) boundary conditions for tethered animals, although we shall not do this here. Three quantities are conserved in the absence of body forces and damping and with time-independent curvature: the total energy H(t), the angular momentum L(t) and the linear momentum M(t). These are defined as follows: 1 H(t) = 2

l ρIϕt2 0

1 2 2 2 M + ρA xt + yt ds, + EI

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l L(t) =

ρIϕt + ρA (x yt − xt y) ds,

0

l ρA rt ds.

M(t) =

(6)

0

Direct computations of their rates of change under (2)–(4) yield dH = (ϕt M + f xt + g yt ) |s=l s=0 + dt dL = (M + g x − f y) |s=l s=0 + dt

l

l δ r˙ · W + M −κt + ϕstt ds, EI 0

x Wy − y Wx ds,

0

s=l l dM f = + Wds, g s=0 dt

(7)

0

showing that for W = 0, δ = κt = 0 and with free boundary conditions the time derivatives all vanish. Since there are two components of linear momentum Equations (6) actually describe four conserved quantities. We use these in Sect. 4.2 to check our simulation code. 2.2 Taylor’s model of hydrodynamic reaction forces and swimming diagrams With fluid loading present the body force W(s, t) depends on the local velocity of the body relative to the fluid. Taylor [44] assumed that body curvatures are relatively small, so that the normal and tangential components of W at a point s at time t are similar to those experienced by a rigid circular cylinder set at the tangent angle ϕ(s, t) to uniform flow at the appropriate relative velocity. He was therefore able to use extensive data and analyses predicting that for a rod of radius a, the force due to perpendicular flow of fluid of density ρf and dynamic viscosity μ with speed v is given by F = CN ρf av2 + CT v3/2 ,

(8)

where the drag coefficient CN varies only between 0.9 and 1.1 in the Reynolds

number range 20 < R < 105 , and CT is closely approximated by CT ≈ 8ρf aμ in the range 10 < R < 105 (cf. [44, Fig. 1]). Implicit in this model are the additional assumptions that normal velocities are sufficiently high, and oscillation frequencies sufficiently low, that the vortex shedding is near to fully-developed, and that the body is long enough to establish a steady longitudinal boundary


849

layer. For the parameter values taken in Sect. 4, based on a 20 cm long and 1–2 cm diameter lamprey swimming freely at about 1 body length per second with a lateral oscillations of ±3 cm at 2 Hz, we can estimate transverse and longitudinal Reynolds numbers of 4,800 and 40,000 respectively, well within the range for to which (8) applies for constant speed motions. However, flow visualizations of Tanaka and others reproduced in [20, e.g. Fig. 62] indicate that asymmetric vortex shedding takes a time of order 4d/Utransverse = 4(1 − 2)/24 = 1/6 − 1/3 sec to develop, similar to the half cycle duration of 1/4 s. The theory would therefore better suit a slower (and/or thinner) swimmer, but the application is not excessively outside its range. Drag forces for smooth oblique cylinders can be decomposed into normal and tangential components in terms of the normal and tangential velocities v⊥ and v at (s, t) as FN = aρf CN |v⊥ |v⊥ +

8ρf aμ|v⊥ | v⊥ ,

FT = 2.7 2ρf aμ|v⊥ | v .

(9)

This relation must be modified for rough cylinders [44], but here we consider only smooth ones. The body forces are given by W = −FN n − FT t,

(10)

where n and t denote the normal and tangential unit vectors to the rod’s centerline at s. In terms of the inertial coordinate basis (êx , eˆ y ): n = eˆ y cos ϕ − eˆ x sin ϕ,

t = eˆ x cos ϕ + eˆ y sin ϕ.

(11)

In [44] Taylor balances body forces at steady state by supposing that the rod assumes a prescribed sinusoidal shape moving at constant speed U parallel to eˆ x in an inertial frame, but backward relative to material points, thus driving the mass center of a one-wavelength piece forward at speed V, also assumed constant. Specifically, the centerline of the rod is described by y = B sin

2π {x + (U − V)t} , λ

(12)

and Taylor seeks V such that the longitudinal (êx ) forces balance over a full wavelength: λ λ FN sin φ ds = FT cos φ ds. (13) 0

0

Motions are characterized by the ratios B/λ of amplitude to wavelength and n = V/U of the speed of the mass center to that of the traveling wave.1 Taylor 1 In the swimming literature the roles of U and V are typically reversed, U denoting forward speed

and V wave speed; and n is referred to as ‘slip.’ In this paper we shall adhere to Taylor’s usage.

850

T. McMillen, P. Holmes 1/2

defines a Reynolds number R1 = U2aρf /μ in terms of U, so that, if CN R1 is fixed, n depends only on the ratio B/λ and the behavior can be characterized in a swimming diagram in which n is plotted against B/λ. For our purposes it is more natural to consider the speed of the traveling wave relative to arc-length s along the body centerline. We will study swimming for a rod of fixed length l that supports a traveling wave of curvature of fixed speed in the body frame, whose amplitude and wavelength λ in the inertial frame vary in concert. Specifically, since l is the arc-length of one wavelength of the sine wave under Taylor’s assumption, l, B and λ are related by 2 l = sec(α)E sin2 (α) , λ π

where tan(α) =

2π B , λ

(14)

and E[·] is the complete elliptic integral of the second kind [44, Eq. (3.12)]. The wave speed u relative to arc-length is related to the speed U in the inertial frame by u = U l/λ. Thus, we seek the dependence of speed on B/λ for waves of the form x − Vt u + t (15) y = B sin 2π λ l in terms of the ratio n = V/u of the speed of the center of mass in an inertial frame to the speed of the backward traveling wave in arc-length. Motions for fixed u are characterized by setting a second Reynolds number R2 = u2aρf /μ constant, and solving for n as a function of B/λ. Fig. 2 shows n and n as functions of B/λ for fixed speeds U and u, respectively. To compute these, as in [44], we equilibrate normal and tangential forces over one full wavelength using the sinusoidal geometry and the expressions (9). For fixed U when R1 is large enough, V increases monotonically with B/λ, but if the arc-length speed u is fixed, V achieves a well-defined maximum as a function of B/λ for each R2 . The limiting behaviors n → 0 as B → 0 and B → l/4 are easy to understand, since in the former case the rod is motionless and λ = l, and in the latter it collapses on itself as λ → 0 (α → 0 in Eq. (14)), so that the forces at points s and l/2 − s or s and l − s cancel. On Fig. 2 we also identify waves with minimum energy output by computing the work done on the fluid by the rod, and matching the resulting curves for constant speed with curves of n and n

for which the speed is held constant. The dashed curves in Fig. 2 thus represent values of B/λ at which the rod does the least work on the surrounding fluid (cf. [44, Sect. 6]). In this sense, they represent the most efficient swimming waves. Taylor’s force balance is most readily understood if one initially neglects tangential forces FT , in which case V = U and the shape (12) remains stationary in the inertial frame while the rod’s material particles move along a fixed sinusoidal ‘tube’ at speed u = U l/λ. Since their velocity is everywhere tangential to the fixed shape, they exert no force on the fluid. When tangential forces are included, V < U and the shape moves backwards in the inertial

851

1

1

0.8

0.8

0.6

0.6

n'=V/u

n=V/U


0.4

0.2

0.4

0.2

0

0 -2

10

-1

10

0

-2

10

-1

10

0

10

B/λ

10

B/λ

Fig. 2 Left panel Swimming diagram, following Taylor [44], showing the ratio n = V/U of the mass center speed relative to the speed of the backward traveling wave in an inertial frame as a 1/2

function of the amplitude/wavelength ratio B/λ. Each curve represents a fixed value of CN R1 : 10000, 1000, 500, 100, 10, 2, 1 (top to bottom). Right panel Ratio n = V/u of the speed of the center of mass relative to the speed of the backward traveling wave in arc-length for fixed values of 1/2

CN R2 = 10000, 1000, 500, 100, 10, 2, 1. The dashed curve denote B/λ values for which work done on the fluid is minimized

frame, leading to a net normal drag force in the forward direction that exactly balances the backward tangential drag force caused by the forward motion of material points. When the mass center speed is less than V, the higher backward speed of the shape creates a larger net normal force forward than the backward tangential force, thus accelerating the body. In fact we need only consider the x-components of the normal and tangential forces, since symmetry in y about the midline leads to automatic cancellation of their y-components (see Fig. 3). Our model treats rods not with a prescribed shape, but rather with prescribed intrinsic curvature. (Rods with a prescribed shape are briefly treated in Sect. 4.1.) The shape that emerges depends on the interaction of activation κ(s, t) with passive elasticity of the rod and the surrounding fluid as determined by Eqs. (1)–(5) and (9)–(10). Key parameters are the stiffness, reference shape and dimensions FN nx

FL t x

4 2 -40

-30

-20

-10

0

10

.

t=3.75

r

t=0.75

-2 -4 -6

Fig. 3 Swimming rod, showing velocity of a material point and x-components of the corresponding normal and tangential forces at two different times. The net x-component of the normal force decreases as the rod accelerates and asymptotic speed is achieved when tangential and normal forces, integrated along the rod, exactly balance

852


of the rod, and its preferred curvature, which represents muscle activity. Our assumption of isotropy is not strictly true, since muscles, tendons, skin, etc. are anisotropic, having different bending stiffnesses in different directions. This can be partially accounted for by taking an elliptical cross-section as in Fig. 1: we can increase muscle forces by increasing the cross sectional width 2b. Moreover, we may taper the rod towards the caudal (tail) end to reflect body geometry. More critically, our planar deformations neglects torsional effects which necessarily cause three-dimensional motions, and which are certainly important in some maneuvers [49]. In calculating W, we consider only the height 2a of the rod, assuming that fluid reaction forces are the same as those on a cylinder of radius a, although in reality the constant CN changes slightly for elliptical rods. Further, we may set CN = 1, since Reynolds numbers for lampreys and eels lie well within the range 20 < Re < 105 ; for example, in their work on the eel Anguilla rostrata, Tytell and Lauder cite Re = 60, 000 based on body length l = 20 cm for a specimen swimming at 1.4 l/ sec. [46], and speeds reported in [47] range from 0.5 to 2 body lengths per second. Translating to Taylor’s body-diameter-based number, we obtain R1 ≈ 2000–8000. Geometric and material parameter values will be specified in Sect. 4. 2.3 Traveling waves in uniform activated rods If we specialise to the case of a uniform rod (ρ, A, E and I constant) in the absence of fluid reaction forces (W = 0), and suppose that κ(s, t) = κ(s − ct) prescribes a uniform traveling wave of preferred curvature, then we may seek a traveling wave solution by assuming that the fields x, y and ϕ depend upon s and t only via ξ = s − ct. Letting (·) denote ∂/∂ξ(·) and appealing to the chain rule to deduce that xtt = c2 x

, ϕs = ϕ , etc., we may integrate Eqs. (2)–(3) once to obtain ρAc2 x (ξ ) = f (ξ ) + k1 ,

ρAc2 y (ξ ) = g(ξ ) + k2 ;

(16)

y = sin ϕ.

(17)

while Eqs. (1) become x = cos ϕ,

Using (5) and substituting (17) into (16), we may eliminate the contact forces f and g from (4) and obtain a single third order ODE in the tangent angle ϕ: −cδ ϕ

+ (E − ρc2 )Iϕ

+ k1 sin ϕ − k2 cos ϕ = EIκ (ξ ).

(18)

Note that E − ρc2 > 0 for cases of physical interest. A similar ODE, lacking third order dissipation and the ‘forcing’ term κ (ξ ), was derived and solved for solitary and periodic waves by Coleman and Dill [17], cf. [18], and we shall start by setting both δ and κ to zero.


853

To solve (18) we must supply appropriate boundary conditions and specify the constants of integration k1 and k2 . Unidirectional traveling waves such as those we seek can exist only in infinitely long rods, or, if they are periodic, in rods with joined ends that form closed (topological) circles [17, 18]. However, while not directly relevant to the study of activated dynamics of finite rods, they will nonetheless allow us to derive exact solutions that will be used to check our numerical schemes in Sect. 4. Motivated by the fact that lampreys and eels exhibit about one full wave along their bodies, we seek periodic solutions of period l for which y (ξ ) = sin ϕ(ξ ) = 0 at ξ = 0, l, i.e. with maxima (or minima) of the space curve (x(ξ ), y(ξ )) at head and tail. Without loss of generality we may also assume that l l x(0) = 0, x(l) = 0 cos ϕ(ξ ) dξ and y(l) = 0 sin ϕ(ξ )dξ = y(0) = 0 (corresponding to ‘swimming’ in the x direction). Since y (0) = 0, we conclude from (16) that g(0) = g(l) = k2 = 0: the contact force normal to the direction of motion must vanish when integrated over one period, otherwise the subset [0, l] would move in the y direction. (But note that material particles do experience net motion in the x direction, so that f does not have zero mean; indeed from (16)–(17) its mean is ρAc2 cos ϕ(0) − k1 .) This leads to the boundary value problem: (E − ρc2 )Iϕ

+ k1 sin ϕ = 0,

ϕ(0) = 0 = ϕ(l),

(19)

which is the classical Euler buckling problem [3], or, as an initial value problem, the simple pendulum [29]. The solutions of interest to us are periodic ‘librations’ which can be expressed in terms of the Jacobian elliptic function sn as ϕ(ξ ) = 2 sin−1 [k sn(αξ , k)],

(20)

having period T(k, α): T(k, α) =

4K(k) . α

(21)

π/2 Here α = k1 /(E − ρc2 )I, K(k) = 0 [1/ (1 − k2 sin2 ψ)] dψ is the complete elliptic integral of the first kind, and k is the elliptic modulus defined in Eq. (62) of the Appendix. To fit one full wave into the interval [0, l], the relationship 4K(k) = αl

(22)

must therefore hold, implicitly determining k in terms of m, l and α, and hence of the material parameters E, I, ρ, the wavespeed c, and integration constant k1 . For the reader’s convenience, some details of the derivation are provided in Appendix A. We consider the effect of small activation, restoring the ‘forcing’ function EIκ (ξ ) to equation (19). Rewriting this in terms of the independent variable

854


via η = αξ as in Appendix A, we obtain d2 ϕ + sin ϕ = εh(η), dη2

(23)

where εh(η) = EIκ (η/α)/α. For small ε we may apply the perturbation theory described in [28, Chap. 4.5–6], and originally due to Melnikov, to determine the fate of the family of periodic orbits of (19) in the presence of a weak traveling wave of activation. Specifically, if (ϕk , dϕk ) denotes an orbit of period 4K(k) for dη (23) with ε = 0, and the Melnikov function 4K(k)

M(η0 ) =

dϕ (η) h(η + η0 ) dη dη

(24)

0

has a simple zero at η0 = η0∗ , then a periodic orbit of period T persists in the neighborhood of the unperturbed orbit for (24) with ε sufficiently small [28, Theorem 4.6.2]. Assuming a sinusoidal traveling wave h(η) = A sin(2π η/l) with one full wavelength in the body (as in Taylor’s work, and see Sects. 3 and 4 below) and substituting the cnoidal function (65) into (24), we obtain ⎡

4K(k)

⎢ M(η0 ) = A sin[2π(η+η0 )/l] dη = 2kA ⎣

⎤

⎥ cn(η, k) cos(2π η/l) dη⎦ sin(2π η0 /l),

0

(25) where we use the fact that cn(η, k) is an even function. Further using the Fourier series representation (66)–(67) and noting that in terms of η (22) implies that 4K = l, we may compute the integral term by term to obtain ⎤ ⎡ l

∞ 2π η 16π A qn+1/2 ⎣ (2n + 1)2π η cos dη⎦ sin(2π η0 /l), cos M(η0 ) = l l l 1 + q2n+1 n=0

0

(26) where the elliptic nome q is defined in (67). Only the n = 0 term survives to yield 8π A sin(2π η0 /l) M(η0 ) = , (27) q1/2 + q−1/2 which clearly has simple zeroes. Hence, unperturbed waves that are in resonance with the applied intrinsic curvature εh(η) = EIκ (η/α)/α are ‘selected’ as periodic solutions in the perturbed problem. This implies that for undamped, uniform rods, small preferred curvatures can maintain a forced traveling wave that approximates the


855

unperturbed cnoidal waveform, but with a fixed phase relationship to the imposed curvature. 3 Discretization of the actuated rod, muscle activation, and curvature We discretize the rod equations with step size h = l/N in the arclength variable s, letting xi (t) = x(ih, t), i = 0, . . . , N, and similarly for the other field variables yi , ϕi and parameters Ai , Ii . We approximate the inextensibility constraints (1) by xi+1 − xi =

h cos ϕi + cos ϕi+1 , 2

yi+1 − yi =

h sin ϕi + sin ϕi+1 ; 2

(28)

Eqs. (2)–(4) are approximated by ρAi h x¨ i = hWxi + fi − fi−1 , ρAi h y¨ i = hWyi + gi − gi−1 , h h gi + gi−1 cos ϕi − fi + fi−1 sin ϕi , ρIi h ϕï = Mi − Mi−1 + 2 2

(29) (30) (31)

and the constitutive relation (5) becomes

Mi = EIi

ϕi+1 − ϕi ϕ˙ i+1 − ϕ˙i − κi + δi . h h

(32)

The force and moment free boundary conditions M = f = g = 0 at s = 0, l become M0 = f0 = g0 = 0 = MN = fN = gN .

(33)

In Sect. 3.2 we will derive the form of the intrinsic curvature and moment resulting from a traveling wave of activation [Eqs. (49)–(50)]. Corresponding to the total energy H(t), angular momentum L(t) and linear momentum M(t) (6) are their discrete analogs. Setting mi = ρAi h and Ji = ρIi h, these are defined as follows: N h 2 1 2 2 2 Ji ϕ˙i + M + mi x˙ i + y˙ i , H(t) = 2 EIi i i=1

L(t) =

N

Ji ϕ˙i + mi (xi y˙ i − x˙ i yi ) ,

i=1

Mx (t) =

N i=1

mi x˙ i ,

My (t) =

N i=1

mi y˙ i .

(34)

856


Analogously to the derivation of conserved quantities in the continuous case (7), direct computations of the rates of change of the above quantities under (28)–(32) yield dH = ϕ˙N+1 MN − ϕ˙ 1 M0 + fN x˙ N − f0 x˙ 1 + gN y˙ N − g0 y˙ 1 dt h gN˙ ϕ N cos(ϕN ) + g0 ϕ˙1 cos(ϕ1 ) − gN˙ ϕ N sin(ϕN ) − f0 ϕ˙1 sin(ϕ1 ) + 2

N δi x˙ i Wxi + y˙ i Wyi + Mi −κ˙ i+1 + +h ϕï+1 − ϕï , (35) hEIi i=1

dL = MN − M0 + gN xN − g0 x1 − fN yN + f0 y1 dt h + (gN cos(ϕN ) + g0 cos(ϕ1 ) − fN sin(ϕN ) − f0 sin(ϕ1 )) 2 N +h xi Wyi − yi Wxi ,

(36)

i=1

dMx = f N − f0 + h Wxi , dt N

i=1

dMy = gN − g0 + h Wyi . dt N

(37)

i=1

As in the continuous case, when W = 0, δ = κt = 0 and with free boundary conditions these time derivatives all vanish. 3.1 Discrete models of angulliform swimmers The finite-difference discretization derived above is related to previous models that represent the body as a planar chain of rigid links hinged at their endpoints and subject to forces and moments generated by passive springs, dashpots, and active elements. Specifically, in modeling lamprey Ekeberg [21, 22] represents the body as a chain of N links each of length h, mass mi and moment of inertia Ji (Bowtell and Williams [5, 6] take a chain of point masses; see Sect. 3.4). The configuration of the ith link is described by its midpoint (xi , yi ) and the angle ϕi between its centerline and the inertial basis vector eˆ x (Fig. 4) and the center of mass of each link is assumed to coincide with its midpoint. Equations (28) then express the condition that the links remain connected at the joints. Letting (fi , gi ) and Mi denote the components of contact force and the torque at the joint connecting link i to link i + 1 and (hWxi , hWyi ) be the body force acting on the midpoint of link i (Fig. 5(a)), balances of linear and angular momenta yield Eqs. (29)–(31) above with mi = ρAi h and Ji = ρIi h. One can argue that animals with relatively rigid vertebrae are better represented by discrete link models, although we note that lampreys have soft vertebral arches, and as we shall show in Sect. 4.3.4, for the large segment numbers N = O(100) typical of eels and lampreys, the behaviors of the discrete


857

y yN yN-1

ϕN ϕ N-1

...

y3 y2 y1

ϕ3 ϕ2 ϕ1

x1

x2

x

xN-1 xN

x3

Fig. 4 Representation of the swimmer as a chain of interconnected links

y

(a) Mi

D

GR

C

α'

β

gi

Wyi Wxi

M i-1

(c)

ψi = ϕi+1− ϕi

fi

h/2 l

w

fi-1

β α

B

GL

A

g i-1

x (d) (b)

f

f

f

D C

α'

β

B ψi

f

f

f

A

Fig. 5 a Forces and moments acting on link i. b Bending moments are determined by muscles on both sides of the body modeled by springs and dashpots with additional active elements. c Forces and moments associated with a single joint. d Joint geometry for |ϕi+1 − ϕi | > π − 2β

and continuum models are very close. However, the discretization reveals how muscle activity determines preferred curvature κ(s, t) and affects bending stiffness EI of the continuum model. Specifically, in the model of [5], the joint connecting each pair of links of length h is actuated by a pair of spring-dashpotactuators with spring constant μ and damping coefficient γ anchored to arms of length w that project normally from the links’ midpoints (Fig. 5(b)). The linear springs and dashpots represent passive muscle and body tissue viscoelasticity, and the actuators generate prescribed contractile muscle forces fLi and fRi on the right and left sides of the body respectively; as we shall see, this is essentially

858


a linearization of a Hill-type muscle model [59]. (In [21] the stiffnesses of passive springs are modulated by the spiking activity of a CPG-motoneuron model: this is not easily related to Hill-type models, so here we adopt the scheme of [5].) Suppressing the dependence on i and denoting the relative extensions of the spring-dashpot-actuators CD and AB as R and L (Fig. 5(c)), the total forces tending to contract these elements may be written ˙ R, GR (t) = fR (t) + νR + γ ˙ L, GL (t) = fL (t) + νL + γ

(38)

where we note that the springs are in tension (and hence generating contractile forces) when R , L > 0. From elementary trignometry we have 2l cos(α ) − h w CD − h = = cos(ψ/2) − 1 − 2 sin(ψ/2) h h h 2l cos(α) − h w AB − h L = = = cos(ψ/2) − 1 + 2 sin(ψ/2), h h h R =

and (39)

where ψ = ϕi+1 − ϕi is the angle between neighboring links. Finally, computing the moment arms LR , LL to the joint along normals from the lines AB and CD on which the forces act (Fig. 5(c)): LR = l sin(α ) = l cos(β − ψ/2) = w cos(ψ/2) + (h/2) sin(ψ/2) LL = l sin(α) = l cos(β + ψ/2) = w cos(ψ/2) − (h/2) sin(ψ/2),

and (40)

we find that the resulting torque at joint i is given at leading order by Mi = −(GRi LRi − GLi LLi ) = fLi (t) − fRi (t) w

ϕi+1 − ϕi ϕ˙i+1 − ϕ˙i h2 +2γ w2 , (41) + 2νw2 − (fLi (t) + fRi (t)) 4 h h where we have used cos(ψ/2) ≈ 1 and sin(ψ/2) ≈ ψ/2. If |ψ| > π − 2β both line segments CD and AB lie on the same side of the joint (Fig. 5(d)) and, rather than cancelling, the contractile forces fLi and fRi sum to produce a moment that tends to collapse the joint. This does not occur for small angles. Comparing (41) with the discretized constitutive relation (32) we see that the link and discretized rod models coincide if the stiffness EIi and intrinsic curvature κi are interpreted as follows: EIi = 2νw2 −

h2 (fRi + fLi ), 4

κi =

4(fRi − fLi )w , 8νw2 − (fRi + fLi )h2

δ = 2γ w2 .

(42)

This reveals that the effect of muscle forces (fRi , fLi ) is twofold: as one might expect, the difference between left and right is responsible for producing nonzero preferred curvature, but the sum has the effect of reducing the passive


859

bending stiffness. In fact this is reasonable, since coactivated muscles produce a distributed axial compressive load, and the classical problem of Euler buckling can be interpreted as cancellation of bending stiffness by an externally imposed axial load. Specifically, for the discretization of Fig. 5(c), once ψi > 0 the moment due to the active force fRi in CD is larger than that due to fLi in AB, and the active forces therefore promote further increase in ψi . We note that in [21, Eq. (7)] the passive stiffness term analogous to the second term of (41) contains the sum of 2νw2 and h2 (fLi + fRi )/4, rather than their difference; we believe that this is incorrect, although, as we shall see, the change due to the latter term is probably insignificant. The number of muscle fibers typically depends upon the cross-sectional area A(s) of the animal, so we expect contractile forces and passive stiffness and damping coefficients to vary similarly. It is convenient to work with parameters that are independent of A. For the elliptical cross section of Fig. 1 we take w = αb where α ∈ (0, 1) is a dimensionless parameter characterizing muscle attachment geometry and set fR,L = αab f¯R,L ,

ν = αab ν, ¯

¯ δ = 2α 2 b2 δ,

(43)

so that the force density f¯R,L , stiffness ν¯ and damping δ¯ have units N/m2 , N/m2 and N s/m2 respectively, and they depend only on activation and elastic properties (and α). This will enable us to separate material and geometric effects. 3.2 Muscle activation and force generation Ventral root recordings such as those of [16], reproduced in [14], show that waves of motoneuronal activity consisting of bursts of closely spaced action potentials (APs or spikes), separated by near-silent interburst periods, travel the length of the lamprey spinal cord. The waves, generated spontaneously by a distributed CPG, are in near-antiphase contralaterally and maintain approximately constant burst/silence ratios and segment-to-segment ipsilateral phase lags, regardless of overall frequency. They exhibit about one full wavelength from head to tail (a phase lag of ∼ 0.01 cycle per segment) and produce muscle activation with similar phasing, evident in electromyograms (EMGs) [51], as follows. Bundles of myofibrils make up the muscle fibers. The AP bursts cause calcium release in the sarcoplasmatic reticulum (SR) that surrounds the myofibrils and is encircled by T-tubuli at repeated intervals. The resulting muscle contraction occurs in three phases [34]. (i) A motoneuronal AP arrives at the neuromuscular junction, producing an AP at the motor end plate and exciting the muscle fiber. (ii) The AP opens gates in the SR and releases Ca2+ ions into the muscle filaments. (iii) Ca2+ causes conformational changes in the thick filament which forms a cross-bridge to a thin filament [37]; a subsequent conformational change then slides the thin filament over the thick one, shortening the muscle. This is followed by relaxation. The process is modelled by a

860


unidirectional linear cascade [30, 31] in which motoneuron output u(t) is converted into T-tubuli depolarisation response β and SR calcium release η via β¨ + c1 β˙ + c2 β = c3 u(t), η¨ + c4 η˙ + c5 η = c6 β(t),

(44) (45)

thereby producing activation A(t) =

a0 + (ρη)2 . 1 + (ρη)2

(46)

In (46) the variable ρ(t) represents muscle fatigue [43] and is governed by a first order linear ODE, but here we neglect this effect and set ρ ≡ 1. Note that A(t) refers to muscle activation, not to be confused with body cross-sectional area A, and we subsequently add subscripts R and L to indicate right and left as in fR,L above. Following [14, 16], we assume that the motoneuronal outputs constitute a traveling wave u(s−ct) of AP bursts along one side of the body, with u(s−l/2−ct) on the other. Since we shall not include proprioceptive feedback that might modulate its behavior, the CPG that generates these coordinated bursts need not be modeled; we may simply take them as prescribed. We also note that the linear dynamics of (44)–(45) imply that the effects of individual spikes superpose, so that an increase in the number of APs in a burst of fixed duration increases the amplitude of A(t). We adopt parameter values that produce activations that peak and collapse within 40–50% of the burst period of 0.5–1 s, specifically: c1 = 6.0, c2 = 0.2, c3 = 0.5, c4 = 20.0, c5 = 1.5, c6 = 5.0 and a0 = 0. The muscle activation A(t) in turn produces contractile forces in the muscle fibres. A widely-used macroscopic model of vertebrate muscle due to Hill [33], cf. [59], expresses this force as a product A(t) Fl (l) Fv (v), where the latter terms describe dependence on muscle length l and shortening velocity v = −˙l. Fl (l) peaks at the optimal length l0 and is approximately parabolic, falling to zero at about 0.5l0 and 1.5l0 ; Fv (v) decays to zero at a maximal shortening velocity vmax and rises to a constant asymptote for lengthening (v < 0). However, for small contractions and small contraction velocities, we may neglect this nonlinear state dependence and assume that force density is proportional to activation, scaled by a factor F¯ that represents the force per unit muscle cross section generated by unit activation: ¯ f¯R,L = AR,L F,

(47)

as was implicitly done in [21, Eq. (7)]. We also adopt this simplification since we wish to separate the effects of activation, passive viscoelastic and resistive (hydrodynamic) effects, which nonlinear muscle dynamics would substantially complicate (cf. [24]). The full Hill model and other models specifically developed


861

for lamprey muscle [55] may be necessary to properly capture the observation that EMG waves travel faster than mechanical waves (body shapes) [56], since this is presumably due both to time delays in muscle force generation (leading to preferred curvature κ), and to mechanical inertia as the rod attempts to follow κ in the presence of constraint and hydrodynamic body forces. We can now derive the curvature resulting from a traveling wave of activation. We consider the case in which the number of links approaches infinity, in which case the model approaches the elastic rod, allowing continuous variation of stiffness that, as noted in [22], is perhaps more representative of the lamprey spinal cord than the relatively small number of rigid links used in previous studies (e.g., 10 links in [21]). From (42)–(43) and (47), as h → 0 stiffness and curvature approach the continuum limits: EI = 2α 3 ν¯ ab3

and

κ=

F¯ (AR − AL ), 2α ν¯ b

(48)

¯ ; also, κ and for an elliptical cross section (I = π ab3 /4) we have E = 8α 3 ν/π is inversely proportional to width. We use the above definitions EI and κ in the computations of Sect. 4, and in Sect. 4.3.4 we verify that inclusion of the neglected O(h2 ) terms has little effect for the numbers of segments typical of eels and lampreys. Figure 6 illustrates the relationship among motoneuronal output u(s − ct), preferred curvature κ and the resulting intrinsic shape determined by Eqs. (44)–(46), (48) and integration of the expression for curvature in the traveling frame. As noted above, the addition of spikes to a burst produces a concomitant amplitude increase in curvature, and, via the nonlinear curvature expression, an increase in amplitude and change in shape. Curvature maxima slightly lag burst midpoints (cf. [25, Fig. 1]), and greater lags are observed near the caudal end. We have already noted that Taylor [44] assumed a sinusoidal traveling wave. Carling et al. [10] also assumed a prescribed shape, but of a sine wave linearly increasing from head to tail. In their Lagrangian formulation [5, 6] Bowtell and Williams defined generalized forces as square, sine and attenuated sine waves, implying, via (42) and (48) similar forms in prescribed curvature, although, as noted in Sect. 3.4, our model yields different linearized equations from theirs. Ekeberg [21] set stiffnesses proportional to activations derived from a CPG model that produces an irregular traveling wave that ramps up and down with a duty cycle of approximately 50%. It is therefore difficult to compare his model directly with ours. To illustrate the effects of motoneuronal and muscle activity in the context of these disparate formulations, in Fig. 7 we show the shapes of a uniform rod resulting from four different intrinsic curvatures. The first produces a sine wave y0 = B sin(2π(x − ct)/λ) and is computed by assuming this shape and deriving its curvature; the second is due to the motoneuronal bursts, calcium dynamics and simplified Hill model described above, and the third and fourth emerge from a sine wave and square wave of curvature, respectively. Figure 7 illustrates

862

T. McMillen, P. Holmes Curvature: κ0(ξ)

Motoneurons: u(ξ) 1

Shape: y0(x)

1.5 1.5 1

0.5

1

0.5 0 –0.5

0.5

0

0

–0.5

–0.5 –1

–1

–1.5 –1

–1.5 0

5

10

1

0

5

10

0

2

4

6

8

1.5 1.5 1

0.5

1

0.5 0 –0.5

0.5

0

0

–0.5

–0.5 –1

–1

–1.5 –1

–1.5 0

5

ξ

10

0

5

ξ

10

1

2

3

4

5

x

Fig. 6 Relationships among motoneural activation, intrinsic curvature and shape for a uniform rod. The left and center panels show activation and intrinsic curvature in the traveling wave variable ξ = s − u t, activity on the left side of the animal being depicted as negative; the corresponding shapes are shown in the right panels. Curvature maxima occur at mid burst and an increase in spike rate within a burst raises the amplitudes of curvature κ0 (ξ ) and shape waves y0 (x), and reduces the wavelength λ of the latter in the inertial (x) frame (right panels). An increase in bursting rate (not shown) would raise the wave speed u and reduce the wavelength

that, even if the body exactly follows the prescribed curvature (as it would for large bending stiffness EI and damping δ), the relations among the various fields are not necessarily simple, although the double integration implicit in determing y0 (x) substantially smooths gradients and discontinuities in κ(s). We note that the motoneuron and muscle model produces an intrinsic shape remarkably close to the sine wave assumed in [44], and we shall adopt such a ‘preferred sinusoid’ activation in the simulations of Sect. 4, although as we shall see the actual shape departs from this due to passive elasticity and hydrodynamic forces, with interesting consequences. Tapering the rod will produce more realistic amplitude variations as in [5, 10, 21]. 3.3 The final model Based on the analysis and preliminary shape studies, in the simulations to follow we shall consider uniform and linearly tapered elliptical rods, activated by


κ(ξ)

sine wave 1

0

0

–2

–1

φ(ξ)

5

10

0

5

10

square wave of curvature

1

1

0

0

1

1 0

5

10

2

2

2

2

0

0

0

0

–2

0

5

10

–2

0

5

10

–2

0

5

10

–2

0

5

10

0

5

10

2

2 y0(x)

sine wave of curvature

Hill type activation

2

0

863

0

0

–2

–2 1 2 3 4 5

1 2 3 4 5

1

1

0

0

–1

–1

–2

0 1 2 3 4 5

–2

1 2 3 4 5

Fig. 7 Intrinsic curvature and associated tangent angle yielding a sine wave y0 (x) = A sin x (left panels); and the tangent angles and shapes resulting from the curvature arising from a Hill type activation, a sinusoidal curvature and a square wave of curvature

prescribed curvatures that yield sinusoidal intrinsic shapes. Specifically, to calculate the moment Mi in (32) we set π EI(s) = E ab3 , 4

¯ R − AL ) def κsin s − u t 4α 2 F(A l κ(s, t) = = π Eb Eb

(49)

where u is the speed of the backward-traveling wave of activation and we have separated the material and geometric parameters E, b from the ‘standardized’ ¯ R − AL ) (with units N/m2 ) that yields the curvature κsin (s − u t/l) = π4 α 2 F(A sinusoidal shape of Fig. 7 (left column) with one full period in the body. The dependence of κ on width b(s) implies that for tapered rods the intrinsic curvature will not produce a sine wave; indeed, for smaller b intrinsic curvature is greater and the force tending to restore the rod to its intrinsic curvature is smaller, causing larger amplitudes in thinner regions. Curvature is also inversely proportional to E, so softer materials will exhibit larger amplitude waves, as well as greater departures from their intrinsic shapes. Summarizing and substituting (49) into (32), we have Mi = E

u 2b2 δ¯ π ab3 ϕi+1 − ϕi − ab2 κsin i h − t + ϕ˙i+1 − ϕ˙i , 4h l h

(50)

864


where the geometrical parameters a, b specify the elliptical cross section and h the discretization (link length), E and δ¯ specify the viscoelastic material, and κsin denotes the standardized curvature. 3.4 Comparison with the linearized model of [6] As noted above, Bowtell and Williams derived discrete [5] and subsequently continuum [6] models of anguilliform body dynamics in the absence of hydrodynamic forces. They assumed a chain of point masses connected by rigid massless rods, hinged at the masses, and subject to active and passive forces generated by springs and dashpots much as in [21] (cf. Fig. 5(c)). It is difficult to compare our PDE model with their single integro-differential equation for curvature [6, Eq. (12)], but we may directly compare the linearized model of [6] with the analogous linearization of (1–5) lacking body forces. For |ϕ| 1 we deduce from (1) that dx/ds = cos ϕ ≈ 1 and dy/ds = sin ϕ ≈ ϕ. Hence x ≈ s and we must assume fs = 0 for consistency, implying f ≡ 0 (from the free boundary conditions). Thus, Eqs. (2)–(3) with Wx = Wy = 0 imply f ≡ 0,

gss = ρA ystt ≈ ρA ϕtt ,

(51)

and from (4)–(5) we have ρI ϕtt ≈ [EI (ϕs − κ) + δϕst ]s + g − f ϕ.

(52)

Differentiating (52) twice with respect to s, using (51) and rearranging the terms, we obtain ρA ϕtt ≈ −(EIϕs + δϕst − κ)sss − (ρI ϕtt )ss .

(53)

The first four terms of (53) coincide with those of [6, Eq. (16)] (if the curvature term of (53) is appropriately identified with Bowtell and Williams’ muscle force), but since we include the effects of rotary inertia we also obtain a term involving I, which is absent in their point mass formulation. Such a term appears in classical Euler–Bernouilli models for uniform straight beams with appreciable thickness, as one can see by using ϕ ≈ yx to replace ϕ by y in (53): ρA ytt + EI yssss + δ ysssst − ρI ysstt ≈ 0,

(54)

cf. [45]. Thus our model generalizes both a classical beam model and the specific anguilliform swimming models of [5, 6, 21]. We observe that neglecting the moment of inertia I in the equations above is formally equivalent to setting the right hand side of (4) equal to zero, thus approximating the rotational dynamics by a pure moment balance: a limit appropriate to the assumption of relatively stiff and strongly damped rods whose curvatures remain close to the preferred curvature.


865

4 Numerical simulations We now explore the effects of activation and material and geometric body parameters on the motions of the rod. Some properties, such as motoneuron firing rates, are under the animal’s control; others, such as fluid properties and its intrinsic geometry and stiffness, are not (or barely so). Our standardized sinusoidal curvature (49), while not strictly correct, will allow us to explore these properties in a transparent way. The bulk of the section will examine the influence of parameters, starting with those under the animal’s control, but first we simulate rods with prescribed shapes as benchmarks for cases in which curvature, instead of shape, is prescribed. We then verify the numerical method and show that the actuated rod, with resistive hydrodynamic forces, does indeed ‘swim’ in a qualitatively realistic manner, before continuing to parameter studies. The numerical method, detailed in Appendix B, employs discrete versions (69)–(72) of the integrated constraint equations (1) that express link positions in terms of that of the head (x1 , y1 ) and the link angles ϕi , thus guaranteeing that the inextensibility constraint is precisely satisfied for the discrete system (28–32) and eliminating the need to solve ODEs for (xi , yi ); i = 2, . . . , N. In contrast to Ekeberg’s algorithm [21], our final second-order system is of dimension N + 2 instead of 3N, and projections are not required to resolve numerical violations of the constraints. We found that simulations using the former method gave significantly larger errors, even if projections were applied at every timestep, and that our algorithm ran 5–10 times faster in the Matlab enviroment employing ode45, allowing us to perform extensive simulations with up to 80 segments. Nonetheless, with careful error control and for lower values of N, Ekeberg’s method and ours gave similar results.

4.1 Rods with prescribed shapes The shape of an inextensible rod is determined by its centerline curvature ϕs , and if the orientation of one point is known, then the tangent angle ϕ(s, t) relative to inertial space is entirely determined. As in [44] we restrict to shapes whose symmetry ensures that the net torque on the body vanishes, so that ϕ is a given function of s and t and we need only solve for translation dynamics. (In general one must also solve for the net torque as well as the translation forces.) The governing equations therefore reduce to the two coupled second order ODE’s (88)–(89) derived in Appendix B, which can be solved with a standard Runge–Kutta algorithm. The moving shape can then be reconstructed via the integrated constraint equations (69)–(72). To compute the terms on the RHS of these equations, we must know ϕ(s, t) and its first and second time derivatives. In cases where curvature is given directly, ϕ follows from a single integration, but if the shape is prescribed in the form y = Y(x, t) (as the sine wave of [44] and the left panels of Fig. 7), then we have

866


ϕ(x, t) = tan−1

∂Y , ∂x

and to obtain ϕ as a function of arc-length s, we must interpolate along this curve using ds2 = dx2 + dY 2 . We performed simulations using a spatial discretization of N = 100 and three prescribed shapes: a sine wave, and the shapes arising from a sine wave of curvature and a square wave of curvature (Fig. 7, left, center-right and right panels). The rod begins at rest at time t = 0, and accelerates until it reaches a constant asymptotic speed. Figure 8 shows the asymptotic speeds and times required to reach them in terms of nondimensional velocities n = V/u versus amplitude to wavelength ratio B/λ, as in Fig. 2 (right). The results for the sine wave coincide with those obtained by the methods of Sect. 2.2 and in all cases the asymptotic (steady) speeds and times required to reach them are similar. As in Taylor’s analysis, the behavior primarily depends on B/λ and the speed u of the backward traveling wave.

4.2 The numerical scheme and code verification The results of Sect. 4.1 show that our discretizations of the constraint and linear momentum balance equations (1)–(3) behave properly. To confirm the code’s performance in the general case, we first verify that it conserves energy and linear and angular momentum in the absence of body forces and dissipation,

0.7

1

1.8

3

3.7

0.5

12

square wave of curvature sine wave sine wave of curvature

9

0.4

0.3 6 0.2

Asymptotic speed (V/u)

Time to reach asymptotic speed (secs)

B (cm) 15

0.1 0

0.035

0.05

0.1

0.2

0.36

B/λ

Fig. 8 Nondimensional asymptotic speeds (solid curves, right axis) and time in seconds to reach asymptotic speed (dashed curves, left axis) for a sine wave and the shapes arising from a square wave and sine wave of curvature. In all cases a circular rod of radius 1 cm and length l = 20 cm carries a traveling wave of speed u = 40 cm s−1 . Since l is fixed, the amplitude B (cm) is a function of B/λ (14), as shown at top


867

and we further check by computing periodic traveling waves both with and without actuation, appealing to the theory of Sect. 2.3. Once the coupled ODE (90) governing the tangent angles and head position is formed, it can be solved to any given degree of accuracy using standard methods. We employ Matlab’s adaptive Runge–Kutta algorithm ‘ode45,’ with time steps bounded by 1 ms. However, even if this equation can be solved with perfect accuracy, errors in the solution still arise due to numerical round-off errors in the formation of the equations, in particular the solution of the matrix equation (86). The discretized energy and momenta (34) serve as useful diagnostics for accuracy. The rate of change of the discretized energy (35) is dH = R(t) + boundary terms, dt

N x˙ i Wxi + y˙ i Wyi + Mi −κ˙ i+1 + where R(t) = h i=1

δ . (55) ϕï+1 − ϕï hEIi

Let H n = H(nτ ) be a sampling of the discrete energy at time t = nτ , for n = 0, 1, . . . , T, where the total simulation is run for Tτ seconds. Then, since the boundary terms vanish, measures of the relative error in the calculation are given by T H n+1 − H n − τ Rn 1 n and E = En = E . (56) Hn T n=0

These quantities are a measure of the ‘pointwise’ error at a given time, and the average error of the entire simulation, respectively. Similar formulations are used for the angular and linear momenta. Throughout, we use a time step of τ = 1 ms. We ran several simulations in which energy and momenta are conserved (R(t) = 0), for example, of an intrinsically straight rod with a motionless but curved initial condition in the absence of body forces. In this case, the rod oscillates around the straight position indefinitely with positive energy and zero linear and angular momenta. Relative fluctuations E n in the discretized energy are less than 2 × 10−7 , the mean error E is less than 10−7 , and fluctuations in the discretized angular and linear momenta are less than 2 × 10−5 . We also verified that the code reproduces simple solutions such as that corresponding to a straight rod given an initial velocity. We set tolerances in the algorithms so that E n was less than 2 × 10−3 for t ≥ 0.5 s and mean error E on 0 ≤ t ≤ 4 s is less than 10−3 . Errors are greatest (∼ 4 × 10−3 for t < 0.2) when the rod starts from rest with ϕt = 0 and ϕs = κ: since κt = 0 it has to “catch up” to its intrinsic shape. Thereafter errors decline considerably. Although not directly relevant to the free boundary condition case, we also verified that the scheme for periodic boundary conditions is correct by reproducing the cnoidal solution of Sect. 2.3 for the unforced, undamped system. We

868


set κ ≡ 0, and chose initial conditions as follows: ϕ(s, 0) = 2 sin−1 k sn(αs, k) ,

∂t ϕ(s, 0) =

−2ckα cn(αs, k)dn(αs, k)

; 1 − k2 sn2 (αs, k)

(57)

the initial speed of the rod must also be consistent with the boundary conditions for a traveling wave, i.e. the time-derivatives of x(s, 0) and y(s, 0) must be such that the rod follows along the path of the curve determined by ϕ(s). With these appropriately set, the numerical scheme faithfully reproduces the cnoidal traveling wave ϕ(s − c t). 4.3 Numerical experiments In the following experiments we actuate the rod with a sinusoidal traveling wave of preferred curvature as in (49), with wavelength (in terms of arc-length) equal to the length l of the rod, and amplitudes that vary as specified below. We generally take a wave that passes down the body in 0.5 s (frequency = 2 Hz) with one period equal to the body length l. Unless otherwise stated, we consider a uniform circular rod of length 20 cm (hence u = 40 cm/s) and radius 1 cm with free boundary conditions. These values are based on observations of lampreys swimming freely in tanks [13]. We found that Young’s modulus values E ∼ 1 suffice for the rod to closely follow its intrinsic shape, and we henceforth adopt E = 0.7 MPa unless otherwise stated. [For comparison, the moduli of some common materials are as follows [4]: rubber: 7 MPa; tendon: 550 MPa; steel: 2 × 105 MPa (1 MPa = 106 N/m2 ).] Such small values of E are consistent with the extreme flexibility of anesthetized lampreys [13] and they are required to obtain results consistent with observations: see Fig. 10. We will discuss the effects of stiffness in Sect. 4.3.3: see especially Fig. 13. We set the damping coefficient δ¯ = 1 and take viscosity and density values of water: μ = 10−3 Pa·s and ρf = 1 g cm−3 The Reynolds number defined in Sect. 2.2 is therefore R2 = u2aρf /μ = 8000. In discretizing we generally take N = 40 links (h = 0.5) and we ignore the effects of the (fRi + fLi ) softening terms in EIi and κi of (42). In Sect. 4.3.4 we shall explore the effects of including these terms as the number of links varies. 4.3.1 The effects of hydrodynamic body forces We first show that the model can produce straight swimming and execute a turn by integrating the equations with body forces present. Figure 9 shows several snapshots of a rod tapered toward the tail as it moves through the fluid, demonstrating the effect of the sign of the spatial average of prescribed curvature. l If 0 κ(s, t) ds = 0, the net motion is along a line, as in the upper panel. If l 0 κ(s, t) ds = 0, the body turns either left or right depending on the sign, as in the lower panel. Thus, as in [21], the animal can turn left or right by increasing the firing rates of motoneurons on its left or right sides.

y (cm)


869

2 0 -2 -50

-40

-30

-20

-10

0

x (cm)

0 -5 -10

y (cm)

-15 -20 -25 -30 -35 -40 -45

-20

-10

0

10

x (cm) Fig. 9 Snapshots of the rod moving through fluid. The top panel shows the result of average intrinsic curvature κ = 0: the mass center travels in an approximately straight line. The bottom panel shows κ < 0: the rod turns to the left. The dashed line is the path followed by the center of mass. The rod tapers linearly toward the tail: b(s) = 1 − 12 sl

When the rod is tapered, as in the previous example, it is not immediately clear how intrinsic curvature should be prescribed. As we have seen Eq. (49), if muscle force is proportional to the product of cross-sectional area and motoneuron output and the latter is uniform along the animal then κ(s, t) ∝ κsin (s − u t/l)/Eb(s), and κ, and hence the amplitude of the intrinsic waveform, varies with b(s). However, neural activity may also vary with body width, (partially) compensating for this effect. We therefore consider two ‘limiting’ cases: (1) constant activation amplitude, so that intrinsic curvature increases with taper; and (2) constant amplitude of curvature. Figure 10 shows successive shapes of the rod in a frame in which the center of mass is fixed. Higher amplitudes are clearly apparent toward the tapered

870


Water forces off

E=0.1

Water forces on 2

2

0

0

–2

–2

–4

–4

E=0.7

E=0.1: Curvature amplitude constant

0

5

10

15

0

5

10

15

2 0

0

–2

–2

–4

–4 0

5

10

15

5

0

0

–2

–2

–4

10

15

–4 0

5

10

x (cm)

15

0

5

10

15

x (cm)

Fig. 10 Successive snapshots of a tapered rod’s centerline in a frame fixed at the center of mass. The left and right panels shows centerlines with and without body forces present. The first and third rows show the effects of different Young’s moduli with uniform activations. The second row shows the effects of constant curvature amplitude. Snapshots are taken at the same times. When Young’s modulus is moderately large (E ≈ 0.7 MPa) hydrodynamic forces have little effect on shape. The bottom row shows tracings of intact lamprey in fluid (left) and on a slippery bench (right), reproduced from [5, Fig. 3]

end. The left and right columns show that inclusion of hydrodynamic forces has little effect on shape when Young’s modulus is of order one; only for E = 0.1 are differences apparent at the head and in the center. The figure also shows that constant curvature amplitude yields smaller motions in the head and central portions, while the tail experiences smaller changes in curvature, and tends to act as a ‘paddle.’ As we will see in Sect. 4.3.3, this activation mode is doubly beneficial, since it not only leads to higher speeds, but also demands lower motoneuronal spike rates and muscle forces near the tail, and thus lower energy. We note the qualitative similarities between our results and the body centerline snapshots and numerical computations of [5, Fig. 3]: in particular the upper


871

right panel of Fig. 10 is similar to frames from a video of a lamprey out of water on a wet bench, and the left second and third panels of Fig. 10 are similar to frames from a film of a swimming lamprey. We reproduce these in the bottom row of Fig. 10. 4.3.2 Acceleration to asymptotic speed and the effects of activation strength A major parameter under the animal’s control is the amplitude of its intrinsic shape, which as we showed in Sect. 3.2 depends on the AP (spike) frequency in a motoneuronal burst. Here we explore the effects of such amplitude changes on a uniform rod while holding the material and geometric parameters constant (Young’s modulus, cross section and length). We begin simulations with the rod at rest in a shape corresponding to its intrinsic curvature and allow curvature to change in time according to (49), solving for positions and speeds of the material points. Figure 11 shows the forward and lateral velocity components computed for a rod swimming in the −x direction. Under the influence of prescribed curvature corresponding to a sinusiodal traveling wave of speed u, the rod accelerates toward a well-defined mean asymptotic speed (MAS) around which it oscillates at the activation period. We express this as a nondimensional ratio by dividing by wavespeed u as in Fig. 2. To compute MAS we run the simulation until the speed averaged over one cycle equilibrates, and we estimate the time necessary to reach the MAS by finding the time to reach a speed within 0.01 of the MAS. Both MAS and time to MAS depend on the intrinsic shape amplitude B, as the B (cm)

1.2 B=3.5

0.4

Forward velocity

0.3

B=2.5

0.2 0.1

Lateral velocity

0 -0.1 0

0.5

1

1.5

2

2.5

Time (sec)

3

3.5

4

2.5

3

3.9 0.55

8 7

0.5

6

0.45

5

0.4

4

0.35

MAS(V/u)

Speed (n'=V/u)

0.5

Time to reach MAS (secs)

0.6

1.8

3 0.3

2

0.25

1

0.2

0 0.06

0.1

0.15

0.2

0.35

B/λ

Fig. 11 Nondimensional speeds n = V/u of the mass center of a uniform circular rod with intrinsic sinusoidal shape y0 = B sin 2π λx − ul t (cf. Fig. 2). The wavespeed u is 2 body lengths per second

(40 cm s−1 ). Left panel Speed vs time for a rod started from rest with activation amplitudes B = 2.5 and 3.5 cm: the rod with greater activation amplitudes has a smaller mean asymptotic speed (MAS), but reaches its MAS sooner than the rod with a smaller amplitude. The dashed curves are the speeds of the center of mass averaged over one period of the backward traveling wave. Right panel MAS (solid line, right axis), and time to reach MAS (dashed line, left axis) as functions of amplitude B (top) and B/λ (bottom). The dash-dot curve is the speed of a rod carrying a sine wave, computed as in Sect. 2.2, with R2 = 8000

872


4 2 0

t = 0.4 sec

4 2 0

t = 0.5 sec

4 2 0

t = 0.9 sec

4 2 0

t = 1.2 sec

4 2 0

y (cm)

t = 2.1 sec

4 2 0

–30

–25

–20

–15

–10

–5

0

5

10

y (cm)

y (cm)

y (cm)

t = 0.0 sec

y (cm)

y (cm)

the left panel suggests. The right panel shows their variations versus B and B/λ (connected via (14)). As in Fig. 2, the MAS exhibits a maximum in B/λ, but the curve lies significantly below the analogous curve from Taylor’s swimming diagram. At steady state the mass center oscillates laterally and in the fore-aft direction with amplitudes that grow with B. This is due to the fact that the elastic rod does not precisely follow its intrinsic shape and departures from it grow with B (for fixed stiffness EI). Figure 12 shows the centerlines of rods with prescribed shape and prescribed intrinsic curvature: the latter’s shape lags the time-dependent sinusoid, leading to slight asymmetries that cause net oscillatory moments and translational forces. Uniform rods carrying traveling sine waves exhibit no such oscillations: due to the symmetry of the curve y = B sin(2π x/λ) under rotation through π about (x, y) = (π , 0), the hydrodynamic forces (10) in the y direction cancel at all times when integrated along a full wavelength. Taylor’s force balance in the x direction also applies [44]: as the wave propagates, pieces of the symmetric curve lost at the head are replaced at the tail, so that (13) holds instantaneously and the mass center translates at constant speed. Numerical solutions with different algorithms and error tolerances (not reported here)

15

x (cm) Fig. 12 Successive snapshots of centerlines of swimming rods carrying a prescribed traveling sinusoid (dashed curve) and prescribed preferred curvature corresponding to a traveling sinusoid (solid curve). The mass center of the rod with prescribed shape swims at constant speed and faster than that with prescribed curvature, and the latter experiences net moments and oscillatory forces due to its departure from the symmetric sinusoidal wave


873

confirm that oscillation amplitudes do not depend on either the scheme or accuracy level. Moreover, our simulations of Sect. 4.1 with prescribed shape yield no oscillations. We therefore believe that they are a consequence of asymmetries in body shape, and not a numerical artifact. Carling et al. [10] observed similar oscillations in two-dimensional Navier–Stokes simulations of swimming with prescribed shape, in their case presumably due to unsteady fluid motions. These results have interesting implications with regard to efficiency. Energy expended in muscle contractions increases with force generated, and this in turn increases body amplitudes B. As Fig. 11 shows, MAS increases with B to a maximum, after which it decreases in spite of higher energy expenditures. This agrees qualitatively with the observation that wave amplitudes in steadily swimming lampreys and eels are not particularly large in comparison to body length. In contrast, the time to reach MAS decreases monotonically as B/λ increases, as might be expected, since higher amplitudes can deliver more power for acceleration. There is evidently a trade-off among acceleration, maximal speed and energy expenditure. Presumably, animals employ the largest amplitudes possible to maximize acceleration in rapid starts and maneuvers, and they might be expected to work near the maxima of the MAS curve during normal swimming.

4.3.3 Effects of material properties and body geometry In this section we explore the effects of properties beyond the animal’s control: specifically, Young’s modulus and taper toward the tail. Presumably, these material and geometric parameters are the result of evolutionary pressures. As we saw in Sect. 3.2, intrinsic curvature is inversely proportional to the product of body width b and Young’s modulus E (Eq. (49)). We first explore the effects of Young’s modulus by fixing the rod geometry and activation strength and varying E, thus causing changes in the amplitude B of the traveling wave. Figure 13 shows MAS and the time to reach MAS over a range of E: since B ∼ κ ∼ 1/E, MAS increases with E for small values of E until a maximum is reached, and time to reach MAS increases monotonically with E. For comparison, we replot the data from Fig. 11 showing MAS for a rod with E = 0.7 and varying amplitude B as a dash–dot curve in the left panel of Fig. 13. The closeness of the MAS curves shows that their concave form and variation from 0.15 to 0.5 is primarily due to the changing amplitude that results from decreasing intrinsic curvature as E increases. We also computed MAS with curvature amplitude, and hence B, held constant by increasing the activation strength with E: Fig. 13 (right panel). Somewhat counter-intuitively, we find that MAS decreases modestly with E (from 0.53 to 0.48 over two decades) while the time to reach MAS increases by 50%, and both approach asymptotes. Together, these results show that wave amplitude is a primary determinant of swimming speed, while stiffness has little qualitative effect. To study the effects of body taper we set b(s) = 1 − r · (s/l), and find MAS and time to reach MAS for the range 0 ≤ r ≤ 0.7. When r = 0 the rod is untapered and when r = 0.7 it tapers linearly tailward to 3/10 of the radius at

874

T. McMillen, P. Holmes Activation amplitude constant B (cm) 3.8

2.5

1.8

1.2

Curvature amplitude constant 0.9 0.45 0.52 0.4

6

2

0.51

0.35 4

0.5

0.3

MAS (V/u)


0.53

3

8

1 0.25

2

0.49

0.2 0

0.4

0.7

1

Youngs Modulus

1.5

2

0 0.1

1

10

0.48 70

Youngs Modulus

Fig. 13 Effects of stiffness on MAS (solid line, scale on right axes) and time to reach MAS (dashed line, scale on left axis) as functions of Young’s modulus E. Left panel The constant activation amplitude producing (κsin (s − u t/l)) is chosen such the amplitude of the intrinsic sine wave is B = 2.5 cm at E = 0.7 MPa. B values are shown on the top axis, decreasing to the right. The MAS of rods with constant E = 0.7 MPa from Fig. 11 is replotted as the dash–dot curve. Right panel: Curvature amplitude (κ = |κsin |/Eb) is held constant across trials

the rostral (head) end. We again consider both constant activation and constant curvature amplitudes and also show results for circular (a = b) and elliptical cross-section rods whose heights (a = 1) do not change with width. We take a preferred curvature κsin (s − u t/l) that yields an intrinsic sinusoidal shape of amplitude B = 2.5 in the constant width case. Figure 14 shows that, for constant activation amplitude, small taper tends to increase MAS, but past a certain point increased taper reduces MAS. With constant curvature amplitude, increasing taper raises MAS monotonically. This is counterintuitive, since for curvature amplitude to remain constant, activation must decrease in proportion to b(s). A tapered body uses smaller muscle forces and expends less energy, while achieving greater speeds. In both cases, increase in taper tends to decrease the time to reach MAS, although accelerations are higher in the constant activation case. Taper thus confers a significant advantage, provided that motoneuronal activation also decreases toward the tail (in addition to decreased caudal muscle forces due to reduced cross-sectional area). Elliptical rods with thinner caudal cross-sections in the plane of the wave exhibit slightly larger MAS and faster acceleration, since the normal hydrodynamic forces are greater in proportion to local body mass. This outweighs the advantage of increased flexibility toward the tail, but if one included added (virtual) mass as suggested by Lighthill [39] the effect would be negated, for the effective cross sectional area would remain circular. The effect is, in any case, small: differences between elliptical and circular tapered rods in Fig. 14 are everywhere less than 10%.


875

3

Curvature amplitude constant 0.55

3

0.45

2

0.55

circular 2

0.45 circular

1

0

0

0.2

0.4

0.6

0.35

1

0.25

0

MAS


Activation amplitude constant

0.35

0

0.2

0.4

r

0.6

0.25

r

Fig. 14 Effects of taper. MAS (solid line, right axis) and time to reach MAS (dashed line, left axis) are shown as functions of r, the degree of taper, specified by b(s) = 1 − r · sl . The left panel shows the effects of constant activation amplitude; the right panel shows the effects of constant shape amplitude. The curves labeled ‘circular’ correspond to a = b; the other curves correspond to a = 1. In all cases κsin is the curvature of a sine wave with amplitude B = 2.5 cm

4.3.4 On discretization and softening effects In the above simulations we have used Eqs. (49) to specify bending stiffness and preferred curvature in our discretized model, thus neglecting the softening terms of Eq. (42). We now confirm that this approximation is acceptable for the number of links used (N = 40) by estimating their magnitudes in terms of N and E. The difference in forces on either side of the animal was approximated by π κsin , f¯R − f¯L = 4α 2 and if we assume that the active regions on the left and right sides do not overlap (Figs. 6, 7), then π |κsin |. f¯R + f¯L = 4α 2

(58)

Interpreting E = 8α 3 ν¯ /π as in Sect. 3.2, the bending stiffness and curvature in (42), including softening can therefore be written as

h2 π 3 ab , EI = E − |κ | sin 2 4 4αb

κ=

κsin E−

h2 |κ | 4αb2 sin

so that the softening term reduces Young’s modulus by E→E−

h2 |κsin |. 4αb2

, b

(59)

876


0.49

0.48

3.1

0.47 softening terms included

3

MAS (V/u)


3.2

0.46 2.9 10

20

30

40

50

60

70

80

N

Fig. 15 Effect of the number of links N. The intrinsic curvature is such that κsin /Eb corresponds to a sine wave with amplitude B = 2.5 cm. The MAS (solid curve, right axis) and time to reach MAS (dashed curve, left axis) are shown both with and without softening terms

Hence stiffness varies along the rod as the wave of curvature travels its length, having greatest effect at maxima of intrinsic curvature, and it increases intrinsic curvatures while decreasing elastic forces that keep the rod close to its intrinsic curvature. For B = 2, max |κsin | ≈ 0.25, and for B = 3, max |κsin | ≈ 0.5. For a 20 cm rod with b = 1, α = 1, and N = 10 (h = 2), the softening terms at maximal curvature are approximately 1 when B = 2 and 0.5 when B = 3. For N = 40 (h = 0.5) they are 1/16 and 1/32 for B = 2 and B = 3, respectively. We therefore expect the softening effect to be negligible except for small values of N and E. To illustrate this we compute MAS and time to reach MAS both with and without the softening terms (fRi + fLi )h2 of Eqs. (42) in EI and κ. We set E = 0.7 MPa (E = 7 g cm−1 ms−2 ); softening effects are even smaller for larger E. Figure 15 shows that MAS increases with N to an asymptote, varying by less that 1% for N > 30 both with and without the softening term. The time to reach MAS is slightly reduced when softening terms are included, but only by 5% for N = 10, and again the difference is negligible for N > 30. Our neglect of softening terms for N = 40 is therefore justified. We can address a further question of biological relevance by examining the numbers of vertebrae in anguilliform swimmers. Eels, lampreys and numerous elongate fish have O(100) segments, well above N = 30 [52, Table 1]. Our results suggests that these relatively long spinal cords have evolved to (modestly) improve speed.

5 Conclusions In a classic paper Taylor [44] characterized the swimming behavior of a uniform circular rod that assumes a traveling sinusoid via a swimming diagram that plots


877

a normalized velocity against the ratio of the sine wave’s amplitude to its wavelength (Fig. 2). Once this has been constructed, the steady speed can be read off for any given set of physical parameters (fluid viscosity and density, wave amplitude and speed, length of the rod, etc.). We have extended Taylor’s kinematic analysis to investigate the behavior of an actuated elastic rod that attempts to follow an intrinsic shape prescribed via its time-dependent curvature. We employ geometrically exact inextensible rod theory with a linear constitutive (bending) relation, and study the dynamics and stability of motions subject to hydrodynamic body forces approximated, as in [44], by passive drag on a uniform cylinder. We show that forced traveling waves exist for small activations in the absence of body forces, and we develop a finite-difference numerical scheme to simulate the full problem. We use this to study acceleration from rest, to show that rods reach a well-defined mean asymptotic speed, and to study the dependence of these on activation and material and geometric properties. By analyzing discretized rigid link and hinge models, we show how a calcium release and linearized Hill-type muscle activation model can create intrinsic shapes remarkably close to sine waves. However, when curvatures corresponding to sinusoidal traveling waves are imposed, passive elasticity and hydrodynamic reaction forces cause the rod’s response to depart from the intrinsic shape, resulting in significant oscillatory moments and forces at steady state. This causes mass center oscillations and reduces the mean speed from Taylor’s prediction while preserving the qualitative dependence of speed on wave amplitude (Fig. 11). The dependence on Young’s modulus E is subtle: increased E implies that the rod approaches its intrinsic shape faster but also that it will overshoot and experience faster oscillations, leading to the counter-intuitive result that speeds decline as E increases while other parameters are held constant. For the rod to adopt its time-dependent intrinsic shape on a fast time scale, both E and the damping coefficient δ must be large. While E certainly affects speed and acceleration (Fig. 13), it does so primarily by modulating bending wave amplitudes. The model effectively produces a set of swimming diagrams consisting of curves representing a normalized mean asymptotic speed in terms of the amplitude to wavelength ratio of the intrinsic shape. The single diagram of [44] is no longer possible, since MAS depends on activation and the rod’s material and geometric properties. However, we find that the qualitative behavior of a rod with intrinsic sinusoidal shapes is similar to that of one with a strictly prescribed shape, so that Taylor’s swimming diagrams remain a good guide. For instance, the Reynolds number R2 = u2aρf /μ, which isolates hydrodynamic parameters along with the wave speed and rod radius, reveals that the ratio of density to viscosity, and the product of wave speed and radius, are determining factors in swimming speed. However, for a particular rod, the net quantitative effects of all the parameters must be obtained through specific simulations. The model also allows us to investigate other qualitative and quantitative properties. We show that turns can be effected by imposing unequal motoneuronal firing rates contralaterally, leading to nonzero average body curvature.

878


We have also explored the effects of taper, which is ubiquitous in nature. We find that taper confers significant advantages for both acceleration and mean asymptotic speed, and that these are accompanied by energy savings, provided that motoneuronal activation decreases in proportion to taper. Finally, we explored the effect of the number of links N in the discretized model, demonstrating that an increase in N, or equivalently, a decrease in the spatial discretization scale, tends to increase speed. This effect attenuates for N > 30 and the discrete and continuum models have very similar behavior in this range. Since many anguilliform swimmers have vertebra numbers of this order (only 1 of the 24 species appearing in Table 1 of [52] has N < 30), our continuum model may be rather generally useful. Much remains to be done. We have not studied the effects of wave speed or the number of wavelengths supported in the body. Our standard ‘sinusoidal’ curvature comprises one wavelength per body length: activations with different wavelength/body length ratios and amplitude patterns could easily be constructed to model other animals, such as eels, which can exhibit 1.5 - 2 wavelengths [46, Table 1]. (Preliminary work suggested that a circular rod of length 2l carrying two wavelengths swam faster than one of length l, when diameters are equal.) Nor have we systematically studied starts from rest, turns and other maneuvers, or the details of phase relationships among motoneuron bursts, muscle activation, and mechanical waves [56]. Our model would also allow studies of the dynamical effects of muscle recruitment, based on electromyogram measurements such as those of [25], by appropriately modulating preferred curvature. To do this a proper nonlinear muscle model, with velocity and stretch dependent terms (as noted in Sect. 3.2) and fast and slow muscle types, needs to be included. Perhaps most critically, our hydrodynamic body forces based on passive drag cannot predict unsteady effects including vortex shedding and generation of the wake patterns seen in [46–48]. A major challenge is to improve this drag model to better account for these effects without going to full Navier–Stokes simulations. Acknowledgements This work was supported by DoE: DE-FG02-95ER25238, NSF EF-0425878 and NIH NS054271. TM was supported by a National Science Foundation Postdoctoral Fellowship and by the Council on Science and Technology of Princeton University. The authors thank Avis Cohen, Thelma Williams, Lex Smits and the referees for comments and information.

Appendix: Computational details and the numerical scheme A Integration of Eq. (19)

Setting α = k1 /(E − ρc2 )I and η = αξ , (19) becomes the dimensionless pendulum equation d2 ϕ + sin ϕ = 0, dη2

(60)


879

which has the first integral 1 2

dϕ dη

2 + (1 − cos ϕ) = (1 − cos ϕ0 ),

(61)

where ϕ0 denotes the maximum value achieved by ϕ. Using trignometrical identities and the change of variables sin(ϕ/2) = sin(ϕ0 /2) sin φ, (61) may be rearranged as a quadrature involving the incomplete elliptic integral with modulus k ∈ [0, 1) [1, 8]: φ1

F(φ1 , k) = 0

k = sin

η1

dφ 1 − k2 sin2 φ

ϕ 0

2

dη = η1 ,

= 0

and sin φ1 =

sin

where

ϕ1 2

k

.

(62)

Appealing to the definition of the elliptic function sn(η1 , k) = sin φ1 , (62) yields sin

ϕ 1

2

= k sn(η1 , k),

(63)

and dropping the subscript 1 and returning to the travelling wave independent variable ξ results in (20). For the period calculation, we use reflection symmetry of orbits of (60) about both axes in the phase plane and the fact that φ1 = π/2 when ϕ1 = ϕ0 to conclude that

F

π 2

π/2

,k = 0

dφ 1 − k2 sin2 φ

= K(k) =

ϒ , 4

where ϒ = αT,

(64)

giving (21). Note that in the limits k = 0 and k = 1 respectively the solution ϕ(η) of (60) approaches a sinusoid and a heteroclinic orbit connecting the equilibria ϕ = ±π : ϕ(ξ ) = ±2 tan−1 (sinh η), and that its period 4ϒ increases monotonically from 2π to infinity as k goes from 0 to 1. It will also be useful to write the expression for the derivative dϕ = 2k cn(η, k), dη

(65)

explicitly, along with the Fourier series representation of the cnoidal elliptic function: cn(η, k) =

∞ (2n + 1)π η 2π qn+1/2 , cos k K(k) 2K(k) 1 + q2n+1 n=0

(66)

880


where q denotes the elliptic nome: q = exp

−π K(1 − k) . K(k)

(67)

Note that lim cn(η, k) = cos η,

k→0

as required, since in this low amplitude limit (60) becomes the harmonic oscillator. See [1, 8] for further details and relations among elliptic functions and their derivatives. B The numerical scheme B.1 Free boundary conditions The constitutive relation (1) allows us to compute positions of points on the body centerline in terms of the head position and the tangent angle: s x(s, t) = x(0, t) +

s

cos(ϕ(s , t))ds ,

y(s, t) = y(0, t) +

0

sin(ϕ(s , t))ds , (68)

0

so it suffices to solve for ϕ and (x(0, t), y(0, t)). We employ the discretization of Sect. 3.1 (Figure 4), which implies that xi = x1 (t) + Ci (t), yi = y1 (t) + Si (t),

(69) (70)

h h cos(ϕ1 ) + h cos(ϕj ) + cos(ϕi ), 2 2 i−1

where Ci (t) =

and

(71)

j=2

h h sin(ϕ1 ) + h sin(ϕj ) + sin(ϕi ), 2 2 i−1

Si (t) =

i = 2, . . . , N,

(72)

j=2

and C1 = S1 = 0. Substituting (69)–(72) into the discretized force balance equations (29)–(30), we obtain ¨ i + hWxi + fi − fi−1 /mi , x¨ 1 = −C y¨ 1 = −S¨ i + hWyi + gi − gi−1 /mi ,

(73) i = 1, . . . , N,

(74)

where mi = ρAi h and the forces Wi depend on the velocities of material points, and are thus functions of x˙ 1 , y˙ 1 , ϕi and ϕ˙ i .


881

Since Eqs. (73)–(74) must hold for each i and the LHS does not depend on i, we may subtract them pairwise to deduce that fi and gi satisfy the matrix equations Af˜ = b and A˜g = d, (75) where T f˜ = f1 , f2 , . . . , fN−1 , T g˜ = g1 , g2 , . . . , gN−1 ,

(76) (77)

hd h h cos(ϕi ) + cos(ϕi+1 ) + Wxi − Wx,i+1 , (78) 2 dt2 mi mi+1 h h d2 h di = Wy,i+1 , i = 1, . . . , N −1, sin(ϕi )+sin(ϕi+1 ) + Wyi − 2 2 dt mi mi+1 (79) 2

bi =

− m1 + m12 1 ⎜ ⎜ 1 ⎜ m2 A=⎜ ⎜ ··· ⎝ ⎛

0

1 m2

⎞

− m12 + m13

0

0

···

0

0

1 m3

0

···

0

0

0

0

0

···

1 mN−1

− m 1 + m1 N N−1

⎟ ⎟ ⎟ ⎟. ⎟ ⎠ (80)

The tridiagonal form of A means it is easily inverted to solve for f˜ and g˜ . Let f = (f1 , . . . , fN )T , g = (g1 , . . . , gN )T , and ϕ = (ϕ1 , . . . , ϕN )T , and denote by cos(ϕ) the N-dimensional vector whose ith component is cos(ϕi ), and similarly for sin(ϕ), ϕ 2 , etc. Furthermore, let A+ be the N × N matrix whose first N − 1 columns and rows are A−1 , and whose Nth column and row consists of zeros: +

A =

A−1 0

0 . 0

(81)

Then, using the boundary conditions f0 = fN = 0 = g0 = gN (contact forces vanish at head and tail), f and g can be solved for in terms of ϕ, its first and second derivatives, and the first derivatives of x1 and y1 that enter the body forces Wx,i , Wy,i : ˜x , f = A+ −H(cos(ϕ))ϕ˙ 2 − H(sin(ϕ))ϕ¨ + W ˜y , g = A+ −H(sin(ϕ))ϕ˙ 2 + H(cos(ϕ))ϕ¨ + W

(82) (83)

882


where ⎞ − mh2 Wx2 ⎟ ⎜ ··· ⎟ ˜x=⎜ W ⎟, ⎜ h ⎝ mN−1 Wx,N−1 − mhN WxN ⎠ 0 ⎛

⎞ − mh2 Wy2 ⎟ ⎜ ··· ⎟ ˜y = ⎜ W ⎟, ⎜ h ⎝ mN−1 Wy,N−1 − mhN WyN ⎠ 0 ⎛

h m1 Wx1

h m1 Wy1

(84) and ⎛

p1 ⎜ 0 h⎜ ··· H(p) = ⎜ 2⎜ ⎝ 0 0

p2 p2

0 p3

0 0

··· ···

0 0

0 0

0 0

0 0

··· ···

pN−1 0

⎞ 0 0 ⎟ ⎟ ⎟. ⎟ pN ⎠ 0

(85)

The moment equation (31) can be written in matrix form as J − G(cos(ϕ))A+ H(cos(ϕ)) − G(sin(ϕ))A+ H(sin(ϕ)) ϕ¨ = −G(cos(ϕ))A+ H(sin(ϕ)) + G(sin(ϕ))A+ H(cos(ϕ)) ϕ˙ 2 ˜ + G(cos(ϕ))A+ W ˜ y − G(sin(ϕ))A+ W ˜ x, +M

(86)

where J is the diagonal matrix with entries Ji = ρIi h, ⎛

p1 ⎜ p2 ⎜ h⎜ 0 G(p) = ⎜ · 2⎜ ⎜ ·· ⎝ 0 0 ⎛

0 p2 p3

0 0 p3

0 0 0

··· ··· ···

0 0 0

0 0

0 0

0 0

··· ···

pN−1 pN

⎞ M1 ⎜ M2 − M1 ⎟ ⎜ ⎟ ⎟ ˜ ··· M=⎜ ⎜ ⎟ ⎝ MN−1 − MN−2 ⎠ −MN−1

0 0 0

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎠ pN

and

(87)

is a vector of joint moments that contains the time-dependent prescribed curva˙ The matrix on the LHS of (86) depends only on ture and depends upon ϕ and ϕ. ˙ x˙ 1 , y˙ 1 ). ϕ, and can be inverted to give an equation for ϕ¨ in the form ϕ¨ = F(ϕ, ϕ, To form explicit ODEs for x1 and y1 , we sum Eqs. (73)–(74) from i = 1 to N


883

and enforce the boundary conditions to obtain 1 ï , hWxi − mi C m

(88)

1 hWyi − mi S¨ i , m

(89)

N

x¨ 1 =

i=1 N

y¨ 1 =

i=1

" where m = N i=1 mi is the total mass of the rod. The full equations can then be written as a second order system in the (N+2)-vector z = (x1 , y1 , ϕ1 , ϕ2 , . . . , ϕN )T : z¨ = G(z, z˙ , t),

(90)

which may be solved using a Runge–Kutta algorithm. B.2 Periodic boundary conditions Here we consider the case of periodic boundary conditions (f0 = fN , f1 = fN+1 , and similarly for g and ϕ), with homogeneous material properties, i.e. mi = m0 , i = 1, . . . , N. In this case we must solve for the contact forces at head and def def tail, fˆ = f0 = fN and gˆ = g0 = gN , in addition to z. This may be done by modifying Eqs.(82)–(83) as follows: ˜x , f = e fˆ + A+ −H(cos(ϕ))ϕ˙ 2 − H(sin(ϕ))ϕ¨ + W ˜y , g = e gˆ + A+ −H(sin(ϕ))ϕ˙ 2 + H(cos(ϕ))ϕ¨ + W

(91) (92)

where ⎛

⎞⎞ ⎛ ⎞ −1/m0 1 ⎜ ⎜ 0 ⎟⎟ ⎜ 1 ⎟ ⎜ −1 ⎜ ⎟⎟ ⎜ ⎟ ⎜A ⎜ ··· ⎟⎟ ⎜ ⎟ ⎜ ⎟⎟ e=⎜ ⎜ ⎝ 0 ⎠⎟ = ⎜···⎟, ⎜ ⎟ ⎝ 1 ⎠ ⎝ −1/m0 ⎠ 1 1 ⎛

(93)

and A and A+ are as before, and making the corresponding changes in the matrix moment equation (86). If the body forces are also periodic, i.e. Wx,N+1 = Wx1 , then the boundary conditions imply N N d2 d2 cos(ϕ ) = 0 = sin(ϕi ). i dt2 dt2 i=1

i=1

(94)

884


Therefore, we have sin(ϕ)T ϕ¨ = − cos(ϕ)T ϕ˙ 2

and

cos(ϕ)T ϕ¨ = sin(ϕ)T ϕ˙ 2 .

(95)

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