Passive swimming of a flexible body.

<p>The wavelength of the velocity change in the flow field is taken to be half the body length. (a) The position of the flexible body at different instants of time. (b) Velocity profile of the center of mass of the flexible body.</p

Parameter values.

<p>Parameter values.</p

(a) The first few fundamental deformation modes of an undulatory swimmer.

<p>These modes represent the underdamped oscillatory eigenvectors of the system (: first deformation mode; : second deformation mode; : third deformation mode). (b) Amplitude or “weights” of various fundamental deformation modes during muscle activated swimming in the linear regime.</p

The trajectory of the center of mass of a rigid link in an external flow during passive swimming.

<p>The trajectory of the center of mass of a rigid link in an external flow during passive swimming.</p

Amplitude or “weights” of Euler-Bernoulli beam deformation modes during muscle activated swimming in the nonlinear regime at Reynolds number 8000.

<p>Amplitude or “weights” of Euler-Bernoulli beam deformation modes during muscle activated swimming in the nonlinear regime at Reynolds number 8000.</p

A Forced Damped Oscillation Framework for Undulatory Swimming Provides New Insights into How Propulsion Arises in Active and Passive Swimming

<div><p>A fundamental issue in locomotion is to understand how muscle forcing produces apparently complex deformation kinematics leading to movement of animals like undulatory swimmers. The question of whether complicated muscle forcing is required to create the observed deformation kinematics is central to the understanding of how animals control movement. In this work, a forced damped oscillation framework is applied to a chain-link model for undulatory swimming to understand how forcing leads to deformation and movement. A unified understanding of swimming, caused by muscle contractions (“active” swimming) or by forces imparted by the surrounding fluid (“passive” swimming), is obtained. We show that the forcing triggers the first few deformation modes of the body, which in turn cause the translational motion. We show that relatively simple forcing patterns can trigger seemingly complex deformation kinematics that lead to movement. For given muscle activation, the forcing frequency relative to the natural frequency of the damped oscillator is important for the emergent deformation characteristics of the body. The proposed approach also leads to a qualitative understanding of optimal deformation kinematics for fast swimming. These results, based on a chain-link model of swimming, are confirmed by fully resolved computational fluid dynamics (CFD) simulations. Prior results from the literature on the optimal value of stiffness for maximum speed are explained.</p></div

Fully resolved simulations of three-dimensional eel for the kinematics cases , , and .

<p>The figure shows normalized axial and lateral velocities (normalized by the wave speed, ) of the center of mass of the eel as a function of normalized time (normalized by ). Upper: axial velocity; Lower: transverse velocity. In the figure . (––) profile for from Kern and Koumoutsakos (K & K) <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003097#pcbi.1003097-Kern1" target="_blank">[24]</a>.</p

Validation of the nonlinear resistive chain-link PM model.

<p>Forward (upper curves) and lateral (lower curves) velocities of the COM of a uniform circular rod following its preferred shape , normalized by the wavespeed ( cm/s) are shown in this figure. fig_B25 shows the case when the amplitude B = 2.5 cm and (b) is for the case of B = 3.5 cm. Solid lines denote the PM model and dashed lines denote the model as described in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1003097#pcbi.1003097-McMillen2" target="_blank">[28]</a>. Taylor's nonlinear resistive drag model is used in these simulations.</p

(a) Normalized swimming speed () (— —, left vertical axis) and the objective function (––, right vertical axis) as a function of Young's modulus (or equivalently bending stiffness for constant ).

<p>The preferred curvature of a traveling wave is used to actuate the swimming motion. (b) Normalized swimming speed () (——, left vertical axis) and the objective function (––, right vertical axis) as a function of Young's modulus (or equivalently bending stiffness for constant ). The curvature at the optimal condition of (a) is used as the preferred curvature to actuate the swimming motion.</p

The swimming velocity of a single rigid link in the presence of external force and torque at various phase differences.

<p>—: analytical; : numerical.</p