article: 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.
  The movement of animals like undulatory swimmers is an emergent behavior that starts from muscle activation [1]. There are two primary issues in undulatory swimming. The first issue is to understand how muscle activation leads to the observed deformation kinematics. The second issue is to understand how deformation kinematics produce movement of the body as a whole, e. g. , forward translational motion. Muscle activation, body mass, elastic properties, and interactions with the environment cause the observed deformation of undulatory swimmers. The deformation kinematics of undulatory swimmers can be complex in general. The question of whether precise and complicated activation of muscles is essential to generate specific deformation kinematics is important to understand how these swimmers control their movement. The deformation of the body has been regarded to have two components. The part that is caused by active muscle forcing is the “active” component, whereas the part that results due to fluid forcing is the “passive” component [2], [3]. The active and passive components are strongly coupled [2], [3]. For example, Liao et al. [4] carried out experiments on a dead trout in the wake of a cylinder and observed its swimming motion. Such swimming caused solely by surrounding fluid forces is referred to as “passive swimming”. They found that the “passive swimming” gait was similar to the “active swimming” gait of the trout. Here, swimming caused by muscle activation is referred to as “active swimming. ” It is challenging, in general, to separate the similar looking active and passive components of the deformation. According to the active-passive decomposition framework, passive swimming has only passive component of deformation. However, active swimming has both active and passive deformation components because muscle activation as well as external fluid forces are present. A detailed mathematical description of passive motion induced by external vorticies can be found in Alben [5]. Work by Tanabe and Kaneko [6] on a falling paper shows that fluttering and rotation of the falling paper can be periodic or chaotic depending upon the external forcing from the fluid. Active swimming can be due to muscle forcing in a swimming animal or due to active motor torques in a robotic swimmer [7]–[12]. The observed body deformations are a response to the combined internal (muscle/motor) and external (fluid) forcing which dictates the swimming behavior of a system. The question of how deformation kinematics lead to forward translational motion is crucial to understand optimal conditions for movement. Some swimming gaits are optimal for fast swimming (during prey capture or predator escape), some are optimal for conserving energy (keeping a fixed station), whereas some optimal gaits combine both these objectives (traveling long distances). Several studies on optimal swimming motion in the noninertial regime have been done in the past [13]–[17]. For example, Spagnollie and Lauga [18] showed that the helical shape for an infinitely long flagellum leads to the fastest swimming motion. Pironneau and Katz [19] studied the optimal flagellar undulations that lead to minimal energy expenditure. Avron et al. [20] used conformal mapping techniques to find optimal two-dimensional microswimmers. Optimized artifical microswimmers have been studied by Dreyfus et al [21]. Wilkening and Hosoi [22] analyzed optimal shapes of a swimming sheet at low Reynolds number using analytical and numerical techniques. A comprehensive review of swimming at low Reynolds number and optimal conditions in the noninertial regime can be found in Lauga and Powers [23]. In comparison, an understanding of optimal conditions for swimming in the inertial regime is much less developed. There are some key differences in swimming in the noninertial and inertial regimes. For example, in the noninertial regime, the power required to generate body deformations at each instant is fully dissipated into the surrounding fluid. However, in the inertial regime some part of the power is required to accelerate the body. Analysis of optimal swimming motion in the inertial regime is further complicated by nonlinear effects. Kern and Koumoutsakos [24] used fully resolved CFD techniques based upon unstructured grids to obtain the optimized kinematics of a two- and three-dimensional eel for fast and efficient swimming. Tytell et al. [1] used an immersed body technique to understand the effect of body stiffness and inertia on the motion of an undulatory swimmer. McMillen et al. [25] used a calcium dynamics model for muscle contractions to understand the swimming motion of an anguilliform swimmer in the inertial regime. In spite of this progress, complicated interactions between muscle forcing, hydrodynamic forces, inertia, and elastic properties of a swimmer make it difficult to identify the key control parameters that lead to optimal swimming. In this work a chain-link model [25]–[28] of undulatory swimming is used as a model system to interrogate the emergence of deformation and movement in response to forcing. We consider swimming to be a forced damped oscillation problem. Deformations due to muscles and due to interaction with the environment (fluid) are not differentiated. The system is forced by muscles (active swimming), the surrounding fluid (passive swimming), or both, among various possibilities. It is shown that a body has fundamental modes of deformation with corresponding natural frequencies. Forcing triggers the first few deformation modes of the body. An understanding emerges of how the deformation in turn generates forward translational motion of the body. Relatively simple forcing patterns can trigger complex deformation kinematics that lead to movement. For a 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. We quantitatively show how the power generated from muscle work is used in creating body deformations and then it is dissipated into the surrounding fluid. A closed form expression for swimming velocity is derived by a leading-order analysis of the equations of motion. Optimal kinematics for fast swimming motion are obtained from the closed form expression. This helps identify the key parameters controlling the optimal conditions for fast swimming. The kinematics obtained for fast swimming, even faster than those identified in [24], are verified by performing fully resolved CFD simulations in the nonlinear regime. This approach also leads to an understanding of optimal body stiffness and forcing frequencies of muscle activation. In the following sections, the forced damped oscillation formulation is presented first. It is then used to elucidate how swimming motion is generated. Examples of active and passive swimming that arise because of different types of forcing of the system are presented including a discussion on why typical swimming motions involves lower deformation modes. The pathway of power transfer during the swimming cycle is discussed thereafter. Optimal deformation kinematics for fast swimming are discussed next, followed by optimal parameters for fast swimming.
  We model the body of an undulatory swimmer as a planar chain of interconnected rigid links (Fig. 1) and use a resistive drag model to account for the hydrodynamic forces acting on the body. The rigid links forming the body follow constraint dynamics, i. e. , the endpoints of the two adjacent links are forced to lie on top of each other at the common joint. To solve the equations of motion, the forces at the link joints, needed to maintain this constraint, can be eliminated from the equations, and the remaining equations can be solved for the independent variables (velocities of each link). This approach is followed by McMillen et al. in [25], [28]. In our present analysis, we implement this constraint by connecting the links by stiff springs at the joints. We will refer to this approach as the penalty method (PM). The stiff springs penalize the system if two links at a common joint try to separate from each other. This approach has been used in the fully resolved immersed body (IB) method of simulations [1], [29]–[32]. Implementing the constraint via stiff springs will help us obtain the leading-order equations of motion (section 2. 4). When we are using the resistive drag model (section 2. 2) with nonlinear equations of motion, we refer to the model as the nonlinear resistive chain-link PM model. When we are using the resistive drag model with leading-order equations of motion, we will refer to it as the leading-order resistive chain-link PM model. To derive the nonlinear equations of motion, we isolate a single link which is assumed to be cylindrical in shape of length and radius. By analyzing the forces and moments on the link, we can write Newton' s law of motion for it. The end links are treated differently from the internal links because they have free boundary conditions. For the internal link, the equations of motion are written as (1) (2) (3) in which and are the axial and transverse directions, respectively. , denote the coordinates of the center of mass (COM), is the angle with the horizontal axis, is the mass, and is the rotational inertia of the link. is the angular velocity, is the axial velocity, and is the transverse velocity of the link. Taylor' s resistive model [33] is used for the hydrodynamic forces and in the axial and transverse directions, respectively. and are the forces at the joint in the axial and the transverse directions, respectively, and are written as (4) (5) The spring stiffness coefficient is assumed to be the same in both axial and transverse directions. in Eq. (1) is the moment produced by the muscles. Following the model for muscle moment in [28], the moment is written as (6) in which denotes the Young' s modulus of the material of the body, is the viscoelastic damping coefficient, is the cross sectional moment of area of the link, and is the preferred curvature at the joint. The first term on the right-hand side of Eq. (6) is a model for the activation by the muscles. It is proportional to the difference between the actual curvature with respect to the preferred curvature. The preferred curvature can be modeled in terms of neuronal models for muscle activation [25]. Those details are not considered in this work. Instead, the preferred curvature is computed based on some given preferred shape of the body. Thus, according to the model, muscle torques are activated when the swimming body tries to match its preferred shape. The drag force depends upon the local velocity of the body relative to the fluid. Depending upon the Reynolds number (), there are different drag models. Taylor [33] used experimental data to propose a drag model for smooth cylinders kept in a perpendicular flow for a range of Reynolds number (). McMillen et al. [28] decomposed the drag force into normal and tangential components for smooth oblique cylinders based upon Taylor' s fitting of drag coefficients [33]. According to [28], drag force (per unit length) acting at the COM of smooth oblique cylinder of radius is decomposed into normal and tangential components in terms of the normal and tangential velocities, and, respectively, and is written as (7) in which is the viscosity of the fluid, is the fluid density, and are the drag forces (per unit length) in the normal and tangential directions, respectively. The value of, used in Eq. (7) varies between 0. 9 to 1. 1 for the Reynolds number range. The normal and tangential drag forces can be written as (8) in which n and t denote the normal and tangential vectors, respectively, to the link. For low flows, using slender body theory, Lighthill [34] was able to decompose the drag forces into tangential and normal components. He derived expressions for drag coefficients in the normal () and tangential () directions as (9) in which is the body length, is the body radius and (with being the wavelength of the undulation). Using this drag model the drag forces on a link at low can be written as (10) We use resistive drag models as given by Eq. (8) for intermediate to high Reynolds number () and Eq. (10) for low Reynolds number (). Tytell et al. [1] showed that the lateral forces calculated by the resistive drag model lag behind the lateral forces when calculated by fully resolved CFD simulations. McMillen [25], [28] showed that the resistive drag model is a good approximation to qualitatively capture the forward swimming velocity profile. For the purposes of this work, we use the resistive drag model. This is because while the resistive lateral forces might lag behind the actual lateral forces as shown by Tytell et al. [1], it is the force balance in the axial direction that determines the axial swimming velocity. Even if the reactive force model is used for the thrust component of the axial motion, it should be equated to the resistive drag force in the axial direction according to the theory by Lighthill [35]. In addition, in the average sense, both the reactive and the resistive force models have similar nonlinear characteristics. Finally, as will be shown in this paper, the results obtained from the resistive model are qualitatively consistent with our fully resolved CFD calculations. Validation of the nonlinear resistive chain-link PM model is done using the parameters reported in [28]. The swimming motion is actuated by a traveling wave of preferred curvature. The preferred shape of the body is assumed to be a backward traveling wave of the form (11) in which is the bodylength of the fish, is the amplitude of the traveling wave, and is the frequency of the traveling wave, which is taken to be 2 Hz. The preferred curvature is computed from this preferred shape. The body is assumed to have a uniform circular cross section of radius 1 cm and length 20 cm. This gives a wavespeed of 40 cm/s. The Young' s modulus of the body is taken to be MPa from [28]. The spring stiffness coefficient is taken to be (N/m). This value proved sufficient to keep the two links together at the common joint. The water viscosity and density are taken to be and. The values of the various parameters are summarized in table 1. For these values, the Reynolds number defined by. For drag forces, we use Taylor' s drag model from Eq. (8), with suitable drag coefficients as in Eq. (7). The body is discretized into 40 links of equal length. To solve the equations of motion (1) – (3) we use MATLAB' s built-in time adaptive ODE solver ode45 with time steps bounded to 1 ms. To confirm that PM approach gives accurate results, we compared its solution with the approach in [28], in which they used constraints at the joints between the links instead of using stiff springs. Fig. 2 shows the comparison of the PM model with the model in [28]. Two amplitudes of the backward traveling wave are considered (Fig. 2 (a) and 2 (b) ). As can be seen in the figure, the two models are in good agreement with each other. To gain insight into how muscle activation or external forcing leads to forward translation motion of an undulatory swimmer, leading-order equations of motion are derived. The derivation is based on the assumption of small body deformations. Thus, angles made by the rigid links with the horizontal axis are assumed to be small. We approximate terms like and. The leading-order axial momentum equation is derived from Eqs. (3) and (4) as (12) The leading-order transverse momentum equation is derived from Eqs. (2) and (5) as (13) The leading-order rotational momentum equation is derived from Eqs. (1), (4), and (5) as (14) The leading-order analysis of the drag terms, and, in Eqs. (12) and (13) is performed to obtain the complete set of leading-order equations. Substituiting and in terms of and, n and t in terms of in Eq. (10), we obtain the component of the drag force as (15) Defining, and substituting and in terms of and in Eq. (15) we get (16) Similarly we can write the leading-order equation for as (17) Eqs. (12), (13) and (14) together with Eqs. (16) and (17) gives us the complete set of leading-order equations of motion. These are summarized below: (18) (19) (20) The muscle moment term in Eq. (18) is linear in. We remark that a nonlinear muscle model with dominant leading-order terms with respect to will also fit within this leading-order framework. From Eqs. (18) – (20), it can be seen that to the leading order, the rotational and transverse momentum equations, Eq. (18) and Eq. (19), are coupled to each other, but are independent of the axial velocity equation. However, the axial velocity equation is dependent on the tranverse and rotational velocities due to the “source term” identified in Eq. (20). The leading-order equations show that the axial velocity is lower order () compared to the transverse and rotational velocities (). Eqs. (18) – (20) constitute the leading-order forced damped oscillation equations for undulatory swimming. To derive an expression for the steady swimming velocity of an undulatory swimmer, we consider the leading-order axial momentum equation for the link (21) Averaging the above equation over links, leads to the cancellation of the internal spring forces, and gives the equation for the axial velocity of the COM of the body, (22) Taking a time-average of the above equation over a single period of the swimming cycle gives (23) in which indicates time-average of a quantity. The left-hand side of Eq. (23) vanishes at steady state, and this gives the steady state swimming velocity of an undulatory swimmer as (24) Eq. (24) indicates that there must be anistropy in drag for the body to swim. This is in agreement with the previous studies on swimming at low Reynolds number [36]. It also shows that the swimming velocity depends upon the time-average of the product of the lateral velocity and the angular position of the body. This term primarily depends on the deformation kinematics. If a nonzero time-averaged lateral velocity and angular position can be provided to the body, then it is able to propel itself forward. The body can swim even if there is no active muscle forcing. For example, if the surrounding fluid can provide the body with appropriate lateral and angular velocities, then it can lead to swimming. This has been observed in the case of a “dead” trout swimming in the wake of a cylinder [4], [37]. Thus, the leading-order equations indicate that the dynamics of swimming are a response to the forcing that goes into the system irrespective of whether the forcing is internal (muscle) or external (surrounding fluid). As will be shown later, the swimming system has fundamental modes of deformation, just like a forced damped oscillator. This is referred to as a unified forced damped oscillation framework for undulatory swimming in this work. To compare the leading-order system of equations with the nonlinear equations of motion, we take a case of anguilliform swimming at low Reynolds number. The physical parameters are summarized in table 2 and they represent the case of a juvenile zebrafish [38]. The initial configuration of the body is taken to be its intrinsic rest state. Fig. 3 compares the three models. The nonlinear resistive chain-link PM model (circles) agrees well with the model described in [28] (solid line). The nonlinear resistive chain-link PM model solved Eqs. (1) – (3) by using the leading-order drag forces. The nonlinear model of [34] also solved Eqs. (1) – (3) with the leading-order drag forces, but the constraint forces were eliminated instead of modeling them as spring forces. Eqs. (18) – (20) were solved for the leading-order resistive chain-link PM model with leading-order drag forces. As shown by dashed lines in Fig. 3, it gives a higher forward swimming velocity compared to the nonlinear models. The steady state velocity for the leading-order resistive chain-link PM model from the simulation and normalized by the wavespeed is 0. 092 (Fig. 3, dashed lines). This is also verified from Eq. (24), in which the computed solution for and from the simulation is used to compute the normalized steady state swimming velocity. For the nonlinear models, the normalized steady state forward swimming velocity is 0. 064 (Fig. 3, dashed and circles). The Reynolds number based upon the swimming speed of juvenile zebrafish is around 20.
  The term passive swimming is used here to imply swimming generated by some external forcing, but with no active muscle forcing. The forced damped oscillation formulation, discussed above, shows that as long as there is forcing of the system that leads to appropriate lateral and angular movement (i. e. , is nonzero), there is swimming. Any type of swimming would fall within this unified framework. To ensure that this is the case we show, in this section, that passive swimming is indeed resolved in this unified framework. Some fundamental scenarios are discussed below. The term active swimming is used here to imply swimming generated by internal forcing such as muscle activation. The goal in this section is to show that active swimming is resolved within the unified framework of a forced damped oscillation formulation. To that end, we consider various aspects of muscle activated swimming. We first discuss how momentum and power is transferred during undulatory swimming. The response of the body to muscle activation in terms of its fundamental deformation modes is discussed thereafter. Parameters like muscle frequency, reference curvature, and muscle stiffness affect swimming motion. In this section we use results from the previous sections to interrogate questions related to optimal conditions that lead to fast swimming.
  In this work our primary thesis is that the deformation is a result of some forcing which can be caused by muscles or the surrounding fluid or both. We have used a forced damped oscillation framework to formulate the problem of swimming. This is in the same spirit as the damped harmonic oscillator framework that has been applied in the past to analyze numerous engineering systems such as the tuned mass damper used in power transmission, automobiles, and buildings to reduce vibrations. The results presented above validate several hypotheses that show the utility of the forced damped oscillation framework. These examples (points 1–4, below), which provide new insights, are discussed below. 1. Our hypothesis was that the forced damped oscillation framework would provide unified insights into how the body deformation emerges and subsequently how the body deformation leads to swimming movement. First, the problem formulation and the results discussed earlier validate this hypothesis where it is seen that body deformations emerge as a superposition of various deformation modes in response to some forcing. This is similar in principle to how various deformation modes emerge on a guitar string or a fluttering flag. Second, insights into how body deformations lead to forward swimming movement is represented by Eq. (24). As hypothesized, this equation provides a unified insight into how forward swimming momentum is obtained, no matter how the system is forced. Thus, active and passive swimming scenarios are understood in a unified way. It is to be noted that although Eq. (24) was based on a simplified model, the results were verified from fully resolved computational fluid dynamic simulations of optimal gaits for fast swimming. 2. Our hypothesis was that the forced damped oscillation framework will provide insights into why relatively simple forcing patterns can trigger seemingly complex deformation kinematics that lead to movement. This was validated from our results where we show how forcing triggers different deformation modes of the body. The presence of multiple deformation modes makes the body deformation appear complex. 3. We hypothesized that the proposed framework will explain why lower deformation modes (i. e. , fewer number of waves on the body) are typically observed in undulatory swimming in nature. This was resolved by our results where we show how the value of the forcing frequency relative to the natural frequencies of the body is important to trigger the various deformation modes of the body (see Fig. 9). Typical forcing frequencies in nature are lower than the typical resonance frequencies of the damped oscillator (i. e. , the fish body). Hence, it is not surprising that lower deformation modes are observed in nature. In anguilliform swimmers, the body is long and slender. As a result the first few deformation modes would be triggered in these swimmers. However, carangiform, sub-carangifrom, and thunniform swimmers have relatively short and stiff bodies. In such swimmers, the body deformations are predominant in the caudal fin region indicating lower deformation modes compared to anguilliform swimmers, as expected from our results. 4. The proposed framework helped identify parameters for which swimming is viable. The specific parameters that were identified include appropriate frequencies for swimming (Fig. 9), appropriate body stiffness for swimming (Figs. 12 (a) and 12 (b) ), and appropriate gaits for fast swimming (Figs. 10 and 11). The forced damped oscillation framework was crucial to obtain key insights into each of these parameters. For example, the forcing frequencies at which swimming would be viable are those that do not trigger the higher deformation modes (Fig. 9). Thus, high frequencies relative to the natural frequencies are not appropriate for swimming. Similarly, for a given forcing frequency the body should be stiff enough for swimming to be possible (Figs. 12 (a) and 12 (b) ). This is because higher stiffness leads to higher natural frequencies compared to the forcing frequency which in turn would not trigger the higher deformation modes that are not desirable for swimming. Finally, the gaits that maximize the forward momentum transfer according to Eq. (24) are most appropriate for fast swimming. It was found that higher deformation modes do not work well according to this metric. These results were confirmed from fully resolved simulations (Fig. 10). The results in point 4, above, are practically the most important since they have biological as well as engineering relevance. The practical utility of the above results in engineering is apparent if one considers the design of a biomimetic underwater robotic vehicle. The above analysis helps identify material properties as well as viable frequencies of an undulator that would drive the vehicle. The above results may also provide useful insights in evolutionary biology. In particular consider the issues pertaining to the evolutionary emergence of certain species. To that end consider the evolution of aquatic animals that use the undulatory mode for propulsion. Assume that only those undulatory animals that can swim effectively would have survived. If so, then it is of interest to know if specific parameters such as body stiffness and activation frequency of the muscles are crucial for survival. Based on the results above, one could hypothesize that only those species with appropriate muscle activation frequency (low; e. g. , Fig. 9) and body stiffness (high; e. g. , Fig. 12 (b) ) would survive and evolve. Thus, this analysis helps identify the key mechanistic parameters that would have been crucial to the evolutionary emergence of undulatory swimmers. While our study certainly does not directly consider evolution, it points to the ways to combine analyses such as ours with evolutionary biology to explore what impact the mechanics of movement may have played on the emergence of certain animals. Similarly, these results could be potentially used to gain insights into the ability to adapt through evolution. As an example, consider a hypothetical scenario where the ability to move is dependent on having a very specific deformation of the body which in turn is very sensitive to the muscle activation pattern. In such a scenario, any change in the parameters would seriously hamper the ability of that animal to move. Such an animal would not be very robust in its ability to move in a changing environment. In addition, it is very likely that any small changes to physical characteristics from one generation to next (and through evolution) would have a detrimental effect on the survival of that species. Our results suggest that in reality the above scenario may not occur. We find that the ability to swim, while dependent on the mechanistic parameters, may not be very sensitive to it. For example, results in Fig. 12 (b) indicate that as long as the body stiffness is above a certain value, the ability to swim fast is insensitive to the value of stiffness. Similarly, as long as the frequency is below certain value (30 Hz for the parameters used in Fig. 9), the wave efficiency is not significantly affected, once again suggesting some degree of robustness to changes in the parameters. Finally, the result that simple forcing patterns can lead to swimming movement is also suggestive of robustness in the sense that it is not essential to have a very specific and complex forcing pattern to make swimming viable. One may therefore hypothesize that the relative insensitivity to parameters, and the noncomplexity of the feasible forcing patterns would be beneficial for adaptation through evolution. Although we do not directly interrogate this issue of adaptation, such hypotheses motivated by the mechanics of movement warrant further investigation. The proposed framework leads to an understanding of many key parameters that are critical to movement. In turn, it has the potential to provide further insights into how movement can be controlled in biomimetic underwater vehicles and analogously to decipher how neuronal control of movement might function in animals. In this work we also quantified the pathway of power transfer from muscles to forward swimming. It is shown that over a steady swimming cycle, the net power generated by the muscles is transferred into the elastic power of the body which is then dissipated into the fluid while undulating the body (Table 3). Identifying this pathway is crucial to obtain a measure for useful work during swimming. This in turn can be used to estimate useful measures for swimming efficiency. This result is important because the efficiency of undulatory swimmers is conventionally defined as the ratio of power spent in overcoming hydrodynamic drag in the direction of swimming to the total muscle power produced. “Gray' s paradox” [44]–[46] was based on this definition of efficiency where it was paradoxically found that the drag power was greater than the muscle power. Although the paradox is now considered resolved, our results suggest that the assumption, underlying Gray' s result, that the useful power is the one spent to overcome drag in the swimming direction may not be appropriate. The useful power is in fact the one spent to undulate the body. Thus, “Gray' s paradox” should be revisited in the context of our results on the pathway of power transfer. In summary, we applied the forced damped oscillation framework to undulatory swimming to understand the emergence of movement due to muscular and/or environmental forcing.
summary: The damped harmonic oscillator framework has been applied to interrogate numerous engineering systems like the tuned mass damper used in power transmission, automobiles, and buildings to reduce vibrations. We apply the same framework to undulatory swimming to understand the emergence of movement due to muscular and/or environmental forcing. It helps elucidate why flexible bodies can indeed be propelled forward by not only the internal muscular forcing but also by external fluid forces as reported earlier in which dead trout were found to swim in the wake of cylinder. We show how forcing triggers the first few deformation modes of the swimmer similar to how the appropriate forcing triggers the fundamental deformation modes on a guitar string. We show how the lower deformation modes produce the best forward propulsion of the body. This insight reveals that swimming is viable for small enough frequencies of neuromuscular activation relative to the natural frequencies of the body and for sufficiently stiff elastic properties. Thus, these results identify the key mechanistic parameters that would have been crucial to the evolutionary emergence of swimming animals. The proposed framework can help understand neural control of movement in swimming as well as control of underwater vehicles.
keywords: biophysics
  biomechanics
  biological fluid mechanics
  biology
title: A Forced Damped Oscillation Framework for Undulatory Swimming Provides New Insights into How Propulsion Arises in Active and Passive Swimming