## Abstract

Aquatic animals commonly oscillate their fins, tails, or other structures to propel and control themselves in water. These elements are not perfectly rigid, so the interplay between their stiffness and the fluid loading dictates their dynamics. We examine the propulsive qualities of a tail-like flexible beam actuated by a dynamic moment over a range of frequencies and flow speeds. This is accomplished using the equations of fluid-immersed beams in combination with a set of tractable expressions for thrust and efficiency. We solve these expressions over the velocity–frequency plane and show that the flexible propulsor has regions of both positive and negative thrust. We also show the behavior of a sample underwater vehicle with fixed drag characteristics as an illustration of a realizable system.

## 1 Introduction

There is tremendous interest in bio-inspired underwater propulsion systems as alternatives to the standard propeller found on most undersea vehicles [1]. Nature has evolved a number of periodic propulsive techniques [2], e.g., thunniform, carangiform, anguilliform, and batoid, for a large range of aquatic creatures [3,4]. The periodic motions of fish allow them to attain tremendous speeds and agility during hunting and evasive maneuvers while also exhibiting impressive efficiencies over long distances. Given this impressive performance, a significant amount of analytical work has been committed to understanding the hydrodynamics of periodic motion, with the intention of adapting it to engineered systems [5–7]. Experimental work has also been conducted with foils and panels which are periodically pitching [8], heaving [9], or both pitching and heaving [10–12]; and using different shapes [13,14], aspect ratios [15], and flexibilities [16,17]. Because these engineered systems can be run repeatably and understood in detail, the analysis, measurement, and control [18] of these systems have provided scientists and engineers with a better understanding of periodic propulsion [4].

A number of robotic implementations have been built. These include the well-known RoboTuna [19], the designs described in Refs. [20–22], and many others reviewed in Ref. [23]. Many of these designs use fully actuated, rigid-link systems for propulsion, but such articulated systems have a drivetrain complexity [24] that increases with the number of joints. For the purpose of simplification, flexible propulsors have been used in place of fully actuated systems. Researchers have studied the effects of flexibility using distinct foils and panels [25,26], as well as continuous tails actuated with a periodic moment [27] or with tail motion generated through instability [28,29].

Systems using a tail-like flexible beam for propulsion are governed by equations modeling fluid-immersed beams [29–32]. Unlike the works cited above, the present communication investigates the forced vibration of a fluid-immersed flexible beam, rather than a beam driven to instability by fluid flow. This model can be studied over a range of frequencies and external flow velocities, treating them as independent variables, to determine the beam motion. The motion of the forced system can be used to evaluate propulsive performance using Lighthill’s equations for slender bodies [5]. These equations can be used to determine the resultant velocity, thrust, and Froude efficiency of an underwater vehicle with a tail driven at a specific frequency and amplitude, based on its drag model and other system parameters. Because our analytical model is computationally inexpensive and accepts a wide range of thin beam geometries, we are able to observe inflection points in the thrust and efficiency behavior that are not easily found experimentally. In particular, we are able to see a decline in thrust production with increased frequency that has not been experimentally observed with shorter tails [33].

The physical system is described in Sec. 2 along with a set of assumptions that simplify the mathematical model. The model is presented in Sec. 3 and the method of solution is discussed in Sec. 4. For a specific set of system parameters, the thrust and efficiency of the underwater vehicle, associated with different driving frequencies, are discussed in Sec. 5. Concluding remarks are provided in Sec. 6.

## 2 System Description and Assumptions

Consider the underwater vehicle immersed in quiescent fluid (Fig. 1). It is composed of a rigid body and a tail-like flexible beam. The center-of-mass of the rigid body is constrained to translate along a channel and the beam is connected to the rigid body by a revolute joint. The vehicle is propelled by the oscillatory motion of the flexible beam, which is generated by actively controlling the revolute joint. We make several simplifying assumptions in modeling the vehicle.

- (A1)
The vehicle is in a state of dynamic equilibrium. The revolute joint of the vehicle translates with constant velocity

*U*_{e}along the negative*X*-axis in the quiescent fluid. We define the reference frame*XY*where the*X*-axis is aligned with the channel and the origin is located at the projection of the revolute joint onto the*X*-axis; this implies that the*XY*frame translates along the channel. The*XY*frame is therefore an inertial reference frame and*U*_{e}denotes the external flow relative to this frame. The underwater vehicle is neutrally buoyant in a fluid of density*ρ*_{f}and its motion is confined to the horizontal plane. - (A2)
The

*x*_{1}*y*_{1}frame is fixed to the rigid body at its center-of-mass. The orientation of the rigid body relative to the*XY*frame is denoted by*α*, which is measured positive about the vertical axis. The*x*_{2}*y*_{2}frame is located at the revolute joint where the*x*_{2}-axis is aligned with the slope of the beam. The angle between the*x*_{1}and*x*_{2}axes is*δ*, which is measured positive about the vertical axis and is assumed to be small. - (A3)
The rigid body is symmetric about the vertical plane that contains the

*x*_{1}-axis. It has mass moment of inertia*J*about its center-of-mass, which is located at a distance ℓ from the revolute joint. For*α*= 0, the projected area of the rigid body in the*YZ*plane is*A*_{r}. The drag coefficient of the rigid body is*C*_{D}and the drag force acts in the positive*X*direction at a distance*l*_{d}from the center-of-mass, measured along the*x*_{1}-axis. - (A4)
The flexible beam satisfies Euler–Bernoulli assumptions. It has length

*L*and a rectangular cross section with flexural rigidity*EI*, where*E*and*I*are Young’s modulus and area moment of inertia, respectively. The fluid volume associated with the motion of the flexible beam yields an added mass per unit length*m*_{e}; this fluid is assumed to have a uniform velocity profile [29] or an infinitely thin boundary layer. This assumption is made for tractability, see, e.g., Ref. [34]. - (A5)
The displacement of a point on the beam at a distance

*x*from the revolute joint is denoted by*y*(*x*,*t*) in the*XY*frame. The revolute joint has a displacement of*y*_{0}≜*y*(0,*t*). The slope of the beam is denoted by*θ*(*x*,*t*) ≜ [∂*y*(*x*,*t*)/∂*x*]. The slope of the beam at*x*= 0, denoted by*θ*_{0}≜*θ*(0,*t*), is small. Since*θ*_{0}and*δ*are both small,*α*≜ (*θ*_{0}−*δ*) is small. - (A6)
The net drag on the underwater vehicle is entirely due to the drag on the rigid body, i.e., the flexible tail produces no drag. The drag force acts at a point on the longitudinal axis that lies behind the center-of-mass; this is consistent with hydrodynamically stable bodies. The net thrust produced by the vehicle is generated by the oscillating motion of the flexible tail.

## 3 Dynamic Model

### 3.1 Rigid Body Dynamics.

*X*and

*Y*axes. The drag force and the reaction force of the channel on the rigid body are denoted by $D$ and $R$. The reaction moment about the

*Z*-axis is $M$. Summing the moments about the center-of-mass of the rigid body, we get

*α*≈ 1 and (ℓ − ℓ

_{d}) sin

*α*≈ 0, we get

### 3.2 Fluid-Immersed Beam Dynamics.

*m*

_{e}and

*m*

_{b}denote the mass per unit length of the surrounding fluid (added mass) and the beam, respectively. Including the wake of the rigid body or a boundary layer over the tail would introduce a constant to the second term of Eq. (4) [34]. For a uniform velocity profile (A4), this constant is 1. The boundary conditions at the free end of the beam are

*a*)

*b*)

*a*) and (6

*b*) provide the boundary conditions at the revolute joint:

*a*)

*b*)

*β*,

*λ*, and

*κ*are non-dimensional parameters:

## 4 Method of Solution

### 4.1 Exact Solution.

*δ*

_{0}and Ω are the driving amplitude and frequency and

*ω*is the non-dimensional excitation frequency. We assume the response of the flexible beam to have the form

*ϕ*(

*u*) is a complex shape function with spatially varying magnitude and phase. Substituting Eq. (12) into Eq. (7) results in

*u*. Applying the boundary conditions in Eqs. (8) and (9), we get

*z*

_{i}=

*z*

_{i}(

*ω*) are the roots of the characteristic equation of (13). Given

*u*

_{e},

*β*, and

*ω*, substituting Eq. (15) into Eq. (14) yields

*η*

_{i},

*ζ*

_{i},

*i*= 1, 2, 3, 4, are given by the relations

*A*

_{i}to determine the shape function

*ϕ*(

*u*). The response of the flexible beam takes the form

*u*is bounded and the second exponential term is periodic as its exponent is imaginary. Each term in Eq. (17) is a traveling wave.

### 4.2 A Case Study.

We consider the rigid body in Fig. 1 to be an ellipsoid with length along the *x*_{1}, *y*_{1}, and *z*_{1} axes equal to 0.4 m, 0.06 m, and 0.12 m, such that *A*_{r} = 5.65 × 10^{−3} m^{2} and ℓ = 0.2 m. Its density is assumed to be the same as that of the surrounding fluid, which is water and equal to *ρ*_{f} = 1000 kg/m^{3}; this yields *J* = 0.0123 kg m^{2}.

The flexible beam has length *L* = 0.45 m and a rectangular cross section of width 0.001 m and height 0.1 m, such that *I* = 8.333 × 10^{−12} m^{4}. The material of the beam is assumed to be Cirlex^{®}, for which *ρ*_{b} = 1420 kg/m^{3} and *E* = 2.7 GPa. For these dimensions, the linear density of the beam is *m*_{b} = 0.142 kg/m; following standard approximations in the literature in Refs. [29,36], the linear density of the added mass of water is *m*_{e} = 7.854 kg/m. This corresponds to the mass of the cylinder of fluid surrounding the flapping beam.

*ω*= 40.0 and

*u*

_{e}= 2.0.

## 5 Investigation of Propulsive Performance

### 5.1 Thrust.

*ν*(

*u*,

*τ*) is solved using the procedure in Sec. 4.1 and has both real and imaginary parts. Since only the real part physically contributes to the thrust, $Re[\nu ]$ is used in place of

*ν*in Eq. (19) and has the form

*δ*

_{0}= 12 deg (≈0.2 rad). This does not universally guarantee small beam displacements due to resonance; to conform with Euler–Bernoulli assumptions, we restrict our investigation to the domain where the maximum elongation of the beam does not exceed 5%, i.e.,

For the case study discussed in Sec. 4.2, Fig. 4 shows a contour plot of the non-dimensional time-averaged thrust produced by the flexible beam in the domain *ω* ∈ (0, 100] and *u*_{e} ∈ (0, 10]. The hatched regions show areas of negative thrust and solid gray regions show areas where the elongation exceeds 5% due to resonance. The general trend of Fig. 4 shows higher thrust at higher frequency with regions of significantly greater thrust due to resonance. In the neighborhood of resonance, denoted by the solid gray areas in Fig. 4, the tip of the tail moves significantly more which results in greater thrust—this follows from Eq. (19). As the frequency increases beyond a region of resonance, the tail movement, and thus the thrust, drop off quickly.

*ω*and

*u*

_{e}are treated as independent variables in the results presented in Fig. 4. In reality,

*u*

_{e}is a consequence of the dynamic equilibrium where the time-averaged thrust $F\xaf$ is equal to the time-averaged drag $D\xaf$ for a given

*ω*. The time-averaged drag is assumed to be

*C*

_{D}= 0.1 [37]. In light of this, we now look at the locus of points where $F\xaf=D\xaf$. A dimensional plot of the locus is shown in Fig. 5(a). For a given Ω in this plot, if

*U*

_{e}lies below (above) the locus, it will accelerate (decelerate) until its velocity reaches the locus. Consider the case of the vehicle accelerating from rest using Ω = 10 rad/s. Because its velocity initially lies below the locus, the vehicle accelerates to the speed on the locus which matches the frequency,

*U*

_{e}= 0.41 m/s. By varying Ω, the vehicle can attain higher and lower

*U*

_{e}for a given amplitude

*δ*

_{0}. The general trend of Fig. 5(a) shows higher

*U*

_{e}at higher frequencies;

*U*

_{e}increases rapidly as the system approaches resonance and decreases post-resonance. This is in accord with the trends observed in Fig. 4. Each point on the locus can also be associated with a specific value of Froude efficiency [5]. The derivation of this efficiency follows.

### 5.2 Efficiency.

*y*(

*x*,

*t*) which produce the thrust:

*η*is defined when the vehicle is moving with a constant speed

*U*

_{e}, so we compute the efficiency only for points on the locus of dynamic equilibrium. Figure 5(b) shows the Froude efficiency as a function of Ω; the corresponding value of

*U*

_{e}can be found from Fig. 5(a). For example, with a driving frequency of Ω = 10 rad/s, the underwater vehicle will have a steady-state velocity of

*U*

_{e}= 0.41 m/s with an efficiency of

*η*= 0.56.

## 6 Conclusion

The dynamics of a flexible tail-like structure, connected to a rigid body by an actively controlled revolute joint, can be analyzed as a fluid-immersed beam in axial flow. The rigid body imposes boundary conditions at one end of the beam while the other end is free. Subject to simplifying assumptions, the dynamics of the flexible beam are analytically tractable and result in traveling waves. These traveling waves produce thrust that can propel the underwater vehicle by overcoming the drag of the rigid body. The efficiency of the thrust varies as a function of the flow velocity and oscillation frequency of the revolute joint. The locus of dynamic equilibrium points, where thrust and drag forces balance each other, was obtained for a sample vehicle; the efficiency values on the locus are found to exceed 50%.

Since the analysis is based on Euler–Bernoulli beam theory, simulations were carried out using a small amplitude of the revolute joint. Nevertheless, the deflection of the flexible tail-like structure becomes large near resonance and such regions were therefore excluded from our investigation. A more accurate model of the fluid–structure interaction is necessary to investigate the behavior of the dynamic system over the complete domain. In certain regions of the velocity–frequency plane, the flexible beam produces negative thrust, implying that it acts as a brake. It should be noted that negative thrust can potentially produce backward motion, but the current model needs to be expanded to account for negative flow velocity. Similar to the thrust, the power can also be negative. While this condition is not sustainable for a self-propelled underwater vehicle, it may be possible to exploit it for energy extraction if the underwater vehicle remains anchored in a flow such as a stream or river.

In addition, there are other avenues for future work. For example, certain parameters of the model could be varied to study their impact on the propulsive performance of the system. The torque and power requirements at the revolute joint can be investigated for feasibility in a real-world system. To accommodate torque and/or power limitations of a physical system, more complex trajectories through the velocity–frequency plane could be designed by changing the amplitude and frequency of the revolute joint.

## Acknowledgment

The support provided by the Office of Naval Research, ONR Grant N00014-19-1-2535, is gratefully acknowledged.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.