Underwater Propulsion Using Forced Excitation of a Flexible Beam

Aspelund, Sanders; Abdullatif, Mahmoud; Mukherjee, Ranjan; Hellum, Aren

doi:10.1115/1.4062450

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 x₁y₁ 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 x₂y₂ frame is located at the revolute joint where the x₂-axis is aligned with the slope of the beam. The angle between the x₁ and x₂ 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 x₁-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 x₁-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 y₀ ≜ 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 θ₀ ≜ θ(0, t), is small. Since θ₀ and δ are both small, α ≜ (θ₀ − 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.

Fig. 1

View large Download slide

An underwater vehicle composed of a rigid body and a tail-like flexible beam

3 Dynamic Model

3.1 Rigid Body Dynamics.

The free-body diagrams of the rigid body and the flexible beam are shown in Fig. 2. The reaction forces at the revolute joint are assumed to be

F

and

S

along the 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

M - S ℓ \cos α + F (ℓ - ℓ_{d}) \sin α = J \ddot{α}

(1)

Substituting cos α ≈ 1 and (ℓ − ℓ_d) sin α ≈ 0, we get

M - ℓ S = J \ddot{α}

(2)

One boundary condition at the revolute joint is geometric:

y_{0} = ℓ \sin α \approx ℓ α = ℓ (θ_{0} - δ)

(3)

The other boundary condition arises from the shear force

S

and the moment

M

⁠, which are dependent on the dynamics of the flexible beam. These will be described in the next section.

Fig. 2

View large Download slide

Free-body diagrams of the rigid body and flexible beam shown in Fig. 1

3.2 Fluid-Immersed Beam Dynamics.

The dynamics of the flexible beam in Fig. 1 is identical to that of a beam in axial flow [28,35]:

E I \frac{\partial^{4} y}{\partial x^{4}} + m_{e} U_{e}^{2} \frac{\partial^{2} y}{\partial x^{2}} + 2 m_{e} U_{e} \frac{\partial^{2} y}{\partial x \partial t} + (m_{e} + m_{b}) \frac{\partial^{2} y}{\partial t^{2}} = 0

(4)

where 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. [34]. For a uniform velocity profile (A4), this constant is 1. The boundary conditions at the free end of the beam are

E I \frac{\partial^{2} y}{\partial x^{2}} |_{x = L} = 0, E I \frac{\partial^{3} y}{\partial x^{3}} |_{x = L} = 0

(5)

Using the moment equilibrium and the pinned condition , the boundary conditions at the revolute joint can be expressed as

E I \frac{\partial^{2} y}{\partial x^{2}} |_{x = 0} - ℓ E I \frac{\partial^{3} y}{\partial x^{3}} |_{x = 0} = J [\frac{\partial^{3} y}{\partial x \partial t^{2}} |_{x = 0} - \frac{d^{2} δ}{d t^{2}}]

(6a)

y |_{x = 0} = ℓ [\frac{\partial y}{\partial x} |_{x = 0} - δ]

(6b)

We introduce the following change of variables:

ν = \frac{y}{L}, u = \frac{x}{L}, u_{e} = U_{e} \sqrt{\frac{m_{e} L^{2}}{E I}}, τ = t \sqrt{\frac{E I}{(m_{e} + m_{b}) L^{4}}}

to obtain the non-dimensional equation of motion of the beam:

\frac{\partial^{4} ν}{\partial u^{4}} + u_{e}^{2} \frac{\partial^{2} ν}{\partial u^{2}} + 2 \sqrt{β} u_{e} \frac{\partial^{2} ν}{\partial u \partial τ} + \frac{\partial^{2} ν}{\partial τ^{2}} = 0

(7)

From Eq. , the non-dimensional boundary conditions at the free end of the beam are

\frac{\partial^{2} ν}{\partial u^{2}} |_{u = 1} = 0 and \frac{\partial^{3} ν}{\partial u^{3}} |_{u = 1} = 0

(8)

while Eqs. and provide the boundary conditions at the revolute joint:

\frac{\partial^{2} ν}{\partial u^{2}} |_{u = 0} - κ \frac{\partial^{3} ν}{\partial u^{3}} |_{u = 0} = λ [\frac{\partial^{3} ν}{\partial u \partial τ^{2}} |_{u = 0} - \frac{d^{2} δ}{d τ^{2}}]

(9a)

ν |_{u = 0} = κ [\frac{\partial ν}{\partial u} |_{x = 0} - δ]

(9b)

In Eqs. and , β, λ, and κ are non-dimensional parameters:

β = \frac{m_{e}}{m_{e} + m_{b}}, λ = \frac{J}{(m_{e} + m_{b}) L^{3}}, κ = \frac{ℓ}{L}

(10)

4 Method of Solution

4.1 Exact Solution.

The system of differential equations – is solved using the procedure from Refs. [28,35]. Since the revolute joint is actively controlled, we assume

δ (t) = δ_{0} e^{i Ω t} = δ_{0} e^{i ω τ}, ω ≜ {(\frac{m_{e} + m_{b}}{E I})}^{1 / 2} Ω L^{2}

(11)

where δ₀ 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

ν (u, τ) = ϕ (u) e^{i ω τ}

(12)

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

ϕ^{⁗} + u_{e}^{2} ϕ^{″} + 2 i u_{e} \sqrt{β} ω ϕ^{'} - ω^{2} ϕ = 0

(13)

where ( )′ denotes the spatial derivative of ( ) with respect to u. Applying the boundary conditions in Eqs. and , we get

\begin{aligned} ϕ^{″} (1) = 0, ϕ^{‴} (1) = 0, ϕ (0) - κ ϕ^{'} (0) = - κ δ_{0} \\ ϕ^{″} (0) - κ ϕ^{‴} (0) + ω^{2} λ ϕ^{'} (0) = ω^{2} λ δ_{0} \end{aligned}

(14)

where forcing terms appear on the right-hand side. We assume the shape function to be of the form

ϕ (u) = A_{1} e^{z_{1} u} + A_{2} e^{z_{2} u} + A_{3} e^{z_{3} u} + A_{4} e^{z_{4} u}

(15)

where z_i = z_i(ω) are the roots of the characteristic equation of . Given u_e, β, and ω, substituting Eq. into Eq. yields

[\begin{matrix} z_{1}^{2} e^{z_{1}} & z_{2}^{2} e^{z_{2}} & z_{3}^{2} e^{z_{3}} & z_{4}^{2} e^{z_{4}} \\ z_{1}^{3} e^{z_{1}} & z_{2}^{3} e^{z_{2}} & z_{3}^{3} e^{z_{3}} & z_{4}^{3} e^{z_{4}} \\ η_{1} & η_{2} & η_{3} & η_{4} \\ ζ_{1} & ζ_{2} & ζ_{3} & ζ_{4} \end{matrix}] [\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ - ω^{2} λ \\ - κ \end{matrix}] δ_{0}

(16)

where η_i, ζ_i, i = 1, 2, 3, 4, are given by the relations

η_{i} = z_{i}^{2} - κ z_{i}^{3} + λ ω^{2} z_{i}, ζ_{i} = 1 - κ z_{i}

We can solve Eq. for the A_i to determine the shape function ϕ(u). The response of the flexible beam takes the form

ν (u, τ) = \sum_{n = 1}^{4} A_{n} e^{z_{n} u} e^{i ω τ} = \sum_{n = 1}^{4} A_{n} e^{Re [z_{n}] u} e^{i (Im [z_{n}] u + ω τ)}

(17)

The forced response in Eq. is the sum of four terms wherein each term is a product of two exponential terms: the first exponential term is bounded because u is bounded and the second exponential term is periodic as its exponent is imaginary. Each term in Eq. 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 x₁, y₁, and z₁ axes equal to 0.4 m, 0.06 m, and 0.12 m, such that A_r = 5.65 × 10⁻³ m² 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/m³; this yields J = 0.0123 kg m².

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 m⁴. The material of the beam is assumed to be Cirlex^®, for which ρ_b = 1420 kg/m³ 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.

For the set of dimensional parameters provided above, the non-dimensional parameters in Eq. are

β = 0.9822, λ = 0.0169, κ = 0.4444

A sequence of images showing the traveling wave nature of the flexible beam under forced vibration is shown in Fig. 3 for the particular case of ω = 40.0 and u_e = 2.0.

Fig. 3

View large Download slide

A sequence of images showing the shape of the flexible beam Re[ν(u, τ)] during forced vibration for ω = 40.0 and u_e = 2.0 over one cycle of oscillation. The revolute joint is at u = 0.

5 Investigation of Propulsive Performance

5.1 Thrust.

The method proposed by Lighthill for “slender fish” [5] can be used to estimate the thrust generated by the flexible beam in fluid flow. As in Refs. [7,28], the more general case of a beam is used rather than the reduced case of a “tapered fish” that has neither mass nor area at its leading edge. The time-averaged net thrust produced by the tail-like flexible beam is

\bar{F} = \frac{1}{2} m_{e} [\bar{{(\frac{\partial y}{\partial t})}^{2} - {(U_{e} \frac{\partial y}{\partial x})}^{2}} |_{x = L} - \bar{{(\frac{\partial y}{\partial t})}^{2} - {(U_{e} \frac{\partial y}{\partial x})}^{2}} |_{x = 0}]

(18)

where the notation

\bar{(\cdot)}

is the cycle-average of (·). The thrust can be non-dimensionalized using the change of variables in Sec. 3.2 and computed over one cycle using

\begin{aligned} {\bar{F}}^{*} & = \frac{\bar{F} L^{2}}{E I} = \frac{ω}{4 π} \int_{0}^{2 π / ω} {{[{(\frac{\partial ν}{\partial τ})}^{2} - {(u_{e} \frac{\partial ν}{\partial u})}^{2}]}_{u = 1} \\ - {[{(\frac{\partial ν}{\partial τ})}^{2} - {(u_{e} \frac{\partial ν}{\partial u})}^{2}]}_{u = 0}} d τ \end{aligned}

(19)

The function 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,

R e [ν]

is used in place of ν in Eq. and has the form

\begin{aligned} R e [ν (u, τ)] & = \sum_{n = 1}^{4} e^{Re [z_{n}] u} {R e [A_{n}] \cos (I m [z_{n}] u + ω τ) \\ - I m [A_{n}] \sin (I m [z_{n}] u + ω τ)} \end{aligned}

(20)

The amplitude of the sinusoidal excitation will determine the magnitude of the thrust; we chose δ₀ = 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.,

sup_{τ \in [0, 2 π / ω]} \int_{0}^{1} \sqrt{1 + {(R e [ν^{'}])}^{2}} d u \leq 1.05

(21)

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.

It should be noted that 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

\bar{F}

is equal to the time-averaged drag

\bar{D}

for a given ω. The time-averaged drag is assumed to be

\bar{D} = (1 / 2) ρ_{f} C_{D} A_{r} U_{e}^{2}

(22)

with C_D = 0.1 [37]. In light of this, we now look at the locus of points where

\bar{F} = \bar{D}

⁠. 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 δ₀. 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.

Fig. 4

Contour plot of the non-dimensional time-averaged thrust F¯* in the ω–ue plane; hatched areas signify regions with negative thrust and the three solid gray areas indicate regions around resonance where the elongation exceeds 5%

View large Download slide

Contour plot of the non-dimensional time-averaged thrust ${\bar{F}}^{*}$ in the ω–u_e plane; hatched areas signify regions with negative thrust and the three solid gray areas indicate regions around resonance where the elongation exceeds 5%

Fig. 5

View large Download slide

(a) Locus of dynamic equilibrium for the underwater vehicle described in Sec. 4.2 and (b) Froude efficiency along the locus of dynamic equilibrium

5.2 Efficiency.

To compute the propulsive efficiency, we use an expression derived by Lighthill for the time-averaged power

\bar{P}

required to generate the displacements y(x, t) which produce the thrust:

\bar{P} = U_{e} m_{e} [\bar{\frac{\partial y}{\partial t} (\frac{\partial y}{\partial t} + U_{e} \frac{\partial y}{\partial x})} |_{x = L} - \bar{\frac{\partial y}{\partial t} (\frac{\partial y}{\partial t} + U_{e} \frac{\partial y}{\partial x})} |_{x = 0}]

(23)

This expression can be non-dimensionalized using the same change of variables as before to get

\begin{aligned} {\bar{P}}^{*} & = \frac{\bar{P} L^{3}}{E I} = \frac{ω}{2 π} \int_{0}^{2 π / ω} {{[\frac{\partial ν}{\partial τ} (\frac{\partial ν}{\partial τ} + u_{e} \frac{\partial ν}{\partial u})]}_{u = 1} \\ - {[\frac{\partial ν}{\partial τ} (\frac{\partial ν}{\partial τ} + u_{e} \frac{\partial ν}{\partial u})]}_{u = 0}} d τ \end{aligned}

(24)

The Froude efficiency [5], defined as the ratio between the power of forward motion through the fluid and the power required to generate the thrust by the motion of fluid, is

η = \frac{\bar{F} U_{e}}{\bar{P}} = \frac{{\bar{F}}^{*} u_{e}}{{\bar{P}}^{*}}

(25)

The efficiency 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.

References

1.

Lauder

,

G. V.

,

2010

, “Swimming Hydrodynamics: Ten Questions and the Technical Approaches Needed to Resolve Them,”

Animal Locomotion

,

Taylor

,

G. K.

,

Triantafyllou

,

M. S.

, and

C.

Tropea

, eds.,

Springer

,

Berlin

, pp.

3

–

15

.

Google Scholar

Crossref

2.

Van Buren

,

T.

,

Floryan

,

D.

, and

Smits

,

A. J.

,

2020

, “Bio-Inspired Underwater Propulsors,”

Bio-Inspired Structures and Design

,

Soboyejo

,

W.

, and

Daniel

,

L.

, eds.,

Cambridge University Press

,

Cambridge

, pp.

113

–

137

.

Google Scholar

Crossref

3.

Lighthill

,

M. J.

,

1969

, “

Hydromechanics of Aquatic Animal Propulsion

,”

Annu. Rev. Fluid Mech.

,

1

(

1

), pp.

413

–

446

.

Google Scholar

Crossref

4.

Lauder

,

G. V.

,

Anderson

,

E. J.

,

Tangorra

,

J.

, and

Madden

,

P. G. A.

,

2007

, “

Fish Biorobotics: Kinematics and Hydrodynamics of Self-Propulsion

,”

J. Exp. Biol.

,

210

(

16

), pp.

2767

–

2780

.

Google Scholar

Crossref

PubMed

5.

Lighthill

,

M. J.

,

1960

, “

Note on the Swimming of Slender Fish

,”

J. Fluid Mech.

,

9

(

2

), pp.

305

–

317

.

Google Scholar

Crossref

6.

Lighthill

,

M. J.

,

1971

, “

Large-Amplitude Elongated-Body Theory of Fish Locomotion

,”

Proc. R. Soc. Lond. Ser. B Biol. Sci.

,

179

(

1055

), pp.

125

–

138

.

Google Scholar

Crossref

7.

Wu

,

T. Y.-T.

,

1971

, “

Hydromechanics of Swimming Propulsion. Part 1. Swimming of a Two-Dimensional Flexible Plate at Variable Forward Speeds in an Inviscid Fluid

,”

J. Fluid Mech.

,

46

(

2

), pp.

337

–

355

.

Google Scholar

Crossref

8.

Koochesfahani

,

M. M.

,

1989

, “

Vortical Patterns in the Wake of an Oscillating Airfoil

,”

AIAA J.

,

27

(

9

), pp.

1200

–

1205

.

Google Scholar

Crossref

9.

Lewin

,

G. C.

, and

Haj-Hariri

,

H.

,

2003

, “

Modelling Thrust Generation of a Two-Dimensional Heaving Airfoil in a Viscous Flow

,”

J. Fluid Mech.

,

492

(

1

), pp.

339

–

362

.

Google Scholar

Crossref

10.

Hover

,

F. S.

,

Haugsdal

,

Ø.

, and

Triantafyllou

,

M. S.

,

2004

, “

Effect of Angle of Attack Profiles in Flapping Foil Propulsion

,”

J. Fluids Struct.

,

19

(

1

), pp.

37

–

47

.

Google Scholar

Crossref

11.

Read

,

D. A.

,

Hover

,

F. S.

, and

Triantafyllou

,

M. S.

,

2003

, “

Forces on Oscillating Foils for Propulsion and Maneuvering

,”

J. Fluids Struct.

,

17

(

1

), pp.

163

–

183

.

Google Scholar

Crossref

12.

Zhu

,

R.

,

Wang

,

J.

,

Dong

,

H.

,

Quinn

,

D.

,

Bart-Smith

,

H.

,

Santo

,

V. D.

,

Wainwright

,

D.

, and

Lauder

,

G.

,

2019

, “

Computational Study of Fish-Shaped Panel With Simultaneously Heaving and Bending Motion

,”

AIAA Scitech 2019 Forum

,

San Diego, CA

,

Jan. 7–11

,

American Institute of Aeronautics and Astronautics

, p.

1655

.

13.

Park

,

H.

,

Park

,

Y.-J.

,

Lee

,

B.

,

Cho

,

K.-J.

, and

Choi

,

H.

,

2016

, “

Vortical Structures Around a Flexible Oscillating Panel for Maximum Thrust in a Quiescent Fluid

,”

J. Fluids Struct.

,

67

(

1

), pp.

241

–

260

.

Google Scholar

Crossref

14.

Yeh

,

P. D.

,

Li

,

Y.

, and

Alexeev

,

A.

,

2017

, “

Efficient Swimming Using Flexible Fins With Tapered Thickness

,”

Phys. Rev. Fluids

,

2

(

10

), p.

102101

.

Google Scholar

Crossref

15.

Dewey

,

P. A.

,

Boschitsch

,

B. M.

,

Moored

,

K. W.

,

Stone

,

H. A.

, and

Smits

,

A. J.

,

2013

, “

Scaling Laws for the Thrust Production of Flexible Pitching Panels

,”

J. Fluid Mech.

,

732

(

1

), pp.

29

–

46

.

Google Scholar

Crossref

16.

Riggs

,

P.

,

Bowyer

,

A.

, and

Vincent

,

J.

,

2010

, “

Advantages of a Biomimetic Stiffness Profile in Pitching Flexible Fin Propulsion

,”

J. Bionic Eng.

,

7

(

2

), pp.

113

–

119

.

Google Scholar

Crossref

17.

Richards

,

A. J.

, and

Oshkai

,

P.

,

2015

, “

Effect of the Stiffness, Inertia and Oscillation Kinematics on the Thrust Generation and Efficiency of an Oscillating-Foil Propulsion System

,”

J. Fluids Struct.

,

57

(

1

), pp.

357

–

374

.

Google Scholar

Crossref

18.

Morgansen

,

K. A.

,

Triplett

,

B. I.

, and

Klein

,

D. J.

,

2007

, “

Geometric Methods for Modeling and Control of Free-Swimming Fin-Actuated Underwater Vehicles

,”

IEEE Trans. Robot.

,

23

(

6

), pp.

1184

–

1199

.

Google Scholar

Crossref

19.

Triantafyllou

,

M. S.

, and

Triantafyllou

,

G. S.

,

1995

, “

An Efficient Swimming Machine

,”

Sci. Am.

,

272

(

3

), pp.

64

–

70

.

Google Scholar

Crossref

20.

Epps

,

B. P.

,

Valdivia y Alvarado

,

P.

,

Youcef-Toumi

,

K.

, and

Techet

,

A. H.

,

2009

, “

Swimming Performance of a Biomimetic Compliant Fish-Like Robot

,”

Exp. Fluids

,

47

(

6

), p.

927

.

Google Scholar

Crossref

21.

Ozmen Koca

,

G.

,

Korkmaz

,

D.

,

Bal

,

C.

,

Akpolat

,

Z. H.

, and

Ay

,

M.

,

2016

, “

Implementations of the Route Planning Scenarios for the Autonomous Robotic Fish With the Optimized Propulsion Mechanism

,”

Measurement

,

93

(

1

), pp.

232

–

242

.

Google Scholar

Crossref

22.

Lu

,

B.

,

Zhou

,

C.

,

Wang

,

J.

,

Fu

,

Y.

,

Cheng

,

L.

, and

Tan

,

M.

,

2022

, “

Development and Stiffness Optimization for a Flexible-Tail Robotic Fish

,”

IEEE Robot. Autom. Lett.

,

7

(

2

), pp.

834

–

841

.

Google Scholar

Crossref

23.

Salazar

,

R.

,

Fuentes

,

V.

, and

Abdelkefi

,

A.

,

2018

, “

Classification of Biological and Bioinspired Aquatic Systems: A Review

,”

Ocean Eng.

,

148

(

1

), pp.

75

–

114

.

Google Scholar

Crossref

24.

Zuo

,

W.

,

Fish

,

F.

, and

Chen

,

Z.

,

2021

, “

Bio-Inspired Design, Modeling, and Control of Robotic Fish Propelled by a Double-Slider-Crank Mechanism Driven Tail

,”

ASME J. Dyn. Syst. Meas. Control

,

143

(

12

), p.

121005

.

Google Scholar

Crossref

25.

Katz

,

J.

, and

Weihs

,

D.

,

1978

, “

Hydrodynamic Propulsion by Large Amplitude Oscillation of an Airfoil With Chordwise Flexibility

,”

J. Fluid Mech.

,

88

(

3

), pp.

485

–

497

.

Google Scholar

Crossref

26.

Hoover

,

A. P.

,

Cortez

,

R.

,

Tytell

,

E. D.

, and

Fauci

,

L. J.

,

2018

, “

Swimming Performance, Resonance and Shape Evolution in Heaving Flexible Panels

,”

J. Fluid Mech.

,

847

(

1

), pp.

386

–

416

.

Google Scholar

Crossref

27.

Valdivia y Alvarado

,

P.

, and

Youcef-Toumi

,

K.

,

2005

, “

Design of Machines With Compliant Bodies for Biomimetic Locomotion in Liquid Environments

,”

ASME J. Dyn. Syst. Meas. Control

,

128

(

1

), pp.

3

–

13

.

Google Scholar

Crossref

28.

Hellum

,

A.

,

Mukherjee

,

R.

, and

Hull

,

A. J.

,

2011

, “

Flutter Instability of a Fluid-Conveying Fluid-Immersed Pipe Affixed to a Rigid Body

,”

J. Fluids Struct.

,

27

(

7

), pp.

1086

–

1096

.

Google Scholar

Crossref

29.

Hellum

,

A.

,

Mukherjee

,

R.

,

Bénard

,

A.

, and

Hull

,

A. J.

,

2013

, “

Modeling and Simulation of the Dynamics of a Submersible Propelled by a Fluttering Fluid-Conveying Tail

,”

J. Fluids Struct.

,

36

(

1

), pp.

83

–

110

.

Google Scholar

Crossref

30.

Paidoussis

,

M. P.

,

1966

, “

Dynamics of Flexible Slender Cylinders in Axial Flow Part 2. Experiments

,”

J. Fluid Mech.

,

26

(

4

), pp.

737

–

751

.

Google Scholar

Crossref

31.

Hannoyer

,

M. J.

, and

Paidoussis

,

M. P.

,

1979

, “

Dynamics of Slender Tapered Beams With Internal or External Axial Flow—Part 1: Theory

,”

J. Appl. Mech.

,

46

(

1

), pp.

45

–

51

.

Google Scholar

Crossref

32.

Gregory

,

R. W.

,

Paidoussis

,

M. P.

, and

Hawthorne

,

W. R.

,

1966

, “

Unstable Oscillation of Tubular Cantilevers Conveying Fluid II. Experiments

,”

Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.

,

293

(

1435

), pp.

528

–

542

.

33.

Quinn

,

D. B.

,

Lauder

,

G. V.

, and

Smits

,

A. J.

,

2014

, “

Scaling the Propulsive Performance of Heaving Flexible Panels

,”

J. Fluid Mech.

,

738

(

1

), pp.

250

–

267

.

Google Scholar

Crossref

34.

Kutin

,

J.

, and

Bajsić

,

I.

,

2014

, “

Fluid-Dynamic Loading of Pipes Conveying Fluid With a Laminar Mean-Flow Velocity Profile

,”

J. Fluids Struct.

,

50

(

10

), pp.

171

–

183

.

Google Scholar

Crossref

35.

Paidoussis

,

M. P.

,

2014

,

Fluid–Structure Interactions: Slender Structures and Axial Flow

, 2nd ed., Vol. 1,

Academic Press

,

Amsterdam

.

36.

Brennen

,

C. E.

,

1982

, “

A Review of Added Mass and Fluid Inertial Forces

,”

Naval Civil Engineering Laboratory

,

Port Hueneme, CA

, Technical CR82.010.

37.

White

,

F. M.

,

2011

,

Fluid Mechanics

, 7th ed.,

McGraw-Hill Series in Mechanical Engineering

,

McGraw-Hill

,

New York

.

Underwater Propulsion Using Forced Excitation of a Flexible Beam

Abstract

1 Introduction

2 System Description and Assumptions

3 Dynamic Model

3.1 Rigid Body Dynamics.

3.2 Fluid-Immersed Beam Dynamics.

4 Method of Solution

4.1 Exact Solution.

4.2 A Case Study.

5 Investigation of Propulsive Performance

5.1 Thrust.

5.2 Efficiency.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Underwater Propulsion Using Forced Excitation of a Flexible Beam

Abstract

1 Introduction

2 System Description and Assumptions

3 Dynamic Model

3.1 Rigid Body Dynamics.

3.2 Fluid-Immersed Beam Dynamics.

4 Method of Solution

4.1 Exact Solution.

4.2 A Case Study.

5 Investigation of Propulsive Performance

5.1 Thrust.

5.2 Efficiency.

6 Conclusion

Acknowledgment

Conflict of Interest

Data Availability Statement

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only