Optimization and simulation of a bionic fish tail driving system based on linear hypocycloid with hydrodynamics

Abstract

The tail driving system based on linear hypocycloid has the advantages of adjustable phase difference, no quick-return, and combining speed reducer with transformation mechanism. The plane complex movement of the driving system was realized via a motion triangle with a linear hypocycloid planetary gear train and a linkage. In this article, we approach the question of which kind of parameter design can make this driving system more efficient in swimming from a hydrodynamic perspective. First, dynamic and hydrodynamic models were established with momentum theorem, Lagrange theorem, and two-dimensional foil theory. And then, hydrodynamic optimization on kinematic parameters (i.e. caudal peduncle’s reciprocation velocity and caudal fin swing angle) and structural parameters (i.e. swing amplitude, V planetary carrier’s angle, and sun gear’s radius) for a better propulsive efficiency was developed in detail. Second, influences of structural parameters on vortex ring were further conducted by numerical simulation in FLUENT. Finally, the prototype and experimental platform of the designed driving system were established, and the theoretical derivation of lift and lateral forces was testified by experiment.

Keywords

Linear hypocycloid tail driving system hydrodynamic visual flow field experiment validation

Introduction

Instead of using propellers, bionic fish accomplished swimming with body deformation and fin movements has stimulated extensive attention by biologists, mathematicians, and engineers due to its remarkable feats such as speed, efficiency, and agility. In particular, carangiform locomotion is one of the most popular swimming modes due to its movements mainly restricted to the last third of the body. For this reason, bio-fish prototypes to imitate real carangiform fish have been developed by researchers just like mushrooms after rain.^1–3

Since the 1980s, the robotic carangiform fish such as RoboTuna, RoboPike, SPC-I/II/III, and Essex-G9 have come out.^4–6 Specially, the researchers of Pasadena City College have designed a robotic fish with springs. The application of the springs could not only act as robotic skeleton but also absorb some energy of the motor.⁷ In Tokyo University of science and technology, the two-joint robotic dolphin has been developed with the crank, rocker mechanism. The robotic mechanism is simple, but its joints were driven by springs, which led to the difficulty in controlling oscillation angles of the caudal fin.⁸ In china, the guide rods and hydraulic system were applied to imitate the swim bladder of the robotic fish.⁹ This structure was good for robotic fish swimming up and down in the water. But it lacked the diving system of propulsion, and it was impossible to propel forward.

In the aspect of theory, “Two-dimensional Foil Theory” is proposed by Wu,¹⁰ in his study on swimming propulsion with utilizing potential flow theory, which has been widely applied by researchers. Computational Fluid Dynamics (CFD) is a reliable approach to evaluate hydrodynamic performance because of the fact that CFD can provide high-fidelity representation of fluid–structure interactions. The tuna caudal fin’s role in controlling and utilizing reverse Karman street to improve the propulsive efficiency has been studied with CFD.^11,12 Furthermore, the influence of the vortex structure and the propulsive efficiency by relevant hydrodynamic parameters, such as Reynolds number (Re), the tail-swing frequency, the instantaneous attack angle, and Strouhal number St, have been investigated.¹³

The two-joint composite motion of the tail driving system based on linear hypocycloid was realized by a composite mechanism with a linear hypocycloid planetary gear train and a linkage. Structural or kinematic parameters such as reciprocation velocity, swing angle, the reference circle of the planetary gear, the angle of the V type planetary carrier, and the joint position on the caudal fin link will play important role in its propulsive performance. In this article, based on hydrodynamic models developed for the driving system, hydrodynamic optimization on kinematic or structural parameters for a better propulsive efficiency was developed in detail. Second, influences of structural parameters on vortex ring were further conducted by numerical simulation in FLUENT. Finally, the prototype and experimental platform of the designed driving system were established, and the theoretical derivation of lift and lateral forces was testified by experiment.

Structure of the two-joint linear hypocycloid tail driving system

The caudal fin of aquatic tuna acts nearly in a planar composite movement with caudal peduncle’s reciprocation and caudal fin’s oscillation. Here, the tail driving system was designed as a motion triangle to imitate the caudal fin’s motion with a linear hypocycloid planetary gear train and a linkage. The structure of the driving mechanism was shown in Figure 1(b), and the detailed working principle, shown as Figure 1(a), is represented as follows: Point A and point B are separately located at the reference circles of the two planetary gears, and the reference circle of the planetary gear was half of the sun gear. When the two planetary gears 1 and 2 were meshing with the sun gear driven by the V planetary carrier, point A and B would then reciprocate along diameter line OA and OB separately with a phase difference decided by the angle of the V planetary carrier. Point A, point B and the connected point C were working together to form a motion triangle ABC. Based on the motion triangle ABC, the caudal fin link AC would not only rotate around the point A to imitate caudal fin’s oscillation but also moved with the point A to imitate caudal peduncle’s reciprocation.

Figure 1.

The working mechanism and the structure of the driving system: (a) the working principle and (b) 3D structure.

Based on the planetary gear train, with the reference circle of the planetary gear half of the sun gear, the motion equations of point A and point B could be described as

{\begin{matrix} S_{A} = 2 r \cos (ω t - β) \\ S_{B} = 2 r \cos (ω t + φ - α) \end{matrix}

(1)

According to geometrical relationship among the length of the motion triangle $Δ ABC$ , the caudal peduncle’s reciprocating displacement $S_{A}$ and the caudal fin’s oscillating angle $θ$ could be deduced as

{\begin{matrix} S_{A} = 2 r \cos (ω t - β) \\ θ = \arcsin (\sin θ_{max} \sin (ω t + \frac{φ}{2} - \frac{α + β}{2})) \end{matrix}

(2)

To gain caudal fin’s oscillation in symmetric, the length of the link must be satisfied with the equation $l_{AC} = l_{BC} = l_{link}$ , and so the maximum oscillating angle should be

\sin θ_{max} = \frac{2 r \sin (\frac{φ}{2} - \frac{α - β}{2})}{l_{link}}

(3)

Here, r is the reference circle radius of planetary gear, which is half to sun gear. $ω$ is the angular velocity of planetary carrier, $φ$ is the angle of V type planetary carrier, and $β$ and $α$ represent dip angle of the diameter line OA and OB, separately.

From equations (1)–(3), the caudal peduncle’s reciprocating with point A is only decided by the structure parameter of the planetary gear and the angular velocity of the V planetary carrier. The maximum oscillating angle is not only decided by planetary gear’s structure parameter but also linkage’s such as the radius of planetary gear r, the angle of V type planetary carrier $φ$ , the length of the link $l_{link}$ , and the motion parameter of angle difference between $β$ and $α$ . In this article, the line OA and line OB were treated as a same line, that is, the dip angle of the diameter line OA and OB are equal to zero. Also of note is that there was a phase difference $Δ φ$ between caudal peduncle reciprocating along Z-axis and caudal fin oscillating around point A, which was an important parameter for tail propulsion. From equation (2), the relationship of the phase difference $Δ φ$ and planetary carrier’s angle $φ$ was derived as $Δ φ = | \frac{π}{2} - \frac{φ}{2} |$ .

Hydrodynamics analysis of the linear hypocycloid tail driving system

To gain the propulsive efficiency of the driving system, the following hypotheses have been done:

The caudal fin was rigid, and the caudal fin was identically moving in a plane.

Without considering the loss energy of the motor, the output power of the motor is identically equal to the total power of the driving system.

The total energy of a system L was the sum of the potential energy P plus the kinematic energy K, that is L = P+K. Here, with the hypothesis that the caudal fin was identically moving in a plane, the potential energy P = 0, so the energy L of the driving system was equal to the kinematic energy K. Based on Lagrange function, the torque equation of the driving system could be defined as

M_{i} = \frac{d}{dt} (\frac{\partial K}{\partial {\overset{\cdot}{q}}_{i}}) - \frac{\partial K}{\partial q_{i}}

(4)

Here, M_i represents the force F of caudal peduncle or the oscillating torque M₂ of caudal fin, $q_{i}$ represents $S_{A}$ or $θ$ , ${\overset{\cdot}{q}}_{i}$ represents motion velocity $v_{A}$ of caudal peduncle reciprocating along Z-axis or angular speed $ω_{θ}$ of caudal fin oscillating, and their equations would be shown as

{\begin{matrix} v_{A} = - 2 r ω \sin (ω t) \\ ω_{θ} = \frac{ω \cos (ω t + \frac{φ}{2})}{\sqrt{4 - \sin^{2} (ω t + \frac{φ}{2})}} \end{matrix}

(5)

With equations (2) and (4), the force of caudal peduncle F and output torque of caudal fin M₂ could be derived as

{\begin{matrix} F = a_{z} (m_{1} + m_{2}) + \frac{m_{2} l}{2} [α_{θ} \cos θ - ω_{θ}^{2} \sin θ] \\ M_{2} = \frac{m_{2} l^{2} α_{θ}}{12} + a_{z} l_{2} ω_{θ} \sin θ \end{matrix}

(6)

where m₁ and m₂ are the mass of caudal peduncle and caudal fin, and centroid positions are located at their geometric centers, respectively; l is the length of caudal fin; a_z is the acceleration of caudal peduncle; and $α_{θ}$ is the angular acceleration of caudal fin.

Here, the total power of the driving system could be deduced by (Figure 2)

P = \frac{\int_{0}^{T} (F v_{z} + M_{2} ω_{θ}) dt}{T}

(7)

Figure 2.

Hydrodynamics analysis view.

Then, the output power could be gained from the product of the efficient thrust and the propulsive velocity. The efficient thrust was along the propulsive direction, which could be obtained by lift force analysis. Here, the caudal fin is still simplified as a two-dimensional (2D) waving plate,⁵ and the lift force and the efficient thrust could be gained by

{\begin{matrix} R_{L} = π ρ l {| v_{\infty} |}^{2} \sin θ \\ R_{T} = R_{L} \cdot \sin γ \\ R_{D} = R_{L} \cdot \cos γ \end{matrix}

(8)

where ρ is the density of water and $v_{\infty}$ is the flow velocity relative to infinity.^14,15 $v_{\infty}$ is equal to the velocity of the caudal fin’s center-of-pressure $v_{p}$ , and ${\vec{v}}_{p} = {\vec{v}}_{0} + {\vec{v}}_{z} + {\vec{v}}_{θ}$ . The lift force R_L is perpendicular to $v_{p}$ , and $γ$ is the angle between lift $R_{L}$ and axle OX.

The propulsive velocity $v_{2}$ of the caudal fin can be derived as

{\vec{v}}_{2} = {\vec{v}}_{θ} + {\vec{v}}_{Z}

(9)

where ${\vec{v}}_{θ}$ is the relative velocity of caudal fin rotating around point A, and $v_{θ} = l ω_{θ} / 2$ .

Above all, the propulsive efficiency $η$ could be derived as

η = \frac{R_{T} v_{2}}{P}

In addition, according to equation (7), the thrust force coefficient of this designed system could be derived as

{\begin{matrix} C_{x} = \frac{2 R_{T}}{ρ {CV}_{m}^{2}} = \frac{2 π ρ v_{p}^{2} \sin θ \sin γ}{ρ {CV}_{m}^{2}} \\ C_{z} = \frac{2 R_{D}}{ρ {CV}_{m}^{2}} = \frac{2 π ρ v_{p}^{2} \sin θ \cos γ}{ρ {CV}_{m}^{2}} \end{matrix}

(10)

where V_m is the maximum flow velocity of the caudal fin, and $V_{m} = \sqrt{v_{0}^{2} + {(2 rw)}^{2}}$ .

Wen et al.¹⁶ indicate that attack angle would also affect thrust, and live fish could achieve a higher propulsive efficiency by an optimal attack angle. In the designed mechanism, the attack angle $α_{k} (t)$ between $v_{p}$ and the geometry axes of caudal fin would be generated when the caudal peduncle worked together with flow fluid in caudal fin. The governing function of attack angle $α_{k} (t)$ is derived as

α_{k} (t) = arctg \frac{v_{z} + \frac{1}{2} l ω_{θ} \cos θ}{v_{0} - \frac{1}{2} l ω_{θ} \sin θ} - θ

(11)

Parameter optimization

Generally speaking, propulsive efficiency $η$ could be improved by increasing swimming velocity v₂ and thrust R_T, or reducing input power P_in. In this designed mechanism, a greater propulsive efficiency was obtained by the optimization of kinematic parameters (such as caudal peduncle’s velocity v_z and oscillation angle $θ$ ) and structural size (such as oscillating amplitude $θ_{max}$ , sun gear’s radius r, and phase difference $ϕ$ ).

Kinematic parameters optimization

Here, the surface velocity of caudal fin and oscillating angle would impact on thrust of bionic fish; as a result, it would lead to the change of propulsive efficiency. In the design, the relationship between these correlated velocities and the thrust $R_{T}$ could be derived as

\begin{array}{l} R_{T} = (v_{θ}^{2} + v_{z}^{2} + v_{0}^{2} + 2 v_{z} \sqrt{v_{z}^{2} + v_{0}^{2}} \sin α) \cdot \\ π ρ l \sin θ \sin (\frac{π}{4} + \frac{α}{2} + θ) \end{array}

(12)

From Figure 3(a), the thrust R_T, propulsive velocity $v_{2}$ , and propulsive efficiency $η$ grew with the increase in caudal peduncle’s velocity v_z. Figure 3(b) illustrated that lift force R_T was greater with the increase in the oscillating angle $θ$ , but the propulsive efficiency $η$ was not. When $θ < 30 \circ$ , the propulsive efficiency $η$ would grow with the increase in the oscillating angle. When $θ > 30 \circ$ , the propulsive efficiency $η$ would decrease with the increase in the oscillating angle. The peak value of propulsive efficiency could be achieved at $θ = 30 \circ$ .

Figure 3.

The influence of kinematic parameters: (a) caudal peduncle’s velocity via $R_{T}$ , $v_{p}$ , and η and (b) the influence of oscillating angle.

The influence of the propulsive efficiency and structure parameters

Based on the optimal result of kinematic parameter, optimal oscillation amplitude $θ_{max}$ could be gained. Figure 4 illustrated that with the increase in oscillation amplitude $θ_{max}$ , the propulsive velocity would reach a maximum value at $30 \circ$ . On the other hand, section “Structure of the two-joint linear hypocycloid tail driving system” has pointed out that when $θ_{max} = 30 \circ$ , analogous-sine motion of swing angle tally with live carangiform fish. So, oscillation amplitude $θ_{max} = 30 \circ$ would be adopted in our design.

Figure 4.

The relationship between swing amplitude and propulsive velocity.

Substituting $θ_{max} = 30 \circ$ , equations (2) and (5) into equation (8), the propulsive velocity of the caudal peduncle was specialized as

v_{2} = \sqrt{[- 2 r \sin (ω t)]_{z}^{2} + \frac{l^{2} ω_{θ}^{2}}{4} - 2 rl ω_{θ} \sin (ω t) \cos θ}

(13)

Figure 5 presented that with the V planetary carrier’s angle $ϕ$ rising, the propulsive velocity would generate a minimum value at V planetary carrier’s angle $ϕ$ near to $90 \circ$ , and the maximum propulsive velocity occurred at $ϕ$ equal to zero. The point is that V planetary carrier’s angle $ϕ = 0 \circ$ , based on the referred relationship in section “Structure of the two-joint linear hypocycloid tail driving system,” a real phase difference $Δ φ$ of caudal peduncle and caudal fin is equal to 90°, which actually accorded with other research. So the optimal value of V planetary carrier’s angle $ϕ$ is zero.

Figure 5.

The relationship between V planetary carrier’s angle and propulsive velocity.

As seen in Figure 6, the propulsive velocity $v_{2}$ would be linear growth with the rise of sun gear’s radius r. Referred to the effects of structural parameters on the linear hypocycloid mechanism’s kinematics in Wang et al.,¹⁷ the sun gear’s radius should be appropriate for ensuring the aspect ratio of bionic fish’s body, here, the sun gear’s radius $r = 150 mm$ was chosen as the optimal size.

Figure 6.

The influence of sun gear’s radius.

According to the optimal structure result, we chose sun gear’s radius r = 150 mm, the planetary carrier’s angle $φ = 0$ which leads to a real phase difference of caudal peduncle and caudal fin $Δ φ = 90 \circ$ and $θ_{max} = 30 \circ$ into next section for numerical simulation study.

Numerical simulations on the tail driving system

In this section, propulsive performance of tail driving system was numerically simulated in FLUENT Software and further influences of structural sizes on vortex rings would be developed to verify the feasibility of the tail driving system.

Numerical solver set-up

Here, the caudal fin with NACA0012 shape was directly adopted for numerical simulation. The size of the computational domain was $3 m \times 6.4 m$ . The computational domain of the unstructured grid was divided into kernel and non-kernel areas. Because the tail oscillating led to severe grid distortion, the area of the tail oscillating was kernel. The cell size of the kernel area was 1/20 or 1/30 of the characteristic chord length of tail. Finally, the computational domain produced 105,260 unstructured cells. As shown in Figure 7, the inlet edge was Inlet-velocity, the outlet edge was Outflow, the surface of the tail was No-moving Wall, and up and down edges was Symmetry Wall. With dynamic mesh technique and UDF package in FLUENT, the tail composite motion was realized using DEFINE_CG_MOTION which could define caudal fin’s oscillating with $θ (t)$ and caudal peduncle’s motion with S_A(t).

Figure 7.

The boundary conditions of the tail oscillating.

Refer to Aureli’s and Tafuni’s research on Reynolds number,^18,19 turbulence model of numerical simulation was determined to Reynolds number. Only when $Re \leq 2300$ , the fluid flow is laminar; otherwise, it is turbulence. The equation of Reynolds number is defined as

Re = \frac{ρ v_{0} L}{μ}

(14)

where L is the characteristic length of the tail, $ρ$ is the density of water, and $μ$ is dynamic viscosity.

Substituting the flow velocity $v_{0} = 0.4 m / s$ , L = 0.01 m, $ρ = 1000 kg / m^{3}$ , and $μ = 1.01 \times 10^{- 3} m^{2} / s$ into equation (14), the value of Reynolds number Re is equal to $3.96 \times 10^{5}$ . So, the fluid flow was turbulence. In the numerical simulation, the viscous model of $k - ω$ was adopted, and the coupling between the pressure and the velocity was achieved by pressure implicit split operator (PISO) algorithm.

Fixed parameters set-up

Essentially, a better propulsive performance came from a high utilization of vortexes. According to the structural optimization and observation results of live fish, when the phase difference $Δ φ$ of caudal fin and caudal peduncle was 90°, two scattered vortexes shedding from caudal fin were formed. If the phase difference $Δ φ$ was not 90° and too small, one ripened vortex would be broken up. Therefore, in order to reduce the workload of the numerical optimization, fixed parameters such as phase difference $Δ φ = 90 \circ$ , flow velocity v₀ = 0.4 m/s, and oscillation frequency f = 1 Hz were directly adopted to verify the influences of sun gear’s size and oscillating amplitude to propulsive efficiency.

Based on proposed fixed parameters, and substituting these parameters into equations (2) and (11), the motion equations of caudal peduncle and caudal fin could be obtained as

{\begin{matrix} S_{A} = 0.3 \cos (2 π t) \\ θ = \arcsin (\frac{1}{2} \sin (2 π t)) \\ α = arctg (\frac{- π \sin (2 π t)}{4}) - \arcsin (\frac{1}{2} \sin (2 π t)) \end{matrix}

(15)

Figure 8 represents the temporal evolution of oscillating angle $θ$ , displacement $S_{A}$ , and attack angle $α_{k}$ . The driving system could obtain the maximum attack angle about 30°, which not only confirmed to Carp’s but also successfully achieved a higher propulsive efficiency. The conclusion on maximum attack angle was also seen in Wen et al.¹⁶ and Triantafyllou et al.²⁰ which indicated that two vortexes would be generated to form a single row in one period with the maximum attack angle less than 36°.

Figure 8.

Temporal evolution of oscillating angle, displacement and resulting attack angle.

The influences of structure sizes on vortexes

The generation of wake was unsteady and instantaneous with one vortex forming and another dissipating. According to equation (14), the wake of the tail oscillating with sun gear’s radius r = 150 mm, oscillating amplitude $θ_{max} = 30 \circ$ , and oscillation frequency f = 1 Hz was obtained, as shown in Figure 9. The reverse Karman streets 1 and 2 interacted to form a single-row ring. A stronger jet was perpendicular to the axis of vortex ring. Caudal fin oscillation generated a lift force R_L due to this jet reacted on the caudal fin. Besides, one component force of R_L in the propulsive direction was thrust R_T, and the other in the Z-axis was lateral force R_D. After relationship analysis of wake and structural sizes, their relationship was presented as follows: the vortex 1 and 2 with perpendicular distance $Δ y$ was about to 4r, the horizon distance $Δ x$ was half of wavelength $λ / 2$ , and the swimming velocity of robotic fish was equal to $λ \times f$ . The equation of Strouhal number could be deduced as $St = 2 r / λ f$ , if the St varied from 0.42 to 0.55, the robotic fish would gain a higher propulsive efficiency of the robotic fish.^14,15

Figure 9.

The wake structure with r = 150, $θ_{max} = 30^{\circ}$ .

The primary cause of robotic fish swimming was ring generated by vortexes. We obtained structural sizes of rings with different structural parameters of the tail driving system and listed their detail information, as shown in Figure 10 and Table 1.

Figure 10.

The numerical vortexes with difference structural parameters: (a) r = 150, θ_max = 15°; (b) r = 150, θ_max = 45°; (c) θ_max = 30°, r = 120 mm; and (d) θ_max = 30°, r = 180 mm.

Table 1.

The measure values of two vortexes with difference structural parameters.

Variables	r = 150, but $θ_{max}$ varied			$θ_{max} = 30^{\circ}$ , but r varied
Variables	15°	30°	45°	120 mm	150 mm	180 mm
Coordinates of vortexes, m	(4.49, 0.1) (5.75, −0.23)	(1.17, −0.38) (1.96, 0.31)	(2.85, −0.92) (3.88, −0.58)	(3.46, −0.37) (4.91,0.11)	(1.17, −0.38) (1.96, 0.31)	(3.22, −0.04) (4.64, 0.77)
4r (Δy), m	0.33	0.69	0.34	0.48	0.69	0.81
$λ / 2 (Δ x)$ , mm	1.26	0.79	1.23	1.45	0.79	1.42
St	0.13	0.44	0.14	0.17	0.44	0.29

Due to the influence of gravity and the velocity of flow fluid, the completed vortex moved down, which led to measure errors and coupled motile errors. From Table 1, the perpendicular distance of the vortex ring was up to sun gear’s radius, and when r = 150 and $θ_{max} = 30 \circ$ , the robotic Strouhal number was 0.44. This number between 0.42 and 0.55 achieves a higher propulsive efficiency.²⁰

Experiments

Based on the above and the optimized parameters r = 150, $θ_{max} = 30 \circ$ , and $φ = 0 \circ$ , the tail driving prototype, the experimental prototype, and platform of the hypocycloid tail driving system have been developed. The experimental prototype was shown as Figure 11(a). The total experimental equipment was composed of measure system, control system, and the hypocycloid tail driving device, as shown in Figure 11(b).

Figure 11.

The experimental device platform: (a) the tail driving prototype and (b) the power driving system.

This measure system was composed of force and torque sensor. The tension sensor and torque sensor were mounted on the up-plate of acrylic carriers. But the servo motor was mounted on the down-plate of acrylic carriers to make its output shaft connect to the input shaft of the tail driving device. The acrylic carriers were hung on the slider which was looped on the circular guide rail, as shown in Figure 12(a). Here, the circular guide rail was adopted to weaken the influences of robotic fish sloshing. Because of the changes in caudal fin’s oscillating angles, the measure system should be rotated with the amplitude of 20°.

Figure 12.

The control platform of the tail driving system: (a) The measure system and (b) The control system.

For the control system, as shown in Figure 12(b), the controller of the servo motor was directly connected to direct-current power (DC power) and a computer by USB. First, the experimental equipment of the tail driving system should be debugged by virtual control software-9010 on the computer. After the tail driving system rotating smoothly, thrust and lateral force could be constantly gathered by force sensors and monitored by software-M400. Then, these gathered data were substituted into equation (10) to obtain thrust and lateral force coefficient.

In order to obtain average values of thrust and lateral force coefficient, eight experiments have been conducted. The scatter diagram is shown in Figure 13.

Figure 13.

The scatter diagram of hydrodynamic coefficients obtained by eight experiments.

The average values of thrust and lateral force coefficient from experiments compared with theoretical derivation are shown in Figure 14. The theoretical hydrodynamic coefficient has a periodical change, which confirms basically to experimental. Regardless of theoretical and experimental hydrodynamic coefficient, the amplitude of lateral force coefficient is larger than that of thrust coefficient, but the time-average lateral force efficient $C_{z}$ approaches to zero, and positive average thrust coefficient $C_{x}$ is obtained.

Figure 14.

Comparison of theoretical and experimental hydrodynamic coefficients.

Conclusion

Based on the designed driving system, dynamic model, hydrodynamics, and optimization of kinematic and structural parameters have been developed comprehensively.

The hydrodynamic model of the two-joint bionic fish has been developed based on caudal peduncle’s reciprocation. Optimal results based on hydrodynamics show that thrust $R_{T}$ , propulsive velocity $v_{2}$ , and propulsive efficiency $η$ increase with the growth of caudal peduncle’s reciprocating velocity $v_{z}$ . In addition, thrust $R_{T}$ increases with the growth of caudal fin’s swing angle $θ$ , but propulsive efficiency $η$ will reach a maximum at $θ = 30 \circ$ . Propulsive velocity $v_{2}$ will be larger with the rise of sun gear’s radius r, instead, $v_{2}$ will reach a maximum value at swing amplitude $θ_{max} = 30 \circ$ , V planetary carrier’s angle $φ = 0 \circ$ , which make phase difference between caudal peduncle and caudal fin $Δ φ = 90 \circ$ .

Numerical simulation of the driving system has been conducted in FLUENT to study the influences of structural sizes on the robotic fish’s propulsive efficiency. In the visible flow field around caudal fin oscillation, structural sizes of the vortex ring have been studied. The influences of structural sizes of the tail driving system on vortex rings have been developed in FLUENT. The result showed that the perpendicular distance of the vortex ring was up to sun gear’s radius, and when r = 150 and $θ_{max} = 30 \circ$ , the robotic Strouhal number was 0.44, which could achieve a higher propulsive efficiency.

With the optimal structure parameters with $θ_{max} = 30 \circ$ , $φ = 0 \circ$ , and $r = 150 mm$ , the tail driving prototype and the experimental platform of the hypocycloid tail driving system have been developed. Compared to the change tend of the further experiments, the time-average thrust coefficients of theoretical and experimental study are all positive when they change periodically, but the average lateral coefficients are zero. The resulting consistency verified the feasibility of the designed driving system, which would hold a solid basis for the future underwater propulsion.

Footnotes

Academic Editor: M Affan Badar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by National Nature Science Foundation of China (No. 51205173).

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