With growing adoption of body-caudal-fin (BCF)-based bionic propulsion robots, optimizing thrust generation has become a critical focus of research. Despite extensive experimental efforts, theoretical studies in this area remain insufficient. This paper attempts to simplify the swing of the caudal fin as a rotation problem of the hub-beam system. Employing the Euler–Bernoulli beam theory to describe the elastic deformation of the beam and the Morison equation to describe the hydrodynamic load, the governing equation of this system is derived based on the principle of virtual work. We then investigate the effects of material properties, motion parameters, and structural parameters on thrust generation. Our model suggests that the flexibility of the beam can greatly increase instantaneous thrust. In addition, the thrust calculated by our model agrees well with the experiment results. Our work can provide guidance on choosing the proper material and rotation mode to improve propulsion performance in the design of BCF bionic robots.
MengYWuZDongH, et al. Toward a novel robotic manta with unique pectoral fins. IEEE Trans Syst Man Cybern Syst2020; 52: 1663–1673.
2.
WenLRenZDi, SantoV, et al. Understanding fish linear acceleration using an undulatory biorobotic model with soft fluidic elastomer actuated morphing median fins. Soft Robot2018; 5: 375–388.
3.
TianQWangTWangY, et al. Robust optimization design for path planning of bionic robotic fish in the presence of ocean currents. J Mar Sci Eng2022; 10: 1109.
4.
LuYCaoYPanG, et al. Effect of cross-joints fin on the thrust performance of bionic pectoral fins. J Mar Sci Eng2022; 10: 869.
5.
KatzschmannRKDelPretoJMacCurdyR, et al. Exploration of underwater life with an acoustically controlled soft robotic fish. Sci Robot2018; 3: aar3449.
6.
LiuYJiangH.Optimum curvature characteristics of body/caudal fin locomotion. J Mar Sci Eng2021; 9: 537.
7.
PicardiGDe LucaMChimientiG, et al. User-driven design and development of an underwater soft gripper for biological sampling and litter collection. J Mar Sci Eng2023; 11: 771.
8.
LauderGV.Fish locomotion: recent advances and new directions. Annu Rev Mar Sci2015; 7: 521–545.
9.
FlammangBEAlbenSMaddenPG, et al. Functional morphology of the fin rays of teleost fishes. J Morphol2013; 274: 1044–1059.
10.
KopmanVPorfiriM.Design, modeling, and characterization of a miniature robotic fish for research and education in biomimetics and bioinspiration. IEEE/ASME Trans Mech2012; 18: 471–483.
11.
KancharalaAKPhilenMK.Study of flexible fin and compliant joint stiffness on propulsive performance: theory and experiments. Bioinspir Biomim2014; 9: 036011.
12.
LucasKNThornycroftPJGemmellBJ, et al. Effects of non-uniform stiffness on the swimming performance of a passively-flexing, fish-like foil model. Bioinspir Biomim2015; 10: 056019.
13.
CuiZJiangH.Design and implementation of thunniform robotic fish with variable body stiffness. Int J Robot Autom2017; 32: 109–116.
14.
HeathcoteSWangZGursulI.Effect of spanwise flexibility on flapping wing propulsion. J Fluids Struct2008; 24: 183–199.
15.
QuinnDBLauderGVSmitsAJ.Scaling the propulsive performance of heaving flexible panels. J Fluid Mech2014; 738: 250–267.
16.
SongJZhongYDuR, et al. Tail shapes lead to different propulsive mechanisms in the body/caudal fin undulation of fish. Proc Inst Mech Eng C J Mech Eng Sci2021; 235: 351–364.
17.
FeilichKLLauderGV.Passive mechanical models of fish caudal fins: effects of shape and stiffness on self-propulsion. Bioinspir Biomim2015; 10: 036002.
18.
ZhongQZhuJFishFE, et al. Tunable stiffness enables fast and efficient swimming in fish-like robots. Sci Robot2021; 6: abe4088.
19.
SteigmannDJ.Finite elasticity theory. Oxford: Oxford University Press, 2017.
20.
SteigmannDBîrsanMShiraniM.Lecture notes on the theory of plates and shells. Cham: Springer, 2023.
21.
ChouraSJayasuriyaSMedickMA.On the modeling, and open-loop control of a rotating thin flexible beam. J Dyn Syst Meas Contr1991; 113: 26–33.
22.
HanagudSSarkarS.Problem of the dynamics of a cantilevered beam attached to a moving base. J Guid Control Dynam1989; 12: 438–441.
23.
YangJBJiangLJChenDC.Dynamic modelling and control of a rotating Euler–Bernoulli beam. J Sound Vib2004; 274: 863–875.
24.
YigitAScottRAUlsoyAG.Flexural motion of a radially rotating beam attached to a rigid body. J Sound Vib1988; 121: 201–210.
25.
NaganathanGSoniAH.Coupling effects of kinematics and flexibility in manipulators. Int J Rob Res1987; 6: 75–84.
26.
KaneTRRyanRRBanerjeerAK.Dynamics of a cantilever beam attached to a moving base. J Guid Control Dynam1987; 10: 139–151.
27.
SimoJCVu-QuocL.The role of non-linear theories in transient dynamic analysis of flexible structures. J Sound Vib1987; 119: 487–508.
28.
TangSHuC-M.Nonlinear modeling and partial linearizing control of a slewing Timoshenko-beam. J Dyn Syst Meas Control1996; 118: 75–83.
29.
SimoJCVu-QuocL. On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comput Methods Appl Mech Eng1988; 66: 125–161.
30.
SimoJCVu-QuocL. On the dynamics of flexible beams under large overall motions—the plane case: part II. J Appl Mech1986; 53: 855–863.
31.
ZhuWDMoteCDJr. Dynamic modeling and optimal control of rotating Euler–Bernoulli beams. J Dyn Syst Meas Control1997; 119: 802–808.
32.
YuanKHuC-M. Nonlinear modeling and partial linearizing control of a slewing Timoshenko-beam. J Dyn Syst Meas Control1996; 118: 75–83.
33.
AnS-QZouH-LDengZ-C, et al. Dynamic analysis on hub–beam system with transient stiffness variation. Int J Mech Sci2019; 151: 692–702.
34.
DangarwalaRKGopalKN.Coupled free vibration analysis of rotating non-uniform cantilever beams by an element-wise Ritz method using local hierarchical functions. Comput Struct2023; 288: 107133.
35.
WrightADSmithCEThresherRW, et al. Vibration modes of centrifugally stiffened beams. J Appl Mech1982; 49: 197–202.
QianLFBatraRCChenLM.Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov–Galerkin method. Compos Part B Eng2004; 35: 685–697.
38.
BatraRC.Finite plane strain deformations of rubberlike materials. Int J Numer Methods Eng1980; 15: 145–156.
39.
MorisonJRJohnsonJWSchaafSA.The force exerted by surface waves on piles. J Pet Tech1950; 2: 149–154.
40.
MorisonJRJohnsonJWO’BrienMP.Experimental studies of forces on piles. Coast Eng Proc1953; 1: 25.
41.
KopmanVLautJAcquavivaF, et al. Dynamic modeling of a robotic fish propelled by a compliant tail. IEEE J Ocean Eng2014; 40: 209–221.
42.
ShahabSErturkA.Electrohydroelastic Euler–Bernoulli–Morison model for underwater resonant actuation of macro-fiber composite piezoelectric cantilevers. Smart Mater Struct2016; 25: 105007.
43.
BaruhHTadikondaSSK. Issues in the dynamics and control of flexible robot manipulators. J Guid Control Dynam1989; 12: 659–671.
44.
LiangLLiuSZhouJ.Quasi-variational principles of single flexible body dynamics and their applications. Sci China Ser G Phys Mech Astron2009; 52: 775–787.
45.
ChenJZhangGLiD, et al. Virtual work principle for piezoelectric semiconductors and its application on extension and bending of ZnO nanowires. Crystals2023; 13: 1368.
46.
QuYPanEZhuF, et al. Modeling thermoelectric effects in piezoelectric semiconductors: new fully coupled mechanisms for mechanically manipulated heat flux and refrigeration. Int J Eng Sci2023; 182: 103775.
47.
QuYPanEZhuF, et al. A thermodynamically consistent theory for flexoelectronics: interaction between strain gradient and electric current in flexoelectric semiconductors. Int J Eng Sci2025; 208: 104165.
48.
QuYXieXZhangS, et al. A rigid-flexible coupling dynamic model for robotic manta with flexible pectoral fins. J Mar Sci Eng2024; 12: 292.
