This paper discusses the mathematical modeling, dynamic analysis, and design optimization of a class of drag-based vibratory swimmers using averaging. A typical swimmer of this class consists of a surface or subsurface main body, an oscillatory mass inside the body, and a rigid fin attached to the main body using torsional springs and immersed in water. This paper takes advantage of third-order averaging techniques for dynamic analysis and design optimization of the swimmer for high frequencies. The averaged dynamics are used for determining the optimum stiffness of the springs for maximizing the average speed of the swimmer for high-frequency motion of the oscillatory mass. Experimental results confirm the general response of the mathematical model.
BurnsJA (2014) Introduction to the calculus of variations and Control with modern applications. In: Applied Mathematics and Nonlinear Science Series. Boca Raton, FL: Chapman and Hall/CRC.
2.
ChernouskoFL (2008) On the optimal motion of a body with an internal mass in a resistive medium. Journal of Vibration and Control14(1-2): 197–208.
3.
ChernouskoFLBolotnikNNFigurinaTY (2013) Optimal control of vibrationally excited locomotion systems. Regular & Chaotic Dynamics18(1-2): 85–99.
4.
ChildressSSpagnolieSETokiedaT (2011) A bug on a raft: recoil locomotion in a viscous fluid. Journal of Fluid Mechanics669: 527–556.
5.
CrouchPE (1981) Geometric structures in systems theory. IEE Proceedings D (Control Theory and Applications)128(5): 242–252.
6.
DillerESittiM (2013) Micro-scale mobile robotics. Foundations and Trends in Robotics2(3): 143–259.
7.
EhlersKMKoillerJ (2011) Micro-swimming without flagella: propulsion by internal structures. Regular & Chaotic Dynamics16(6): 623–652.
8.
GuckenheimerJHolmesP (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences. New York, NY: Springer-Verlag.
9.
KellySDXiongH (2010) Self-propulsion of a free hydrofoil with localized discrete vortex shedding: analytical modeling and simulation. Theoretical and Computational Fluid Dynamics24: 45–50.
10.
KhalilHK (1996) Nonlinear Systems. New Jersey: Prentice-Hall, Inc.
11.
MaggiaMEisaSATahaHE (2020) On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques. Nonlinear Dynamics99: 813–836.
12.
NayfehAH (1981) Introduction to Perturbation Techniques. New York: John Wiley and Sons.
13.
QuillenACAskariHKelleyDH, et al. (2016) A coin vibrational motor swimming at low Reynolds number. Regular and Chaotic Dynamics21(7-8): 902–917.
SandersJAVerhulstF (1985) Averaging Methods in Nonlinear Dynamical Systems. New York, NY: Springer-Verlag.
16.
SarychevA (2001) Stability criteria for time-periodic systems via high-order averaging techniques. In: ThomaMAllgowerFMorariM (eds) Nonlinear Control in the Year 2000. Springer, pp. 365–377.
17.
StrogatzSH (2015) Nonlinear Dynamicas and Chaos. 2nd edn.Westview Press.
18.
TahmasianS (2021) Dynamic analysis and optimal control of drag-based vibratory systems using averaging. Nonlinear Dynamics104: 2201–2217. DOI: 10.1007/s11071-021-06440-0.
19.
TahmasianSKatrahmaniA (2020) Vibrational control of mechanical systems with piecewise linear damping and high-frequency inputs. Nonlinear Dynamics99: 1403–1413. DOI: 10.1007/s11071-019-05361-3.
20.
TahmasianSJafaryzadABulzoniNL, et al. (2020) Dynamic analysis and design optimization of a drag-based vibratory swimmer. Fluids5(1). DOI: 10.3390/fluids5010038.
21.
TallapragadaPKellySD (2015) Self-propulsion of free solid bodies with internal rotors via localized singular vortex shedding in planar ideal fluids. The European Physical Journal224: 3185–3197.
22.
TsakalisKSIoannouPA (1993) Linear Time-Varying Systems. New Jersey: Prentice-Hall.
23.
VelaPABurdickJW (2003a) Control of underactuated mechanical systems with drift using higher-order averaging theory. Maui, Vol. HI, 3111–3117.Proc. IEEE Conference on Decision and Control.
24.
VelaPABurdickJW (2003b) A general averaging theory via series expansions. In: Proc American Control Conference, pp. 1530–1535. Denver, CO.
25.
VetchaninEVMamaevISTenenevVA (2013) The self-propulsion of a body with moving internal masses in a viscous fluid. Regular & Chaotic Dynamics18(1-2): 100–117.
26.
WigginsS (2003) Introduction to applied nonlinear dynamical systems and chaos. In: Texts in Applied Mathematics. 2nd edn.New York: Springer.
