On the Existence, Uniqueness, and Stability of Periodic Modes of Motion of a Locomotion System with a Mobile Internal Mass
Tóm tắt
In this paper, we consider the rectilinear motion of a locomotion system consisting of a hull and an internal mass in a resisting medium during periodic movement of the internal mass relative to the hull. The periodic modes of the system’s movement, in which the hull’s speed is also a time periodic function, are studied. The questions of the existence and uniqueness of periodic modes of system motion, their stability with respect to the initial conditions, and the rate of convergence of arbitrary movements in relation to them are studied. The periodic mode of motion of the locomotion system is shown to exist, be unique, and be exponentially stable if the medium resistance is monotonic and unlimitedly increases with increasing speed and if the speed of the internal mass relative to the hull is continuous. A two-sided assessment of the hull’s speed in a periodic mode of motion is obtained. In particular cases of linear and piecewise linear medium resistance, a periodic mode of the system’s motion is constructed, and the rate of exponential convergence is calculated.
Tài liệu tham khảo
F. L. Chernous’ko, “Analysis and optimization of the rectilinear motion of a two-body system,” J. Appl. Math. Mech. 75, 493–500 (2011).
N. N. Bolotnik, T. Yu. Figurina, and F. L. Chernous’ko, “Optimal control of the rectilinear motion of a two-body system in a resistive medium,” J. Appl. Math. Mech. 76, 1–14 (2012).
G. Wagner and E. Lauga, “Crawling scallop: friction-based locomotion with one degree of freedom,” J. Theor. Biol. 324, 42–51 (2013).
A. Nunuparov, F. Becker, N. Bolotnik, I. Zeidis, and K. Zimmermann, “Vibration-driven capsubot with an opposing spring: an experimental study,” in ROMANSY 22, Robot Design, Dynamics and Control, Ed. by V. Arakelian and P. Wenger (Springer Int., Rennes, France, 2019), pp. 126–133.
T. Yu. Figurina, “Optimal motion control for a system of two bodies on a straight line,” J. Comput. Syst. Sci. Int. 46, 227 (2007).
B. Bardin and A. Panev, “On dynamics of a rigid body moving on a horizontal plane by means of motion of an internal particle,” Vibroeng. Proc. 8, 135–141 (2016).
D. Yu. Knyaz’kov and T. Yu. Figurina, “On periodic modes of motion of a system of two interacting bodies,” in Proceedings of the 61st All-Russia Conference (MFTI, Moscow, 2018), pp. 20–22.
D. Yu. Knyaz’kov and T. Yu. Figurina, “Periodic modes of rectilinear motion of a two-body system,” in Proceedings of the International Conference on Modern Problems of Mathematics and Mechanics (MAKS Press, Moscow, 2019), pp. 717–719.
I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations (Mosk. Gos. Univ., Moscow, 1984) [in Russian].