Abstract

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.

中文翻译：

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具有移动内部质量的运动系统的周期性运动模式的存在，唯一性和稳定性

摘要

在本文中，我们考虑内部质量相对于船体的周期性运动过程中，由船体和内部质量组成的运动系统在抵抗性介质中的直线运动。研究了系统运动的周期模式，其中船体的速度也是时间周期函数。研究了系统运动的周期性模式的存在和唯一性，它们相对于初始条件的稳定性以及与它们有关的任意运动的收敛速度的问题。如果介质阻力是单调的并且随着速度的增加而无限增加，并且内部质量相对于船体的速度是连续的，则运动系统的周期性运动模式已存在，唯一且呈指数稳定。获得了在周期性运动模式下船体速度的双向评估。在线性和分段线性介质电阻的特定情况下，构建系统运动的周期性模式，并计算指数收敛的速率。