On oscillations of a frictional pendulum

Abstract

Conditions are established under which a standard limit cycle occurs in the system under consideration, or the trajectory closes under the influence of a stagnation domain. It is pointed out that when the solution falls into the stagnation domain it makes no sense to use the asymptotic method because of a large error Introduction. A dry-friction problem is considered in [3,4] as an example of a standard statement for problems solved by the averaging method. In these problems, the representative point falls into the discontinuity domain of the right-hand side of the equation of motion. Otherwise, the authors point out that the discontinuity domain is missed [5,6]. Let us show that the first case of averaging may be nontrivial and that the possibility of averaging should sometimes be proved. Frictional oscillation problems have recently taken on a new relevance [10]. A simple frictional pendulum, which was used by Strelkov [7] to detect auto-oscillations experimentally, can exhibit several modes of nonlinear oscillations. These modes depend on the parameter values and the method of modeling the interaction. One of the modes is that in which the phase trajectory closes because of the stagnation point caused by dry friction and the representative point falls into the discontinuity domain of the right-hand side of the equations of motion. The closed trajectory is asymmetric in this case, as in [13]. In another mode, a limit cycle occurs, which may be identified with the help of the Bogolyubov theorem [2].The present paper is a continuation of [13], since it also addresses the statement of dry-friction problems and examines the possibility of applying the averaging method. We will consider conditions under which a standard limit cycle described in [1] occurs or a circular trajectory closes under the influence of the stagnation domain. In solving engineering problems, the averaging method is applied in both cases. We will also consider the case where the trajectory gets into a stagnation point, and the asymptotic approach can be used only up to a certain value of the parameter.1. Preliminaries. Consider a sleeve with a pendulum loosely fit onto a smooth shaft rotating with constant speed (Fig. 1a). The friction torque between the shaft and the sleeve depends on their relative speed of rotation ω ϕ = − d dt/ Ω, where

Elements of Lyapunov stability theory for dynamic equations on time scale

“…Estimate (3.8) allows establishing new conditions for stability of the unperturbed motion of the quasilinear system (3.1) on a time scale in much the same way as for a system of ordinary differential equations (see Theorems 1-3 in [28] …”

“…Of obvious interest is to generalize the approaches proposed to vibrating systems [28,29] and hybrid systems [32] that have continuous and discrete components.…”

“…A prototype of this statement is Lemma 1 from[28], which was proved for the time scale T R = .Lemma 3.4. Assume that condition (3.3) for m > 1 and condition (3.4) are satisfied.…”

“…If y C Î rd , p ³ 0, and a ÎR, statement was obtained for the time scale T R = (see Lemma 2 and Theorem 5 in[28]). Lemma 3.1.…”

Some model problems of dynamics for a rigid body interacting with a medium

“…The parameters of the cylinders are such that inequality (27) holds and inequality (28) The value of the index In allows us to judge whether a weak focal point [11,13,18,20,30] at the origin of coordinates of system (24), (25) is stable or not and, thereby, to conclude whether the system undergoes self-oscillations. The index In depends on higher derivatives of the functions F and s whose signs are unknown.…”