Nonlinear Oscillations in a System with Dry Friction

Abstract

The case is examined where the right-hand side of the equations of motion is discontinuous. Attraction only in the stick domain ensures existence of periodic oscillations. Sufficient stability conditions for the periodic solution of a nonlinear system with dry friction are establishedIn this paper, we examine stable nonlinear oscillations that occur in a system with dry friction when the representative point falls into the domain where the right-hand side of the equations of motion is discontinuous. Averaging is admissible in both systems with hydrodynamic friction [5] and systems with dry friction. An averaged system makes it possible to establish the conditions for stable motion. The frictional force in a system with dry friction has a discontinuity. In one case, the representative point does not reach the discontinuity point during motion; then the attracting trajectory may produce self-oscillations and the average procedure is pertinent. In the other case, the closure of the phase trajectory is due to the attraction only in the neighborhood of the discontinuity point. This neighborhood is called the stick domain. By averaging, we can show that the solution is repelled on almost entire closed trajectory. This is an anbormal case of stable motion of a nonlinear system. The case is also known where the limit cycle of the Van der Pol equation with large parameter also has a repelling trajectory [6].The objective of the study is to demonstrate that the stable periodic oscillations of a nonlinear system can be due to sticking near the discontinuity point of the right-hand side of the equations of motion rather than due to the attraction of the phase trajectory. We will establish the quality of a stable closed trajectory using well-known results from the theory of oscillations [1].

On oscillations of a frictional pendulum

“…We assume that the conclusions become evident after determining the quality of the solutions on the trajectory using formulas (10). The qualitative analysis and approximate solution are based on the approach outlined in [12,13].…”

“…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].…”

Dynamic response of objects with shock-absorbers to seismic loads

“…The design of buildings is improved to enhance their capability of resisting loads due to vibrations of the foundation (ground). Dynamic loads on various mechanical systems can significantly be reduced by using shock-absorbing devices with flexible, frictional, and other elements [7,[13][14][15]. Currently, various shock-absorbers are intensively used to reduce seismic loads on buildings and above-ground structures [2,9,11,16,17].…”

Chaotic frictional vibrations excited by a quasiperiodic load

“…Results from studies on nonlinear frictional vibrations are reviewed in [14]. Compound vibrations and nonlinear vibrations due to dry friction are addressed in [15,17], and the dynamic processes associated with vibration damping are examined in [13,18].The present paper is concerned with a quasiperiodically excited Duffing oscillator frictionally interacting with a moving belt. We will use the multiple-scales method to derive a system of nonautonomous equations to describe modulation of oscillations.…”

“…Results from studies on nonlinear frictional vibrations are reviewed in [14]. Compound vibrations and nonlinear vibrations due to dry friction are addressed in [15,17], and the dynamic processes associated with vibration damping are examined in [13,18].…”