Forced vibrations of an oscillator in the presence of dry friction

Abstract

We study the vibrations of an oscillator with two degrees of freedom in the presence of dry friction. We compare the nature of the damping of the free oscillations in a straight line with the general case. For the forced vibrations we determine the way in which the critical values of friction at which there exist periodic motions depend on the parameters of external action in resonance mode. Five figures. Bibliography: 5 titles.It is a problem of considerable interest in automatic control theory and in the calculation of vibrographs, regulators, and the like to take account of the Coulomb frictional forces in solving the problem of the forced vibrations of an oscillator. An analogous problem arises in the theory of seismoinsulation in the analysis of the forced vibrations of devices on a kinematical basis [1,2].The first to solve the problem of the forced vibrations of an oscillator with dry friction was Eckolt [5]. A more complete solution for the case of a sinusoidal external force was given by Den Hartog [4]. Den Hartog [3] gave the solution for the whole range of frequencies and values of the resistance and obtained the general equation of motion of the system for vibrations both with and without pauses. He showed that in all cases the vibrations can have at most one pause per half-period.However, in all the works just mentioned only systems with one degree of freedom are considered, when the motion of the oscillator is along a line. In the present paper we consider the motion of a system with two degrees of freedom in resonance mode in the presence of dry friction.The equations of motion. A model of an oscillating system with dry friction is provided by a particle of mass M that is able to slide along a horizontal surface. The coefficient of sliding friction is f. There are other forces acting on the body besides the force of dry friction: an elastic force F = -j, directed toward a fixed center O (taken as the origin of the coordinate system) and proportional to the distance ~' of the mass from the center, and a harmonic perturbing force/~p given by components Fp~ = F~ sinpt and Fpy = Fy sin(pt + 90). The mechanical model just described is shown in the upper part of Fig. 1.The differential equations of motion of the mechanical system being studied can now be written in vector form as Mr-'= -j-*~I + Fp,where the dry frictional force is defined by the equationsI~'[ r 0 (~'= ~; -min{fMg, lRl}R[R] -1, iv'l=0 (/~=r~-cr-).Before we take up the analysis of the equations of motion of the system (1), we simplify them by dividing both sides by M and writing them in coordinate form ii +co2z = Asinpt-ff~, ~1 +co2y = Bsin(pt + ~o)-fly,where { r 0, = -min(k,l-~l])/~l}Rl1-1, = 0, /~1 = (A sin pt -co 2 x )F + (B sin(pt + qo) -002 y)y, k = fg, A = F~M -1, B = FyM -1, 00 2 = cM -1.Simferopol' University.

Evolution of the resonance vibrations of an oscillator with dry friction

“…This dependence, which is determined when the condition max x/(Asint) 2 + (Bsin(t + a)) 2 = 1, is the same as that given in [3]. t We remark that the nature of the rotational vibrations caused by the noncentral location of the dampers at subcritical values of the damping differs only slightly from the rotational vibrations at frictional values above the critical.…”

“…The authors [3] have studied in detail the question of the forced vibrations of a system with a centrally located damper. It was established that the critical value of the damping parameter k at which there are bounded vibrations in the system is determined by ~s from…”

“…(1), in which one must put xb = 0, Yb = 0. The case of a centrally located sliding-friction damper (l = 0) was studied in detail in [3]. It was established that Eq.…”