Evolution of the resonance vibrations of an oscillator with dry friction

Abstract

In analyzing the vibrations of structures on kinematic foundations the study of the influence of Coulomb frictional forces on the forced vibrations of an oscillator is of great interest. It has been established [2] that in first approximation a structure on a kinematic foundation is a mechanical system with three degrees of freedom, whose position is determined by the horizontal displacement of the center of mass and the angle of revolution about the vertical axis. For that reason we take as a model of a structure on a kinematic foundation a rigid body undergoing plane-parallel motion under the action of a central force proportional to the displacement of the center of mass and directed toward a center (the equilibrium position of the seismic isolation system), which is prevented from rotating about the vertical axis by a moment proportional to the angle of rotation. The damping of the vibrations is accomplished by a point dry-friction damper D (Fig. 1).In these variables the equations of motion of this system have the form Free vibrations. The free vibrations of this system can be described by Eqs. (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. (1) cannot be solved in closed form, and hence the analysis was carried out by modeling on an analog computer. We integrated Eqs. (1) on an analog computer and established that the trajectory of the center of mass in the case l r 0 has the same shape as in the case of a centrally located damper (l = 0, see Fig. 2), up to the time when the sliding damper finally stops (v= 0).After the damper stops a rotational oscillation of the center of mass about the damper occurs. The graph of variation of the coordinate ~o is shown in Fig. 3 (curve 1 ). Just as in the case of a centrally located damper, the maximal elongation Of the center of mass from the origin over a period decreases by 4k.Forced Vibrations. Under resonance conditions the forced vibrations of the system can be described by Eqs.(1) if we set Xb = -Asint, Yb = -Bsin(t + c~). 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