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

No abstract

We study the dynamics of a rigid body on rockers with a nonspherical contact surface. I~ is shown that in the case of small oscillations the equations of motion contain strong nonlinearity, which makes it possible to avoid resonance (nonperiodic) oscillations. We study free and forced oscillations under a harmonic force on the plane of whose parameters bifurcation curves are constructed separating the periodic and nonoscillatory processes.With the development of seismically stable construction for the kinematic foundations of various structures [1,2] it became necessary to study the dynamics of a building as a rigid body (the carried body) on rockers (carriers) that are rigid cylindrical bodies with axial symmetry, having spherical contact surfaces. Such a mathematical model was studied in [3], where the small deviations of the "building + support" system from the equilibrium position are described by harmonic functions. The frequency of the oscillations of such a system is determined by the support parameters, i.e., it is regulated constructively. Under a harmonic external force with a frequency coinciding with the frequency of free oscillations the system goes into oscillatory resonance, requiring a way of providing damping, for example using dry friction dampers [3] of significant mass and contradicting the idea of seismic insulation of buildings. For that reason the problem arises of synthesizing rockers so as to assure that the small oscillations of the system are nonlinear. The solution of this problem can be achieved by varying the supporting surface of the columns.Consider a mechanical system consisting of a carried rigid body of mass M supported by n carrier bodies (supports), whose mass is negligible in comparison with M. The supports are cylindrical columns of height H with supporting surfaces obtained by rotating a certain curve 77 = 7/(~). In what follows we shall confine ourselves to the class of surfaces obtained by rotating the curve r] = a~ 2+~ (~ = [~l/d, e > 0), which provides zero curvature at the point ~ = 0 (here d is the radius of the column and a the height of the segment, see Fig. 1).We shall study the initial motion of the carried body, confining ourselves to its motion in the xy-plane. We determine the position of the support relative to a coordinate system rigidly attached to the base whose axis deviates from the vertical by the angle ~o (see Fig. 1).We now write the condition for absence of slipping between the support, the body, and the base, which imposes restrictions on the coordinates of the system: VB = 5x AB.( 1) Since we are studying the initial motion of the carried body, all the supports will have the same velocity 17 = 17B, and the coordinates xs at different supports will differ by a constant. Thus the position of the system can be determined by the coordinates x and y of the center of mass of the carried body and the angle of deviation of the supports from the vertical ~o. In view of (1) (with 17 = xi"+ y~, the system has one degree of freedom, which we choose to be the an...