No abstract

We study the dynamics of a building with a nonlinear seismic insulation system whose motion is described by the equation of transverse oscillations of a rod. We obtain the connection between the displacement of the lower section of the rod and the forces and moments in the section. We propose a numerical procedure for solving a nonlinear Volterra integral equation of second kind with dry friction damping. We determine the region of variation of the parameters of the building and the seismic insulation system in which it is possible to use the rigid-body model of the building. Two figures. Bibliography: 5 titles.

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...

We study the dynamics of seismically isolated buildings. We analyze the influence of the parameters of the seismic isolation system on the magnitude of mazimal horizontal displacements of a piece of equipment modeled as a rigid body, using the accelerograms of actual earthquakes.Numerous seismic isolating devices are used to protect buildings and equipment from seismic action. Among these devices the class of kinematic foundations stands out as comparatively simple in technical execution and satisfying industrial requirements for construction [1]. These devices allow objects to move relative to a stationary supporting foundation, which is rigidly attached to the ground. This goal is achieved by special elements that attach the support foundation to the movable equipment.In the present article we consider a kinematic foundation ( Fig. 1) consisting of supporting elements: uprights located in supporting wells. Damping of the vibrations is accomplished by dry-friction dampers located in the upper part of the supporting wells.Spatial motion of a seismically isolated building. As an acceptable physical model of a seismically isolated building we use the rigid-body model. It is assumed that the kinematic foundation uprights are symmetrically located, so that their geometric center coincides with the projection of the center of mass of the building onto its foundation. The dynamic system formed by a body of large mass M and the kinematic foundation is a mechanical system with three degrees of freedom, allowing motion in the xyplane and rotation about the z-axis. In the absence of external torque~ and the assumption of symmetry there is no rotation of the system. In this case the system has two degrees of freedom (motion in the plane) and it is possible to take account of three components of external action--the accelerations of the foundation along the respective axes.The motion of the system in the case of small deviations from the equilibrium position is described by a system of differential equations ( Here x = ~./H, y = f//H, u = u/v~, v = 6/v/ff~ are the dimensionless displacements and velocities of the center of mass of the building along the respective axes; H is the characteristic size (the height of an upright); g is the acceleration of gravity; ,. = t~/-~ is dimensionless time; t is actual time; a = FJMg is the dimensionless force of dry friction in the dampers; w 2 ,~ e = (2R -H)/H is the dimensionless stiffness; e is the eccentricity, which provides a quasi-elastic force; R is the radius of the spherical surfaces of the ends of the uprights; we~ = ~/g, weu = ~Deu/g, and Wez = wez/g are the dimensionless accelerations.To carry out numerical experiments we used an equivalent integral form for representing the unknowns via the Duhamel integral