Seismic impacts create the possibility of parametric resonances, i.e. the possibility of the appearance of intense transverse vibrations of structure elements (in particular, of high-rise structures) from the action of periodic longitudinal forces. As a design model of a high-rise structure, a model is used which adopted in the calculation of high-rise structures for seismic effects, - a weightless vertical rod (column) rigidly restrained at the base with a system of concentrated masses (loads) located on it (Fig. 1). By solving the differential equation of the curved axis influence function for a rod is constructed by means of which influence coefficients are determined for the rod points, in which the concentrated masses are situated. These coefficients are elements of the compliance matrix . Next, the elements of the stiffness matrix are determined by inverting the matrix . Using a diagonal matrix of the load masses and matrix a system of differential equations of free vibrations of a mechanical system, consisting of concentrated masses, is constructed, and the frequencies and forms of these vibrations are determined. From the vertical component of the seismic impact, its most significant part is picked out in the form of harmonic vibrations with the predominant frequency of the impact. Column vibrations are considered in a moving coordinate system, the origin of which is at the base of the column. The forces acting on the points of the mechanical system (concentrated masses) are added by the forces of inertia of their masses associated with the translational motion of the coordinate system. The forces of the load weights and forces of inertia create longitudinal forces in the column, periodically depending on time. Further, the integro-differential equation of the dynamic stability of the rod, proposed by V. V. Bolotin in [8], is written. The solution to this equation is sought in the form of a linear combination of free vibration forms with time-dependent factors. Substitution of this solution into the integro-differential equation of dynamic stability allows it to be reduced to a system of differential equations with respect to the mentioned above factors with coefficients that periodically depend on time. For some values of the vertical component parameters of the seismic action, namely the frequency and amplitude, the solutions of these equations are infinitely increasing functions, i.e. at these values of the indicated parameters, a parametric resonance arises. These values form regions in the parameter plane called regions of dynamic instability. Next, these regions are being constructed. A concrete example is considered.
Numerical simulation of oscillations of a nonlinear mechanical system with a spring-loaded viscous friction damper
