Dynamic-model reduction for a multimass system with nonideal constraint in a seismic damper

Abstract

A dynamic model for a multimass system with a seismic damper as a nonlinear inertial component is reduced via balancing. The seismic damper is subject to a nonideal constraint. As an example, the order of the system of differential equations is reduced by one for a dynamic three-mass system. A comparison of the processes in the original and reduced systems confirms that it is possible to use balancing to simplify such models Introduction. Most kinematic pairs are subjected, to a greater or lesser extent, to friction, which in certain cases has a strong effect on dynamic processes in various engineering devices, machines, and mechanisms [2,8,10,14,18]. The friction in seismic dampers and structural elements of buildings and other structures contributes to the damping of seismic vibrations and to the decrease of the extreme dynamic loads [9,11,12,15]. Theoretical studies of dynamic processes play an important role in designing seismic isolation systems. This is why there is an objective need for mathematical models that would adequately describe their major properties and meet the requirements of numerical integration (see, e.g., [17]) such as that the system of differential equations must not be stiff [5,6,16,19], which frequently leads to the need to reduce the system.Due to the increasing interest to the seismoprotection of buildings and other structures [9, 20], we will outline the model of an in-line system that includes a passive seismic damper as a nonlinear inertial via balancing [16,19] for not only linear and nonlinear systems [5,13], the latter including inertialess and inertial components, but also systems that have a nonlinear inertial component with a nonideal constraint. The balancing method is detailed in [5,6,16,19].It is natural that the approach outlined here can be applied to not only buildings subject to seismic effects, but also mining, metallurgy, and farm machinery, machining tools, etc. subjected to their typical loads. Mathematical Model of a System with a Seismic Damper that Includes a Nonideal Constraint.First, we will briefly outline a model for a system with seismic dampers that include ideal constraints. With such an assumption, the reactions R i at the contact points between the bodies I and O are normal to the surface, i.e., the angles of friction r i between the normal and total reactions R are zero (Fig. 1).The body O undergoes plane translation and has only one degree of freedom because two of the three possible degrees of freedom of a free body in plane motion are excluded by the two normal constraints to the surfaces at the contact points between the bodies I and O. Each additional damper of the same shape and size (third, fourth, …, nth) would introduce a repeated (passive) constraint [4] not affecting the law of motion of the system, but making it (singly, doubly, (n -2)-tuply) statically indeterminate. The reactions in each damper cannot be found through kinetostatics. For the translating system in Fig. 1, it is convenient to use two equations of motion of the body O and an ...