Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7384940, 10 pages

https://doi.org/10.1155/2017/7384940

## Stochastic Response Characteristic and Equivalent Damping of Weak Nonlinear Energy Dissipation System under Biaxial Earthquake Action

Civil Engineering and Architecture Department, Guangxi University of Science and Technology, Liuzhou 545006, China

Correspondence should be addressed to Yu Xia; moc.361@niar-mmus

Received 24 January 2017; Accepted 12 April 2017; Published 10 May 2017

Academic Editor: Roman Lewandowski

Copyright © 2017 Yu Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The random response characteristic of weak nonlinear structure under biaxial earthquake excitation is investigated. The structure has a SDOF (single degree of freedom) with supporting braces and viscoelastic dampers. First, it adopts integral constitutive relation and establishes a differential and integral equations of motion. Then, according to the principle of energy balance, the equation is linearized. Finally, based on the stochastic averaging method, the general analytical solution of the variance of the displacement and velocity response and the equivalent damping is deduced and derived. At the same time, the joint probability density function of the amplitude and phase and displacement and velocity of the energy dissipation structure are also given. The dynamic characteristics of a structure with viscoelastic dampers are determined as a solution to the variance of displacement response, so the equivalent damping is taken into consideration as a solution to replace the original nonlinear damping. It means it has established a unified analytical solution of stochastic response analysis and equivalent damping of a SDOF nonlinear dissipation structure with the brace under biaxial earthquake action in this paper.

#### 1. Introduction

In addition to the two seismic force components in the horizontal direction, there is still vertical seismic force components. Actual earthquake structure is always subject to vertical and horizontal earthquake actions. Under a larger action of earthquake, the response of structure is further increased. At this time, the vertical earthquake action can not be ignored. Therefore, it is important to research structure response in the horizontal and vertical earthquake. As land in big cities is limited, buildings are located close to each other. To reduce the seismic responses of buildings, adjacent buildings are linked together by connecting dampers, such as the Triple Towers in Downtown Tokyo [1]. Researchers have proposed different types of connecting devices to connect adjacent buildings. These devices include passive dampers [2–5], semiactive dampers [6–8], and active dampers [9, 10]. It is now well recognized that seismic responses of adjacent buildings can be mitigated by connecting dampers. Biaxial earthquake action will aggravate the vibration of the structure; the devices of passive control will reduce the structural vibration. The passive control techniques, such as viscous and viscoelastic dampers, have been widely used [11]. Linear viscoelastic damper is a kind of excellent performance of energy dissipation device and is widely used in seismic engineering. The integral model is the most general model of viscoelastic dampers [12]. Other models, such as the complex modulus model [13], the fractional derivative model [14–16], and the general differential model, are all approximate model. Analytical modeling of a novel type of passive friction damper for seismic hazard mitigation of structural systems is present [17]; numerical results show that the proposed damper is more efficient in dissipating input seismic energy than a passive linear viscous damper with same force capacity. The equivalent linearization of the motion equations with Maxwell dampers will effectively solve the problem of nonlinear equations. A system with nonlinear dampers is usually replaced by an equivalent linear system, with its properties determined by using different methods, like equating the energy dissipated [18], equating power consumption [19], replacing the nonlinear viscous damping by an array of frequency and amplitude-dependent linear viscous models [20], and other random vibration theories [21]. Malone and Connor [22] have reported a method setting a new degree of freedom at mass-less point between a dashpot and a stiffness spring of the Maxwell model and then apply a common numerical integration scheme. In this method, it is necessary to consider twice the degrees of freedom of the original system. It has been mainly applied to the analyses of material stress-strain relationships. Kitagawa et al. [23] have reported the analysis of reinforced concrete elements by considering the effect of strain speed. They treated the Maxwell model as a supplementary restoring force on the equation of motion of the system discretized by a central difference method, which is categorized into an explicit integration scheme. It can play a better role of shock absorption by adding brace to the viscoelastic damper. The brace is widely used in damper. The integral model is a typical type; this kind of damper [12] can be used to describe the instantaneous elasticity, creep, relaxation, and strain memory of viscoelastic dampers. Park et al. [24] and Singh et al. [25] describe the use of gradient-based optimization algorithms to obtain the optimal parameters of dampers and their supporting braces in structures subjected to seismic motions. More recently, Chen and Chai [26] also proposed a gradient-based numerical procedure for determining the minimum brace stiffness together with a set of optimal damper coefficients to meet a target response reduction. They used Maxwell model-based brace-damper systems and concluded that brace stiffness equal to the first storey stiffness would be adequate for the desirable levels of response reduction in typical applications. Since the structure is installed with the damper and then turned into the energy dissipation structure, its design can not be directly applied to the response spectrum method. At the same time, this makes the design of actual engineering very inconvenient. The damping ratio of the dissipation system is the sum of the damping ratio of the structure itself and the equivalent damping ratio of the damper. The linear response spectrum method can be used to calculate the equivalent damping ratio of the damper [27, 28]. Therefore, it is greatly significant to establish a equivalent structure. Then, the response spectrum method can be used directly used to structural analysis and engineering design. The relationships between equivalent damping and ductility for the direct displacement-based seismic design (DBSD) method are proposed [29]. As the concept of the DBSD is addressed to highlight the importance of the proper determination of equivalent damping, in the DBSD, the equivalent stiffness is taken as the secant stiffness at maximum deformation, so the appropriate equivalent damping should be determined based on such a prescription. And twenty-one SDOF systems are designed according to the DBSD procedure and analyzed to indicate that the proposed equivalent damping relationships are suitable for the DBSD. In addition, stochastic averaging method is an effective approximation method for predicting the stochastic response of a structure. The basic assumption is small damping and weak broadband excitation. Compared with the modal strain energy method, it is easy to understand and obtain the general analytical solution under a close theoretical basis; the same result of decoupling method of the forced vibration mode under the case of linear small damping can be concluded. In fact, in recent years, the important theoretical results of linear and nonlinear random vibration are obtained by using the stochastic averaging method. It investigates the stochastic response of vibroimpact system with fractional derivative under Gaussian white noise excitation; the nonsmooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system [30]. The first-passage statistics of Duffing-Rayleigh-Mathieu system under wide-band colored noise excitation are studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging [31]. The equivalent linearization can solve the problem of nonlinear structure; a nonlinear stochastic optimal control strategy for single degree of freedom viscoelastic system with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programing principle. As the viscoelastic system is converted into an equivalent nonlinear nonviscoelastic system by replacing the viscoelastic force with amplitude-dependent stiffness and damping [32], in this paper, it has used the equivalent linearization method and the stochastic averaging method, and it has also used the general integral model of viscous and viscoelastic dampers. Considering the comprehensive effect of brace, strain history of damper, dynamic characteristics of structure, and excitation, it establishes a complete analytical solution of stochastic response analysis and equivalent damping of a SDOF nonlinear dissipation structure with the brace under biaxial earthquake action. The new approach can be directly applied to damping engineering design with the response spectrum method.

#### 2. Constitutive Equation of Damper with Brace

##### 2.1. Motion Equation of Maxwell Damper with Braces

The mass matrix, stiffness matrix, and damping matrix of the structure are , , and , respectively. A viscoelastic damper () of the general integral type is equipped between floors. The modified damper with supporting braces () is . The complex modulus, storage modulus, and energy dissipation modulus of and are , , and , , , respectively. The relaxation modulus, equilibrium modulus, and relaxation function of and are , , and , , , respectively. The displacement vector of the structure with respect to the ground is when the horizontal and vertical ground motion are and ; the relative displacement of damper and its supporting braces () are and , respectively; two dampers mentioned above are shown in Figures 1 and 2.