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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.

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 [

The mass matrix, stiffness matrix, and damping matrix of the structure are

Calculation diagram of structure.

The original calculation diagram of structure

The modified calculation diagram of structure with brace

Calculation diagram of damper.

The original calculation diagram of damper

The modified calculation diagram of damper with brace

The motion equation can be expressed as follows:

Considering the weak nonlinear SDOF system, the general energy dissipation structural equation can be expressed as follows (see [

According to article (see [

To get a relative precise result, the error

In order to choose the best equivalent damping

According to the method of multivariate function, the necessary and sufficient condition (see [

Equations (

The required parameters can be obtained simultaneously as follows:

It is known from the paper (see [

The motion equation of equivalent linear structure with viscoelastic dampers (

According to the seismic code [

So

Assume that

So the time domain dynamic equation of the energy dissipation structure of a single degree of freedom with linear viscoelastic damper could be expressed in the following form:

According to the stochastic averaging theory, the standard Van-der-Pol transform is introduced:

The stochastic averaging equations that fit the amplitude

Assume that the state variables of

Because (

The initial conditions of (

Comparing with (

Meanwhile, we obtain the transient joint probability density function of the original weak nonlinear structure from transient displacement

When the expression of

The solution of (

Assume that the form of

Equation (

Then (

According to (

The actual ground motion is highly random characteristics. Because of the rationality and practicality of the earthquake, the ground motion model still needs to be further improved. So the the response spectrum method is adopted in most countries. Once the structure is installed with the damper and it turns into an energy dissipation structure, the response spectrum method can not be directly applied to these structures. Therefore, it is greatly significant to establish the equivalent structure which can be used directly with the response spectrum method. The calculation diagram is shown in Figure

Calculation diagram.

Where

In this case, (

According to the stochastic averaging method, it is known that the probability density function of the amplitude response (

The amplitude probability density function of the original structure can be applied to (

That is, the equivalent structure can be used as a total equivalent ratio of

It shows a SDOF nonlinear generalized Maxwell damper energy dissipation structure and the equivalent structure in Figure ^{2}/s^{3}), ^{−2}, element damping coefficient ^{−1} and ^{2}/s^{3}. According to the equivalent damping ratio formula, when

Calculation diagram.

According to (

Hence,

The total coefficient of the parallel spring group is equal to the sum of the coefficients of each spring:

According to (

From (

Hence, we can obtain the following parameters values:

According to the frequency domain method, frequency response function and the variance of displacement are obtained, respectively.

The relative error can be calculated.

It is known that we have calculated the maximum displacement standard deviation by frequency domain method and equivalent structure. The results of maximum displacement standard deviation are given in Table

Results comparison of the frequency domain method and the proposed method in this paper.

Damping coefficient |
Approximate calculation formula of frequency domain method | Method proposed in this paper | |
---|---|---|---|

Displacement standard deviation/m ( |
Displacement standard deviation/m ( |
Relative error/% | |

10 | 2.7 | 3.897 | 44.3 |

15 | 2.950 | 3.867 | 31.1 |

20 | 2.950 | 3.838 | 30.1 |

25 | 3.032 | 3.809 | 25.6 |

30 | 3.095 | 3.778 | 22.1 |

35 | 3.145 | 3.751 | 19.3 |

40 | 3.186 | 3.724 | 16.9 |

45 | 3.223 | 3.695 | 14.6 |

In this paper, a weak nonlinear structural system with one degree of freedom is researched and a systematically research on the random response characteristic of structure was conducted, which is under biaxial earthquake action. First, integral constitutive relation is adopted; it then establishes a differential and integral equations of motion of SDOF weak nonlinear structure containing the general integral model viscoelastic dampers and the braces. And, then, the motion equation is linearized according to the principle of energy balance. Finally, based on the stochastic averaging method, the general analytical solution of the variance of the displacement, velocity response, and equivalent damping is deduced and derived. The joint probability density function of the amplitude and phase and displacement and velocity of the energy dissipation structure are also given at the same time. Numerical example shows the availability and accuracy of the proposed method. It means it has established a complete 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. The proposed method provides a beneficial reference for the engineering design of this kind of structure.

The authors declare that they have no conflicts of interest.

This study is supported by the National Natural Science Foundation of China (51569005, 51468005, and 51469005), Guangxi Natural Science Foundation of China (2015GXNSFAA139279 and 2014GXNSFAA118315), Innovation Project of Guangxi Graduate Education in China (GKYC201628, GKYC201711, and YCSZ2015207), and Innovation Team of Guangxi University of Science and Technology 2015.