#### 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.

#### 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.

The motion equation can be expressed as follows:where is the mass, is the damping, is the stiffness, and and are horizontal and vertical inertial force vector. is the viscoelastic dampers force. Relevant parameters are listed as follows:

#### 3. The Vibration Equation of Weak Nonlinear System with Single Degree of Freedom and Its Linearization

##### 3.1. The Transfer of the Weak Nonlinear System Equation

Considering the weak nonlinear SDOF system, the general energy dissipation structural equation can be expressed as follows (see [33, 34]):where is the mass, is the damping, is the stiffness, is the weak nonlinear force including the nonlinear damping and the spring forces, is the modified damper with supporting forces, and is a biaxial excitation. The main aim is to replace (3) with an equivalent linear one (see [35]).

According to article (see [35]), can be expressed as follows:where and are the equivalent damping and stiffness, respectively; then the error between solutions of these two systems is minimized with the mean-square method. The difference between (3) and (4) is shown in the following:

To get a relative precise result, the error should be approximating to minimum. It is better to solve the following instead of (6):

In order to choose the best equivalent damping and the equivalent stiffness , it is necessary to minimize the error with statistical procedure, which requires (7) to be approximating to minimum.where denotes the mathematical expectation.

According to the method of multivariate function, the necessary and sufficient condition (see [36]) for the minimum of is obtained; it requires that

Equations (10) lead to two linear equations and determine the optimal values of and .

The required parameters can be obtained simultaneously as follows:

It is known from the paper (see [37, 38]) that and determined by the above formula lead to the minimum value of . It is important to note that it has to solve the linear random vibration system (4) to obtain the optimal values of and .

#### 4. Statistical Characteristics of Displacement and Velocity Response of Weak Nonlinear Energy Dissipation System under Biaxial Earthquake Action

##### 4.1. The Transform of the Time Domain Dynamic Equation

The motion equation of equivalent linear structure with viscoelastic dampers (4) could be written in the following form:wherewhere the symbols , , and are structure self-vibration frequency, damping ratio, and the reciprocal of structure mass, respectively. Moreover, and are the equivalent damping and stiffness, respectively.

According to the seismic code [39], should be ascertained by the maximum between the following:

So can be determined by the following:where and are the horizontal and vertical acceleration, respectively.

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:

##### 4.2. Stochastic Averaging Equation

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

The stochastic averaging equations that fit the amplitude are shown in the following:where and are Wiener process of independent units and is the power spectrum function of in the value of ; the expression of is shown in (22).where , .

##### 4.3. The Transient Joint Probability Density Function of Each Mode Shape of the Nonlinear Structure with Braces

Assume that the state variables of and are and , respectively. Probability density function of is . The transient joint probability density function of and is and the transient joint probability density function of and is , where is structure displacement and is the velocity. According to Itô equation (21), the transient joint probability density function that fits the FPK equation is shown in the following:

Because (20) does not depend on , the probability density function determined by FPK equation is as follows:

The initial conditions of (24) and (25) are, respectively, as follows:

Comparing with (24) and (25), we obtain the relationship of solution under the static initial conditions the following:

Meanwhile, we obtain the transient joint probability density function of the original weak nonlinear structure from transient displacement and transient velocity under the static initial condition.where .

When the expression of is obtained, the original structure of random response characteristics can be fully determined.

The solution of (22) and (25) should also fit under the static initial condition. could be written as follows:where .

Assume that the form of is described as follows:where is the undetermined function.

Equation (31) is substituted into (28); we transform the system of (31) into the following form.

Then (32) is substituted into (31); we can obtain the analytical solution of .

According to (29) and (32), we can obtain the response variance of the structural displacement and velocity, respectively.

#### 5. Equivalent Damping of Weak Nonlinear Structure with the Viscoelastic Damping and the Braces

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 3.

Where is the equivalent to a damping force of from (4), the motion equation of the structure can be described as follows:

In this case, (35) may be written as the following form:where , .

According to the stochastic averaging method, it is known that the probability density function of the amplitude response () of the equivalent structure is . The probability density function fitting the FPK equation is as follows:

The amplitude probability density function of the original structure can be applied to (30); the amplitude probability density function of the equivalent structure is appropriate for (37). We will know the difference by comparing with (30) and (37). After the following processing, the expression can be expressed as follows:where is the equivalent damping ratio of damper; it is consistent with the equivalent damping ratio of the Maxwell damper with the general integral model. For arbitrary random biaxial earthquake excitations and , all stochastic response characteristics calculated with the proposed method in equivalent structure are the same as these of the original structure. The equivalent damping ratio of the whole weak nonlinear dissipation structure is established as follows:

That is, the equivalent structure can be used as a total equivalent ratio of instead of the original structure damping ratio ; then we can use response spectrum method for structural analysis and engineering design.

#### 6. Numerical Example

It shows a SDOF nonlinear generalized Maxwell damper energy dissipation structure and the equivalent structure in Figure 4; the earthquake intensity is 8 degrees (0.2 g); its mass, stiffness, damping, and damping ratio are, respectively,  kg,  N/m,  N·s/m, and . The nonlinear structure is subjected to transient forces under biaxial earthquake.   (m2/s3),  s. The performance parameters of Maxwell damper in parallel are listed as follows: the brace  N/m, equilibrium modulus  N/m,  s−2, element damping coefficient  N·s/m, and the stiffness  kN/m. The excellent frequency and damping ratio of the site are  s−1 and , respectively. Spectral intensity factor  m2/s3. According to the equivalent damping ratio formula, when  s, the attached equivalent damping ratio of damper and the response variance of equivalent structural displacement are calculated; the response variance of original structure is also obtained by the frequency domain method.

According to (2), (14), and (35), we can obtain the value of the following parameters:

Hence,

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

According to (36), can be calculated as follows:

From (32) and (34), we can conduct the following calculations:

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.where

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 1. Results of the two methods are gradually approaching with the increase of the damping coefficient. The maximum displacement relative error is gradually reduced with the increase of the damping coefficient. When increases to a certain value, the results have a higher precision accuracy.

#### 7. Conclusions

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.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

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.