中文摘要：

对弱非线性单自由度非自由振动系统结构在白噪声地震激励下的随机响应问题进行了研究。首先,建立了结构的弱非线性响应方程;然后根据能量平衡原理对运动方程进行了等效线性化;最后获得了金井清谱（Kanai-Tajimi谱）下等效线性化方程的位移响应及速度响应的方差,将弱非线性随机响应问题简化为求解线性随机响应问题,并给出了算例验证。结果表明,该方法可以准确得到弱非线性随机响应问题的解。同时,建立了单自由度结构弱非线性随机响应频域分析的统一解析解法。

英文摘要：

In this paper, the stochastic response of the structure of the weakly nonlinear single degree of freedom system under the excitation of white noise excitation is studied. Firstly, the weakly nonlinear motion equation of the structure is established. Then according to the principle of energy balance, the equation of motion of the equivalent is linearized. Finally, the displacement response and the variance of the velocity response of the Tajimi-Kanai（Tajimi-Kanai spectrum） equivalent linearization equation are obtained. The complex weak nonlinear stochastic response is simplified to the problem of linear stochastic response, and a numerical example is given. Thus a unified analytical solution to the frequency domain equation of the single degree of freedom structure with weak nonlinear response is established.