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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。
The problem of random response of weakly nonlinear single degree of freedom non-free vibration system under white noise excitation is studied. Firstly, the weakly nonlinear response equation of the structure is established. Then, the equations of motion are equivalently linearized according to the principle of energy balance. Finally, the displacement response and the velocity of the equivalent linearization equation under the Kanai-Tajimi spectrum are obtained Response variance, the problem of weakly nonlinear stochastic response is simplified to solve the problem of linear stochastic response, and a case study is given. The results show that the proposed method can obtain the exact solution to the weakly nonlinear stochastic response problem. At the same time, a unified analytic solution to the frequency domain analysis of weakly nonlinear single-degree-of-freedom structure is established.