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本文在等价线性化法的基础上提出了一种用振型分解法分析小阻尼、弱非线性多自由度体系受平稳高斯宽频带随机荷载作用的动力可靠性的完整的解析方法。先在平稳高斯白噪声的荷载条件下证明了等价正规反应是近似独立的,并由此求出正规反应的联合概率密度函数,建立了确定等价阻尼和等价固有圆频率的公式;进一步求出体系质点相对位移及其导数的联合概率密度函数后,提出了一种基于首通破坏机制和“最弱链环”(weakest link)模型的非线性多自由度体系的动力可靠性的计算方法,求出了以质点相对位移量为界限的非线性多自由度体系的平稳反应的动力可靠性的解析表达式。其次,把上述方法推广到了荷载是平稳高斯宽频带过程的情况。最后把本文提出的理论应用到了拟线性多自由度体系。对于两个自由度的情况求出了数字结果,并与Monte Carlo模拟结果进行了比较,验证了本文提出的方法的正确性。在结论中还指出了本文的方法用来处理过渡反应时应该注意的一些问题。
Based on the equivalent linearization method, this paper proposes a complete analytical method for analyzing the dynamic reliability of a small damped, weakly nonlinear multi-degree-of-freedom system subjected to a stationary Gaussian wide-band random load using the mode-decomposition method. Firstly, under the condition of stationary Gaussian white noise, it is proved that the equivalence regularity is approximately independent, and the joint probability density function of normal response is obtained. The formula to determine the equivalent damping and the equivalent natural frequency is established. After calculating the joint probability density function of the relative displacement and its derivative of the system particle, a dynamic reliability calculation of a nonlinear multi-degree-of-freedom system based on the first-pass destruction mechanism and the weakest link model is proposed. Methods The analytical expression of the dynamic reliability of a smooth multi-degree-of-freedom system with bounded relative displacements of particles was obtained. Second, the above method is extended to the case where the load is a stationary Gaussian broadband process. Finally, the theory proposed in this paper is applied to the quasi-linear multi-degree of freedom system. The numerical results are obtained for the two-degree-of-freedom case and compared with the Monte Carlo simulation results, which verifies the correctness of the proposed method. The conclusions also point out some of the issues that should be noted when this method is used to handle transitional reactions.