非独立泊松白噪声激励下随机非线性系统的响应研究

Random excitations are widespread in the nature, engineering and other fields. Accurate characterization for these random excitations produces great scientific significance. Generally, these random excitations are assumed to be independent of each other in the previous work. However, in the real world some random excitations are correlated to each other. Excited by these random excitations, there can be much discrepancy between the obtained results when the correlation of the external and parametric excitations is and is not taken into account, which directly influences the prediction of the safety and reliability for the structures. Based on the previous work, this project is to study the response of random nonlinear systems under correlated Poisson white noise excitations from two aspects of correlated impulse amplitudes and correlated impulse arrival time. Exponential-polynomial closure (EPC) method is improved to solve the corresponding Fokker-Planck-Kolmogorov (FPK) equations, which are derived and built under different correlated conditions. With the obtained probability density function of response, the statistical properties of random nonlinear systems under correlated Poisson white noise excitations can be revealed. The combined effect of structural internal force and response is taken into account to study the transmission and distribution of vibration energy in structure under correlated Poisson white noise excitations. With this project, it is supposed to give a deepened insight into the response analysis of systems under the combined action of various random loadings. It is fundamental for analyzing the probabilistic solutions of large nonlinear stochastic dynamical systems or multi-degree-of-freedom systems under correlated random excitations. It further provides theoretical proofs to predict the safety and reliability of the real structures in engineering. Therefore, this project is of important theoretical significance and practical value.

随机激励普遍存在于自然及工程等领域中，对其进行准确地表征具有重要的科学意义。已有的研究通常忽略随机激励之间的非独立性，使得系统的响应跟实际情况有着很大差别，直接影响对结构安全性的评估。本课题拟研究非独立泊松白噪声激励下随机非线性系统的响应，考虑非独立的脉冲幅值和非独立的脉冲到达时刻两种情形，推导并构建不同非独立条件下的FPK方程，基于指数多项式闭合法发展求解系统响应概率密度函数的方法，阐明非独立泊松白噪声激励下随机非线性系统响应的统计特性；考虑结构内力和变形的综合影响，揭示非独立泊松白噪声激励下结构中振动能量的传递和分布规律。本研究旨在发展和完善非独立白噪声激励下随机非线性系统响应的理论和方法，为多种随机荷载共同作用下系统响应的研究提供指导，为关联随机激励下多自由度或者大系统随机响应问题的解决奠定基础，为实际工程中结构的安全可靠性评估提供依据，具有重要的理论意义和工程实用价值。

随机激励普遍存在于自然及工程等领域中，对其进行准确的表征具有重要的科学意义。已有的研究为了简化分析，通常将外部荷载模拟为典型的高斯白噪声过程，忽略了随机激励的其他重要特性，使得系统的响应与实际情况有着很大差别，直接影响对结构安全性的评估。.本课题拟基于指数多项式闭合法，分别考虑随机激励的非高斯特性、随机激励的有限带宽性、随机激励的非平稳特性这三种情形，推导并构建不同情形下的FPK方程，进一步发展求解系统响应概率密度函数的方法，分别阐述不同情形下随机非线性系统响应的统计特性。考虑随机激励的非高斯特性，研究了非零脉冲幅值对动力系统响应统计特性的影响，引入了Ito方程新的修正项来考虑泊松激励的非高斯特性；考虑了随机激励的有限带宽性和时间相关性，研究了彩噪声激励下随机非线性系统的响应。将彩噪声处理为额外的系统响应，使得原有的彩噪声激励下的一维系统转化为白噪声激励下的多维系统，考虑了过滤高斯白噪声激励、Ornstein–Uhlenbeck (OU) 彩噪声激励、联合白噪声和过滤高斯白噪声激励三种情形；考虑随机激励的非高斯性和有限带宽性，研究过滤泊松白噪声下随机非线性系统的响应，将过滤泊松白噪声处理为额外的系统响应，使得原有的过滤泊松白噪声激励下的一维系统转化为白噪声激励下的多维系统，考虑了线性过滤和非线性过滤两种情形；考虑随机激励的非平稳性，研究了非线性随机动力系统，包括迟滞非线性系统暂态响应的统计特性及分布规律，同时考虑了参数激励和外部激励间的相关性对系统瞬态响应的影响。.本研究进一步改进指数多项式闭合法，使其可以研究多维状态变量，同时可以考虑时间变量，研究系统的瞬态响应。基于该研究，为多种随机荷载共同作用下系统响应的研究提供指导，为随机激励下多自由度或者大系统随机响应问题的解决奠定基础，为实际工程中结构的安全可靠性评估提供依据，具有重要的理论意义和工程实用价值。