非光滑随机系统动力学研究

Non-smooth factors, such as impact, friction and unilateral constraint, are ubiquitous in modern engineering structures. Non-smooth factors can dramatically change the dynamic mechanical behaviors, and even induce the structural insecurity. So far, the researches on non-smooth systems concentrate on the cases with deterministic excitations, and the studies on the cases with stochastic excitations are very few. According to the different mechanisms where the non-smooth factors entering into, the non-smooth systems are reclassified. The three subclasses are the conservative- and dissipative-mechanisms non-smooth system, the inertia- mechanism non-smooth system and the constraint-mechanism non-smooth system, respectively. Based on the above classification, this project investigates the stochastic dynamics of non-smooth systems and establishes the unified approaches for each subclass. The traditional procedures which are developed in smooth stochastic systems, such as stochastic averaging, equivalent linearization technique and equivalent nonlinear system method, are generalized to analyzing the non-smooth stochastic systems.The applicability and accuracy of the proposed analytical and semi-analytical procedures will be validated through numerical simulation for the original non-smooth systems and the corresponding experiments. Research results can guide the engineering application of the typical non-smooth systems, and in theoretical aspect, extend the research field of stochastic dynamics.

碰撞、摩擦、单边约束等非光滑因素普遍存在于现代工程结构中，这些因素可显著改变结构动态力学行为，并可能引发结构安全性问题，迄今对非光滑系统的研究主要集中于确定性激励情形，在随机激励方面的研究较少且不够系统，因此很有必要系统研究非光滑系统的随机动力学行为。本项目以工程中几类典型的非光滑系统为研究对象，依据非光滑因素出现于何种力学机制中，将非光滑系统分为保守与耗散机制非光滑类、惯性机制非光滑类和约束机制非光滑类，推广传统的光滑随机系统分析方法，诸如随机平均法、等效线性化方法、等效非线性系统法等使之适用于非光滑系统的分析，进而发展各类非光滑系统随机动力学行为的统一分析方法。这些解析与半解析方法的适用性和有效性，将通过对原始非光滑系统的数值模拟和相关实验手段验证。研究成果可用于指导各类典型非光滑系统的实际工程应用，并将拓展随机动力学理论的研究领域。

复杂工程结构包含着各类非光滑因素，它们显著影响结构动态力学行为，并可能危及结构运行安全，研究非光滑系统的动力学行为意义重大。本项目从物理机制视点，将典型非光滑因素严格划分为惯性机制非光滑、保守与耗散机制非光滑、单边约束机制非光滑等三大类，分别发展各类非光滑系统的随机响应、稳定性和可靠性的解析或半解析分析方法，并进一步拓展到以响应最小、稳定性最大和可靠性最大为目标的随机最优控制研究。此外，首次提出并发展了物理系统变分律识别的数据驱动方法，并成功应用于自动化识别非光滑系统的变分方程和边界条件，首次提出并发展了确定显含系统和激励参数的稳态响应概率密度显式表达式的数据驱动方法。在本项目资助期间，项目组共培养博士生15人，硕士生9人，发表SCI期刊论文95篇，圆满完成项目研究目标。