几类非线性随机动力学系统的近似瞬态响应

The stationary response of the nonlinear stochastic dynamical system has been studied systematically and profoundly. However, for a considerable number of stochastic dynamical systems relating to national security, such as launch vehicle and so on, the transient response is more of interest. It is an important basis of design, safety assessment and control of the corresponding system. Thus, the prediction of transient response for nonlinear stochastic dynamical system has important significances and engineering application values. In the present project, approximate transient solutions of non-resonant/resonant single/multi-degree-of- freedom nonlinear stochastic systems are studied, where some significant factors, including external resonance, time delay, viscoelasticity, hystereticity and so on, are considered. The main procedures are as follows. Firstly, by using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker-Planck-Kolmogorov equation can be derived for the non-resonant and resonant cases. Then, the solution of the equation is represented by an exponential compound function. For non-resonant case, the internal function is approximately expressed as a series expansion in terms of a set of multivariable polynomials with time-dependent coefficients, while, for resonant case, it is expressed as a series expansion in terms of two different sets of basic functions. Finally, according to the Galerkin method, the problem can be solved from a set of linear first-order ordinary differential equations and the transient respons of the system can be obtained.

非线性随机动力学系统的平稳响应已经得到了深入系统的研究，然而相当部分涉及国家安全的随机动力学系统，例如运载火箭等，更加关注其瞬态响应的分析。瞬态响应是这些系统设计、安全评估及性能控制的重要依据，因此研究非线性随机动力学系统的瞬态响应具有重要的理论意义及工程应用价值。本项目以非共振/共振单/多自由度非线性随机系统为研究对象，研究随机系统的近似瞬态解，并考虑了外共振、时滞、粘弹性、滞迟等重要因素的影响。基本研究方法是：推广基于广义谐和函数的随机平均法，对非共振与共振情形分别得到相应的平均Fokker-Planck-Kolmogorov方程；将上述方程的解表示为指数形式的复合函数，对非共振情形，将内函数近似表示为变系数的多元正交基函数的级数和，对共振情形，则采用变系数的两组不同的基函数的级数和；应用Galerkin法将上述问题转化为一阶线性常微分方程组问题，进而得到系统的近似瞬态响应。

非线性随机动力学系统的平稳响应已得到深入研究，然而在很多情况下，人们更关注系统的瞬态响应行为。例如，对运载火箭而言，工程师必须了解从发射到达到预定轨道的过程中，箭体对噪声的瞬态响应行为，瞬态响应是这类涉及国计民生的系统设计、安全及性能评估的重要依据。因此，研究随机动力学系统的瞬态响应行为具有重要的理论意义及工程应用价值。本项目以多自由度非线性随机系统为研究对象，建立系统瞬态响应的近似分析方法。应用基于广义谐和函数的随机平均法降低系统维数，导出相应的平均Fokker-Planck- Kolmogorov方程，并发展两类技术研究之。其一，将瞬态响应概率密度近似表示为多重正交基函数的级数和，其时间依赖系数由Galerkin技术给出，并推广应用于滞迟、粘弹性、时滞系统及非高斯白噪声激励情形；其二，将瞬态解近似表示为复分数矩的展开式，基于Mellin变换得到系统近似瞬态响应。在瞬态响应分析的基础之上，初步研究了最优瞬态控制及可靠性控制问题，建立了最优瞬态及可靠性控制策略。数值结果表明瞬态响应分析技术及控制策略的有效性。此外，初步探讨了变质量系统随机响应及随机最优控制和非线性随机振动能量收集技术，拓展了瞬态响应方面的后继研究方向。