基于格林函数的参数不确定非线性颤振系统分析方法研究

Based on the Green function, this project aims to investigate new computational methods for high-dimensional nonlinear flutter systems with uncertain parameters by combining the harmonic balancing, homotopy analysis and stochastic collocation methods. Under the assumption of potential flow, the velocity potential function will be expanded as Fourier series with the coefficients as functions in space coordinates. The potential equations governing the coefficient functions can be deduced using harmonic balancing or homotopy analysis techniques. These equations are further solved by applying the Green function, harmonic balancing and homotopy analysis. By combining harmonic balancing with incremental process and minimum optimal techniques, a new method will be proposed for the nonlinear aeroelastic systems with multiple uncertain parameters. This method can be used to search solutions in the considered parametric region. Based on an iteration scheme, the homotopy analysis method will be extended, so that it is capable of analyzing nonsmooth dynamical systems. And, it will be further improved to analyze high-dimensional uncertain flutter systems with both smooth and nonsmooth nonlinearities. The nonsmooth nonlinearities result from the clearances of the linkages between the airfoil and control surfaces. Wind tunnel tests will be performed. The validity and feasibility of applying the classical Theodorsen's unsteady aerodynamics theory to nonlinear flutter will be investigated by using the above mentioned methods and the wind tunnel test data. The results obtained by the above approaches will be employed to evaluate the statistics of vibration responses via stochastic collocation methods. These methods could be widely applicable to the nonlinear aeroelastic analysis and design of aircraft, large suspension bridges, high-speed vehicles, and etc.

本项目拟基于格林函数，研究含不确定参数的非线性颤振系统分析的新方法。在势流假设下，将流体速度势和结构位移展开为时间Fourier级数，系数分别为待定空间函数和参数。应用谐波平衡和同伦分析分别推导待定系数控制方程组，它由位势方程和代数方程组成。基于格林函数法求解该方程组：结合谐波平衡、增量过程和最小值优化，提出一种新的求解方法，其突出优点是能在整个不确定参数空间内搜索极限环响应范围；在同伦分析中引入迭代过程，使之能有效求解非光滑的高维不确定性颤振系统。采用随机配置法求得极限环振动及其分岔等响应的相关统计量，分析不确定参数对颤振特性的影响规律。结合风洞试验和直接数值模拟验证所提方法，并分析Theodorsen非定常气动力理论应用于非线性颤振分析的有效性和计算精度。本项研究不仅具有重要的理论意义，而且在飞行器、大型桥梁、高速车辆等结构系统的非线性气动弹性力学设计中具有广阔的工程应用前景。

本项目研究了基于格林函数的非线性振动分析新方法，用于分析了含不确定参数的非线性颤振系统和分数阶非线性系统。结合谐波平衡、增量过程和最小值优化提出的求解方法，其突出优点是能在整个不确定参数空间内搜索极限环响应范围。用所提方法分析带外挂质量和操作面的机翼非线性颤振系统，研究了极限环的鞍结分岔、周期倍化和对称性破缺等现象。求解了含分段线性和迟滞非线性等非光滑因素的高维系统；探讨了操纵面间隙或迟滞非线性的机翼颤振系统的极限环分岔等动力学现象，研究了不确定参数对系统响应的影响。在所提方法的基础上引入多个不可约时间尺度，用于描述不可约频率，推导了半解析求解格式，获得了高精度的准周期稳态响应，揭示了准周期分岔出现的机理。预测了分段非线性颤振系统的亚临界分岔点，解释了亚临界分岔出现的原因。在本项目的资助下，项目组发表学术论文41篇，其中SCI收录27篇，包括力学、振动和气动弹性力学领域的重要期刊ASME JAM一篇、AIAA J三篇、NLM三篇、JSV一篇以及ND两篇。