基于时滞惯性流形的二维平面壁板非线性气动弹性分析

详细研究了二维平面壁板的非线性气动弹性现象。采用平板的Von Karman几何大变形理论以及气动力的一阶活塞理论，推导出系统的非线性偏微分控制方程。然后，运用基于时滞惯性流形的非线性Galerkin方法求解方程，将高阶屈曲模态用低阶模态来表示并引入时间滞后，这样既保留计算精度又大幅度地节约计算时间。最后，详细数值分析了其中的气动弹性行为，分别以无量纲动压和无量纲压缩内力为分岔参数，无量纲幅值为响应给出了分岔图，发现系统存在阵发性通往混沌的途径，以及混沌区域周期窗口和自相似的特征。进一步，通过对系统的相图、位移的FFT频谱以及Lyapunov指数的分析，发现系统的动力学行为存在稳定、屈曲、谐调和非谐调运动四种典型类型，而非谐调运动又表现出倍周期运动、准周期运动和混沌运动等丰富的非线性响应，所研究结果为识别和进一步控制此类非线性气动弹性现象提供了理论依据。

Nonlinear aeroelastic analysis of a two-dimensional panel based on inertial manifolds with delay

Von Karman’s large deformation plate theory was used to describe the panel deformation,and the aerodynamic loads were obtained by using the first order piston theory.The nonlinear partial differential equation of the system was derived.Then,the nonlinear Galerkin method based upon inertial manifolds with delay(IMD) was applied to the approaching of the governing equations.By this method,the higher-order modes were expressed by the lower-order modes and a time delay was introduced.Thus the same precision is kept and long computation time is saved.The numerical examples were given,and the dimensionless dynamic pressure and the dimensionless compressive internal force were considered as bifurcation parameters,respectively,to study the stability and bifurcation of the response.In particular,the route to chaos by intermittent transition was studied,and the periodic windows and self-similarity phenomena were captured in the chaos region.Through the phase portraits,FFT analysis and Lyapunov exponent,it demonstrates that there exist four distinct regions,representing stable,buckling,synchronous and non-synchronous motions of the system.In the non-synchronous region,a rich variety of nonlinear responses,such as double-period motion,quasi-period motion and chaotic motion,are found.The results can gain a fundamental understanding and developing of the nonlinear phenomena.