A complete, self-contained mathematical framework for modelling the coupled aeroelastic and flight dynamic behaviour of free-flying flexible aircraft subject to atmospheric gust encounters is presented. The framework integrates three physical disciplines: geometrically-exact nonlinear beam theory for structural dynamics, unsteady two-dimensional strip aerodynamics based on Theodorsen thin-aerofoil theory with indicial functions for shed-wake and gust-penetration effects, and quaternion-based rigid-body flight dynamics for singularity-free attitude propagation. The coupled system is assembled into a first-order state-space form amenable to time-domain simulation, model order reduction, and control design. Detailed derivations of all coupling terms, including coordinate transformations between aerodynamic and structural frames, the Jacobian block structure, and gust input matrices, are provided. Two gust models are treated: the certification-standard discrete gust and the Von Karman continuous turbulence spectrum. The framework is verified against published benchmarks, including high-altitude long-endurance aircraft configurations and a very flexible flying-wing, demonstrating close agreement in structural frequencies, flutter speed, and static aeroelastic deflections. This paper serves as a self-contained reference for researchers implementing coupled aeroelastic-flight dynamic analysis tools for very flexible aircraft.

翻译：本文提出了一个完整、自洽的数学框架，用于模拟自由飞行柔性飞机在大气阵风遭遇中的气动弹性与飞行动力学耦合行为。该框架整合了三个物理学科：几何精确非线性梁理论用于结构动力学、基于西奥多森薄翼型理论并采用指示函数处理尾迹和阵风穿透效应的非定常二维条带空气动力学，以及基于四元数的刚体飞行动力学以实现无奇点的姿态传播。耦合系统被组装成一阶状态空间形式，适用于时域仿真、模型降阶和控制设计。详细推导了所有耦合项，包括气动与结构坐标系之间的坐标变换、雅可比块结构以及阵风输入矩阵。文中处理了两种阵风模型：适航标准离散阵风和冯·卡门连续湍流谱。该框架已针对已发表的基准算例进行验证，包括高空长航时飞机配置和一种非常灵活的飞翼，在结构频率、颤振速度和静态气动弹性变形方面展现出良好的一致性。本文为研究人员实现针对超柔性飞机的耦合气动弹性-飞行动力学分析工具提供了完整的自洽参考。