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：本文提出了一种完整且自洽的数学框架，用于模拟遭受大气突风干扰的自由飞行柔性飞机的耦合气动弹性与飞行动力学行为。该框架整合了三个物理学科：基于几何精确非线性梁理论的结构动力学、基于西奥多森薄翼型理论并引入指示函数以表征尾迹泄出与突风穿透效应的非定常二维条带气动力学，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。耦合系统被组装为适用于时域仿真、模型降阶与控制设计的一阶状态空间形式。详细推导了所有耦合项，包括气动坐标系与结构坐标系之间的坐标变换、雅可比块结构以及突风输入矩阵。本文处理了两种突风模型：认证标准离散突风与冯·卡门连续湍流谱。该框架通过与已发表的基准数据验证，包括高空长航时飞机构型与极柔性飞翼，在结构频率、颤振速度与静态气动弹性变形方面表现出高度一致性。本文为研究人员实现极柔性飞机耦合气动弹性-飞行动力学分析工具提供了自洽的参考。