中文摘要：

过载是火箭的真实环境．考虑纵向过载环境，根据Hamilton变分原理，得到了欧拉梁变系数横向自由振动微分方程，证明了该变系数系统存在满足刚度和质量正交条件的本征解，揭示了纵向过载环境下欧拉梁本征函数的非对称性．用本征向量叠加方法和改进欧拉中点直接积分算法求解了该变系数系统的动态响应，二者结果的一致性说明该理论推导的正确性和方法的可行性．

英文摘要：

Overload is the real environment of launch vehicles. The variable-coefficient transverse differential equation for the free vibration of Euler beam under longitudinal overload was derived on the basis of the Hamiltonian principle. For the variable-coefficient dynamic system, the existence of eigensolutions satisfying the orthogonal conditions of mass and stiffness was proved, and the asymmetry of such eigenfunction was re- vealed. A modified Euler symplectic midpoint difference method and a modified mode superposition method were employed to solve the equation of variable-mass systems. The numerical results of the two methods agree well, and validate the correctness of present theory and feasibility of the two methods.