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中文摘要：
杆截面形状和质量以某些特定函数（如多项式函数、指数函数及三角函数等）变化时，给出了其弹性撞击问题的闭合解。使用直接模态叠加法（DMSM）研究了不等截面杆及含附加弹簧-质量系统杆的弹性撞击问题。把撞击物和被撞击物作为一个整体系统振动，从而把撞击问题转化为振动问题。基于各段杆上纵向位移的闭合解，得到撞击系统的频率方程、广义质量，并由Duhamel积分得到动力响应。与常规有限元法比较，所提方法能够减少划分单元的总数和CPU运行时间。算例验证了本文方法的有效性。
英文摘要：
When a rod＇s shape and mass vary with certain functions such as polynomial, exponential and trig functions, the closed form solutions for the elastic impact problems are presented. （This article presents the closed form solutions for the elastic impact problems when a rod＇s shape and mass vary. with such functions as polynomial, exponential and trig functions. ） Using direct mode superposition method （DMSM）, the impact problems of non-uniform rods and structures with lumped masses and spring supports are discussed. （It also discusses the impact problems of non-uniform rods and structures with lumped masses and spring supports using direct mode superposition method （DMSM）. The impacting mass and the impacted object are considered as one integrated vibration system; thus the impact problem is transformed into a vibration problem. The longitudinal displacement of every step of the rod is supposed according to the closed-form solution of a free vibration problem, and then the frequency equation and generalized mass of the impacted system are obtained, and the dynamic response is also yielded in the form of Duhamel integration. Compared with the conventional finite element method, the proposed method can reduce the total number of elements and CPU running time. The numerical examples show the advantages of the present method.
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