弹性碰撞问题的模态叠加法及哈密顿体系下的计算研究
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摘要
结构撞击问题有着广泛的工程背景和重要的理论与学术价值。基于直接模态叠加法(BMSM)，文中针对几种不同的杆或梁结构的弹性撞击问题和哈密顿体系下的数值计算开展了研究，主要工作集中在以下几个方面：
1 给出细长圆锥形的截面杆受到质点纵向弹性碰撞时的解析解。使用DMSM方法用于分析质点—圆锥形杆碰撞，即由叠加法给出杆端不含有及含有弹簧时的响应。其结果可验证数值解和其他解析解。算例显示，所提出方法的优点之一是响应解的解析形式简洁。算例表明一些描述杆几何形状的变量在撞击分析中具有重要作用。
2 研究了非均匀截面杆或多段杆结构碰撞问题，把杆的质量函数和刚度函数作为两个独立的函数，进行适当的函数变换后，基本方程转化为可解的常微分方程。得到满足正交条件的基本解，并且使用DMSM方法建立了非均匀截面单杆或多段杆结构碰撞时的频率方程和撞击响应。而对于受载杆的撞击问题，即一端固支，一端自由杆与一个弹簧—质量系统耦合，使用DMSM方法，考虑系统的一些参数对系统频率的影响，并且给出此杆结构受撞击后的动态响应。
3 研究了质点撞击到Euler-Bernoulli梁上任意位置的问题。对撞击点把梁分成的两部分分别假设位移函数，利用DMSM得到响应的解析解。对于复合材料梁端部受撞击甸题，把质量块看成质点，基于弯扭耦合梁模型，使用模态叠加法给出动力响应与撞击力的结果。算例表明此方法是有效的。另外，对于小球撞击Euler-Bernoulli梁问题，引入撞击力—时间模型，得到如下两种预报撞击力的方法。与已有方法相比，方法一简化了计算过程，得到近似解，且此法可以推广到四边简支板中去。方法二能够较好地描述撞击力和撞击响应，同时可分析各种因素对接触撞击力的影响。
4 提出一种确定恢复系数的方法：即首先使用DMSM方法得到撞击结束时间，再得到恢复系数的步骤。算例表明，本文方法能够从理论上得到弹性碰撞恢复系数的表达式，且结果是有效的。
5 研究了哈密顿体系下的薄板自由振动问题。推导出相应的对偶方程组，对于不同边界条件，把x方向模拟为时间，得到振动频率的辛求解方法。而对两对边简支的中厚板静力弯曲问题，在全状态下建立了Mindlin板的哈密顿正则方程，进而采用直接法给出了两对边简支中厚板静力弯曲的解析解，算例结果验证方法的有效性。
6．对于杆结构和梁结构的弹性碰撞问题，分别使用有限元离散后，使用精细
The impact problems of structures have wide engineering background and are of important theoretical significance and research worthiness. This dissertation mainly deals with .the dynamic impact problems of several structures based on Direct Mode Shape Superposition Method (DMSM), and some related problems in Hamiltonian system are also discussed. The main concerned contents of the results may be summarized as follows:
1. The paper presents exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle. A new method is proposed for analysis of impact between particle and conical rod with or without supports of spring, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical examples show that one of the advantages of the present method is that the analytical form is much simple. It is also found that some variables describing the geometrical shape of rods play an important role in impact dynamic system.
2. Impact problem of non-uniform rod or multi-step rod structure are systematically analyzed. The mass and the stiffness are considered as two independent functions. For several formations of the function, the fundamental equation is transformed into constant differential equation after proper transformation, and the basic solutions are obtained. Then the frequency equation and impact forced are gained by using DMSM. Futermore, for the impact problem of a structure of a loaded rod coupled with a spring-mass system, the response of the system is revealed. Some factors have effect on values of the system's frequency and the effects are shown, then the responses of the rod structure are given at different moments.
3. The problem of particle colliding at an aribitray location on the beam is analyzed. Suppose that the displacement function of the two parts separated by the impact point, and the analytical solution is obtained by using DMSM. The impact problem of a mass colliding with a composite beam is also considered. Based on the bending-torsion coupled dynamic model of Timoshenko beam, the results of dynamic response and impact force are presented by using DMSM for the problem of a moving rigid-body impacting at a cantilever's end. The rigid-body and the beam are regarded
1 给出细长圆锥形的截面杆受到质点纵向弹性碰撞时的解析解。使用DMSM方法用于分析质点—圆锥形杆碰撞，即由叠加法给出杆端不含有及含有弹簧时的响应。其结果可验证数值解和其他解析解。算例显示，所提出方法的优点之一是响应解的解析形式简洁。算例表明一些描述杆几何形状的变量在撞击分析中具有重要作用。
2 研究了非均匀截面杆或多段杆结构碰撞问题，把杆的质量函数和刚度函数作为两个独立的函数，进行适当的函数变换后，基本方程转化为可解的常微分方程。得到满足正交条件的基本解，并且使用DMSM方法建立了非均匀截面单杆或多段杆结构碰撞时的频率方程和撞击响应。而对于受载杆的撞击问题，即一端固支，一端自由杆与一个弹簧—质量系统耦合，使用DMSM方法，考虑系统的一些参数对系统频率的影响，并且给出此杆结构受撞击后的动态响应。
3 研究了质点撞击到Euler-Bernoulli梁上任意位置的问题。对撞击点把梁分成的两部分分别假设位移函数，利用DMSM得到响应的解析解。对于复合材料梁端部受撞击甸题，把质量块看成质点，基于弯扭耦合梁模型，使用模态叠加法给出动力响应与撞击力的结果。算例表明此方法是有效的。另外，对于小球撞击Euler-Bernoulli梁问题，引入撞击力—时间模型，得到如下两种预报撞击力的方法。与已有方法相比，方法一简化了计算过程，得到近似解，且此法可以推广到四边简支板中去。方法二能够较好地描述撞击力和撞击响应，同时可分析各种因素对接触撞击力的影响。
4 提出一种确定恢复系数的方法：即首先使用DMSM方法得到撞击结束时间，再得到恢复系数的步骤。算例表明，本文方法能够从理论上得到弹性碰撞恢复系数的表达式，且结果是有效的。
5 研究了哈密顿体系下的薄板自由振动问题。推导出相应的对偶方程组，对于不同边界条件，把x方向模拟为时间，得到振动频率的辛求解方法。而对两对边简支的中厚板静力弯曲问题，在全状态下建立了Mindlin板的哈密顿正则方程，进而采用直接法给出了两对边简支中厚板静力弯曲的解析解，算例结果验证方法的有效性。
6．对于杆结构和梁结构的弹性碰撞问题，分别使用有限元离散后，使用精细
The impact problems of structures have wide engineering background and are of important theoretical significance and research worthiness. This dissertation mainly deals with .the dynamic impact problems of several structures based on Direct Mode Shape Superposition Method (DMSM), and some related problems in Hamiltonian system are also discussed. The main concerned contents of the results may be summarized as follows:
1. The paper presents exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle. A new method is proposed for analysis of impact between particle and conical rod with or without supports of spring, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical examples show that one of the advantages of the present method is that the analytical form is much simple. It is also found that some variables describing the geometrical shape of rods play an important role in impact dynamic system.
2. Impact problem of non-uniform rod or multi-step rod structure are systematically analyzed. The mass and the stiffness are considered as two independent functions. For several formations of the function, the fundamental equation is transformed into constant differential equation after proper transformation, and the basic solutions are obtained. Then the frequency equation and impact forced are gained by using DMSM. Futermore, for the impact problem of a structure of a loaded rod coupled with a spring-mass system, the response of the system is revealed. Some factors have effect on values of the system's frequency and the effects are shown, then the responses of the rod structure are given at different moments.
3. The problem of particle colliding at an aribitray location on the beam is analyzed. Suppose that the displacement function of the two parts separated by the impact point, and the analytical solution is obtained by using DMSM. The impact problem of a mass colliding with a composite beam is also considered. Based on the bending-torsion coupled dynamic model of Timoshenko beam, the results of dynamic response and impact force are presented by using DMSM for the problem of a moving rigid-body impacting at a cantilever's end. The rigid-body and the beam are regarded
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