机械系统刚—柔—液耦合多体动力学递推建模研究
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摘要
随着现代科技的不断发展，在航空航天、车辆、机器人等领域均出现了大量刚-柔-液耦合的多体系统。针对柔性多体系统和充液多体系统这两类多体动力学问题，本文在空间算子代数递推理论的基础上，推导出了刚柔耦合多体系统空间算子代数理论以及计及液体小幅晃动的空间算子代数理论的基本方程，并提出了基于区间算法的多体系统动力学求解方法，编制了相应的计算程序，主要包括以下五个方面内容：
（1）、针对目前对多体系统动力学发展中提出的“提高计算速度和精度”的要求，总结了当前国际上三种重要的具有O（n）次计算效率的动力学方法：空间算子代数方法，李群李代数方法和哈密顿递推形式的动力学建模方法。在此基础上采用基于完全笛卡尔坐标的多刚体系统动力学方法，推导出了杆件机构O（n）次的正向动力学模型。
（2）、探讨了柔性多体系统的空间算子代数建模问题。首先对柔性多体系统进行了刚体近似处理，采用有限段方法对该柔体进行了近似刚体的等效力学模型建模，并通过空间算子代数对该系统进行了动力学分析。其次，对刚柔耦合树形拓扑结构的多体系统进行了建模研究，并通过编制了该多体系统动力学计算程序。最后，通过具体算例与传统拉格朗日建模方法进行了分析比较，验证了空间算子代数方法是一种高精度的建模方法。
（3）、采用空间算子代数理论对小幅液体晃动的动力学递推算法进行了研究。利用有限元模态分析技术得到了液体晃动的模态特征，通过动量定理和动量矩定理得到液体贮箱间晃动作用力的表达，在此基础上采用空间算子代数推导出了该小幅晃动充液多体系统的递推动力学模型，并编制了小幅晃动充液多体系统的动力学计算程序。
（4）、采用区间算法对多体系统动力学模型进行了求解研究。探讨了基于Newmark法和区间算法联合求解多体系统动力学微分方程问题，并对含有欠驱动关节系统的空间算子代数动力学模型进行求解分析。通过具体算例验证了该算法的有效性和正确性。
（5）、利用解耦的自然正交补矩阵得到递推形式的闭环多体系统的李群李代数表达。采用分度、分段圆法设计了共轭凸轮轮廓，并以凸轮-滚子（自由滚动）凸轮减速机为对象，采用基于递推形式的虚拟样机技术对其动力学特性进行了研究。
With the development of technology, there are large numbers of rigid-flexible-liqued couplingmultibody system in the field of aerospace, vehicles and robot. To solve these kinds of problems, thespatial operator algebra theory of rigid-flexible coulping and considering small amplitude liquidsloshing multibody system are deduced as well as the solving method of multibody system dynamicsbased on Interval Arithmetic is obtained. The main content of this thesis include five aspects:
(1)To meet the demand of improving calculate velocity and precision during the developmentprocess of multibody system dynamics, the spatial operator algebra theory, Lie Groups and Liealgebraic method and the recursive Hamilton dynamics equations which are most important of O(n)times dynamics computational efficiency are summarized. Meanwhile, the dynamics of rod linkagesis deduced by multibody system dynamics method described by fully cartesian coordinate.
(2)The dynamics of flexible multibody system is studied. Firstly, the flexible object is treated asapproximate rigid object, the Mathematical modeling for this approximate rigid object is founded bythe finite segment method and the flexible multibody system is dynamical analyzed by spatialoperator algebra theory. Next, the rigid-flexible coupling multibody system dynamics is modeled,meanwhile, the numerical code for this system which is based on Mathematical software is obtained.Finally, the calculational procedure is compared between the spatial operator algebra dynamicsmodeling and tradition Lagrange modeling method by a calculating case.
(3)The dynamic characteristic of liquid-filled tank is researched by spatial operator algebratheory. The force expression for the tank and liquid is obtained by the momentum theorem and themomentum moment theorem. The mode of liquid sloshing is achieved by Finite element modalanalysis and the mode results are utilized for the recursive dynamics modeling of small amplitudeliquid sloshing multibody system which is accomplished by spatial operator algebra theory. Thedynamic calculational procedure for small amplitude liquid sloshing multibody system is achieved byMathematical software.
(4)The study on the problem of multibody system dynamics is deduced by Interval Arithmetic.Based on the Newmark method and Interval Arithmetic to solve the multibody system dynamicsdifferential equations and the spatial operator algebra theory modeling with underactuated jointsystem. The corresponding numerical computation program is compiled. The specific example is usedto demonstrate the effectiveness and correctness for the new algorithm.
(5) Based on natural orthogonal complement and decoupling of natural orthogonal complement,the recursive dynamics for the closed loop systems are discussed. A new plane cam reducer such assimple parallel cam reducer and planetary cam reducer are designed and the cam curve are designedby means of divided degree and divided segment method for the conjugate cams. The dynamiccharacteristic of these two reducers is studied by Virtual Prototyping Technology based on recuisivedynamics.
（1）、针对目前对多体系统动力学发展中提出的“提高计算速度和精度”的要求，总结了当前国际上三种重要的具有O（n）次计算效率的动力学方法：空间算子代数方法，李群李代数方法和哈密顿递推形式的动力学建模方法。在此基础上采用基于完全笛卡尔坐标的多刚体系统动力学方法，推导出了杆件机构O（n）次的正向动力学模型。
（2）、探讨了柔性多体系统的空间算子代数建模问题。首先对柔性多体系统进行了刚体近似处理，采用有限段方法对该柔体进行了近似刚体的等效力学模型建模，并通过空间算子代数对该系统进行了动力学分析。其次，对刚柔耦合树形拓扑结构的多体系统进行了建模研究，并通过编制了该多体系统动力学计算程序。最后，通过具体算例与传统拉格朗日建模方法进行了分析比较，验证了空间算子代数方法是一种高精度的建模方法。
（3）、采用空间算子代数理论对小幅液体晃动的动力学递推算法进行了研究。利用有限元模态分析技术得到了液体晃动的模态特征，通过动量定理和动量矩定理得到液体贮箱间晃动作用力的表达，在此基础上采用空间算子代数推导出了该小幅晃动充液多体系统的递推动力学模型，并编制了小幅晃动充液多体系统的动力学计算程序。
（4）、采用区间算法对多体系统动力学模型进行了求解研究。探讨了基于Newmark法和区间算法联合求解多体系统动力学微分方程问题，并对含有欠驱动关节系统的空间算子代数动力学模型进行求解分析。通过具体算例验证了该算法的有效性和正确性。
（5）、利用解耦的自然正交补矩阵得到递推形式的闭环多体系统的李群李代数表达。采用分度、分段圆法设计了共轭凸轮轮廓，并以凸轮-滚子（自由滚动）凸轮减速机为对象，采用基于递推形式的虚拟样机技术对其动力学特性进行了研究。
With the development of technology, there are large numbers of rigid-flexible-liqued couplingmultibody system in the field of aerospace, vehicles and robot. To solve these kinds of problems, thespatial operator algebra theory of rigid-flexible coulping and considering small amplitude liquidsloshing multibody system are deduced as well as the solving method of multibody system dynamicsbased on Interval Arithmetic is obtained. The main content of this thesis include five aspects:
(1)To meet the demand of improving calculate velocity and precision during the developmentprocess of multibody system dynamics, the spatial operator algebra theory, Lie Groups and Liealgebraic method and the recursive Hamilton dynamics equations which are most important of O(n)times dynamics computational efficiency are summarized. Meanwhile, the dynamics of rod linkagesis deduced by multibody system dynamics method described by fully cartesian coordinate.
(2)The dynamics of flexible multibody system is studied. Firstly, the flexible object is treated asapproximate rigid object, the Mathematical modeling for this approximate rigid object is founded bythe finite segment method and the flexible multibody system is dynamical analyzed by spatialoperator algebra theory. Next, the rigid-flexible coupling multibody system dynamics is modeled,meanwhile, the numerical code for this system which is based on Mathematical software is obtained.Finally, the calculational procedure is compared between the spatial operator algebra dynamicsmodeling and tradition Lagrange modeling method by a calculating case.
(3)The dynamic characteristic of liquid-filled tank is researched by spatial operator algebratheory. The force expression for the tank and liquid is obtained by the momentum theorem and themomentum moment theorem. The mode of liquid sloshing is achieved by Finite element modalanalysis and the mode results are utilized for the recursive dynamics modeling of small amplitudeliquid sloshing multibody system which is accomplished by spatial operator algebra theory. Thedynamic calculational procedure for small amplitude liquid sloshing multibody system is achieved byMathematical software.
(4)The study on the problem of multibody system dynamics is deduced by Interval Arithmetic.Based on the Newmark method and Interval Arithmetic to solve the multibody system dynamicsdifferential equations and the spatial operator algebra theory modeling with underactuated jointsystem. The corresponding numerical computation program is compiled. The specific example is usedto demonstrate the effectiveness and correctness for the new algorithm.
(5) Based on natural orthogonal complement and decoupling of natural orthogonal complement,the recursive dynamics for the closed loop systems are discussed. A new plane cam reducer such assimple parallel cam reducer and planetary cam reducer are designed and the cam curve are designedby means of divided degree and divided segment method for the conjugate cams. The dynamiccharacteristic of these two reducers is studied by Virtual Prototyping Technology based on recuisivedynamics.
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