主惯量表示及其在刚体系统仿真中的应用

发布时间：2018-05-27 23:25

本文选题：约束系统 + 质量矩阵　；参考：《大连理工大学》2017年博士论文

【摘要】：约束多体系统的运动学和动力学是CAD(计算机辅助设计)和CAE(计算机辅助工程)的重要组成部分。由于约束条件的存在,约束系统的动力学方程呈现出多种表达形式。这些不同的数学描述在理论和数值上都体现出不同的特征。过往几十年,已经有大量的研究工作致力于约束系统的各种不同的理论表述以及相关的数值仿真研究。尽管如此,在目前已有的数值仿真方面的著作中,还没有不同表述对精度的影响这方面较为透彻的研究工作。本文对约束多体动力学方程惯量(质量)矩阵的各种表达式以及数值仿真中不同惯量表示对于精度的影响进行了详细的研究,主要的研究内容可以概括为如下三个部分:(1)以Schur分解为基础,对约束多体系统质量矩阵的增广表达进行了详细分析,然后通过定义广义角速度,提出了质量矩阵的一种特殊表述形式,被称为主惯量表示。在标准形式中,主惯量表示将质量矩阵分为两部分:一个单位矩阵和一个位移相关矩阵,两矩阵相互之间的比例由伸缩参数σ控制,其中伸缩参数σ可以为任意常数。通过建立质量矩阵的增广表达式与主惯量表示在数学上的等价性,确保了主惯量表示对于一般约束动力学系统的普遍存在性。(2)在主惯量表示的框架下,详细推导了约束动力学系统的离散动能误差估计。误差估计表明:在Lagrange框架下,动能的离散误差是伸缩参数的线性函数;在Hamilton框架下,动能的离散误差是伸缩参数倒数的线性函数。并且当质量矩阵为广义位移的函数时,误差函数的斜率与离散误差同阶。与之相反,动能的离散误差与质量的增广表达式的具体形式无关。因此,主惯量表示与质量的增广表达式在数值上是不等价的。根据误差分析结果,本文进一步建议惯量主值的算术平均以及调和平均可以作为伸缩参数σ的合理的预条件值,以得到较小的离散误差。(3)以主惯量表示为基础,针对约束系统提出了一种通过确定伸缩参数的(最佳)预条件值来改进数值积分精度的新方法,并且将其应用于二维和三维刚体的数值仿真。数值结果验证了误差分析结论的正确性,在主惯量表示下数值积分的精度得到了大幅度的改善。在基于对流基矢量或单位四元数的三维刚体旋转的算例中,采用惯量主值的算术平均作为伸缩参数预条件值的数值积分,其精度提高了一个量级以上。根据上述研究结果,主惯量表示作为约束动力系统质量矩阵的一种具体的表示形式,为改善数值积分的精度提供了一种新方式,其相关的理论和数值研究亟待进一步展开。
[Abstract]:Kinematics and dynamics of constrained multibody systems are important components of CAD and CAE. Because of the existence of constraint conditions, the dynamic equations of constrained systems take on many forms of expression. These different mathematical descriptions show different characteristics both theoretically and numerically. In the past few decades, a great deal of research work has been devoted to the various theoretical representations of constrained systems and related numerical simulation. However, in the existing works on numerical simulation, there is no more thorough research on the effect of different expressions on accuracy. In this paper, the effects of various expressions of inertia (mass) matrix of constrained multi-body dynamics equations and different inertia representations in numerical simulation on the accuracy are studied in detail. The main research contents can be summarized as follows: 1) based on the Schur decomposition, the augmented expression of the mass matrix of constrained multibody systems is analyzed in detail, and then the generalized angular velocity is defined. A special representation of mass matrix is presented, which is called principal inertia representation. In the standard form, the principal inertia representation divides the mass matrix into two parts: a unit matrix and a displacement correlation matrix. The proportion between the two matrices is controlled by the scaling parameter 蟽, where the scaling parameter 蟽 can be an arbitrary constant. By establishing the mathematical equivalence between the augmented expression of the mass matrix and the representation of the principal inertia, the universal existence of the representation of the principal inertia for the general constrained dynamical system is ensured under the framework of the representation of the principal inertia. The estimation of discrete kinetic energy error for constrained dynamical systems is derived in detail. The error estimates show that the discrete error of kinetic energy is a linear function of the scaling parameter under Lagrange framework, and the discrete error of kinetic energy is a linear function of the inverse of the scaling parameter in the Hamilton framework. When the mass matrix is a function of generalized displacement, the slope of the error function is the same as the discrete error. On the contrary, the discrete error of kinetic energy is independent of the specific form of the mass augmentation expression. Therefore, the expression of principal inertia and the augmented expression of mass are not numerically equivalent. According to the results of error analysis, it is further suggested that the arithmetic average and harmonic average of the principal value of inertia can be regarded as the reasonable preconditioned value of the extensional parameter 蟽, so as to obtain a smaller discrete error, which is based on the representation of the principal inertia. A new method to improve the numerical integration accuracy by determining the (optimal) preconditioned values of the stretching parameters for constrained systems is proposed and applied to the numerical simulation of 2D and 3D rigid bodies. Numerical results verify the correctness of the error analysis, and the accuracy of numerical integration is greatly improved under the representation of principal inertia. In the example of 3D rigid body rotation based on convection basis vector or unit quaternion, the arithmetic average of the principal value of inertia is used as the numerical integral of the preconditioned value of the telescopic parameter, and its precision is improved by more than one order of magnitude. Based on the above results, as a concrete representation of the mass matrix of constrained dynamic systems, the principal inertia representation provides a new way to improve the accuracy of numerical integration. The related theory and numerical research need to be further developed.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O313.7

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