弹性细杆非线性动力学的有限元方法及应用

发布时间：2018-07-04 19:05

本文选题：纤维 + 弹性细杆　；参考：《东华大学》2017年硕士论文

【摘要】：气流辅助纺纱是指采用气流直接加捻或采用气流辅助加捻的工艺过程方案。在气流辅助纺纱加捻过程中,气流对纤维束的运动,即分散和加捻,起到决定性的作用。所以研究纤维在气流场中的运动规律和行为表现的理论分析和数值模拟方法是研究气流辅助纺纱及其改进工艺的必由之路。本文研究的是纺织纺纱过程中的纤维在流场中的运动,为纤维建立合适有效的物理模型和数学模型,数值模拟纤维在一般受力状态下几何大变形的静力学和动力学问题。因为纤维具有超大的长径比,纤维整体即使发生大位移几何变形,纤维材料的应变并不一定很大,本文假设其材料仍处于小应变状态,将纤维看作弹性细杆进行研究。对于本文中探讨的弹性细杆的非线性大变形相关的静力学与动力学问题,采用有限元方法解决。杆单元有限元方法是结构力学中的概念,用于求解杆件在静力作用下作小变形的线性问题的力或位置的求解。尚未在文献中查阅到应用此方法求解非线性与大变形问题。本文研究得出的有限元求解方法可用于弹性细杆大变形非线性静力学和动力学问题的数值计算,因此,在某种程度上,本文所作的工作是创新。本文所探讨的内容可以分为两部分:弹性细杆大变形的非线性问题的静力学分析与弹性细杆大变形的非线性问题的动力学分析。静力学部分主要探讨了以下内容:1.杆单元弹性细杆整体大变形考虑其轴力对于弯曲的影响。求得了基于艾尔米特插值和抛物线插值,考虑杆件单元拉压变形、扭转变形和弯曲变形的杆单元刚度矩阵。2.在每个单元的每一加载步的空间位置设立随体坐标,即局部坐标,以确定弹性细杆在力作用下的杆单元的姿态。随体坐标相对于整体坐标的位置采用欧拉四元数来表示,并导出了角位移的微小欧拉四元数增量与笛卡尔坐标系下的角位移增量之间的坐标变换关系。3.关于载荷分步加载的计算方法,对于对应于每一载荷增量步的系统总体静力学平衡分析的总体位移增量方程,提出了一种方程求解的迭代加速方法。动力学部分主要探讨了以下内容:1.分别探讨弹性细杆的质量与弹性。弹性细杆的弹性的探讨参照静力学部分。对于弹性细杆的质量,杆单元的质量简化为节点处的微段刚体质量单元。为保证模型的精确度,杆单元划分足够小。2.刚体质量单元的一般运动由转动与平动构成。转动参考欧拉动力学方程,平动运用质心运动定理。二者组合推导出刚体质量单元一般运动的在世界坐标系下的运动方程。3.本文导出了弹性细杆有限元总体动力学分析的运动微分方程,并介绍了该运动微分方程的显式差分格式的有限元总体运动微分方程数值解法和隐式差分格式的有限元总体运动微分方程数值解法。4.对于弹性细杆的接触问题,将其按刚度足够大的弹性接触约束处理。本文还对若干算例,采用MATLAB软件编程进行数值模拟计算,验证了本文方法的可行性和准确性。
[Abstract]:Air flow auxiliary spinning is the process plan of using air direct twisting or airflow auxiliary twisting. During the twisting process of airflow auxiliary spinning, the air flow plays a decisive role in the movement of the fiber bundle, that is, dispersion and twisting, so the theoretical analysis and numerical simulation of the motion law and behavior of the fiber in the airflow field are studied. The method is the only way to study the airflow auxiliary spinning and its improvement process. This paper studies the movement of fiber in the flow field in the textile spinning process, establishes a suitable and effective physical model and mathematical model for the fiber, and simulates the static and dynamic problems of the large geometric shape of the fiber in the general state of force. With a large length to diameter ratio, the strain of the fiber material is not necessarily very large even if the fiber has large displacement. This paper assumes that the material is still in a small strain state, and the fiber is considered as a thin elastic rod. The finite element method is the concept of structural mechanics, which is used to solve the force or position of the linear problem with small deformation under static action. This method has not been found in the literature to solve the problem of nonlinear and large deformation. The finite element method in this paper can be applied to the elastic thin rod. The numerical calculation of the nonlinear static and dynamic problems of large deformation, therefore, to some extent, the work of this paper is a innovation. The contents of this paper can be divided into two parts: the static analysis of the nonlinear problem of the large deformation of the elastic thin rod and the dynamic analysis of the nonlinear problem of the large deformation of the elastic thin rod. The following contents are discussed: the effect of the axial force on the bending is considered by the large deformation of the 1. bar elastic thin rod, and the bar element stiffness matrix.2., which considers the tensile and pressure deformation of the element element, the torsion deformation and the bending deformation, is established on the basis of the axial force and parabolic interpolation. The body coordinate, that is, the local coordinate, is used to determine the posture of the rod element under the force of the elastic rod. The coordinate transformation relation between the small Euler four element increment of the angular displacement and the angular displacement under the Descartes coordinate system.3. is derived by the coordinate transformation between the angular displacement and the Descartes coordinate system. An iterative acceleration method for solving the total displacement equation for the overall static equilibrium analysis of the system overall static equilibrium analysis corresponding to each load increment step is proposed. The dynamics part mainly discusses the following contents: 1. the mass and elasticity of the elastic thin rod are discussed respectively. The elasticity of the elastic thin rod is discussed and the reference static of the elastic thin rod is discussed. Mechanical part. For the mass of an elastic rod, the mass of the rod element is simplified as a micro rigid body mass unit at the node. To ensure the accuracy of the model, the general motion of a small.2. rigid body mass unit is composed of a rotational and translational motion. The reference Euler dynamic equation is rotated and the motion theorem of the centroid is used for translation. Two combination pushes are applied. This paper derives the motion equation of the general motion of a rigid body mass in the world coordinate system.3. this paper derives the motion differential equation of the finite element dynamic analysis of the elastic thin rod finite element, and introduces the finite element numerical solution of the finite element general motion differential equation and the finite element general of the implicit difference scheme for the explicit difference scheme of the differential equation of motion. The numerical solution of the differential equations of motion (.4.) is used to deal with the contact problem of the elastic thin rod, which is treated with elastic contact constraints with large stiffness. In this paper, a number of examples are given, and the numerical simulation calculation is carried out by MATLAB software programming. The feasibility and accuracy of this method are verified.
【学位授予单位】：东华大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O343

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