多体系统中几何精确大变形梁建模研究
发布时间：2018-08-31 11:16
【摘要】：多体系统动力学建模时,可以将系统中的运动副和物体简化为刚体或柔性体。其中,梁是多体系统中广泛应用的柔性构件,其建模是多体领域的一个研究热点,刚柔耦合多体系统中梁的几何非线性问题显著。高速旋转柔性梁呈现动力刚化现象,基于小变形假设的梁单元能够在附加动力刚化项的条件下解决动力刚化问题,却不适用于大变形分析。几何精确梁除刚截面外不对模型做其他假设,它的位移-应变关系适用于任意大小的位移和转动,适合梁的几何非线性分析。传统几何精确梁单元大多对位移和转动独立插值,应用在细长梁上将面临剪切闭锁的问题,而多体系统中的柔性梁多为细长梁,使用欧拉-伯努利梁符合其受力特征。欧拉-伯努利梁中截面与形心线垂直,这一约束为形心位移与截面转动的插值增加了难度。对节点转动矢量插值的梁单元引起了许多数值求解方面的困难,容易丢失单元应变对刚体位移的客观性。一类几何精确梁单元直接对应变离散,由此能够避开转动插值引起的麻烦,但对于空间梁不易由应变导出节点位移和转动。针对以上几何非线性梁的建模问题,本文基于几何精确欧拉-伯努利梁弹性虚功率和形心线与曲率的关系,先通过单元形心曲线插值得到端面曲率及其对弧长的变化率,进而对单元域内的曲率进行拟合,提出了一种以整体节点位移和转动矢量为基本变量,既可避免转动矢量插值,同时又不增加节点参数的几何非线性空间梁单元。单元对任意刚体位移保有应变客观性,具备可积的显式节点力表达式,在小变形情况下能够退化到线性单元。单元考虑了轴向变形与弯、扭变形的耦合效应,能够完全计及“动力刚化”项,数值算例验证了该单元处理几何大变形,以及动力刚化问题的能力,证实了单元的收敛性与精确性。以上位移-应变混合插值梁单元具有可积的弹性节点力表达式,这对于几何非线性问题是个难得的优点。这一优点得益于单元内做了几何假设。同样基于几何精确梁模型,本文构造了一种严格满足截面与形心线约束和保持应变客观性的几何非线性欧拉-伯努利梁单元。欧拉-伯努利梁截面在运动中始终与形心线垂直,可认为截面转动四元数令初始直梁的截面法线转到变形后形心线切线上,它是一组4变量的3个四元数方程。插值形心线曲线,使其满足边界连续性和位移非线性要求。由形心线切线和四元数方程的通解辅助确定截面转动场,它能够自动满足截面与形心线约束,有效避免转动参数的奇异性。根据该建模策略构造的几何非线性梁单元形式简洁,能够保证单元应变由位移场和转动场一致地导出。此外,多体系统动力学建模要求提供柔性梁截面以及刚性物体的质心、质量、转动惯量等参数,为自动化建模带来不便。对于复杂形状和不规则几何体来说,确定这些参数需要很大的工作量,而且不易计算准确。随着多体系统动力学的推广,多体理论和商业软件的发展趋向于减少用户的人工操作,以便降低输入参数有误的几率。为此,本文基于虚功率原理推导了一种自动集成多体系统中刚体动力学方程的新方法。建模流程是先对刚体采用网格剖分,以组成物体的刚体单元为基本元素,该建模方法与有限元法一样自动集成系统的动力方程。根据需要具体构造了刚性四面体单元和刚性梁单元,对复杂形状的梁截面,构造了对平面形状逼近性较好的三角形单元。以刚体单元为基础并内嵌网格剖分模块的分析软件能够自动获得刚体的几何参数和惯性参数,从而具备了独立处理任意复杂形状系统的能力。针对多体系统中大变形细长梁的动力学建模,本文基于几何精确梁理论,提出了两种大变形空间欧拉-伯努利梁单元。针对系统中刚体的动力学建模存在的问题,本文构造了刚体单元。
[Abstract]:In multi-body system dynamics modeling, the motion pairs and objects in the system can be simplified as rigid or flexible bodies. Beams are widely used flexible components in multi-body systems, and their modeling is a research hotspot in multi-body field. The beam element based on the hypothesis of small deformation can solve the dynamic stiffness problem under the condition of additional dynamic stiffness term, but it is not suitable for large deformation analysis. Most of the traditional geometrically accurate beam elements are interpolated independently for displacement and rotation, so they will face the problem of shear locking when applied to slender beams, while the flexible beams in multi-body systems are mostly slender beams. The Euler-Bernoulli beams are used to fit the mechanical characteristics of the beams. A class of geometrically exact beam elements are directly discrete to the strain, thus avoiding the trouble caused by the rotation interpolation, but it is not easy to derive the node from the strain for the spatial beam. In this paper, based on the geometrically accurate Euler-Bernoulli beam's elastic virtual power and the relationship between the center line and the curvature, the end-to-end curvature and its rate of change of the arc length are interpolated by the element center curve, and then the curvature in the element domain is fitted, and a global method is proposed. The nodal displacement and rotation vectors are the basic variables, which can avoid the interpolation of rotation vectors without adding the nodal parameters. The element retains the strain objectivity for any rigid body displacement, has an integrable explicit nodal force expression, and can degenerate to a linear element under small deformation. The coupling effect of deformation, bending and torsion can fully take into account the term of "dynamic stiffness". Numerical examples show that the element is capable of dealing with large geometric deformation and dynamic stiffness problems, and the convergence and accuracy of the element are verified. The above displacement-strain hybrid interpolated beam element has an integrable elastic nodal force expression, which is suitable for geometry. This advantage is due to the geometric assumptions made in the element. Also, based on the geometric exact beam model, a geometrically nonlinear Euler-Bernoulli beam element is constructed which strictly satisfies the constraints of section and centroid and maintains the objectivity of strain. The section of Euler-Bernoulli beam is always in motion with the center of shape. Line perpendicular, it can be considered that the section rotation quaternion makes the section normal of the initial straight beam turn to the tangent of the centroid after deformation. It is a set of three quaternion equations with four variables. The geometrically nonlinear beam element based on this modeling strategy is simple and can ensure that the element strain is uniformly derived from the displacement field and the rotation field. With the development of multi-body dynamics, the development of multi-body theory and commercial software tends to reduce the user's manual operation in order to reduce the transmission. Based on the principle of virtual power, a new method for automatically integrating the dynamic equations of rigid bodies in multibody systems is presented in this paper. Rigid tetrahedron element and rigid beam element are constructed according to the requirement, and triangular element with good approximation to plane shape is constructed for beam section with complex shape. In this paper, based on the geometric exact beam theory, two large deformation spatial Euler-Bernoulli beam elements are proposed for the dynamic modeling of slender beams with large deformation in multibody systems.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O313.7
[Abstract]:In multi-body system dynamics modeling, the motion pairs and objects in the system can be simplified as rigid or flexible bodies. Beams are widely used flexible components in multi-body systems, and their modeling is a research hotspot in multi-body field. The beam element based on the hypothesis of small deformation can solve the dynamic stiffness problem under the condition of additional dynamic stiffness term, but it is not suitable for large deformation analysis. Most of the traditional geometrically accurate beam elements are interpolated independently for displacement and rotation, so they will face the problem of shear locking when applied to slender beams, while the flexible beams in multi-body systems are mostly slender beams. The Euler-Bernoulli beams are used to fit the mechanical characteristics of the beams. A class of geometrically exact beam elements are directly discrete to the strain, thus avoiding the trouble caused by the rotation interpolation, but it is not easy to derive the node from the strain for the spatial beam. In this paper, based on the geometrically accurate Euler-Bernoulli beam's elastic virtual power and the relationship between the center line and the curvature, the end-to-end curvature and its rate of change of the arc length are interpolated by the element center curve, and then the curvature in the element domain is fitted, and a global method is proposed. The nodal displacement and rotation vectors are the basic variables, which can avoid the interpolation of rotation vectors without adding the nodal parameters. The element retains the strain objectivity for any rigid body displacement, has an integrable explicit nodal force expression, and can degenerate to a linear element under small deformation. The coupling effect of deformation, bending and torsion can fully take into account the term of "dynamic stiffness". Numerical examples show that the element is capable of dealing with large geometric deformation and dynamic stiffness problems, and the convergence and accuracy of the element are verified. The above displacement-strain hybrid interpolated beam element has an integrable elastic nodal force expression, which is suitable for geometry. This advantage is due to the geometric assumptions made in the element. Also, based on the geometric exact beam model, a geometrically nonlinear Euler-Bernoulli beam element is constructed which strictly satisfies the constraints of section and centroid and maintains the objectivity of strain. The section of Euler-Bernoulli beam is always in motion with the center of shape. Line perpendicular, it can be considered that the section rotation quaternion makes the section normal of the initial straight beam turn to the tangent of the centroid after deformation. It is a set of three quaternion equations with four variables. The geometrically nonlinear beam element based on this modeling strategy is simple and can ensure that the element strain is uniformly derived from the displacement field and the rotation field. With the development of multi-body dynamics, the development of multi-body theory and commercial software tends to reduce the user's manual operation in order to reduce the transmission. Based on the principle of virtual power, a new method for automatically integrating the dynamic equations of rigid bodies in multibody systems is presented in this paper. Rigid tetrahedron element and rigid beam element are constructed according to the requirement, and triangular element with good approximation to plane shape is constructed for beam section with complex shape. In this paper, based on the geometric exact beam theory, two large deformation spatial Euler-Bernoulli beam elements are proposed for the dynamic modeling of slender beams with large deformation in multibody systems.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O313.7
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