由大变形引起的有限转动的描述

极分解定理[1]中的R是一个正交张量。一个正交张量对不共面的三个矢量的作用,可以使这三个矢量发生刚性转动而不改变各矢量的大小和它们之间的夹角[2]。但是,在大变形中,变形体上一点的任意三个不共面的物质线索矢量,变形后,每一个矢量的大小和它们之间的夹角都要发生变化。所以用一个正交张量只能描述变形体主向的有限转动,而不可能描述非主向的有限转动。但是,对于每一个物质线素矢量,却可以用正交张量或有限转动矢量来描述[3],i=1、2、3。本文将这三个彼此有联系的正交张量或三个有限转动矢量进行适当的组合,就可以构成一个二阶张量H,张量H就描述了于由变形引起的三个非主向的有限转动。 此外,文中还证明,当小变形时,这三个转动张量退化为由剪切变形确定的反对称张量。     

基面力概念在几何非线性余能有限元中的应用

以基面力为基本未知量描述一个弹性系统的应力状态并表征单元的余能,将大变形的余能分解为变形余能部分和转动余能部分,采用Lagrange乘子法放松单元的平衡方程,利用已有的弹性大变形余能原理建立了一种几何非线性显式有限元模型,编制了相应的几何非线性余能原理有限元程序. 数值算例表明:该方法具有较好的收敛性和计算精度,可进行大载荷步的大位移、大转动计算.      

多万向联轴节传动比的波动特性

本文导出了中间轴两端叉面不共面的多万向联轴节传动的传动比公式和转动方程,定量地描述了传动比的波动特性,提出了选择三万向联轴节传动最优结构参数的方法.     

基于有限变形理论的数值流形方法研究

原有数值流形方法通过积累每一时步的小变形而得到结构最终的大变形,然而,当结构发生大变形、大转动时往往产生较大计算误差. 针对该问题,从动量守恒方程以及应力边界条件的积分弱形式出发,引入流形方法的插值函数,建立了基于有限变形理论的数值流形方法. 通过对比改进前后流形方法的计算迭代格式,指出了原有流形方法计算大变形问题时的误差来源. 最后,通过大变形悬臂梁和旋转块体算例对有限变形流形方法进行了验证. 数值结果表明,改进后的流形方法能够很好地处理大变形大转动问题,消除了转动所带来的计算误差,其计算结果与解析解及ABAQUS 软件求得的数值解相吻合.      

转动基中斜拉索的非线性动力学分析

给出了转动基中柔性斜拉索的运动学描述方法，建立了该系统的运动学控制方程。利用多尺度摄动方法，得到了斜拉索的内共振模式。利用Melnikov方法和留数定理，分析了转动基中斜拉索的全局发岔与混沌性 质，并用数值方法模拟了该系统的混沌运动。     

考虑变形耦合的几何非线性空间梁单元

以"精确几何模型梁单元"为代表的很多几何非线性梁单元,在构造过程中分别对描述截面转动的转角和描述截面形心位置的位移进行了独立插值,由此引起了诸如运动学描述冗余和剪切闭锁等困难。其根本原因在于单元形函数没能体现细长梁中的变形耦合关系。本文对这类传统单元进行了改造,通过深入研究单元变形之间的内在联系,提出了一种变形场完全满足Bernoulli梁变形耦合关系的新单元,避免了构造过程中对转动矢量的插值,并通过数值算例检验了单元的有效性。     

横波引起的转动及其孪生剪切变形1)

弹性介质中局部刚性转动的传播服从波动方程,其速度取决于介质的剪切模量和密度. 但是刚性转动不是介质的变形,为什么其速度依赖于表征形状变形恢复能力的剪切模量?本文证明横波传播过程中局部刚性旋转与剪切变形一一对应,互为依存. 发生刚性转动与剪切变形之后,微元体凭借剪切模量恢复其扰动前的形状,同时转回到扰动前的位置,这是转动可以传播且速度依赖于剪切模量的原因.     

关于几种新的极分解计算方法

对变形梯度极分解的计算方法进行了分析,给出极分解计算的四种新方法：（１）增量叠加法;（２）基于伸长张量不变量（近似）计算法;（３）确定主转动轴计算法;（４）坐标变换法．增量叠加极分解计算方法将为建立以伸长张量为应变度量的大变形大转动有限元分析方法提供基础．本文还给出了伸长张量物质时间导数的简洁表达式     

刚柔耦合约束多体系统的动力分析

本文提出了描述柔性多体系统的牵连坐标系统。该系统由惯性参考系，牵连坐标系，物体坐标系及单元坐标系组成，实现了对刚体平动。刚体转动及弹性运动的连续分解，最大限度地消除了由于铡体大角度转动导致的非线性特性。以有限元法为基础，应用拉格朗日方程建立了在该坐标下的刚柔耦合约束多体系统的动力学控制方程，该方程具有耦合程度小，易于推导，编程及求解等优点，为大规模约束多体系统的动力分析提供了新的途径。本文还讨论了     

杆系结构的几何非线性分析──Ⅱ．三维问题

以三维连续体的虚功增量方程为基础，采用平动、转角位移分别插值的方法，导出了空间Ｔｉｍｏｓｈｅｎｋｏ梁大位移、大转动问题内力分析的ＵＬ法（ＵｐｄａｔｅｄＬａｇｒａｎｇｉａｉ１Ｆｏｒｍｕｌａｔｉｏｎ）。本文考虑了轴向、剪切、弯曲和扭转效应，提出了新的几何刚度矩阵，并建立了描述大转动规律的坐标转换矩阵。算例表明，依本文方法编制的程序具有分析结构强几何非线性行为的能力；在满足本文假定（即每次加载增量中转角增量是小量）的条件下，可以描述任意大角度的刚体转动。     

基于非线性梁理论的有限质点法

有限质点法是以向量式力学为基础,用有限数量的质点来模拟结构的变形行为,质点的运动由牛顿运动定律来计算。在有限质点法中,质点通过构件相连,构件约束着质点的运动,并且其内力由质点的运动变量来描述。基于向量式力学的基本思想和非线性梁理论,提出了一种新的有限质点法,该方法在共旋单元坐标系中描述梁的非线性变形。以空间梁系结构为例,推导了计算构件内力的非线性公式,并考虑了弯扭耦合变形。通过两个连续欧拉角的变换公式得到共旋坐标系的旋转矩阵。与传统的有限质点法相比,本文提出的方法避免了刚体虚转动分析。通过四个结构的数值求解,验证了本文方法在计算结构大变形响应时具有较高的精度。     

A quaternion-based formulation of Euler–Bernoulli beam without singularity

This paper proposes a singularity-free beam element with Euler–Bernoulli assumption, i.e., the cross section remains rigid and perpendicular to the tangent of the centerline during deformation. Each node of this two-nodal beam element has eight nodal coordinates, including three global positions and one normal strain to describe the rigid translation and flexible deformation of the centerline, respectively, four Euler parameters or quaternion to represent the attitude of cross section. Adopting quaternion instead of Eulerian angles as nodal variables avoids the traditionally encountered singularity problem. The rigid cross section assumption is automatically satisfied. To guarantee the perpendicularity of cross section to the deformed neutral axes, the position and orientation coordinates are coupled interpolated by a special method developed here. The proposed beam element allows arbitrary spatial rigid motion, and large bending, extension, and torsion deformation. The resulting governing equations include normalization constraint equations for each quaternion of the beam nodes, and can be directly solved by the available differential algebraic equation (DAE) solvers. Finally, several numerical examples are presented to verify the large deformation, natural frequencies and dynamic behavior of the proposed beam element.     

Hoberman's Sphere, Euler Parameters and Lagrange's Equations

In this classroom note, we explore how the Euler parameters can be used to represent a particular homogeneous deformation of a continuum. One possible application is Hoberman's sphere. With the assistance of the theory of a pseudo-rigid body, we show how the motion of the continuum can be determined. We also present a new derivation of Lagrange's equations for the rotational dynamics of a rigid body where the rotation tensor is parameterized using Euler parameters. This revised version was published online in August 2006 with corrections to the Cover Date.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation, LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率. 有限元方法中,闭锁问题是导致单元收敛性能低下的主要原因, 例如梁单元的剪切以及泊松闭锁.多变量变分原理是缓解梁、板/壳单元闭锁的有效手段. 该方法不仅离散位移场,同时离散应力场或应变场, 可提高应力与应变的计算精度. 本文基于上述局部标架,研究几类梁单元的闭锁处理方法, 包括几何精确梁(geometrically exact beam formulation, GEBF)与绝对节点坐标(absolute nodal coordinate formulation, ANCF)梁单元. 其中, 采用Hu-Washizu三场变分原理缓解几何精确梁单元中的剪切闭锁,采用应变分解法缓解基于局部标架的ANCF全参数梁单元中的泊松闭锁. 数值算例表明,局部标架的梁单元在描述高转速或大变形柔性多体系统时,可消除刚体运动带来的几何非线性, 极大地减少系统质量矩阵和刚度矩阵的更新次数.缓解闭锁后的几类局部标架梁单元收敛性均得到了明显提升.      

A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

A method for relaxing parameter constraints in rigid body dynamics

As attractive alternatives to a set of three Euler angles, the rotation of a rigidly deforming body is often represented using four or more parameters. The accompanying parameter constraints introduce generalized constraint forces in the equations of motion which can often negate the benefits of a particular parameterization. In this paper, we discuss situations where the parameter constraints are not imposed. Thus, although the body no longer deforms rigidly, it does deform homogeneously. This allows the theory of a Cosserat point (or, equivalently, the theory of a pseudo-rigid body) to be used to establish equations governing its motion. Earlier work on this topic by O’Reilly and Varadi considered the four Euler parameters and the single Euler parameter constraint. Here, we consider Poincaré's six parameter representation of a rotation tensor, and, complementing earlier work, discuss numerical implementation and representative simulations. One of the contributions of this paper is the development of a viscoelastic Cosserat point, whose equations of motion are free from parameter constraints and singularities, that can be used to approximate the motion of a rigid body.     

大变形轴向运动梁的精确动力学模型

轴向运动梁的横向振动是具有实际工程背景的动力学问题.该文应用Cosserat弹性杆模型讨论圆截面轴向运动梁的动力学建模及其运动稳定性.以沿梁中心线的弧坐标代替方向固定的坐标轴,根据梁截面的姿态随弧坐标和时间的变化确定梁的变形过程.从欧拉的速度场概念出发,考虑梁截面转动的惯性效应和剪切变形,建立大变形轴向运动梁的动力学方程.其小变形特例为轴向运动的三维Timoshenko梁.基于该模型分析了轴向运动梁准稳态运动的静态和动态稳定性,导出可导致失稳的临界轴向速度.证明空间域内的欧拉稳定性条件是时间域内的Lyapunov稳定性的必要条件.      

Geometrically exact beam theory without Euler angles

In modeling highly flexible beams undergoing arbitrary rigid–elastic deformations, difficulties exist in describing large rotations using rotational variables, including three Euler angles, two Euler angles, one principal rotation angle plus three direction cosines of the principal rotation axis, four Euler parameters, three Rodrigues parameters, and three modified Rodrigues parameters. The main problem is that such rotational variables are either sequence-dependent and/or spatially discontinuous because they are not mechanics-based variables. Hence, they are not appropriate for use as nodal degrees of freedom in total-Lagrangian finite-element modeling. Moreover, it is difficult to apply boundary conditions on such discontinuous and/or sequence-dependent rotational variables. This paper presents a new geometrically exact beam theory that uses no rotation variables and has no singular points in the spatial domain. The theory fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, exact coordinate transformations, and orthogonal virtual rotations. The derivations are presented in detail, fully nonlinear governing equations and boundary conditions are presented, a finite element formulation is included, and the corresponding governing equations for numerically exact analysis using a multiple shooting method is also derived. Numerical examples are used to illustrate the problems of using rotational variables and to demonstrate the accuracy of the proposed geometrically exact displacement-based beam theory.     

Advancements in subsea riser analysis using quasi-rotations and the Newton–Raphson method

Beam structures undergoing finite deflections and rotations in space have extensive application in the subsea industry particularly for the analysis of holistic systems with larger numbers of mooring and riser components. In using the finite element analysis approach, there is an increasing requirement for large element sizes which preserve accuracy with regard to the coupling of axial, bending and torsion response.The authors outline a method for improving the current state of practice for the analysis of riser systems. The approach draws on the convected coordinates method, Euler–Bernoulli beam theory, the principle of virtual work and the finite element method. Two quasi-rotation measures are developed including a quasi-material rotation definition for rotational deformation relative to the convected axis of a beam and a quasi-space rotation definition to deal with the path dependent nature of rotations in three dimensions.The novel aspect of this work is to relate the rate of change of the quasi-material rotation vector along the beam axis to a linear transformation of the beam axis rate-of-rotation vector through utilising the convected coordinates axes system. In this way, incremental values of quasi-material rotation are directly linked to incremental values of nodal quasi-space rotation and a global Newton–Raphson solution technique for interconnecting beam elements is straightforward to assemble.Furthermore, this leads to accurate definitions of coupled axial, bending and torque response for beams with significant deflection. The approach has particular advantages in the analysis of subsea riser sections. Also, the accuracy of the solution is preserved for a fewer number of elements compared to alternative solutions for computationally sensitive load cases with highly non-linear loading regimes.