Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body

A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler–Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler’s equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 133–143, February 2009.     

Nonstructural geometric discontinuities in finite element/multibody system analysis

Existing multibody system (MBS) algorithms treat articulated system components that are not rigidly connected as separate bodies connected by joints that are governed by nonlinear algebraic equations. As a consequence, these MBS algorithms lead to a highly nonlinear system of coupled differential and algebraic equations. Existing finite element (FE) algorithms, on the other hand, do not lead to a constant mesh inertia matrix in the case of arbitrarily large relative rigid body rotations. In this paper, new FE/MBS meshes that employ linear connectivity conditions and allow for arbitrarily large rigid body displacements between the finite elements are introduced. The large displacement FE absolute nodal coordinate formulation (ANCF) is used to obtain linear element connectivity conditions in the case of large relative rotations between the finite elements of a mesh. It is shown in this paper that a linear formulation of pin (revolute) joints that allow for finite relative rotations between two elements connected by the joint can be systematically obtained using ANCF finite elements. The algebraic joint constraint equations, which can be introduced at a preprocessing stage to efficiently eliminate redundant position coordinates, allow for deformation modes at the pin joint definition point, and therefore, this new joint formulation can be considered as a generalization of the pin joint formulation used in rigid MBS analysis. The new pin joint deformation modes that are the result of C ⁰ continuity conditions, allow for the calculations of the pin joint strains which can be discontinuous as the result of the finite relative rotation between the elements. This type of discontinuity is referred to in this paper as nonstructural discontinuity in order to distinguish it from the case of structural discontinuity in which the elements are rigidly connected. Because ANCF finite elements lead to a constant mass matrix, an identity generalized mass matrix can be obtained for the FE mesh despite the fact that the finite elements of the mesh are not rigidly connected. The relationship between the nonrational ANCF finite elements and the B-spline representation is used to shed light on the potential of using ANCF as the basis for the integration of computer aided design and analysis (I-CAD-A). When cubic interpolation is used in the FE/ANCF representation, C ⁰ continuity is equivalent to a knot multiplicity of three when computational geometry methods such as B-splines are used. C ² ANCF models which ensure the continuity of the curvature and correspond to B-spline knot multiplicity of one can also be obtained. Nonetheless, B-spline and NURBS representations cannot be used to effectively model T-junctions that can be systematically modeled using ANCF finite elements which employ gradient coordinates that can be conveniently used to define element orientations in the reference configuration. Numerical results are presented in order to demonstrate the use of the new formulation in developing new chain models.     

On the stability of the rotational motion of a rigid body having a liquid filled cavity under finite initial disturbance

In this paper the problem of the stability of rotational motion of a rigid body which has a liquid filled cavity and a fixed point is investigated without any approximation. Criteria of stability and instability under finite disturbance are obtained. The region of stability is found out explicitly.     

On the distributed Lagrange multiplier/fictitious domain method for rigid‐particle‐laden flows: a proposition for an alternative formulation of the Lagrange multipliers

The distributed Lagrange multiplier/fictitious domain method proposed for the direct numerical simulation of particle‐laden flows is considered in this work. First, it is demonstrated that improved accuracy is obtained with a coupled numerical scheme, whereby the pressure and the Lagrange multiplier fields enforcing incompressibility and rigid body motion, respectively, are calculated and applied together. However, the convergence characteristics of the iterative solution of the coupled scheme are poor because symmetric but indefinite and poorly conditioned matrices are produced. An analysis is then presented, which suggests that the cause for the matrix pathologies lies in the interaction of the respective matrix operators enforcing incompressibility and rigid body motion. On the basis of this analysis, an alternative formulation is developed for the Lagrange multipliers, being now composed of a set of forces distributed only on the particle boundary together with a set of couples distributed within the particle core. The new formulation is tested with several types of flows with stationary or moving particles under creeping or finite Reynolds number conditions and it is demonstrated that it produces correct results and better conditioned matrices, thus enabling faster and more reliable convergence of the conjugate gradient method. The analysis and tests, therefore, support the expectation that the proposed formulation is promising and worthy of further study and improvement. Copyright © 2011 John Wiley & Sons, Ltd.     

用给定转角分步实现刚体任意有限转动的研究

刚体的任意有限转动可以分解为3次绕指定转轴的定轴转动,但在实际应用中,可能出现难以绕某轴实现连续任意角度旋转的情况。本文探究的问题是,刚体的任意有限转动是否可以分解为几次旋转角大小给定、转轴方向任意的定轴转动,以及如何实现这种分解。本文以姿态四元数为主要工具,构造性地证明了上述分解的可行性,并给出了具体的分解方法。     

结构有限元分析中阶跃型弹性约束的一种有效处理方法

本文通过引入弹性约束刚度矩阵和结构位移约束列阵，提出了结构有限元分析中处理阶跃型弹性约束的一种有效方法。该法通过改变弹性约束系数及位移非约束量大小，可方便有效地处理结点常弹性约束，刚性约束，阶跃型弹性的约束及阶跃型刚性约束等问题。     

Validation and application of three-dimensional discontinuous deformation analysis with tetrahedron finite element meshed block

In the last decade, three dimensional discontinuous deformation analyses (3D DDA) has attracted more and more attention of researchers and geotechnical engineers worldwide. The original DDA formulation utilizes a linear displacement function to describe the block movement and deformation, which would cause block expansion under rigid body rotation and thus limit its capability to model block deformation. In this paper, 3D DDA is coupled with tetrahedron finite elements to tackle these two problems. Tetrahedron is the simplest in the 3D domain and makes it easy to implement automatic discretization, even for complex topology shape. Furthermore, element faces will remain planar and element edges will remain straight after deformation for tetrahedron finite elements and polyhedral contact detection schemes can be used directly. The matrices of equilibrium equations for this coupled method are given in detail and an effective contact searching algorithm is suggested. Validation is conducted by comparing the results of the proposed coupled method with that of physical model tests using one of the most common failure modes, i.e., wedge failure. Most of the failure modes predicted by the coupled method agree with the physical model results except for 4 cases out of the total 65 cases. Finally, a complex rockslide example demonstrates the robustness and versatility of the coupled method.     

非结构动网格在三维可动边界问题中的应用

研究用于非结构动网格的弹簧近似方法，采用顶点弹簧描述，导出并讨论了弹簧倔强系数的取值。通过引入边界修正和扭转效应修正，对标准弹簧近似方法进行了改进，转动翼型算例的结果表明，改进后的方法大大提高了网格变形能力和网格质量，应用该动网格方法耦合求解基于（Arbitrary Lagrangian-Eulerian,ALE）描述的三维Euler方程，模拟了作俯抑振动的矩形机翼绕流，计算结果与实验数据及文献计算结果十分一致，作为多个自由刚与流体耦合运动问题的简单例证，耦合刚体动力学方程，模拟了激波与双立方体的相互作用，得到了非定常流场结构，研究表明，基于弹簧近似的非结构动网格与有限体积流式流场解算器相结合，是模拟包含运动边界的非定常流动问题的有效方法。     

一类刚-柔耦合系统的建模与稳定性研究

对于由中心刚体带有柔性梁附件组成的这一类简单刚 柔耦合系统，目前文献广泛采用的Ｅｕｌｅｒ Ｂｅｒｎｏｕｌｉ梁模型中考虑的刚 柔运动耦合项有严重的缺陷．本文对于物理本构关系线性的有限变形梁，分别采用微元法和变分法建立了该系统大挠度非线性动力学方程组．本文使用严格的方法来研究此非线性耦合动力学模型，采用能量 动量矩组合方法构成Ｌｉａｐｕｎｏｖ函数，严格证明了此非线性系统平凡解的积分范数稳定性以及具有鲜明物理意义的最大模范数稳定性．本文对文献中引用的三类线性化模型，采用假设模态法，对中心刚体匀速转动时梁的振动作了数值仿真，进一步验证了本文的结论．上述结果，对选择刚 柔耦合系统正确的动力学模型是有益的．     

基于SE(3)群局部标架的5/6 Dofs CB壳单元

基于李群局部标架(local frame of Lie group,LFLG)的多柔体系统动力学建模方法可自然消除刚体运动带来的几何非线性,使系统的广义弹性力、广义惯性力及其雅可比矩阵满足刚体运动的不变性.本文融合李群局部标架思想和基于连续体(continuum-based,CB)的壳理论,提出基于SE(3)群局部标架...     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.     

非网格重剖分模拟宏观裂纹体的扩展有限单元法(2:数值实现)

探讨了扩展有限单元法的具体实现过程,包括裂纹体几何结构的拓扑分析、广义节点的选取及详细的单元数值计算。并针对前文提出的扩展有限单元平衡方程的统一矩阵实现模式,提出了采用虚拟层合元的思想来处理被裂纹横贯单元的子域积分问题,自然地解决了原方法中由于特殊的位移插值场在裂纹两侧不连续造成的单元刚度阵求解困难。同时依托比较成熟的虚拟层合单元法,可以方便地考虑域内及裂纹面上分布载荷影响。此外,一、二维算例较高精度的数值结果验证了本文算法的有效性和精度。     

A nonlinear finite-element approach for kineto-static analysis of multibody systems

Methods that treat rigid/flexible multibody systems undergoing large motion as well as deformations are often accompanied with inefficiencies and instabilities in the numerical solution due to the large number of state variables, differences in the magnitudes of the rigid and flexible body coordinates, and the time dependencies of the mass and stiffness matrices. The kineto-static methodology of this paper treats a multibody mechanical system to consist of two collections of bulky (rigid) bodies and relatively flexible ones. A mixed boundary condition nonlinear finite element problem is then formulated at each time step whose known quantities are the displacements of the nodes at the boundary of rigid and flexible bodies and its unknowns are the deformed shape of the entire structure and the loads (forces and moments) at the boundary. Partitioning techniques are used to solve the systems of equations for the unknowns, and the numerical solution of the rigid multibody system governing equations of motion is carried out. The methodology is very much suitable in modelling and predicting the impact responses of multibody system since both nonlinear and large gross motion as well as deformations are encountered. Therefore, it has been adopted for the studies of the dynamic responses of ground vehicle or aircraft occupants in different crash scenarios. The kineto-static methodology is used to determine the large motion of the rigid segments of the occupant such as the limbs and the small deformations of the flexible bodies such as the spinal column. One of the most dangerous modes of injury is the amount of compressive load that the spine experiences. Based on the developed method, a mathematical model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II (Part 572) anthropomorphic test dummy. The lumbar spine model is then incorporated into a gross motion occupant model. The analytical results are correlated with the experimental results from the impact sled test of the dummy/seat/restraint system. With this extended occupant model containing the lumbar spine, the gross motion of occupant segments, including displacements, velocities and accelerations as well as spinal axial loads, bending moments, shear forces, internal forces, nodal forces, and deformation time histories are evaluated. This detailed information helps in assessing the level of spinal injury, determining mechanisms of spinal injury, and designing better occupant safety devices.     

The asymptotic solution of a dynamic buckling problem in elastic columns

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Assessment of crystal plasticity based calculation of the lattice spin of polycrystalline metals for FE implementation

When texture is incorporated in the finite element simulation of a metal forming process, much computer time can be saved by replacing continuous texture and corresponding yield locus updates by intermittent updates after strain intervals of e.g. 20%. The hypothesis that the evolution of the anisotropic properties of a polycrystalline material during such finite interval of plastic deformation can be modelled by just rotating the initial texture instead of continuously updating it by means of a polycrystal deformation model is tested in this work. Two spins for rotating the frame have been assessed: the classical rigid body spin and a crystal plasticity based “Mandel spin” (calculated from the rotated initial texture) which is the average of the spins of all the crystal lattices of the polycrystal. Each of these methods was used to study the evolution of the yield locus and the r-value distribution during the 20% strain interval. The results were compared to those obtained by simulating the texture evolution continuously using a polycrystal deformation model. When the texture was not updated during deformation, it was found that for most initial textures the Mandel spin does not perform better than the rigid body spin, except for some special initial textures for which the Mandel spin is much better. The latter ones are textures which are almost stable for the corresponding strain mode. When the texture was updated after each strain interval of e.g. 20% the Mandel spin performed much better than the rigid body spin.     

New control laws for stabilization of a rigid body motion using rotors system

This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced.     

考虑刚弹耦合作用的柔性多体连续系统动力学建模

基于Ｈａｍｉｌｔｏｎ原理建立起一般柔性体连续系统的动力学建模方法,进而以水平面内作大范围回转运动的柔性梁为例,在Ｅｕｌｅｒ－Ｂｅｒｎｏｕｌｌｉ梁模型的假设前提下,根据轴向不可伸长的柔性梁的几何约束条件;推导出作大范围刚体运动的柔性梁连续系统的一致线性化振动微分方程．采用假设模态法对其离散化,导出考虑刚弹耦合作用的柔性梁有限维离散化动力学模型．最后给出仿真算例,验证了该方法的有效性．