Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics
Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics
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DOI：
10.1023/a:1008072517368
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发表时间：
1998-07-01
影响因子：
5.6
通讯作者：
Shabana, AA
中科院分区：
文献类型：
--
作者：
Shabana, AA
多体系统中的可变形部件受到代表机械关节和指定运动轨迹的运动学约束。一般来说，这些约束可以用一组依赖于系统广义坐标和时间的非线性代数方程来描述。当运动学约束扩展到系统的运动微分方程时，为了能够利用有效的稀疏矩阵算法，希望有一个公式可以导致未知加速度和约束力的非零系数的最小数量。本文介绍了柔性多体应用中绝对节点坐标公式的计算机实现程序。在绝对节点坐标公式中，不使用无穷小或有限旋转作为节点坐标。使用全局位移坐标和斜率来定义有限元的结构。利用这一混合坐标集，梁单元和板单元可被视为等参单元。因此，这些广泛使用的单元的动态公式使用绝对节点坐标公式导致一个恒定的质量矩阵。本研究的目的是开发利用这一特征的计算程序。在其中一种方法中，利用QR分解获得了可变形体的最优稀疏矩阵结构。利用单元质量矩阵为常数的事实，对可变形体的修正常数连通性雅可比矩阵进行QR分解。采用等速变换得到与广义坐标二阶导数相关的单位广义惯性矩阵，从而使柔性多体系统运动方程增广拉格朗日公式中出现的系数矩阵的非零分量最小化。本文还提出了一种基于Cholesky分解的替代计算方法。这种替代方法与基于QR分解的方法具有相同的计算优势，可以得到平方速度变换矩阵。本研究提出的计算方法可用于处理柔性多体系统中的大变形问题。它们还具有基于浮动参照系公式的算法的优点，因为它们允许容易地添加一般非线性约束和力函数。
Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.