A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters
A New Singularity-Free Formulation of a Three-Dimensional Euler–Bernoulli Beam Using Euler Parameters
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DOI：
10.1115/1.4031769
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发表时间：
2016-07
影响因子：
2
通讯作者：
W. Fan;Wei-dong Zhu;H. Ren
中科院分区：
文献类型：
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作者：
W. Fan;Wei-dong Zhu;H. Ren
在这项研究中，一个新的奇性自由制定的三维欧拉-伯努利梁的大变形和大的旋转。由梁的斜率积分得到梁的形心线位置，该位置可以用欧拉参数表示。采用超球面插值函数保证欧拉参数的归一化约束方程始终满足。一个梁单元的每个节点只有四个节点坐标，这是显着少于那些在绝对节点坐标公式（ANCF）和有限元法（FEM）。利用带约束系统的拉格朗日方程推导了梁的控制方程和约束方程，并采用微分代数方程（DAE）求解器求解。目前的提法可以用来计算静态平衡和线性和非线性动力学的欧拉-伯努利梁下任意，集中和分布的力量。虽然质量矩阵比ANCF中的更复杂，但刚度矩阵和广义力更简单，这适合于计算梁的平衡。几个数值例子来证明当前配方的性能。结果表明，目前的公式可以达到相同的精度ANCF和有限元与较少的坐标。
In this investigation, a new singularity-free formulation of a three-dimensional Euler–Bernoulli beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope, which can be easily expressed by Euler parameters. The hyperspherical interpolation function is used to guarantee that the normalization constraint equation of Euler parameters is always satisfied. Each node of a beam element has only four nodal coordinates, which are significantly fewer than those in an absolute node coordinate formulation (ANCF) and the finite element method (FEM). Governing equations of the beam and constraint equations are derived using Lagrange's equations for systems with constraints, which are solved by a differential-algebraic equation (DAE) solver. The current formulation can be used to calculate the static equilibrium and linear and nonlinear dynamics of an Euler–Bernoulli beam under arbitrary, concentrated, and distributed forces. While the mass matrix is more complex than that in the ANCF, the stiffness matrix and generalized forces are simpler, which is amenable for calculating the equilibrium of the beam. Several numerical examples are presented to demonstrate the performance of the current formulation. It is shown that the current formulation can achieve the same accuracy as the ANCF and FEM with a fewer number of coordinates.