[1]张志刚,马新旋,王才东,等.基于三次样条插值的几何精确曲梁单元[J].郑州大学学报(工学版),2023,44(06):61-67.[doi:10. 13705/ j. issn. 1671-6833. 2023. 03. 002]
　ZHANG Zhigang,MA Xinxuan,WANG Caidong,et al.A Geometrically Exact Curved Beam Element Based on Cubic Spline Interpolation[J].Journal of Zhengzhou University (Engineering Science),2023,44(06):61-67.[doi:10. 13705/ j. issn. 1671-6833. 2023. 03. 002]
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基于三次样条插值的几何精确曲梁单元()
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《郑州大学学报(工学版)》[ISSN:1671-6833/CN:41-1339/T]

卷:
44
期数:
2023年06期
页码:
61-67
栏目:
出版日期:
2023-12-25

文章信息/Info

Title:
A Geometrically Exact Curved Beam Element Based on Cubic Spline Interpolation
作者:
张志刚; 马新旋; 王才东; 郑华栋; 王良文
郑州轻工业大学 河南省机械装备智能制造重点实验室,河南 郑州 450002
Author(s):
ZHANG Zhigang;  MA Xinxuan;  WANG Caidong;  ZHENG Huadong;  WANG Liangwen
Henan Key Laboratory of Intelligent Manufacturing of Mechanical Equipment, Zhengzhou University of Light Industry, Zhengzhou 450002, China
关键词:
几何精确梁;  三次样条;  曲梁;  几何非线性;  刚柔耦合
Keywords:
geometrically exact beam;  cubic spline;  curved beam;  geometric nonlinearity;  rigid-flexible coupling
分类号:
O302
DOI:
10. 13705/ j. issn. 1671-6833. 2023. 03. 002
文献标志码:
A
摘要:
为了对包含大变形梁的柔性多体系统进行精确建模和动力学仿真分析,基于三次样条插值构造了一种总 体参数少且方便与 CAD 几何模型融合的大变形平面曲梁单元。 首先,选取单元节点位置矢量和首末两端节点处 位置矢量的弧长导数为整体参数,采用三次样条插值函数对梁的形心线运动进行近似;其次,在充分考虑细长梁变 形基础上,利用 Euler-Bernoulli 梁变形假设由插值得到的形心线切向确定了梁截面的转动;最后,依据几何精确梁 理论推导出了梁的轴向应变和曲率,基于虚功率原理推导出了平面曲梁单元的质量矩阵、节点力列阵和广义外力 列阵,并得到了总体切线刚度矩阵。 相较于现有大变形平面梁单元,所提单元总体参数大幅降低。 此外,由于构造 的单元耦合变形场严格保证了梁截面与形心线切向的垂直关系,这从根本上避免了剪切闭锁现象在应用中的出 现。 在数值算例部分,通过对平面梁静力学几何非线性问题和动力学刚柔耦合问题典型算例求解及对比表明:所提出的几何精确曲梁单元在保证计算精度的同时提高了计算效率。
Abstract:
To accurately model and simulate the multibody system including large deformation beams, a 2D large deformation curved beam element, which had few element parameters and can be integrated with the CAD geometric model, was proposed based on cubic spline interpolation. Firstly,by taking the position vector at each node and the axial position gradient vector at the two-end nodes as the global parameters, the motion of the beam centroid line was approximated using the cubic spline interpolation. Secondly, on the basis of fully considering the deformation of the slender beam, the rotation of the beam cross-section was determined by the centroid tangent according to the deformation assumption of Euler-Bernoulli beam. Finally, the axial strain and curvature of the beam were derived based on the geometrically exact beam theory, the mass matrix, nodal force and generalized external force of the planar curved beam element were derived based on the virtual power principle, and the tangent stiffness matrix was obtained. Compared with the existing large deformation planar beam element, the number of the global parameters of the proposed beam element was greatly reduced. In addition, because the vertical relationship between the beam cross-section and the tangent direction of the centroid line was guaranteed in the proposed coupling deformation field, the shear locking phenomenon in the application could be avoided. In the numerical example, through the calculation and comparison of typical examples including the static geometric nonlinear problems and dynamic rigid flexible coupling problems, it was shown that the calculation efficiency of the proposed geometrically exact curved beam element could be greatly improved while not losing the calculation accuracy.

参考文献/References:

[1] HOWELL L L. Compliant mechanisms[ M] . New York: Wiley, 2001.

 [2] 张志刚, 周翔, 房占鹏, 等. 基于绝对节点坐标方法 的柔顺机 构 动 力 学 建 模 与 仿 真 [ J] . 郑 州 大 学 学 报 (工学版) , 2020, 41(2) : 50-55.
 ZHANG Z G, ZHOU X, FANG Z P, et al. Dynamics modeling and simulation of compliant mechanisms using absolute nodal coordinate formulation [ J ] . Journal of Zhengzhou University ( Engineering Science) , 2020, 41 (2) : 50-55.
 [3] REISSNER E. On one-dimensional finite-strain beam theory: the plane problem [ J] . Zeitschrift Für Angewandte Mathematik Und Physik ZAMP, 1972, 23 ( 5 ) : 795 -804. 
[4] SIMO J C. A finite strain beam formulation. The threedimensional dynamic problem. Part I [ J ] . Computer Methods in Applied Mechanics and Engineering, 1985, 49(1) : 55-70.
 [5] SIMO J C, VU-QUOC L. A three-dimensional finitestrain rod model. part II: computational aspects [ J ] . Computer Methods in Applied Mechanics and Engineering, 1986, 58(1) : 79-116.
 [6] SHABANA A A. ANCF consistent rotation-based finite element formulation [ J ] . Journal of Computational and Nonlinear Dynamics, 2016, 11(1) : 014502. 
[7] CRISFIELD M A, JELENI G. Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation [ J ] . Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 1999, 455 (1983) : 1125-1147.
 [8] ZUPAN E, SAJE M, ZUPAN D. The quaternion-based three-dimensional beam theory[ J] . Computer Methods in Applied Mechanics and Engineering, 2009, 198( 49 / 50 / 51 / 52) : 3944-3956.
 [9] GHOSH S, ROY D. Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam[ J] . Computer Methods in Applied Mechanics and Engineering, 2008, 198(3 / 4) : 555-571.
 [10] SHABANA A A. Uniqueness of the geometric representation in large rotation finite element formulations[ J] . Journal of Computational and Nonlinear Dynamics, 2010, 5 (4) : 044501. 
[11] ZHAO Z H, REN G X. A quaternion-based formulation of Euler-Bernoulli beam without singularity [ J] . Nonlinear Dynamics, 2012, 67(3) : 1825-1835.
 [12] FAN W, ZHU W D, REN H. A new singularity-free formulation of a three-dimensional Euler-bernoulli beam using Euler parameters[ J] . Journal of Computational and Nonlinear Dynamics, 2016, 11(4) : 041013.
 [13] ZHU W D, REN H, XIAO C. A nonlinear model of a slack cable with bending stiffness and moving ends with application to elevator traveling and compensation cables [ J ] . Journal of Applied Mechanics, 2011, 78 ( 4 ) : 041017.
 [14] FAN W. An efficient recursive rotational-coordinate-based formulation of a planar Euler-Bernoulli beam[ J] . Multibody System Dynamics, 2021, 52(2) : 211-227.
 [15] SHABANA A A. An absolute nodal coordinate formulation for the large rotation and deformation analysis of flexible bodies: technical report[R] . Chicago: Department of Mechanical Engineering, University of Illinois, 1996.
 [16] 兰朋, 於祖庆, 赵欣. B􀆧zier 和 B-spline 曲线的绝对结 点坐标 列 式 有 限 元 离 散 方 法 [ J] . 机 械 工 程 学 报, 2012, 48(17) : 128-134.
 LAN P, YU Z Q, ZHAO X. Using absolute nodal coordinate formulation elements to model B􀆧zier and B-spline curve for finite element analysis[ J] . Journal of Mechanical Engineering, 2012, 48(17) : 128-134.
 [17] MATTIASSON K. Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals[ J] . International Journal for Numerical Methods in Engineering, 1981, 17(1) : 145-153. 
[18] OMAR M A, SHABANA A A. A two-dimensional shear deformable beam for large rotation and deformation problems[ J ] . Journal of Sound and Vibration, 2001, 243 (3) : 565-576.

更新日期/Last Update: 2023-10-22