基于三次样条插值的几何精确曲梁单元  

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 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)

机构地区：[1]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

出　　处：《郑州大学学报（工学版）》2023年第6期61-67,共7页Journal of Zhengzhou University（Engineering Science）

基　　金：国家自然科学基金资助项目(52075500,11602228);河南省高校科技创新人才支持计划项目(22HASTIT023)。

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。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.

关 键 词：几何精确梁 三次样条 曲梁 几何非线性 刚柔耦合 

分 类 号：O302[理学—力学]