ANCF方法在小变形问题中的应用研究  

Application of ANCF Method in Small Deformation Problem

作　　者：韩凌[1] 刘英[1] 

机构地区：[1]南京林业大学机电学院,江苏南京210037

出　　处：《力学季刊》2017年第4期755-764,共10页Chinese Quarterly of Mechanics

摘　　要：绝对节点坐标法(Absolute Nodal Coordinate Formulation,ANCF)具有不存在小变形、小转动假设,质量矩阵为常数矩阵等优点,但由于弯曲应变与轴向应变不一致,带来剪切闭锁问题.本文基于小变形假设,利用ANCF方法得到两节点梁的刚度矩阵,进而将该方法推广到多节点梁,解得多节点梁的刚度矩阵.利用Maple软件编制求解程序,求解矩形截面梁在无约束条件下的固有频率及外伸梁的末端静挠度.通过与ABAQUS仿真结果及解析解对比发现:当梁上节点数较少时,用ANCF方法得到的结果相较于仿真结果、解析解较为"刚硬","剪切闭锁"现象较为严重.随着节点数的增加,ANCF方法得到的计算结果与仿真结果、解析解趋于一致.当梁上节点数增加到31时,对于自由模态的前4阶固有频率,ANCF方法的求解结果与解析解之间的误差均小于3‰;对端部带有集中质量的外伸梁的末端静挠度,ANCF方法的求解结果与解析解之间的误差均小于0.5‰,剪切闭锁问题得到有效解决.Absolute Nodal Coordinate Formulation（ANCF） method has the advantage of no small deformation assumption, no small rotation assumption. In addition, the mass matrix obtained by ANCF is a constant matrix. However, by ANCF method, the problem of shear locking occurs since the bending strain is not consistent with the axial strain. Based on the small deformation assumption, the stiffness matrix of two-nodal beam is obtained by the ANCF method, and this method is extended to get the stiffness matrix of multi-nodal beam. The solution was used to solve the natural frequency of the rectangular section beam under unconstrained conditions and the static deflection of the end point of the extensional beam. By comparing with ABAQUS simulation results and analytical results, it is found that when the number of nodes on the beam is few, the results obtained by the ANCF method are more ＂rigid＂ than the simulation results and the analytical results, indicating serious ＂shear locking＂ issue. With increase of the number of nodes, the calculated result obtained by the ANCF method tends to be the same as the simulation results and the analytical results. When the number of nodes on the beam increases to 31, the error between the result calculated by ANCF and the analytical solution is less than 3‰ for the first four natural frequencies. For the static deflection of the overhanging beam with the concentrate mass at end point, the error between the results calculated by ANCF and the analytical solution is less than 0.5‰. The shear locking issue is effectively addressed.

关 键 词：绝对节点坐标法 小变形 剪切闭锁 固有频率 挠度 刚度矩阵 

分 类 号：O321[理学—一般力学与力学基础] O326[理学—力学]