基于绝对节点坐标法的大变形柔性梁几种动力学模型研究

章孝顺; 章定国; 陈思佳; 洪嘉振

doi:10.7498/aps.65.094501

摘要

对在平面内大范围转动的大变形柔性梁动力学进行了研究, 基于绝对节点坐标法建立了一种新的大变形柔性梁的非线性动力学模型. 该动力学模型中考虑了柔性梁的轴向拉伸变形和横向弯曲变形, 利用Green-Lagrangian应变张量计算柔性梁的轴向应变及应变能, 利用曲率的精确表达式计算柔性梁的横向弯曲变形能. 运用拉格朗日恒等式给出了柔性梁横向弯曲变形能新的表达式, 该变形能表达式更加简洁, 通过新的变形能表达式得到了新的弹性力模型, 由此得到的动力学方程可以精确地描述柔性梁的几何大变形问题. 通过与高次耦合模型以及ANSYS中BEAM188非线性梁单元模型的比较, 验证了本模型在计算大变形时的正确性以及高次耦合模型在处理大变形问题时的不足. 进一步研究发现, 新的广义弹性力模型可以适当地简化, 给出了两种简化模型, 根据不同模型的计算效率以及计算精度的比较确定了不同模型的适用范围.

关键词:

Abstract

With the development of space technology, flexible appendages such as lightweight manipulators and satellite antennas, often appear in spacecrafts. Usually, the large overall motion of the flexible appendage will bring about large deformation problem. And there is a strong nonlinear coupling between the large overall motion and deformation of the flexible appendage, which brings about a large challenge to the precise control of the spacecraft. Dynamics of a rotating flexible planar beam with large deformation is investigated in this paper. A new nonlinear dynamic model of a flexible beam with large deformation is established based on an absolute node coordinate formulation (ANCF). The longitudinal and bending deformations of the flexible beam are both considered in the model. The longitudinal strain energy and bending strain energy of the beam can be calculated by using Green-Lagrangian strain tensor and the exact expression of the flexible beam curvature, respectively. A new concise expression of the bending deformation energy can be obtained by using the Lagrange identical equation. The new elastic force model is derived from the new expression of the deformation energy. The dynamic equations of the present model can precisely deal with the large deformation problem of flexible beams. Then, simulation results from three dynamic models, including the ANCF model, the high order coupling model (HOC model), and the BEAM188 model in ANSYS, are compared to prove the validity of the ANCF model proposed in this paper. And we can also find the deficiency of the HOC model from the simulation. Further study shows that the new generalized elastic force model can be simplified properly. Two simplified models are presented in this paper. The applicabilities of the simplified models are pointed out from the viewpoints of computational efficiency and accuracy. A dimensionless parameter denoted as is introduced to describe the extent to which a flexible beam pendulum undergoing free falling motion is deformed. The deformation of the flexible beam increases as increases. Considering the calculating efficiency of the dynamic model, when is small, simplified model I is chosen preferentially; when is big, simplified model Ⅱ is adopted preferentially.

Keywords:

作者及机构信息

1.
南京理工大学理学院, 南京 210094;

2.
浙江大学宁波理工学院, 宁波 315100;

3.
上海交通大学工程力学系, 上海 200240

通信作者: 章定国, zhangdg419@mail.njust.edu.cn

基金项目: 国家自然科学基金(批准号: 11272155, 11302192, 11132007)和江苏省333工程(批准号: BRA2011172)资助的课题.

Authors and contacts

1.
School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China;

2.
Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China;

3.
Department of Engineering Mechanics, Shanghai Jiaotong University, Shanghai 200240, China

Corresponding author: Zhang Ding-Guo, zhangdg419@mail.njust.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272155, 11302192, 11132007), and the 333 Project of Jiangsu Province, China (Grant No. BRA2011172).

参考文献

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施引文献

[1]	Kane T R, Ryan R R, Banerjee A K 1987 J. Guid. Control Dyn. 10 139
[2]	Banerjee A K, Kane T R 1989 J. Appl. Mech. 56 887
[3]	Liu J Y, Hong J Z 2004 J. Sound Vib. 278 1147
[4]	Liu J Y, Li B, Hong J Z 2006 Chin. J. Theor. Appl. Mech. 38 276 (in Chinese) [刘锦阳, 李彬, 洪嘉振 2006 力学学报 38 276]
[5]	Cai G P, Hong J Z Yang S X 2005 Mech. Res. Commun. 32 173
[6]	He X S, Deng F Y, Wu G Y, Wang R 2010 Acta Phys. Sin. 59 25 (in Chinese) [和兴锁, 邓峰岩, 吴根勇, 王睿 2010 59 25]
[7]	He X S, Li X H, Deng F Y 2011 Acta Phys. Sin. 60 024502 (in Chinese) [和兴锁, 李雪华, 邓峰岩 2011 60 024502]
[8]	Chen S J, Zhang D G, Hong J Z 2013 Chin. J. Theor. Appl. Mech. 45 251 (in Chinese) [陈思佳, 章定国, 洪嘉振 2013 力学学报 45 251]
[9]	Fan J H, Zhang D G 2014 Acta Phys. Sin. 63 154501 (in Chinese) [范纪华, 章定国 2014 63 154501]
[10]	Simo J C, Quoc L V 1986 Comput. Methods Appl. Mech. Engineer. 58 79
[11]	Qi Z H, Fang H Q, Zhang Z G, Wang L 2014 Appl. Math. Mech. 35 498 (in Chinese) [齐朝晖, 方慧青, 张志刚, 王刚 2014 应用数学和力学 35 498]
[12]	Shabana A A, Hussien H A, Escalona J L 1998 J. Mech. Design 120 188
[13]	Berzeri M, Shabana A A 2000 J. Sound Vib. 235 539
[14]	Gerstmayr J, Irschik H 2008 J. Sound Vib. 318 461
[15]	Peng L, Shabana A A 2010 Nonlinear Dyn. 61 193
[16]	Tian Q, Zhang Y, Chen L, Yang J J 2010 Nonlinear Dyn. 60 489
[17]	Liu C, Tian Q, Hu H Y 2012 Nonlinear Dyn. 70 1903
[18]	Nachbagauer K, Gruber P, Gerstmayr J 2013 J. Computat. Nonlinear Dyn. 8 021004
[19]	Shabana A A 2015 J. Computat. Nonlinear Dyn. 10 024504
[20]	Chung J, Yoo H H 2002 J. Sound Vib. 249 147

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