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## A meshfree method based on point interpolation for dynamic analysis of rotating cantilever beams

Du Chao-Fan, Zhang Ding-Guo
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• #### 摘要

将基于多项式点插值的无网格方法用于旋转悬臂梁的动力学分析. 利用无网格点插值方法对柔性梁的变形场进行离散, 考虑梁的纵向拉伸变形和横向弯曲变形, 并计入横向弯曲变形引起的纵向缩短, 即非线性耦合项, 运用第二类Lagrange方程推导得到系统刚柔耦合动力学方程. 与有限元法相比, 该方法只需节点信息, 无需定义单元, 具有前处理简单的优势; 构造的形函数采用更多的节点插值, 具有高阶连续性. 将无网格点插值方法的仿真结果与有限元和假设模态法进行比较分析, 验证了该方法的正确性, 并表明其作为一种柔性体离散方法在刚柔耦合多体系统动力学的研究中具有可推广性.

#### Abstract

A meshfree method based on polynominal point interpolation method (PIM) is proposed for dynamic analysis of a rotating cantilever beam; PIM is used to describe the deformation of the flexible beam. Its longitudinal and transverse deformations are both taken into consideration, and the coupling term of the deformation which is caused by the transverse deformation is included in the longitudinal deformation of the beam. The rigid-flexible coupling dynamic equations of the system are derived via employing Lagrange's equations of the second kind. Compared with the finite element method (FEM), only individual nodal data are required in the current method so that its preprocessor is easier, and the shape functions constructed have high-order continuities due to more nodes to be interpolated. Simulation results of the meshfree method are compared with those obtained by using FEM and assumption of modes method. It is demonstrated that the meshfree method can be extended to the field of multibody system dynamics.

#### 作者及机构信息

###### 1. 南京理工大学理学院, 南京 210094
• 基金项目: 国家自然科学基金(批准号: 11272155, 11132007)、江苏省“333”工程(批准号: BRA2011172)和高校基本科研业务专项资金(批准号: 30920130112009)资助的课题.

#### Authors and contacts

###### 1. School of Sciences, Nanjing University of Science and Technology, Nanjing 210094, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11272155, 11132007), the 333 Project of Jiangsu Province, China (Grant No. BRA2011172), and the Fundamental Research Funds for the Central Universities of China (Grant No. 30920130112009).

#### 参考文献

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#### 施引文献

•  [1] Hong J Z, You C L 2004 Journal of Dynamics and Control. 2 1 (in Chinese) [洪嘉振, 尤超蓝 2004 动力学与控制学报 2 1] [2] Dwivedy S K, Eberhard P 2006 Mechanism and Machine Theory 41 749 [3] Fang J S, Zhang D G 2013 Acta Phys. Sin. 62 044501 (in Chinese) [方建士, 章定国 2013 物理学报 62 044501] [4] Zhang X, Liu Y 2004 Meshless Method (Beijing: Tsinghua University Press) pp1-6 (in Chinese) [张雄, 刘岩 2004 无网格法(北京: 清华大学出版社)第1–6页] [5] Liu G R, Gu Y T (translated by Wang J M, Zhou X J) 2007 An Introduction to Meshfree Methods and Their Programming (Shandong: Shandong University Press) pp24-40 (in Chinese) [刘桂荣, 顾元通著 (王建明, 周学军译) 2004 无网格法理论及程序设计(山东: 山东大学出版社)第24–40页] [6] Sanborn G G, Shabana A A 2009 Multibody Syst Dyn. 22 181 [7] Sanborn G G, Shabana A A 2009 Nonlinear Dyn. 58 565 [8] Fan J H, Zhang D G 2012 Journal of Mechanical Engineering 48 59 (in Chinese) [范纪华, 章定国 2012 机械工程学报 48 59] [9] Fan J H, Zhang D G 2014 Acta Phys. Sin. 63 154501 (in Chinese) [范纪华, 章定国 2014 物理学报 63 154501] [10] Fan J H, Zhang D G, Hong J Z 2012 Journal of Dynamics and Control. 10 347 (in Chinese) [范纪华, 章定国, 洪嘉振 2012 动力学与控制学报 10 347] [11] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl. Mech. Eng. 19 3 [12] Liu G R, Gu Y T 1999 Proceedings of 4th Asia-Pacific Conference on Computational Mechanics Singapore, Dec, 1999 p1009 [13] Liu G R, Gu Y T 2001 Int. J. Muner. Meth. Engng. 50 937 [14] Liu G R, Dai K Y, Lim K M, Gu Y T 2002 Computational Mechanics 29 510 [15] Wang J G, Liu G R 2003 Int. J. Numer. Meth. Eng. 54 1623 [16] Belytscko T, Lu Y Y, Gu L 1994 Int. J. Numer. Meth. Eng. 37 229 [17] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059 [18] Atluri S N, Zhu T L 1998 Comput. Mech. 22 117 [19] Chen L, Liew K M 2010 Comput. Mech. 47 455 [20] Liu W K, Jun S, Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081 [21] Cheng R J, Liew K M 2009 Comput. Mech. 45 1 [22] Li X G, Dai B D, Wang L H 2010 Chin. Phys. B 19 120202 [23] Liu J Y, Li B, Hong J Z 2006 Chinese Journal of Theoretical and Applied Mechanics 38 276 (in Chinese) [刘锦阳, 李彬, 洪嘉振 2006 力学学报 38 276] [24] Chung J, Yoo H H 2002 Journal of Sound and Vibration 249 147 [25] 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 物理学报 60 024502]
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##### 出版历程
• 收稿日期:  2014-08-20
• 修回日期:  2014-09-18
• 刊出日期:  2015-02-05

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