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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.
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Keywords:
- flexible beam /
- point interpolation method /
- meshfree method /
- natural frequency
[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 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 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 60 024502]
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[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 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 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 60 024502]
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