Skip to main content
SpringerLink
Log in

A recursive formulation based on corotational frame for flexible planar beams with large displacement

基于共旋坐标的平面大变形梁递推方法

  • Published:
Journal of Central South University Aims and scope Submit manuscript

Abstract

A forward recursive formulation based on corotational frame is proposed for flexible planar beams with large displacement. The traditional recursive formulation has been successfully used for flexible mutibody dynamics to improve the computational efficiency based on floating frame, in which the assumption of small strain and deflection is adopted. The proposed recursive formulation could be used for large displacement problems based on the corotational frame. It means that the recursive scheme is used not only for adjacent bodies but also for adjacent beam elements. The nodal relative rotation coordinates of the planar beam are used to obtain equations with minimal generalized coordinates in present formulation. The proposed formulation is different from absolute nodal coordinate formulation and the geometrically exact beam formulation in which the absolute coordinates are used. The recursive scheme and minimal set of dynamic equations lead to a high computational efficiency in numerical integration. Numerical examples are carried out to demonstrate the accuracy and validity of this formulation. For all of the examples, the results of the present formulation are in good agreement with results obtained using commercial software and the published results. Moreover, it is shown that the present formulation is more efficient than the formulation in ANSYS based on GEBF.

摘要

基于共旋坐标法提出了一种大变形梁递推建模方法, 用于解决含有大变形梁的柔性多体系统的动力学问题。 传统递推方法与浮动做标法相结合, 已应用于柔性多体系统的小变形问题。 本文将递推方法与共旋坐标相结合, 把传统递推方法中物体之间的递推拓展到单元之间的递推, 利用共旋坐标法能高效地解决大变形问题的优点, 使得所提的递推方法可应用于梁的大变形问题。 递推方法优点为获得的是一个 O(N)阶的动力学方程, N 为系统的自由度, 这意味着动力学方程求解时间随着 N 线性增长, 使得该方法能在解决大体量的问题时具有较高的计算效率。 通过静力学以及动力学的算例验证所提方法的正确性与效率。 研究结果表明, 所提方法在解决平面大变形梁问题上具有较高的精度和效率。

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. PARK S, YOO H H, CHUNG J. Vibrations of an axially moving beam with deployment or retraction [J]. AIAA Journal, 2012, 51(3): 686–696. DOI: 10.2514/1.J052059.

    Article  Google Scholar 

  2. BELYTSCHKO T, HSIEH B J. Non-linear transient finite element analysis with convected co-ordinates [J]. International Journal for Numerical Methods in Engineering, 1973, 7(3): 255–271. DOI: 10.1002/nme.1620070304.

    Article  MATH  Google Scholar 

  3. SIMO J C, VU-QUOC L. On the dynamics in space of rods undergoing large motions—A geometrically exact approach [J]. Computer Methods in Applied Mechanics and Engineering, 1988, 66(2): 125–161. DOI: 10.1016/0045-7825(88)90073-4.

    Article  MathSciNet  MATH  Google Scholar 

  4. LIU Zhu-yong, HONG Jia-zhen, LIU Jin-yang. Complete geometric nonlinear formulation for rigid-flexible coupling dynamics [J]. Journal of Central South University, 2009, 16(1): 119–124. DOI: 10.1007/s11771-0090020-8.

    Article  Google Scholar 

  5. FU Zhong-qiu, JI Bo-hai, ZHU Wei, GE HAN-BING. Bending behaviour of lightweight aggregate concrete-filled steel tube spatial truss beam [J]. Journal of Central South University, 2016, 23(8): 2110–2117. DOI: 10.1007/s11771-016–3267-x.

    Article  Google Scholar 

  6. LIU Zhu-yong, LIU Jin-yang. Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system [J]. Multibody System Dynamics, 2017, 40(3): 303–326. DOI: 10.1007/s11044-016-9539-2.

    Article  MathSciNet  MATH  Google Scholar 

  7. WASFY T M, NOOR A K. Computational strategies for flexible multibody systems [J]. Applied Mechanics Reviews, 2003, 56(6): 553–613. DOI: 10.1115/1.1590354.

    Article  Google Scholar 

  8. CHACE M A. Analysis of the time-dependence of multifreedom mechanical systems in relative coordinates [J]. Journal of Manufacturing Science and Engineering, 1967, 89(1): 119–125. DOI: 10.1115/1.3609982.

    Google Scholar 

  9. BOOK W J. Recursive lagrangian dynamics of flexible manipulator arms [J]. The International Journal of Robotics Research, 1984, 3(3): 87–101. DOI: 10.1016/B978-0-08-029357–8.50007-X.

    Article  Google Scholar 

  10. CHANGIZI K, SHABANA A A. A recursive formulation for the dynamic analysis of open loop deformable multibody systems [J]. Journal of Applied Mechanics, 1988, 55(3): 687–693. DOI: 10.1115/1.3125850.

    Article  MATH  Google Scholar 

  11. KIM S S, HAUG E J. A recursive formulation for flexible multibody dynamics, part I: Open-loop systems [J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71(3): 293–314. DOI: 10.1016/0045-7825(88)90037-0.

    Article  MathSciNet  MATH  Google Scholar 

  12. BAE D S, HAUG E J. A recursive formulation for constrained mechanical system dynamics: Part I. Open loop systems [J]. Mechanics of Structures and Machines, 1987, 15(3): 359–382. DOI: 10.1080/08905458708905124.

    Article  Google Scholar 

  13. BAE D S, HAUG E J. A recursive formulation for constrained mechanical system dynamics: Part ii. Closed loop systems [J]. Mechanics of Structures and Machines, 1987, 15(4): 481–506. DOI: 10.1080/08905458708905130.

    Article  Google Scholar 

  14. KIM S S, HAUG E J. A recursive formulation for flexible multibody dynamics, part II: Closed loop systems [J]. Computer Methods in Applied Mechanics and Engineering, 1989, 74(3): 251–269. DOI: 10.1016/0045-7825(89)90051-0.

    Article  MathSciNet  MATH  Google Scholar 

  15. ZNAMENáČEK J, VALáŠEK M. An efficient implementation of the recursive approach to flexible multibody dynamics [J]. Multibody System Dynamics, 1998, 2(3): 227–251. DOI: 10.1023/A:1009761925675.

    Article  MathSciNet  MATH  Google Scholar 

  16. JAIN A, RODRIGUEZ G. Recursive flexible multibody system dynamics using spatial operators [J]. Journal of Guidance, Control, and Dynamics, 1992, 15(6): 1453–1466. DOI: 10.1016/B978-0-444-89856-2.50036-3.

    Article  MATH  Google Scholar 

  17. ARGYRIS J, KELSEY S, KANEEL H. Matrix methods of structural analysis: A precis of recent developments [M]. New York.: MacMillan, 1964.

    MATH  Google Scholar 

  18. BELYTSCHKO T, SCHWER L, KLEIN M J. Large displacement, transient analysis of space frames [J]. International Journal for Numerical Methods in Engineering, 1977, 11(1): 65–84. DOI: 10.1002/nme.1620110108.

    Article  MATH  Google Scholar 

  19. YAKOUB R Y, SHABANA A A. Three dimensional absolute nodal coordinate formulation for beam elements: Implementation and applications [J]. Journal of Mechanical Design, 2001, 123(4): 614–621. DOI: 10.1115/1.1410099.

    Article  Google Scholar 

  20. SHABANA A A, YAKOUB R Y. Three dimensional absolute nodal coordinate formulation for beam elements: Theory. Journal of Mechanical Design [J], 2001, 123(4): 606–613. DOI: 10.1115/1.1410100.

    Article  Google Scholar 

  21. SIMO J C, VU-QUOC L. A three-dimensional finite-strain rod model. Part II: Computational aspects [J]. Computer Methods in Applied Mechanics and Engineering, 1986, 58(1): 79–116. DOI: 10.1016/0045-7825(86)90079-4.

    Article  MATH  Google Scholar 

  22. SIMO J C, TARNOW N, DOBLARE M. Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms [J]. International Journal for Numerical Methods in Engineering, 1995, 38(9): 1431–1473. DOI: 10.1002/nme.1620380903.

    Article  MathSciNet  MATH  Google Scholar 

  23. GERSTMAYR J, SHABANA A. Analysis of thin beams and cables using the absolute nodal co-ordinate formulation [J]. Nonlinear Dynamics, 2006, 45(1): 109–130. DOI: 10.1007/s11071-006-1856-1.

    Article  MATH  Google Scholar 

  24. YU Lei, ZHAO Zhi-hua, TANG Jia-li, REN Ge-xue. Integration of absolute nodal elements into multibody system [J]. Nonlinear Dynamics, 2010, 62(4): 931–943. DOI: 10.1007/s11071-010-9775-6.

    Article  MathSciNet  MATH  Google Scholar 

  25. SOUH B. Absolute nodal coordinate plane beam formulation for multibody systems dynamics [J]. Multibody System Dynamics, 2012, 30(1): 1–11. DOI: 10.1007/s11044-012-9335–6.

    Article  MathSciNet  Google Scholar 

  26. BERZERI M, CAMPANELLI M, SHABANA A A. Definition of the elastic forces in the finite-element absolute nodal coordinate formulation and the floating frame of reference formulation [J]. Multibody System Dynamics, 2001, 5(1): 21–54. DOI: 10.1023/a:1026465001946.

    Article  MATH  Google Scholar 

  27. von DOMBROWSKI S. Analysis of large flexible body deformation in multibody systems using absolute coordinates [J]. Multibody System Dynamics, 2002, 8(4): 409–432. DOI: 10.1023/a:1021158911536.

    Article  MathSciNet  MATH  Google Scholar 

  28. LIU Zhu-yong, HONG Jia-zhen, LIU Jin-yang. Finite element formulation for dynamics of planar flexible multi-beam system [J]. Multibody System Dynamics, 2009, 22(1): 1–26. DOI: 10.1007/s11044-009-9154-6.

    Article  MathSciNet  MATH  Google Scholar 

  29. BAUCHAU O A, TRAINELLI L. The vectorial parameterization of rotation [J]. Nonlinear Dynamics, 2003, 32(1): 71–92. DOI: 10.1023/a:1024265401576.

    Article  MathSciNet  MATH  Google Scholar 

  30. ZUPAN E, SAJE M, ZUPAN D. Quaternion-based dynamics of geometrically nonlinear spatial beams using the runge–kutta method [J]. Finite Elements in Analysis and Design, 2012, 54: 48–60. DOI: 10.1016/j.finel.2012.01.007.

    Article  MathSciNet  MATH  Google Scholar 

  31. BATTINI J M. Large rotations and nodal moments in corotational elements [J]. Cmes-Computer Modeling in Engineering & Sciences, 2008, 33(1): 1–15. DOI: 10.3970/cmes.2008.033.001.

    MathSciNet  MATH  Google Scholar 

  32. CRISFIELD M A, JELENIC G. Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation [J]. Proceedings of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences, 1999, 455(1983): 1125–1147. DOI: 10.1098/rspa.1999.0352.

    Article  MathSciNet  MATH  Google Scholar 

  33. ROMERO I. The interpolation of rotations and its application to finite element models of geometrically exact rods [J]. Computational Mechanics, 2004, 34(2): 121–133. DOI: 10.1007/s00466-004-0559-z.

    Article  MathSciNet  MATH  Google Scholar 

  34. ROMERO I, ARMERO F. An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics [J]. International Journal for Numerical Methods in Engineering, 2002, 54(12): 1683–1716. DOI: 10.1002/nme.486.

    Article  MathSciNet  MATH  Google Scholar 

  35. YANG Cai-jin, HONG Di-feng, REN Ge-xue, ZHAO Zhi-hua. Cable installation simulation by using a multibody dynamic model [J]. Multibody System Dynamics, 2013, 30(4): 433–447. DOI: 10.1007/s11044-013-9364-9.

    Article  MathSciNet  Google Scholar 

  36. BAUCHAU O A, HAN Shi-lei, MIKKOLA A, MATIKAINEN M K. Comparison of the absolute nodal coordinate and geometrically exact formulations for beams [J]. Multibody System Dynamics, 2013, 32(1): 67–85. DOI: 10.1007/s11044-013-9374-7.

    Article  MathSciNet  Google Scholar 

  37. HONG Jia-zhen. Computational dynamics of multibody systems [M]. Beijing: Higher Education Press, 1999. (in Chinese)

    Google Scholar 

  38. ARNOLD M, BRüLS O. Convergence of the generalized-α scheme for constrained mechanical systems [J]. Multibody System Dynamics, 2007, 18(2): 185–202. DOI: 10.1007/s11044-007-9084-0.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Tan-hui Wu  (吴坛辉), Zhu-yong Liu  (刘铸永) & Jia-zhen Hong  (洪嘉振)

Authors
  1. Tan-hui Wu  (吴坛辉)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhu-yong Liu  (刘铸永)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jia-zhen Hong  (洪嘉振)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhu-yong Liu  (刘铸永).

Additional information

Foundation item: Projects(11772188,11132007,11202126) supported by the National Natural Science Foundation of China; Project(11ZR1417000) supported by the Natural Science Foundation of Shanghai, China

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, Th., Liu, Zy. & Hong, Jz. A recursive formulation based on corotational frame for flexible planar beams with large displacement. J. Cent. South Univ. 25, 208–217 (2018). https://doi.org/10.1007/s11771-018-3730-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11771-018-3730-y

Key words

关键词

Search

Navigation