Skip to main content
Log in

Eulerian and lagrangian descriptions for the vibration analysis of a deploying beam

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

In this study, the equations of motion derived from the Eulerian and Lagrangian descriptions of the vibration analysis of a deploying beam are compared and discussed. After transforming the equations to their corresponding variational equations, the discretized equations for the two descriptions are derived and their equivalence is verified. The numerical time responses obtained from the equations of both descriptions are the same. We recommend use of the Lagrangian description over the Eulerian one when analyzing the vibration of a deploying beam that includes the entirety of the beam inside and outside a rigid wall.

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

Access this article

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

Instant access to the full article PDF.

References

  1. P. K. C. Wang and J. D. Wei, Vibrations in a moving flexible robot arm, Journal of Sound and Vibration, 116(1) (1987) 149–160.

    Article  Google Scholar 

  2. J. Yuh and T. Young, Dynamic modeling of an axially moving beam in rotation: simulation and experiment, Journal of Dynamic Systems, Measurement and, Control, Transactions of the ASME, 113(1) (1991) 34–40.

    Article  Google Scholar 

  3. M. Stylianou and B. Tabarrok, Finite element analysis of an axially moving beam, part I: time integration, Journal of Sound and Vibration, 178(4) (1994) 433–453.

    Article  Google Scholar 

  4. M. Stylianou and B. Tabarrok, Finite element analysis of an axially moving beam, part II: stability analysis, Journal of Sound and Vibration, 178(4) (1994) 455–481.

    Article  Google Scholar 

  5. B. O. Al-Bedoor and Y. A. Khulief, Finite element dynamic modeling of a translating and rotating flexible link, Computer Methods in Applied Mechanics and Engineering, 131(1–2) (1996) 173–189.

    Article  MATH  Google Scholar 

  6. B. O. Al-Bedoor and Y. A. Khulief, Vibrational motion of an elastic beam with prismatic and revolute joints, Journal of Sound and Vibration, 190(2) (1996) 195–206.

    Article  Google Scholar 

  7. B. O. Al-Bedoor and Y. A. Khulief, An approximate analytical solution of beam vibrations during axial motion, Journal of Sound and Vibration, 192(1) (1996) 159–171.

    Article  Google Scholar 

  8. K. Behdinan, M. Stylianou and B. Tabarrok, Dynamics of flexible sliding beams non-linear analysis part I: formulation, Journal of Sound and Vibration, 208(4) (1997) 517–539.

    Article  Google Scholar 

  9. B. O. Al-Bedoor and Y. A. Khulief, General planar dynamics of a sliding flexible link, Journal of Sound and Vibration, 206(5) (1997) 641–661.

    Article  Google Scholar 

  10. R. F. Fung, P. Y. Lu and C. C. Tseng, Non-linearly dynamic modelling of an axially moving beam with a tip mass, Journal of Sound and Vibration, 218(4) (1998) 559–571.

    Article  Google Scholar 

  11. F. Gosselin, M. P. Paidoussis and A. K. Misra, Stability of a deploying/extruding beam in dense fluid, Journal of Sound and Vibration, 299(1–2) (2007) 123–142.

    Article  Google Scholar 

  12. W. Lin and N. Qiao, Vibration and stability of an axially moving beam immersed in fluid, International Journal of Solids and Structures, 45(5) (2008) 1445–1457.

    Article  MATH  Google Scholar 

  13. M. T. Piovan and R. Sampaio, Vibrations of axially moving flexible beam made of functionally graded materials, Thin-Walled Structures, 46(2) (2008) 112–121.

    Article  Google Scholar 

  14. J. R. Chang, W. J. Lin, C. J. Huang and S. T. Choi, Vibration and stability of an axially moving Rayleigh beam, Applied Mathematical Modelling, 34(6) (2010) 1482–1497.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Park, H. H. Yoo and J. Chung, Longitudinal and transverse vibrations of an axially moving beam with deployment or retraction, AIAA Journal, 51(3) (2013) 686–696.

    Article  Google Scholar 

  16. J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-alpha method, Journal of Applied Mechanics, Transactions of the ASME, 60(2) (1993) 371–375.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jintai Chung.

Additional information

Recommended by Associate Editor Jun-Sik Kim

Sungpil Park received his B.S. degree in 2008 from the Department of Mechanical Engineering at Hanyang University. He is currently a Ph.D. candidate in the Department of Mechanical Engineering at Hanyang University. His research interests are the dynamics of robot manipulator and planetary reduction gears.

Hong Hee Yoo received his B.S. and M.S. degrees from the Department of Mechanical Engineering at Seoul National University in 1980 and 1982, respectively. He obtained his Ph.D. degree from the Department of Mechanical Engineering at University of Michigan, Ann Arbor in 1989. He is currently a professor in the School of Mechanical Engineering at Hanyang University.

Jintai Chung received his B.S. and M. S. degrees from the Department of Mechanical Engineering at Seoul National University in 1984 and 1986, respectively. He obtained his Ph.D. degree from the Department of Mechanical Engineering at University of Michigan, Ann Arbor in 1992. He is currently a professor in the Department of Mechanical Engineering at Hanyang University. His research interests are vibration and noise reductions of rotating machines, vehicles and home appliances.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Park, S., Yoo, H.H. & Chung, J. Eulerian and lagrangian descriptions for the vibration analysis of a deploying beam. J Mech Sci Technol 27, 2637–2643 (2013). https://doi.org/10.1007/s12206-013-0708-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-013-0708-4

Keywords

Navigation