Transactions of the Korean Society for Noise and Vibration Engineering (한국소음진동공학회논문집)

The Korean Society for Noise and Vibration Engineering (한국소음진동공학회)
권호 표지
DOI QR코드

DOI QR Code

Stability Analysis of Beck's Column with a Tip Mass Restrained by a Spring

스프링으로 지지된 자유단에 집중질량을 갖는 Beck 기둥의 안정성 해석

  • Published : 2005.11.01
Download PDF
⟨ Previous Next ⟩

Abstract

The purpose of this paper is to investigate free vibrations and critical loads of the Beck's columns with a tip spring, which carry a tip mass. The ordinary differential equation governing free vibrations of Beck's column subjected to a follower force is derived based on the Bernoulli-Euler beam theory Both the divergence and flutter critical loads are calculated from the load-frequency corves that are obtained by solving the differential equation numerically. The critical loads are presented in the figures as functions of various non-dimensional system parameters such as the subtangential parameter, mass ratio and spring parameter.

이 논문은 자유단이 스프링으로 지지되고 자유단 집중질량을 갖는 Beck 기둥의 안정성 해석에 관한 연구이다. Bernoulli-Euler보 이론을 이용하여 경사종동력을 받는 Beck 기둥의 자유진동을 지배하는 미분방정식과 경계조건을 유도하였다. 이 미분방정식을 수치해석하여 하중-고유진동수 곡선을 얻고 이로부터 발산임계하중 및 동요임계하중을 산출하였다. 수치해석의 결과로부터 경사변수, 집중질량 및 스프링 강성이 임계하중에 미치는 영향을 고찰하였다.

Keywords

References

  1. Bokaian, A., 1988, 'Natural Frequencies of Beams Under Compressive Axial Loads,' Journal of Sound and Vibration, Vol. 126, pp. 49-65 https://doi.org/10.1016/0022-460X(88)90397-5
  2. Ryu, B. J., Yim, K. B., Lee, J. W. and Han, J. S., 1999, 'Vibration and Stability of Non-uniform Tapered Beams Resting on a Two-layered Elastic Foundation', Transactions of the Korean Society for Noise and Vibration Engineering, Vol. 9, No. 4, pp. 828-834
  3. Koiter, W. T., 1999, 'Unrealistic Follower Forces', Journal of Sound and Vibration, Vol. 194, 1996, pp. 636-638 https://doi.org/10.1006/jsvi.1996.0383
  4. Sugiyama, Y., Langthjem, M. A. and Ryu, B.J., 1999, 'Realistic Follower Forces', Journal of Sound and Vibration, Vol. 225, pp. 779-782 https://doi.org/10.1006/jsvi.1998.2290
  5. Beck, M., 1952, 'Die Knicklast des Einseitig Eingespannten, Tangential Gedruckten Stabes,' Zeitschrift fur Angewandte Mathematik und Physik, Vol. 3, pp. 225-228 https://doi.org/10.1007/BF02008828
  6. Kounadis, A and Katsikadelis, J. T., 1976, 'Shear and Rotatory Inertia Effect on Beck's Column', Journal of Sound and Vibration, Vol. 49, pp. 171-178 https://doi.org/10.1016/0022-460X(76)90494-6
  7. Sankaran, G. V. and Rao, G. V., 1976, 'Stability of Tapered Cantilever Columns Subjected to Follower Forces', Computers and Structures, Vol. 6, pp. 217-220 https://doi.org/10.1016/0045-7949(76)90033-X
  8. Pedersen, M., 1977, 'Influence of Boundary Conditions on the Stability of a Column Under Non-conservative Load', International Journal of Solids and Structures, Vol. 13, pp. 445-455 https://doi.org/10.1016/0020-7683(77)90039-7
  9. Yoon, H. I. and Kim, K. S. 1984, 'The Influence of Inertial Moment of Tip Mass on the Stability of Beck's Column', Transactions of the Korean Society of Mechanical Engineers, Vol. 8, No. 2, pp. 119-126
  10. Yoon, H. I. and Kim, K. S., 1985, 'Influence of Spring Constant at Fixed end on Stability of Beck's Column with Tip Mass', Transactions of the Korean Society of Mechanical Engineers, Vol. 9, No. 5, pp. 606-612
  11. Chen, L. W. and Ku, D. M., 1992, 'Eigenvalue Sensitivity in the Stability Analysis of Beck's Column with a Concentrated Mass at the Free end', Journal of Sound and Vibration, Vol. 153, pp. 403-411 https://doi.org/10.1016/0022-460X(92)90373-6
  12. Kuo, S. R. and Yang, Y. B., 1994, 'Critical Load Analysis of SUndamped Non-Conservative Systems Using Bi-eigenvalue Curves', AIAA Journal, Vol. 32, pp. 2462-2468 https://doi.org/10.2514/3.12314
  13. Sato, K., 1996, 'Instability of a Clamped-elastically Restrained Timoshenko Column Carrying a Tip Load, Subjected to a Follower Force', Journal of Sound and Vibration, Vol. 194, pp. 623-630 https://doi.org/10.1006/jsvi.1996.0381
  14. Yoon, H. I., Lim, S. H. and Yu, J. S., 1997, 'Stability of Beck's Column wit a Rotatory Spring Restraining its Free End', Transactions of the Korean Society of Mechanical Engineers A, Vol. 21, No. 9, pp. 1385-1391
  15. Ryu, B. J., Sugiyama, Y. and Lee, G. S., 1997, 'The Influence of an Intermediate Support on the Dynamic Stability of Cantilevered Timoshenko Beams Subjected to Subtangential Follower Forces', Proceedings of Asia-Pacific Vibration Conference '97, Kyongju, Korea, pp. 163-168
  16. Langthjem, A. and Sugiyama, Y., 1999, 'Optimum Shape Design Against Flutter of a Cantilevered Column with an End-mass of Finite Size Subjected to a Non-conservative Load', Journal of Sound and Vibration, Vol. 226, pp. 1-23 https://doi.org/10.1006/jsvi.1999.2211
  17. Andersen, S. B. and Thomsen, J. J., 2002, 'Post-critical Behavior of Beck's Column with a Tip Mass', International Journal of Non-linear Mechanics, Vol. 37, pp. 135-151 https://doi.org/10.1016/S0020-7462(00)00102-5
  18. Dentiko, F. M., 2003, 'Lumped Damping and Stability of Beck Column with a tip mass', International Journal of Solids and Structures, Vol. 40, pp. 4479-4486 https://doi.org/10.1016/S0020-7683(03)00298-1
  19. Rao, B. N. and Rao, G. V., 2004, 'Post-critical Behaviour of Euler and Beck Columns Resting on an Elastic Foundation', Journal of Sound and Vibration, Vol. 276, pp. 1150-1158 https://doi.org/10.1016/j.jsv.2003.11.007
  20. Lee, B. K., Carr. A. J., Lee, T. E. and Ahn, D. S., 2005, 'Elasticas and Buckling Loads of Shear Deformable Tapered Columns', International Journal of Structural Stability and Dynamics, Vol. 5, No. 3, pp. 317-335 https://doi.org/10.1142/S0219455405001684