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

Structural Damping Effects on Stability of a Cantilever Column under Sub-tangentially Follower Force

종동력을 받는 외팔기둥의 동적 안정성에 미치는 구조감쇠 효과

  • Received : 2016.03.03
  • Accepted : 2016.06.24
  • Published : 2016.11.20
Download PDF
⟨ Previous Next ⟩

Abstract

A stability theory of a damped cantilever column under sub-tangential follower forces is first summarized based on the stability map. It is then demonstrated that internal and external damping can be exactly transformed to Rayleigh damping so that the damping coefficients can be effectively determined using proportional damping. Particularly a parametric study with variation of damping coefficients is performed in association with flutter loads of Beck's column and it is shown that two damping coefficients can be correctly estimated for real systems under the assumption of Rayleigh damping. Finally a frequency equation of a cantilever beam subjected to both a sub-tangentially follower force and two kinds of damping forces is presented in the closed-form and its stability maps are constructed and compared with FE solutions in the practical range of damping coefficients.

안정성 지도(stability map)을 이용하여 부분 종동력(sub-tangentailly follower force)를 받는 외팔 기둥의 동적안정성 이론을 요약한다. Rayleigh 감쇠를 가정하여 내적 및 외적 감쇠효과를 2개의 감쇠비를 통하여 반영하고, 감쇠비 변화에 따른 플러터하중의 변화와 관련된 매개변수 연구를 수행한다. 또한, 종동력을 받는 외팔기둥에 대한 진동수 방정식의 엄밀해를 유도하고, 특정 감쇠비 범위에 대한 안정성 지도를 유한요소 해석결과와 함께 비교/분석한다.

Keywords

Acknowledgement

Grant : BK21플러스

Supported by : 성균관대학교

References

  1. Beck, M., 1952, Die Knicklast des Einseitig Eingespannten, Tangential Gedrückten Stabes, Zeitschrift Für Angewandte Mathematik und Physik, Vol. 3, No. 6, pp. 476~477. https://doi.org/10.1007/BF02025577
  2. Ziegler, H., 1968, Principles of Structural Stability, Blaisdell Publishing Company, Waltham, Massachussetts.
  3. Bolotin, V. V., 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, NewYork.
  4. Leipholz, H. H. E., 1980, Stability of Elastic Systems, SijthotI & Noordhoff, Rijn, The Netherlands.
  5. Rao, B. N. and Rao, G. V., 1990, Stability of a Cantilever Column under a Tip-concentrated Sub-tangential Follower Force with Damping, Journal of Sound and Vibration, Vol. 138, No. 2, pp. 341~344. https://doi.org/10.1016/0022-460X(90)90548-E
  6. Vitaliani, R. V., Gasparini, A. M. and Saetta, A. V., 1997, Finite Element Solution of the Stability Problem for Nonlinear Undamped and Damped System sunder Non-conservative Loading, International Journal of Solids and Structures, Vol. 34, No. 19, pp. 2497~2516. https://doi.org/10.1016/S0020-7683(96)00115-1
  7. Sugiyama, Y. and Langthjem, M. A., 2007, Physical Mechanism of the Destabilizing Effect of Damping in Continuous Non-conservative Dissipative Systems, Physical Mechanism of the Destabilizing Effect of Damping in Continuous Non-conservative Dissipative Systems, Vol. 42, No. 1, pp. 132~145. https://doi.org/10.1016/j.ijnonlinmec.2006.11.011
  8. Kirillov, O. N. and Verhulst, F., 2010, Paradoxes of Dissipation-induced Destabilization or Who Opened Whitney’s Umbrella?, Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 90, No. 6, pp. 462~488. https://doi.org/10.1002/zamm.200900315
  9. Luongo, A., D’Annibale, F., 2014, On the Destabilizing Effect of Damping on Discrete and Continuous Circulatory Systems, Journal of Sound and Vibration, Vol. 333, No. 24, pp. 6723~6741. https://doi.org/10.1016/j.jsv.2014.07.030
  10. Lee, S. Y., Chen, T. Y. and Wang, W. R., 1980, Non-conservative Instability of a Timoshenko Beam Subjected to a Partially Tangential Follower Force, Journal of Sound and Vibration, Vol. 188, No. 1, pp. 25~38. https://doi.org/10.1006/jsvi.1995.0576
  11. Sundararamaiah, V. and Venkateswara Rao, G., 1983, Stability of Short Beck and Leipholz Column on Elastic Foundation, AIAA Journal, Vol. 21, No. 7, pp. 1053~1054. https://doi.org/10.2514/3.8201
  12. Chen, L.-W. and Ku, D.-M., 1991, Stability Analysis of a Timoshenko Beam Subjected to Distributed Follower Forces Using Finite Elements, Computers and Structures, Vol. 41, No. 4, pp. 813~819. https://doi.org/10.1016/0045-7949(91)90190-W
  13. Ryu, B. J., Katayama, K. and Sugiyama, Y., 1998, Dynamic Stability of Timoshenko Columns Subjected to Sub-tangential Forces, Computers and Structures, Vol. 68, No. 5, pp. 499~512. https://doi.org/10.1016/S0045-7949(98)00069-8
  14. Attard, M. M., Lee, J.-S. and Kim, M.-Y., 2008, Dynamic Stability of Shear-flexible Beck’s Columns based on Engesser’s and Haringx’s Buckling Theories, Computers and Structures, Vol. 86, No. 21-22, pp. 2042~2055. https://doi.org/10.1016/j.compstruc.2008.04.012
  15. Langthjem, M. A. and Sugiyama, Y., 2000, Dynamic Stability of Columns Subjected to Follower Loads: a Survey, Journal of Sound and Vibration, Vol. 238, No. 5, pp. 809~851. https://doi.org/10.1006/jsvi.2000.3137
  16. Elishakoff, I., 2005, Controversy Associated with the So-called Follower Forces: Critical Overview, Appl. Mech. Reviews, Vol. 58, No. 2, pp. 117~142. https://doi.org/10.1115/1.1849170
  17. Lee, B. K., Li, G., Oh, S. J. and Kim, G. S., 2005, Stability Analysis of Beck’s Column with a Tip Mass Restrained by a Spring, Transactions of the Korean Society for Noise and Vibration Engineering, Vol. 15, No. 11, pp. 1287~1294. https://doi.org/10.5050/KSNVN.2005.15.11.1287
  18. Andersen, S. B. and Thomsen, J. J., 2002, Post-critical Behavior of Beck's Column with a Tip Mass, International Journal of Nonlinear Mechanics, Vol. 37, No. 1, pp. 135~151. https://doi.org/10.1016/S0020-7462(00)00102-5
  19. Lee, J.-S., Kim, N.-I. and Kim, M.-Y., 2007, Sub-tangentially Loaded and Damped Beck’s Columns on Two-parameter Elastic Foundation, Journal of Sound and Vibration, Vol. 306, No. 3-5, pp. 766~789. https://doi.org/10.1016/j.jsv.2007.06.017
  20. Kim, M.-Y., Lee, J.-S. and Attard, M. M., 2013, Stability of Damped Columns on a Winkler Foundation Under Sub-tangential Follower Forces, International Journal of Structural Stability and Dynamics, Vol. 13, No. 2, pp. 1~27.
  21. Detinko, F. M., 2003, Lumped Damping and Stability of Beck Column with a Tip Mass, International Journal of Solids and Structures, Vol. 40, No. 17, pp. 4479~4486. https://doi.org/10.1016/S0020-7683(03)00298-1
  22. Wolfram, S., 1991, MATHEMATICA, 2nded, Addison-Wesley, Reading, MA.
  23. Elishakoff, I., Kaplunov, J. and Nolde, E., 2015, Celebrating the Century of Timoshenko’s Study of Effects of Shear Deformation and Rotary Inertia, Applied Mechanical Reviews, Vol. 67, No. 6, 060802-1–060802-11. https://doi.org/10.1115/1.4031965