Forced Vibration Modeling of Rail Considering Shear Deformation and Moving Magnetic Load

전단변형과 시간변화 이동자기력을 고려한 레일의 강제진동모델링

  • 김준수 (한양대학교 기계공학부) ;
  • 김성종 (한양대학교 기계공학부) ;
  • 이혁 (한양대학교 기계공학부) ;
  • 하성규 (한양대학교 기계공학부) ;
  • 이영현 (국방과학연구소)
  • Received : 2013.06.07
  • Accepted : 2013.09.27
  • Published : 2013.12.01


A forced vibration model of a rail system was established using the Timoshenko beam theory to determine the dynamic response of a rail under time-varying load considering the damping effect and stiffness of the elastic foundation. By using a Fourier series and a numerical method, the critical velocity and dynamic response of the rail were obtained. The forced vibration model was verified by using FEM and Euler beam theory. The permanent deformation of the rail was predicted based on the forced vibration model. The permanent deformation and wear were observed through the experiment. Parametric studies were then conducted to investigate the effect of five design factors, i.e., rail cross-section shape, rail material density, rail material stiffness, containment stiffness, and damping coefficient between rail and containment, on four performance indices of the rail, i.e., critical velocity, maximum deflection, maximum longitudinal stress, and maximum shear stress.


Critical Velocity;Timoshenko Beam Theory;Euler Beam Theory;Beam On Elastic Foundation;Dynamic Response


Supported by : 국방과학연구소


  1. Lee, Y.-H., 2012, "Experimental Tests of a 25mm Square-Bore Railgun," 16th International EML Symposium, Vol 49.
  2. Kenney, J. T., 1954, "Steady-State Vibrations of Beams on Elastic Foundations for Moving Load," J. Appl. Mech. Trans ASME, Vol. 21, 76, No. 4, pp. 359-364.
  3. Steele, C. R., 1967 "The Finite Beam with a Moving Load," J. Appl. Mech. Trans. ASME, ser. E, Vol. 34, 89, No. 1, pp. 111-118.
  4. Steele, C. R., 1968, "The Timoshenko Beam with a Moving Load," J. Appl. Mech., Vol. 35, No. 3, pp. 481-488.
  5. Timoshenko, S., Young, D. H. and Weaver, W. Jr., 1974, "Vibration Problems in Engineering," 4th ed. New York: Wiley, pp. 432-435.
  6. Han, S. M., Benaroya, H. and Wei, T., 1999, "Dynamics of Transversely Vibrating Beams Using Four Engineering Theories," J. Sound Vib., Vol. 225, No. 5, pp. 938-988.
  7. Tzeng, J. T., 2005, "Structural Mechanics for Electromagnetic Railguns," IEEE Trans. Magn., Vol. 41, No. 1, pp. 246-250.
  8. Hutchinson, J. R., 2001, "Shear Coefficients for Timoshenko Beam Theory," Transactions of the ASME, Vol. 68.
  9. Crandall, S. H., 1957, "The Timoshenko Beam on an Elastic Foundation," in Proc. 3rd Midwestern Conf. Solid Mechanics, Ann Arbor, MI, pp. 146-159.
  10. Fryba, L., 1999, "Vibration of Solids and Structures Under Moving Loads," 3rd ed. London, U.K.: Telford, pp. 357-387.
  11. Chen, Y.-H. and Huang, Y.-H., 2000, "Dynamic Stiffness of Infinite Timoshenko Beam on Viscoelastic Foundation in Moving Coordinate," Int. J. Numer. Methods Eng., Vol. 48, pp. 1-18.<1::AID-NME858>3.0.CO;2-G
  12. Fang, Q. and Du, M., 2006, "Dynamic Responses of an Elastically Supported Beams with Damping Subjected to Blast Loads." Mechanics in Engineering, 28(2), 53-56.
  13. Hopkins, D. A., Stefani, F., Hsieh, K.-T. and Kim, B.-K., 1999, "Analysis of Startup Behavior in a C-Shaped Armature Using Linked EMAP3D/DYNA3D Finite Element Codes," IEEE Trans. Magn., Vol. 35, No. 1, pp. 59-64.
  14. Ping, L. and Qingyuan, Z., 2003, "Finite Element Analysis of Infinitely Long Beam Resting on Continuous Viscoelastic Foundation Subjected to Moving Loads," Journal of Traffic and Tran sportation Engineering, Vol. 3, pp. 1-6.
  15. Lewis, K. B. and Nechitailo, N. V., 2007, "Transient Resonance in Hypervelocity Launchers at Critical Velocities," IEEE Transactions on Magnetics, Vol. 43, No. 1.
  16. Daneshjoo, K., Rahimzadeh, M., Ahmadi, R. and Ghassemi, M., 2007, "Dynamic Response and Armature Critical Velocity Studies in an Electromagnetic Railgun" IEEE Transactions on Magnetics, Vol. 43, No. 1.
  17. Nechitailo, N.V. and Lewis, K.B., 2006, "Critical Velocity for Rails in Hypervelocity Launchers," International Journal of Impact Engineering, 33, 485-495.
  18. Johnson, A. J. and Moon, F. C., 2007, "Elastic Waves in Electromagnetic Launchers," IEEE Transactions on Magnetics, Vol. 43, No. 1.