Critical velocities are investigated for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. The determination of critical velocities ultimately comes down to discrimination of the existence of multiple real roots of an algebraic equation with real coefficients of the 4th degree, which can be solved by employing Descartes sign method and complete discrimination system for polynomials. Numerical calculations for the European high-speed rail show that there are at most four critical velocities for an infinite Timoshenko beam, which is very different from those gained by others. Furthermore, the shear wave velocity must be the critical velocity, but the longitudinal wave velocity is not possible under certain conditions. Further numerical simulations indicate that all critical velocities are limited to be less than the longitudinal wave velocity no matter how large the foundation stiffness is or how high the loading frequency is. Additionally, our study suggests that the maximum value of one group velocity of waves in Timoshenko beam should be one “dangerous” velocity for the moving load in launching process, which has never been referred to in previous work. 1. Introduction Moving-load problems have received a great deal of attention worldwide in the past several decades. The earliest moving-load problems are about railway bridges excited by traveling trains. Then the application areas gradually have been extended to high-speed commuter trains, missile sled test tracks, high-speed projectile launchers, and so on. “Critical velocity” is a phenomenon that has been found in beams subjected to moving loads. A resonant wave in a beam can be induced when a load moves at the critical velocity, which results in an unbounded increase of the displacements, rotation, and bending moments of the beam for an undamped case. In reality, damping always exists, but even so, a very large deflection of the beam may occur if the moving load approaches the critical velocity [1–3]. Researches show that the “critical velocity” phenomenon may shorten life of launchers due to high stress, fatigue, premature wear, and gouging of rails [4]. Therefore, it is very necessary to determine the critical velocity for a beam subjected to a moving load. Timoshenko [3] firstly derived an expression for the critical velocity of a concentrated load moving along the Bernoulli-Euler beam resting on a continuous elastic foundation. It was proposed that the critical velocity is analogous to an additional longitudinal compressive force
References
[1]
J. T. Kenney, “Steady state vibrations of beam on elastic foundation for moving load,” Journal of Applied Mechanics, vol. 21, pp. 359–364, 1954.
[2]
V. V. Krylov, A. R. Dawson, M. E. Heelis, and A. C. Collop, “Rail movement and ground waves caused by highspeed trains approaching track-soil critical velocities,” Proceedings of the Institution of Mechanical Engineers F Journal of Rail and Rapid Transit, vol. 214, no. 2, pp. 107–116, 2000.
[3]
S. Timoshenko, “Method of analysis of statical and dynamical stresses in rail,” in Proceedings of the 2nd International Congress of Applied Mechanics, pp. 1–12, Zurich, Switzerland, 1927.
[4]
K. B. Lewis and N. V. Nechitailo, “Transient resonance in hypervelocity launchers at critical velocities,” IEEE Transactions on Magnetics, vol. 43, no. 1, pp. 157–162, 2007.
[5]
S. H. Crandall, “The Timoshenko beam on an elastic foundation,” in Proceedings of the 3rd Midwestern Conference on Solid Mechanics, pp. 146–159, 1957.
[6]
A. L. Florence, “Traveling force on a Timoshenko beam,” ASME Journal of Applied Mechanics, vol. 32, no. 2, pp. 351–359, 1965.
[7]
C. R. Steele, “The Timoshenko beam with a moving load,” Journal of Applied Mechanics, vol. 35, pp. 481–488, 1968.
[8]
S. Chonan, “Moving harmonic load on an elastically supported Timoshenko beam,” Journal of Applied Mathematics and Mechanics, vol. 58, no. 1, pp. 9–15, 1978.
[9]
Y.-H. Chen, Y.-H. Huang, and C.-T. Shih, “Response of an infinite tomoshenko beam on a viscoelastic foundation to a harmonic moving load,” Journal of Sound and Vibration, vol. 241, no. 5, pp. 809–824, 2001.
[10]
I. R. McNab, F. Stefani, M. Crawford et al., “Development of a naval railgun,” IEEE Transactions on Magnetics, vol. 41, no. 1, pp. 206–210, 2005.
[11]
L. Yang, X. Hou, and Z. Zeng, “A complete discrimination system for polynomials,” Science in China E, vol. 39, no. 6, pp. 628–646, 1996.
[12]
L. Yang, X. R. Hou, and Z. B. Zeng, “An alternative algorithm for determining the number of real roots of a polynomial,” in Proceedings of the International Workshop on Logic and Software Engineering, World Scientific, Singapore, 1996.
[13]
M. H. Kargarnovin and D. Younesian, “Dynamics of Timoshenko beams on Pasternak foundation under moving load,” Mechanics Research Communications, vol. 31, no. 6, pp. 713–723, 2004.
[14]
J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, New York, NY, USA, 1973.
[15]
P. M. Fitzpatrick, Advanced Calculus, Thomson, Brooks/Cole, 2nd edition, 2006.
[16]
A. V. Metrikine and H. A. Dieterman, “Lateral vibrations of an axially compressed beam on an elastic half-space due to a moving lateral load,” European Journal of Mechanics A/Solids, vol. 18, no. 1, pp. 147–158, 1999.
