Abstract
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.
Highlights
Moving-load problems have received a great deal of attention worldwide in the past several decades
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
By analyzing a Timoshenko beam resting on an elastic foundation and subjected to a concentrated load traveling along its length, Crandall [5] indicated that the Timoshenko beam model produced a total of three critical velocities
Summary
Moving-load problems have received a great deal of attention worldwide in the past several decades. By analyzing a Timoshenko beam resting on an elastic foundation and subjected to a concentrated load traveling along its length, Crandall [5] indicated that the Timoshenko beam model produced a total of three critical velocities. Further numerical simulations showed that the critical velocity increases as the foundation stiffness increases for a given loading frequency. To address the three problems mentioned above, this paper focuses on critical velocities for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. Numerical simulations for the European highspeed rail indicate that the longitudinal wave velocity is the ultimate velocity limit of the critical velocity for the Timoshenko beam regardless of how large the foundation stiffness is or how high the frequency of the harmonic moving load is.
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