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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.

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 [

Timoshenko [

According to previous statements, the knowledge about critical velocities for an infinite Timoshenko beam under a moving load is still not completely clear. First, no consensus has been reached yet even for the number of the critical velocities for the beam subjected to a harmonic moving load. Second, it is still worth to be discussed whether or not the shear and longitudinal wave velocities are the critical velocities. Third, the velocity of moving load traveling on a Timoshenko beam in previous study was just limited to less than 1.1 km/s [

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. By using the Fourier transform method, the determination of critical velocities for Timoshenko beam ultimately comes down to discrimination of the existence of multiple real roots of an algebraic equation with real coefficients of 4th degree. The main tools used here are Descartes sign method and complete discrimination system for polynomials [

The rest of the paper is organized as follows. In Section

Given a polynomial

Consider an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. Its governing equation for an undamped case can be expressed by [

Eliminating

Applying the following positive and inverse Fourier transforms to (

Our goal is to determine critical velocities of the moving load at which the steady-state beam displacement

Consider

Equation (

If

If

(2) Consider

Equation (

The discriminant sequence of (

The shear wave velocity

(3) Consider

Equation (

The discriminant sequence of (

It is easily checked that

(4) Consider

The discriminant sequence of (

Based on complete discriminant system for polynomials, (

To verify the validity of our analysis results on critical velocities, the parameters of an European high-speed rail are used to carry out numerical simulations: ^{2}, ^{2}, ^{4}, ^{2},

First, consider that the foundation stiffness is taken as ^{2}. The change curves of the phase and group velocities of naturally propagating waves (

If loading frequency

The shear wave velocity

The longitudinal wave velocity

If

The change curves of the phase and group velocities of naturally propagating waves versus wave number ^{2}.

The curves of the phase velocities

The curves of the group velocities

The plot of critical velocity

Critical velocity ^{2}.

From Figure ^{2}.

Next, consider that the foundation stiffness ^{2}. One cutoff frequency ^{2}, respectively, are shown in Figures

Critical velocity ^{2}.

Critical velocity

The phase velocity

Critical velocity ^{2}.

Critical velocity

The phase velocity

Critical velocity ^{2}.

Critical velocity

The phase velocity

Critical velocity ^{2}.

Critical velocity

The phase velocity

In accordance with Figures ^{2}.

To demonstrate the correctness of our numerical results, the influences of the change of foundation stiffness ^{2},

Critical velocity

Critical velocity phenomenon in the Timoshenko beam has received extensive concern over the past decades. In the past research, there is no unified understanding of the number of critical velocities for the Timoshenko beam. Some scholars considered that the number of critical velocities is three; however, others argued that it should be one or two. Furthermore, there is no general agreement whether or not the shear and longitudinal wave velocities in the Timoshenko beam are the critical velocities. Additionally, an important form of damage in launchers is gouging of rails, which occurs when the velocity of the moving load is in the range of 1.5–1.8 km/s. Nevertheless, few researches focus on critical velocities faster than 1.1 km/s for the Timoshenko beam. To clarify these problems mentioned above, in this paper, critical velocities are analyzed for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. Some conclusions drawn from our study are given as follows.

Numerical simulations for the European high-speed rail show that the maximum number of critical velocities for the Timoshenko beam is four if the lowest phase velocity of waves is less than the shear wave velocity, while the maximum number of critical velocities is two if the lowest phase velocity of waves is equal to the shear wave velocity. The number of critical velocities changes with the frequency of the moving load from zero to four.

There exists only one critical velocity equal to the lowest phase velocity of waves in the Timoshenko beam if the loading frequency is zero, provided that the lowest phase velocity of waves is less than the shear wave velocity. There is no critical velocity for the loading frequency which is equal to zero if the lowest phase velocity of waves is equal to the shear wave velocity.

The shear wave velocity

Numerical simulations for the European high-speed rail indicate that the maximum value of the group velocity

According to (

The slope of the Timoshenko beam subjected to a moving load can be sought in the following form:

From (

The authors declare that there is no conflict of interests regarding the publication of this paper.

All authors carried out the proofreading of the paper. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final paper.

This work is supported by the National Natural Science Foundation of China (Grant nos. 11002103 and 11302119), the State Key Program of National Natural Science of China (Grant no. 11032009), and the National Natural Science Foundation of China (Grant no. 11272236).