Analysis of Critical Velocities for an Infinite Timoshenko Beam Resting on an Elastic Foundation Subjected to a Harmonic Moving Load

Critical velocities are investigated for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonicmoving load.The determination of critical velocities ultimately comes down to discrimination of the existence ofmultiple real roots of an algebraic equation with real coefficients of the 4th degree, which can be solved by employing Descartes signmethod 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 nomatter 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.


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][2][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 causing beam buckling.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.The study of Florence [6] found that a semiinfinite Timoshenko beam under a moving concentrated load exhibited a singular behavior when the load speed was equal to the shear wave speed or longitudinal wave speed.Steele [7] investigated the same problem but showed that the solution was bounded when the load speed was equal to the shear 2 Shock and Vibration or longitudinal wave speed.Furthermore, his study indicated that the behavior of the Timoshenko beam is quite similar to that of the Bernoulli-Euler beam for load speed somewhat less than the shear wave speed.Chonan [8] showed that the frequency of the moving load has considerable effect on the dynamical behavior of the Timoshenko beam.Then, later researchers usually investigated critical velocity problem by considering a harmonic load moving on the Timoshenko beam.Chen et al. [9] discussed the critical velocity of an infinite Timoshenko beam to a harmonic moving load by establishing its dynamic stiffness matrix.The dynamic response of a European high-speed railway subjected to a harmonic moving load was calculated to show that there is only a critical velocity when the frequency of the harmonic load  is equal to zero.Moreover, two critical velocities approached one as the loading frequency approached zero, and they separated from the critical velocity for  = 0 more and more as  increases; one is decreasing and the other is increasing from the critical velocity for  = 0, respectively.Further numerical simulations showed that the critical velocity increases as the foundation stiffness increases for a given loading frequency.
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 [9].However, one concern in the launching process is gouging, a form of damage that occurs in the range of 1.5-1.8km/s, which limits the rail life [10].Thus, the distribution of critical velocities faster than 1.1 km/s needs to be analyzed for optimization of the railgun barrel design.
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 [11].Our research suggests that there may be at most four critical velocities if the lowest phase velocity of waves in the Timoshenko beam is less than the shear wave velocity.Moreover, the shear wave velocity must be one critical velocity, but the longitudinal wave velocity is not possible for the Timoshenko beam under certain conditions.Additionally, 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.
The rest of the paper is organized as follows.In Section 2, complete discrimination system for polynomials is introduced briefly.In Section 3, the critical velocities for an infinite Timoshenko beam resting on an elastic foundation under a harmonic moving load are investigated.Numerical simulations are carried out to verify the correctness of analysis results in Section 4. Conclusions are drawn in Section 5.

Complete Discrimination System for Polynomials
Given a polynomial () with real coefficients, its derivative is written as Discr() denotes the Sylvester matrix of () and   () [11,12] and   ();  = 1, 2, . . ., , denotes the determinant of the submatrix of Discr() formed by the first 2 rows and the first 2 columns.If the number of the sign changes of the discriminant sequence is V, then the number of the pairs of distinct conjugate imaginary roots of () equals V. Furthermore, if the number of nonvanishing members of the list is , then the number of the distinct real roots of () equals  − 2V.

Analysis of Critical Velocities for an Infinite Timoshenko Beam under a Harmonic Moving Load
Consider an infinite Timoshenko beam resting on a Winklertype elastic foundation subjected to a harmonic moving load.Its governing equation for an undamped case can be expressed by [13]: in which  = (, ) and  = (, ) are beam deflection and slope., , , , , , , and  are the modulus of elasticity, shear modulus, cross-sectional area of the beam, cross-sectional moment of inertia, the stiffness of the foundation per unit length, sectional shear coefficient, beam material density, and the mass of the beam per unit length, respectively.  ( − V) represents a harmonically varying load with the frequency  moving at a constant velocity V,  = √ −1, and (⋅) is the Dirac's delta function.
Eliminating , (4) becomes (the analysis result is the same if  is eliminated in (4)) . ( Applying the following positive and inverse Fourier transforms to ( 5) where  is the transformed field of , then the stable-state response (slope) of the Timoshenko beam can be expressed by where V 1 = √/, V 2 = √/ are the shear and longitudinal wave velocities, respectively;  1 = √/,  2 = √/ are two "cutoff " frequencies [14] in the Timoshenko beam.The descriptions about the phase and group velocities of waves in the Timoshenko beam are given in Appendix below.
Our goal is to determine critical velocities of the moving load at which the steady-state beam displacement  is infinite.Therefore, the load velocity related to the divergence of integral (7) is the critical velocity.The integrand in (7) decreases as  → ±∞, so the divergence of the integral cannot be related to integration at infinity.Thus, the only possibility for the divergence of integrand in (7) is that Φ() has real zeros.Furthermore, the real zeros have to be of the second or higher order, since for the simple real zero the integral converges in the Cauchy principle sense [15,16].A detailed discussion for Φ() possessing real zeros of at least the second order is as follows.
(1) Consider  = 0. Equation ( 8) can be written as where  1 is defined by (8), 9) can only have simple real zeros.Thus, the shear and longitudinal wave velocities V 1 , V 2 cannot be critical velocities when  = 0.
If V ̸ = V 1 , V 2 , based on complete discrimination system for polynomials, the discriminant sequence of ( 9) is given by where Then, the multiplicities of real zeros of ( 9) are greater than or equal to 2 if and only if  1 =  2 = 0 and  1  1 < 0. Solving the equation  1 =  2 = 0, one has where . The critical velocity defined by ( 11) also has to satisfy the condition (2) Consider  > 0 and V = V 1 .
The discriminant sequence of ( 8) is given by where . Based on complete discriminant system for polynomials, (8) has multiple real zeros if and only if  3 = 0.The critical velocity can be derived by solving  3 = 0 for a given loading frequency .
First, consider that the foundation stiffness is taken as  = 1.6 × 10 7 N/m 2 .The change curves of the phase and group velocities of naturally propagating waves (V 1,2 ,  1,2 ) versus wave number  in Timoshenko beam are presented in Figure 1 based on (A.5) in Appendix.As shown, the minimum value of the phase velocity V 1 is V 1 min = 558.9,and the maximum value of the group velocity  1 is  1 max = 1512.9.In addition, the shear and longitudinal wave velocities in Timoshenko beam are V 1 = √/ = 1399.7 and V 2 = √/ = 5047.3,and the two cutoff frequencies are  1 = √/ = 514.9and  2 = √/ = 22182.7.
(1) If loading frequency  = 0, critical velocity V cr = V 1 min .Furthermore, shear and longitudinal wave velocities cannot be the critical velocity for  = 0.
(2) The shear wave velocity V 1 is the critical velocity if loading frequency satisfies  = 1835.5 or 21372.1.
(3) The longitudinal wave velocity V 2 cannot be the critical velocity for any loading frequency .
The plot of critical velocity V cr versus loading frequency  in Timoshenko is shown in Figure 2, in which   is the resonant loading frequency obtained by solving equation  3 = 0 (defined by ( 19)) with V =  1 max .
From Figure 2, there is only one critical velocity at V cr = V 1 min when loading frequency  = 0; as  is increasing, two critical velocities separate from V 1 min more and more; one decreases and vanishes when  >  1 and the other increases and vanishes when  >   .Beyond that, two other critical velocities approach the shear and longitudinal wave velocities V 1 , V 2 , respectively, when  approaches zero.As  is growing, one critical velocity increases from V 1 and disappears when  >   ; the other decreases from V 2 and disappears when  >  2 .It is shown that there are at most four critical velocities for the Timoshenko beam when  = 1.6 × 10 7 N/m 2 .
Next, consider that the foundation stiffness  has a wide range changing from 5 × 10 6 to 1 × 10 10 N/m 2 .One cutoff frequency  1 will increase with increasing the foundation stiffness , while in the other cutoff frequency  2 , the shear and longitudinal wave velocities V 1 , V 2 remain unchanged.The relationships between the critical velocity V cr and the loading frequency  when  = 5 × 10 6 , 1 × 10 8 , 1 × 10 9 , 1 × 10 10 N/m 2 , respectively, are shown in Figures 3(a In Figures 3-6, V 1 min and  1 max still represent the minimum value of the phase velocity V 1 and the maximum value of the group velocity  1 , respectively.
In accordance with Figures 3-6, V 1 min increases and  1 max decreases with the increase of the foundation stiffness  in the Timoshenko beam.V 1 min is always the critical velocity for loading frequency  = 0 provided that V 1 min is less than the shear wave velocity Once the shear wave velocity V 1 becomes the minimum phase velocity (or  1 max < V 1 ), there is no critical velocity for  = 0 and there are at most two critical velocities for a given loading frequency (see Figure 6).From Figures 3(b)-5(b), it should be noted that the critical velocity, which approaches the longitudinal wave velocity V 2 when  = 0 and decreases with increasing  and then disappears when  >  2 , does not change with the foundation stiffness in the range 5 × 10 6 ≤  ≤ 1 × 10 10 N/m 2 .
To demonstrate the correctness of our numerical results, the influences of the change of foundation stiffness  on critical velocity V cr and loading frequency  are shown in Figure 7, in which , V cr , and  are restricted in the region of 5 × 10 6 ≤  ≤ 1 × 10 10 N/m 2 , V cr < 2 km/s, and  < 4 × 10 3 rad/s.Comparing our results which are presented in Figure 7 with that obtained by Chen et al. [9], our calculation results for V cr < 1.1 km/s and  < 500 rad/s are exactly the same as that derived by them.However, their research paper did not give any details about the distribution of critical velocities in the range of V cr > 1.1 km/s and  > 500 rad/s.

Conclusions
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 1.5-1.8km/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.(1) 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.
(2) 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.
(3) The shear wave velocity V 1 is always one critical velocity and the longitudinal wave velocity V 2 cannot be the critical velocity for any loading frequency if V 1 /V 2 < 1/2 and  1 / 2 < √ 2/2, where  1 and  2 are two cutoff frequencies in the Timoshenko beam.

Figure 1 :
Figure 1: The change curves of the phase and group velocities of naturally propagating waves versus wave number  in the Timoshenko beam for  = 1.6 × 10 7 N/m 2 .
)-6(a).The dependencies of phase and group velocities (V 1 ,  1 ) of naturally propagating waves in the Timoshenko beam on wave number  are shown in Figures 3(b)-6(b) accordingly.