The time domain moving Green function of a railway track and its application to wheel–rail interactions

Journal of Sound and Vibration 377 (2016) 133–154

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The time domain moving Green function of a railway track and its application to wheel–rail interactions X. Sheng n, X. Xiao, S. Zhang State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu, Sichuan, China

a r t i c l e i n f o

abstract

Article history: Received 30 January 2016 Received in revised form 12 April 2016 Accepted 5 May 2016 Handling Editor: L. G. Tham Available online 24 May 2016

When dealing with wheel-rail interactions for a high-speed train using the time domain Green function of a railway track, it would be more reasonable to use the moving Green function associated with a reference frame moving with the train, since observed from this frame wheel/rail forces are stationary. In this paper, the time domain moving Green function of a railway track as an infinitely long periodic structure is defined, derived, discussed and applied. The moving Green function is defined as the Fourier transform, from the load frequency domain to the time domain, of the response of the rail due to a moving harmonic load. The response of the rail due to a moving harmonic load is calculated using the Fourier transform-based method. A relationship is established between the moving Green function and the conventional impulse response function of the track. Properties of the moving Green function are then explored which can largely simplify the calculation of the Green function. And finally, the moving Green function is applied to deal with interactions between wheels and a track with or without rail dampers, allowing nonlinearity in wheel-rail contact and demonstrating the effect of the rail dampers. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Periodic structure Moving load Moving Green function Track dynamics Wheel-rail interaction

1. Introduction Train-track dynamic interactions are responsible for many issues concerning the railway industry, such as roughness on wheel-rail rolling surfaces, rail corrugation, wheel out-of-round and wheel-rail noise. Simulation tools are useful to gain insights into these issues and several packages are commercially available. In case of high-speed train, train-track interactions are of high frequencies, and involve complex vibrational resonances and propagations in the track due to the high train speed. Regular impacts between wheel and rail occur if discrete wheel irregularities are present. Different approaches to modelling train-track interactions have been established. To get deeper insights, especially for high speed and high frequency, modelling approaches are still being improved, and even innovated. One of the important tasks is to model the track and predict track dynamics of high frequency. For a modern railway track with sleepers, it is normally idealised to be an infinitely long periodic structure. The period is equal to the sleeper spacing for a conventional ballasted track, or equal to the length of a slab plus the small slab-slab gap if the track is a nonballasted and pre-cast slab track such as those used for high-speed railways in China. An important aspect in dealing with track dynamics is to calculate the response to moving or stationary harmonic loads. Results from such calculations can

n

Corresponding author. Tel.: þ 86 28 87634209; fax: þ 86 28 87600868. E-mail address: [email protected] (X. Sheng).

http://dx.doi.org/10.1016/j.jsv.2016.05.011 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

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Nomenclature q Q Q^ j G f^ ðΩÞ P k0 mk wW k ðtÞ zk L _W w k ðt n Þ Ck

displacement vector of the rail receptance matrix of the rail inverse Fourier transform of Q~ j moving Green function matrix Fourier transform of fðtÞ half the axle load of the kth wheel mass of the kth wheel vertical displacement of the kth wheel wheel-rail irregularity at the kth wheel-rail contact point sleeper spacing velocity of the kth wheel at instant tn constant in the contact model for the kth whee-rail contact

p0 Q~ j G fðtÞ F k ðtÞ f k ðtÞ ak wRk ðtÞ Gw M wRk0 ðt n Þ Rk

amplitude vector of harmonic loads applied on the rail jth Fourier coefficient matrix of the receptance matrix of the rail stationary Green function matrix moving force vector applied on the rail the kth vertical wheel-rail force the kth dynamic wheel-rail force initial position of the kth wheel vertical displacement of the rail at the kth wheel-rail contact point moving Green function associated with rail vertical displacement number of wheels rail displacement at the kth wheel-rail contact point due to the moving axle loads radius of the kth wheel

reveal dynamic characteristics (including modal resonances and wave propagations) of the track and at the same time provide a basis, either in the time domain or in the frequency domain, for dealing with train-track interactions. It is always attractive to deal with high-frequency vibration problems in the frequency domain. In many earthlier wheel– rail interaction models, e.g. [1], the motion of the wheels is replaced by the motion of a roughness strip, resulting in a linear, time-invariant system. Since wheels are forced to be stationary in the track direction, this ‘moving-roughness approach’ is able to consider the roughness excitation without including the effect of the wheel speed, but totally excludes excitations from moving axle loads. Furthermore, for some roughness wavelengths this approach changes the vibration propagation characteristics of the rail, i.e., it replaces a propagating vibration mode with a non-propagating one, or vice versa [2]. For high-speed train, this approach seems to be inappropriate. Train-track interactions can be studied in the time domain by solving differential equations of motion of the train-track system. This approach requires the track to be truncated into a finite length. To minimise wave reflections from the two ends, the track model must be sufficiently long. For a high-speed train, the entire train may be considered and the track model must be much longer than the train [3]. This is due to the strong inter-vehicle couplings and long-distance propagating vibration waves in the track induced by the fast-moving train. Normally vibration of the rail is modelled using either the modal superposition method [3] or the finite element method [4]. Both would generate a large number of differential equations of time-varying coefficients. In order to account for high frequency and high train speed, a very small time step is required to solve such a large number of differential equations. Recently, impact noise from wheel flats based on measured profiles is predicted in Ref. [5] using the modelling approach of [4] with a detailed account for the contact mechanics. The second time domain approach is based on the idea of ‘mass on time-varying spring’ [6–8]. Here the time-varying spring represents the track which provides varying dynamic stiffness as a wheel rolls over a sleeper bay. In this approach, receptances of the rail are obtained using a model in the frequency domain and rational fraction polynomials are chosen to best fit these receptances. Then the inverse Laplace transform is employed to transform the dynamics of the track into a small set of ordinary differential equations. These differential equations and those for vehicles are coupled via wheel-rail contact conditions to constitute a set of differential equations governing vehicle-track dynamic interactions. It is noticed that in Refs. [7,8], the calculation of rail receptances is carried out in a quasi-static manner. Another approach is the time domain Green function method [7,9,10], in which integral, in addition to differential, equations are solved. According to Refs [9,10], the time domain Green function, which is the response of the rail at a position due to an impulsive force (a delta-function) applied at the same or another position, is first calculated. Then the classic Duhamel integral equation is applied to express responses of the rail at wheel-rail contact points due to wheel-rail forces, and at the same time differential equations are set up for the wheels. These equations are then coupled using wheel-rail contact conditions. Travelling along the rail of a wheel is accounted for by changing the response and loading positions properly in the Green function. To do so, Green functions have to be calculated for every loading positions within a sleeper bay. Refs. [11,12] have used the time domain Green function method to deal with wheel-rail impact and noise caused by a wheel flat. It is observed that Green functions used in [9,10] are calculated without the effect of load speed. Therefore these Green functions are ‘stationary’ ones. It is also noticed that the time-domain Green function in Refs. [11,12] is also calculated in a ‘stationary’ sense based on the finite element model developed in Ref. [4] of a finite length of track. It would be more reasonable to use the Green function which is associated with a reference frame moving with the train, since observed from this frame wheel-rail forces are stationary. Such a Green function may be termed the time-domain moving Green function. This idea has already been explored recently in Refs. [13–15]. It should be noted that in these papers,

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the Green's function is calculated for a homogeneous infinite one-dimensional elastic structure in a moving reference frame, without the effect of periodic discrete supports. A time domain Green function is normally computed as the inverse Fourier transform of the corresponding frequency response function. Therefore it is essential to be able to calculate the response of an infinitely long periodic railway track to a moving harmonic load of high frequency. A Fourier-transform based method [2,16], developed by the first author of the current paper, can efficiently calculate track vibrations excited by a harmonic load of high frequency and moving at high speed, if the track is idealised to be an infinitely long periodic structure. In that method, the rail can be described using either a multi-beam model as in Ref. [17] or a 2.5D FE model [18]. The main feature of the method is that, it expresses the response of the rail observed from the moving load to be a Fourier series, and each Fourier coefficient is calculated as Fourier transform from the wavenumber domain to the spatial domain, both in the track direction. Therefore this method is Fouriertransform based. Based on work of Ref. [2], the so-called Fourier-series approach is developed in Ref. [19] to study steady-state wheel–rail interactions generated by a single, or multiple wheels moving along a railway track as an infinitely long periodic structure. It is shown in Ref. [19] that wheel-rail forces can be determined by solving a set of linear algebraic equations, and therefore this approach is quite accurate and computationally efficient. Similar approach is used in Ref. [20] to investigate dynamic instability of a wheel moving on a discretely supported infinite rail. It must be pointed out that the approach of Ref. [19] is based on two simplifications: (1) the vertical irregular profile (roughness) of the railhead is assumed to be periodic in the track direction with the period equal to a multiple of the sleeper spacing; and (2) linear dynamics is assumed for the wheel/ track system. If any of these two simplifications is not fulfilled, then other approaches such as the time domain Green function method have to be applied. The potential of the time domain Green function approach has not been fully explored. This may be due partly to the complexity of calculating the Green function for a periodic structure. In this paper, the time domain moving Green function of a railway track as an infinitely long periodic structure is defined, derived, discussed and applied. The moving Green function is defined as the Fourier transform of the response of the rail due to a moving harmonic load, from the load frequency domain to the time domain. The response of the rail due to a moving harmonic load is calculated using the Fourier transform-based method detailed in Refs. [2,16]. This paper is a further development and extension of Ref. [21]. Expressions based on the Fourier transform-based method for the response of the rail due to a moving harmonic load is presented in Section 2. The time domain moving Green function is defined and derived in Section 3. This section also presents a relationship between the moving Green function and the stationary Green function, and other properties. An integral equation, which is slightly different from the classical Duhamel integral equation, is also established in this Section. Application of the moving Green function to deal with wheel-rail interactions is formulated in Section 4. Results are given in Section 5 for a conventional ballasted track with or without rail dampers, and finally, this paper is concluded in Section 6.

2. Responses of the rail to moving harmonic loads As an example, Fig. 1 shows a conventional ballasted track with rail dampers at mid-span as an infinitely long periodic structure with the period being L [16]. The externally applied loads on the rail are harmonic with radian frequency Ω and moving in the x-direction at speed c. At t ¼ 0, the loads are applied at the x0 cross-section measured from the 0th sleeper. pffiffiffiffiffiffiffiffi The corresponding force vector distributed along the rail is given by δðx x0 ctÞp0 eiΩt , where, i ¼  1, δ(∙) is the deltafunction, an p0 denotes the amplitude vector of the loads, the dimension of which is identical to the degrees of freedom of the cross-section of the rail. If observation is made from a reference frame moving with the loads, then the displacement vector of the rail is given by qðx0 ; x0 ; tÞ ¼ Q ðx0 ; x0 þ ct; ΩÞp0 eiΩt ; 0

0

(1)

where, x is the coordinate measured from the moving load (Fig. 1). When x 4 0, it is said that the response position is ahead the load, and when x0 o 0, then the response position is behind the load. As in the case of stationary harmonic loads, the

c x0

ct

p0eiΩt

x´

kP mS kB L Fig. 1. Track with rail dampers as a periodic structure.

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matrix, Q ðx0 ; x0 þ ct; ΩÞ, may be termed the receptance matrix of the rail at position defined by x0 , with dependence on load frequency Ω and load position x0 þct being explicitly indicated. It is shown in Refs. [2,16] that, for a given load frequency Ω, the receptance matrix of the rail is not temporally constant, but instead, it is a periodic function of x0 þ ct with period being L. It therefore can be expressed as a Fourier series, given by 1 X

Q ðx0 ; x0 þ ct; ΩÞ ¼

0 Q~ j ðx0 ; ΩÞe  i2π jðx0 þ x þ ctÞ=L ;

(2)

j ¼ 1

where, the Fourier coefficient matrix, Q~ j ðx0 ; ΩÞ, is given by Z 1 1 0 Q~ ðβ; ΩÞeiβx dβ: Q~ j ðx0 ; ΩÞ ¼ 2π  1 j

(3)

This is an inverse Fourier transform of matrix Q^ j ðβ; ΩÞ from the wavenumber, β, in the track direction, to the spatial coordinate, x´. Detailed expressions for this matrix can be found in Ref. [2] for a conventional ballasted track, and in Ref. [16] for the track with rail dampers.

3. Time domain moving and stationary Green functions 3.1. The moving Green function matrix The time domain moving Green function matrix, as a function of time τ, is defined as the inverse Fourier transform with respect to Ω of the receptance matrix Q ðx0 ; x0 þ ct; ΩÞ, i.e. Z 1 1 Gðx0 ; x0 þ ct; τÞ ¼ Q ðx0 ; x0 þ ct; ΩÞeiΩτ dΩ: (4) 2π  1 According to Eqs. (2) and (3), 1 Gðx ; x0 þ ct; τÞ ¼ 2π 0

¼

1 X

Z

1

0 @

1

e  i2πjðx0 þ x

0

1

1 X

0 Q~ j ðx0 ; ΩÞe  i2πjðx0 þ x þ ctÞ=L AeiΩτ dΩ

j ¼ 1

þ ctÞ=L

j ¼ 1

Z

1 ð2πÞ2

1 1

Z

1

0 Q^ j ðβ; ΩÞeiβx eiΩτ dβdΩ:

1

(5)

By defining Gj ðx0 ; τÞ ¼

Z

1 ð2πÞ

2

1 1

Z

1 1

0 Q^ j ðβ; ΩÞeiβx eiΩτ dβdΩ;

(6)

Eq. (5) can be written as Gðx0 ; x0 þ ct; τÞ ¼

1 X

e  i2πjðx0 þ x

0

þ ctÞ=L

Gj ðx0 ; τÞ:

(7)

j ¼ 1

From Eqs. (6) and (7) it can be seen that: (1) The time domain moving Green function matrix can be produced by calculating the double integral expressed in Eq. (6), which is the two-dimensional inverse Fourier transform of Q^ j ðβ; ΩÞ with respect to wavenumber β and frequency Ω. (2) The time domain moving Green function matrix has two time arguments, t and τ, and this is different from a conventional, or ‘stationary’, Green function which has only one time argument, as shown in Section 3.3. The physical meaning of the time domain moving Green function will be discussed in Section 3.5 by comparing it with the corresponding stationary Green function. 3.2. Rail responses to an arbitrary moving force vector-Duhamel integral Now it is assumed that the externally applied moving force vector,fðtÞ, is an arbitrary function of time t and the Fourier transform of fðtÞ is denoted by f^ ðΩÞ. It is located at x0 when t ¼ 0. In other words, Z 1 1 f^ ðΩÞeiΩt dΩ: (8) fðtÞ ¼ 2π  1

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137

By assuming that the track system satisfies the requirements of the superposition principle, then the response of the rail can be mathematically expressed, according to Eqs. (1), (4) and (8), as Z 1 1 qðx0 ; x0 ; tÞ ¼ Q ðx0 ; x0 þ ct; ΩÞf^ ðΩÞeiΩt dΩ 2π  1  Z 1  Z 1 1 f^ ðΩÞeiΩt ¼ Gðx0 ; x0 þct; τÞe  iΩτ dτ dΩ 2π  1 1  Z 1 Z 1  1 0 f^ ðΩÞeiΩðt  τÞ dΩ dτ ¼ Gðx ; x0 þ ct; τÞ 2π  1 1 Z 1 0 ¼ Gðx ; x0 þ ct; τÞfðt  τÞdτ Z 11 ¼ Gðx0 ; x0 þct; t  τÞfðτÞdτ; 1

that is, qðx0 ; x0 ; tÞ ¼

Z

1 1

Gðx0 ; x0 þ ct; t  τÞfðτÞdτ;

(9)

This, according to Eq. (7), becomes 1 X

qðx0 ; x0 ; tÞ ¼

e  i2πjðx0 þ x

0

Z

þ ctÞ=L

1

Gj ðx0 ; t  τÞfðτÞdτ:

1

j ¼ 1

(10)

Eqs. (9) and (10) are similar to, but not exactly the same as, the classical Duhamel integral equation. These integral equations provide a basis for dealing with wheel-rail interactions, as illustrated in Section 4 using the rigid wheel model. 3.3. The stationary Green function matrix The stationary Green function matrix can be calculated straightforwardly by setting c ¼ 0 in Eq. (5), i.e. Z 1 Z 1 1 X  1 0 0 G ðx0 ; x0 ; τÞ ¼ e  i2πjðx0 þ x Þ=L Q^ j ðβ; ωÞeiβx eiωτ dβdω; 2 ð2 π Þ  1  1 j ¼ 1

(11) 

where G ðx0 ; x0 ; τÞ is the stationary Green function matrix with excitation being at x0 and response at x0 þ x0 , and Q^ j ðβ; ωÞ is obtained by setting c ¼ 0 in Q^ j ðβ; ΩÞ (Note: ω ¼ Ω  βc ¼ Ω [2,16]). By defining Z 1 Z 1  1 0 Gj ðx0 ; τÞ ¼ (12) Q^ j ðβ; ωÞeiβx eiωτ dβdω; 2 ð2πÞ  1  1 Eq. (11) can be written as G ðx0 ; x0 ; τÞ ¼

1 X

e  i2πjðx0 þ x Þ=L Gj ðx0 ; τÞ: 0

(13)

j ¼ 1

Some properties of the stationary Green function are discussed in Ref. [9]. One is the Maxwell–Betty reciprocity, which, using the notations of the current paper, is G ðx0 ; x0 ; τÞ ¼ ðG ð  x0 ; x0 þx0 ; τÞÞT ;

(14)

where, the upper subscript ‘T’ represents matrix transpose. Another property is the periodicity of the Green function, that is G ðx0 ; x0 þ L; τÞ ¼ G ðx0 ; x0 ; τÞ:

(15)

3.4. Relationship between the moving and stationary Green function matrices Since ω ¼ Ω  βc, it can be shown that, Z Z 1 Z 1 0 Q^ j ðβ; ΩÞeiβx eiΩτ dβdΩ ¼ 1

1

1 1

Z

1 1

 0 Q^ j ðβ; ωÞeiβðx þ cτÞ eiωτ dβdω;

thus, according to Eqs. (5), (12) and (13), Gðx0 ; x0 þ ct; τÞ ¼

1 X

e  i2πjðx0 þ x

0

þ ctÞ=L

j ¼ 1

¼

1 X j ¼ 1

e  i2πj½x0 þ cðt  τ

Þ þ x0

þ cτ=L

1

1

Z

1

Z

ð2πÞ2  1 Z 1 Z 1

ð2πÞ2

1

1

1 1

0 Q^ j ðβ; ΩÞeiβx eiΩτ dβdΩ

 0 Q^ j ðβ; ωÞeiβðx þ cτÞ eiωτ dβdω

(16)

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¼

1 X

e  i2πj½x0 þ cðt  τÞ þ x

0

þ cτ=L

Gj ðx0 þ cτ; τÞ

j ¼ 1

¼ G ðx0 þ cτ; x0 þcðt  τÞ; τÞ;

(17)

which shows that, the moving Green function matrix, Gðx ; x0 þct; τÞ, can be obtained by replacing x with x þ cτ, and x0 with x0 þcðt  τÞ, in the stationary Green function matrix, G ðx0 ; x0 ; τÞ. By letting τ ¼ t  τ0 in Eq. (17), one has, 0

0

0

Gðx0 ; x0 þ ct; t  τ0 Þ ¼ G ðx0 þ cðt  τ0 Þ; x0 þcτ0 ; t  τ0 Þ; or alternatively, Gðx0 ; x0 þ ct; t  τÞ ¼ G ðx0 þcðt  τÞ; x0 þcτ; t  τÞ:

(18)

3.5. Physical meaning of the moving Green function matrix Physically, the stationary Green function, G ðx0 ; x0 ; τÞ, is the response of the rail at position x0 þ x0 and at the moment τ due to a unit impulse applied at position x0 at the zero instant. Thus the physical meaning of the moving Green function, Gðx0 ; x0 þct; τÞ, may be understood, according to Eq. (17), to be the response of the rail at position x0 þcðt  τÞ þ x0 þ cτ ¼ x0 þ ct þ x0 and at the moment τ due to a unit impulse applied at position x0 þ cðt  τÞ at the moment zero. Illustrations are shown in Fig. 2 for τ ¼ 0, 0 o τ o t, and τ ¼ t. It can be seen from Fig. 2 that, for a given t, as τ increases from zero to t, the position of the impulse moves to the left, from x0 þ ct to x0 . The black point indicates the values which should be taken from the corresponding stationary Green functions at τ0 ¼ 0, τ0 ¼ τ and τ0 ¼ t, respectively. In the middle part of the figure,τ00 indicates the time required for the response position to receive the disturbance from the impulse. The distance (that is x0 þ cτ) between the position of the impulse and that of the response divided by τ00 gives the speed of the wave in the track. It is well known that G ðx0 ; x0 ; τÞ ¼ 0 for τ r 0, therefore from Eq. (17), Gðx0 ; x0 þ ct; τÞ ¼ 0; for τ r 0:

Stationary Green function

Impulse

O

x0+ct

(19)

τ´

x´

Rail

τ=0

Stationary Green function

Impulse

τ´

x0+ c(t-τ) O

x0+ct

x´

Rail

0<τ
Impulse

τ´

x0 O

x0+ct

x´

Rail

τ=t Fig. 2. Illustration of the time domain moving Green function as function of τ. (a) τ ¼ 0; (b) 0 oτ o t; (c) τ¼ t .

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Thus, Eq. (10) can be simplified to be qðx0 ; x0 ; tÞ ¼ Z ¼ ¼

1 X

Z

t 1

1

139

Gðx0 ; x0 þct; t  τÞfðτÞdτ

1

Gðx0 ; x0 þ ct; t  τÞfðτÞdτ

e  i2πjðx0 þ x

0

þ ctÞ=L

Z

t 1

j ¼ 1

Gj ðx0 ; t  τÞfðτÞdτ:

(20)

In Eq. (20), fðτÞ does not vanish when τ r 0 for wheel–rail interaction problem since the track is infinitely long. This means that integration in Eq. (20) must be carried out from the remote past. This issue will be discussed further in Section 4. If the stationary Green function is used, then according to Eqs. (18) and (20) becomes Z t qðx0 ; x0 ; tÞ ¼ G ðx0 þ cðt  τÞ; x0 þ cτ; t  τÞfðτÞdτ: (21) 1

This is the same as Eq. (31) in Ref. [9] which is derived based directly on the stationary Green function. It is noticed that the lower limit of integration in Eq. (31) of Ref. [9] is zero, meaning that fðτÞ ¼ 0 for τ r0 has been assumed by the author. 3.6. Properties of the moving Green function matrix Some properties of the moving Green function matrix are explored as below. (1) The time domain moving Green function matrix vanishes for any non-positive τ, as indicated in Eq. (19). (2) According to Eq. (5), the time domain moving Green function matrix is a periodic function of x0 þ ct with the period being L, and Eq. (7) is the Fourier series form of that period function. In other words, Gðx0 ; x0 þ ct þ L; τÞ ¼ Gðx0 ; x0 þ ct; τÞ;

(22)

and once the Fourier coefficient matrices, Gj ðx ; τÞ, where j ¼ 1; :::;  1; 0; 1; :::; 1, are worked out, then the time domain moving Green function matrix can be easily determined using Eq. (7) for any load position x0 þ ct. This also applies to the stationary Green function, as indicated in Eq. (13). (3) Since the stationary Green function matrix is real, so is the moving Green function matrix, as indicated by Eq. (17). Thus, according to Eq. (7), Gj ðx0 ; τÞ and G  j ðx0 ; τÞ are conjugates of each other, and G0 ðx0 ; τÞ is real. This property can be used to increase computational efficiency. (4) From the discussions in Section 3.5 and Fig. 2, it can be seen that 0

lim Gðx0 ; x0 þ ct; τÞ ¼ 0;

(23)

 lim  Gðx'; x0 þ ct; τÞ ¼ 0: x'→∞

(24)

τ-1

(5) From the discussions in Section 3.5 and Fig. 2, it can be concluded that, if x0 a 0, then there is τ0 such that Gðx0 ; x0 þ ct; τÞ ¼ 0 for 0 r τ r τ0 . τ0 may be termed the time-delay of the moving Green function matrix Gðx0 ; x0 þct; τÞ at x0 and t. The time-delay is caused by the finite wave speed in the track structure. It should be noted that, the bending waves in the rail are dispersive ones and the wave velocity increases with the frequency. Subsequently, the time-delay may depend on the highest frequency taken into account for the numerical simulation.  0 0 0   (6) It can  be seen from Fig. 2 that, for given x (x a 0) and t, the time-delay of Gð  x ; x0 þ ct; τÞ is shorter than that of 0 Gðx ; x0 þ ct; τÞ. In other words, the time-delay of the moving Green function matrix with the response point being behind the loading point is shorter than that of the moving Green function matrix with the response point being ahead the loading point by the same distance. (7) A moving Green function with the response point being behind the loading point is much stronger in magnitude than that of the moving Green function with the response point being ahead the loading point by the same distance.

4. Application of the time domain Green function to wheel–rail interaction analysis It is assumed that there are a number, M, of wheels as rigid body moving along the rail. The mass of the kth wheel is denoted by mk, and its initial position is defined by ak , where k ¼ 1, …, M. The vertical wheel-rail force between the kth wheel and the rail is denoted by F k ðtÞ which contains the component of half the axle load, P k0 , and a dynamic component,f k ðtÞ. The vertical displacement of the kth wheel at the wheel-rail contact point is denoted by wW k ðtÞ, and that of the rail is denoted by wRk ðtÞ, both directed downwards. Wheel-rail irregularity experienced by the kth wheel at position x is denoted by zk(x). The sign of the rail irregularity is defined as positive if the actual rail surface is at a higher level than the

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nominal level. The sign of the wheel irregularity is defined as positive if the radius of the wheel is larger than its nominal one. The vertical motion of the wheel is governed by W

€ k ðtÞ ¼ F k ðtÞ þ P k0 ¼  f k ðtÞ: mk w

(25)

If the moving Green function associated with rail vertical displacement is denoted by G ðx ; x0 þ ct; τÞ, then according to Eq. (20), the rail displacement at the kth wheel-rail contact point can be calculated as w

M Z X

wRk ðtÞ ¼

t 1

j¼1

0

Gw ðak aj ; aj þ ct; t  τÞF j ðτÞdτ:

(26)

If the Hertzian contact theory is used to model wheel-rail contact, then a third equation can be established for F k ðtÞ 40, 2=3 R wW k ðtÞ wk ðtÞ þ zk ðak þctÞ ¼ C k ðF k ðtÞÞ

(27)

where, Ck is a constant determined by the rolling radius, Rk, of the kth wheel, as well as other radii of the wheel and the rail at the contact point. For typically used rail and wheel, Ck may be approximated to be [22],   C k ¼ 3:86Rk 0:115  10  8 m=N 2=3 : (28) References [11,12] have demonstrated that roughness on the wheel and rail rolling surfaces has effect on wheel-rail contact mechanics, however, this effect will not be considered here in this paper. Wheel-rail interactions are governed by Eqs. (25)–(27) from which wheel-rail forces and displacements of the wheel and rail at the contact points can be worked out. It is seen that, the total number of unknowns is 3 times the number of wheels, much smaller than that would be required in traditional methods. The importance of the moving Green function of the rail to wheel–rail interaction is clearly shown in Eq. (26). It is seen that Eq. (25) is a differential equation while Eq. (26) is an integral equation, and wheel-rail contact nonlinearity is taken into account in Eq. (27). Eqs. (25)–(27) are solved as follows. First of all, a time step, Δt, is chosen, and wheel-rail forces and displacements of the wheel and rail at the contact points are evaluated at t n ¼ nΔt, where n ¼ 0; 1; 2; … with t 0 ¼ 0. Secondly, Eq. (25) is expressed alternatively as Z tn 1 _W _W f ðτÞdτ; (29) w k ðt n Þ ¼ wk ðt n  1 Þ  mk t n  1 k Z W wW k ðt n Þ ¼ wk ðt n  1 Þ þ

tn

tn  1

_W w k ðτ Þdτ :

(30)

It is assumed that, dynamic components of the wheel-rail forces appear only for positive time, thus at tn, Eq. (26) becomes wRk ðt n Þ ¼

M Z X j¼1

0 1

Gw ðak  aj ; aj þ ct n ; t n  τÞP j0 dτ þ

M Z X j¼1

tn 0

Gw ðak  aj ; aj þ ct n ; t n  τÞF j ðτÞdτ;

which can be written as wRk ðt n Þ ¼ uk ðt n Þ þ

M X n Z X j¼1i¼1

ti ti  1

Gw ðak  aj ; aj þct n ; t n  τÞF j ðτÞdτ;

(31)

where, uk ðt n Þ ¼

M Z X

1

j¼1

¼

M Z X j¼1

¼

0

1

0

M X I Z X j¼1i¼1

Gw ðak  aj ; aj þct n ; t n  τÞP j0 dτ

Gw ðak  aj ; aj þct n ; t n þ τÞP j0 dτ

ti ti  1

Gw ðak  aj ; aj þct n ; t n þ τÞP j0 dτ;

(32)

and I is an integer large enough to make the truncation made in Eq. (32) valid (see Eq. (23)). Integrations in Eqs. (29)–(32) may be evaluated by adopting the approximation that in any time interval defined by ½t n  1 ; t n  or ½t i  1 ; t i , the wheel-rail force and the Green function vary linearly with time. As a result, the following algebraic equations yield, _W _W w k ðt n Þ ¼ wk ðt n  1 Þ 

Δt f k ðt n Þ þf k ðt n  1 Þ ; mk 2

(33)

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

W _W wW k ðt n Þ ¼ wk ðt n  1 Þ þ wk ðt n  1 ÞΔt 

uk ðt n Þ ¼

M X I Z X

ti

ti  1

j¼1i¼1

ðΔtÞ2 f k ðt n Þ þ 2f k ðt n  1 Þ ¼ αk ðt n Þ þ βk ðt n Þf k ðt n Þ; 3 2mk

(34)

Gw ðak aj ; aj þ ct n ; t n þ τÞP j0 dτ

M X

I G þ ðt ; t Þ þ G þ ðt ; t X kj n i kj n i  1 Þ

j¼1

i¼1

¼ Δt

141

!

2

P j0 ;

(35)

wRk ðt n Þ ¼ uk ðt n Þ h i 3 M X n Gkj ðt n ; t i Þ  Gkj ðt n ; t i  1 Þ F j ðt i Þ  F j ðt i  1 Þ X Gkj ðt n ; t i ÞF j ðt i  1 Þ þ Gkj ðt n ; t i  1 ÞF j ðt i Þ 4 5 þ þ Δt 2 3 j¼1i¼1 2

(36)

where _W αk ðt n Þ ¼ wW k ðt n  1 Þ þ wk ðt n  1 ÞΔt 

βk ðt n Þ ¼ 

ðΔtÞ2 f k ðt n  1 Þ ; 3mk

(37)

ðΔtÞ2 ; 6mk

(38)

and Gkjþ ðt n ; t i  1 Þ ¼ GW ðak  aj ; aj þ ct n ; t n þ t i  1 Þ;

(39a)

Gkjþ ðt n ; t i Þ ¼ GW ðak  aj ; aj þ ct n ; t n þ t i Þ:

(39b)

Gkj ðt n ; t i  1 Þ ¼ GW ðak  aj ; aj þ ct n ; t n  t i  1 Þ;

(40a)

Gkj ðt n ; t i Þ ¼ GW ðak  aj ; aj þ ct n ; t n  t i Þ:

(40b)

Based on Eqs. (39) and (40), two matrices may be defined for each combination of j and k: 2 3 GW ðak  aj ; aj þct 0 ; t 0 þ t 0 Þ GW ðak  aj ; aj þct 0 ; t 0 þ t 1 Þ ⋯ GW ðak  aj ; aj þct 0 ; t 0 þ t I Þ 6 7 6 GW ða  a ; a þct ; t þ t Þ GW ða  a ; a þct ; t þ t Þ ⋯ GW ða  a ; a þct ; t þ t Þ 7 I 7 1 1 0 1 1 1 1 1 j j j j j j 6 k k k Gkjþ ¼ 6 7 7 6 ⋯ ⋯ ⋯ ⋯ 5 4 W W W G ðak  aj ; aj þct N ; t N þ t 0 Þ G ðak  aj ; aj þct N ; t N þ t 1 Þ ⋯ G ðak  aj ; aj þct N ; t N þ t I Þ 2

Gkj

GW ðak  aj ; aj þ ct 0 ; t 0 t 0 Þ 6 6 GW ða  a ; a þ ct ; t t Þ 1 1 0 j j 6 k ¼6 6 ⋯ 4 GW ðak aj ; aj þ ct N ; t N  t 0 Þ

0

⋯

0

GW ðak  aj ; aj þ ct 1 ; t 1 t 1 Þ

⋯

0

⋯

⋯

⋯

GW ðak aj ; aj þ ct N ; t N  t 1 Þ

⋯

GW ðak aj ; aj þ ct N ; t N  t N Þ

(41)

3 7 7 7 7 7 5

(42)

where, N is the number of time steps. It can be seen that matrix Gkjþ is of order ðN þ 1Þ  ðI þ1Þ, and Gkj is of order ðN þ1Þ  ðN þ 1Þ. Since F j ðtÞ ¼ P j0 þ f j ðtÞ, Eq. (36) may be manipulated as wRk ðt n Þ ¼ uk ðt n Þ þ Δt

 M X n G  ðt ; t ÞP þ G  ðt ; t X kj n i j0 kj n i  1 ÞP j0 2

j¼1i¼1

ih i3 M X n Gkj ðt n ; t i Þ  Gkj ðt n ; t i  1 Þ f j ðt i Þ f j ðt i  1 Þ X Gkj ðt n ; t i Þf j ðt i  1 Þ þ Gkj ðt n ; t i  1 Þf j ðt i Þ 4 5 þ Δt þ 2 3 j¼1i¼1 2

¼ uk ðt n Þ þ þ Δt

M X

n X

j¼1i¼1

hh

h

M n h X Δt X

2

Gkj ðt n ; t i Þ þGkj ðt n ; t i  1 Þ

j¼1

! i P j0

i¼1

i h i i Gkj ðt n ; t i Þ=6 þGkj ðt n ; t i  1 Þ=3 f j ðt i  1 Þ þ Gkj ðt n ; t i Þ=3 þ Gkj ðt n ; t i  1 Þ=6 f j ðt i Þ

(43)

142

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

i.e. wRk ðt n Þ ¼ wRk0 ðt n Þ þ γ k ðt n Þ þ

M X

μkj ðt n Þf j ðt n Þ;

(44)

j¼1

where, wRk0 ðt n Þ is rail displacement associated with the moving axle loads, given by wRk0 ðt n Þ ¼ uk ðt n Þ þ

γ k ðt n Þ ¼ Δt

M n h X Δt X

2

j¼1

Gkj ðt n ; t i Þ þ Gkj ðt n ; t i  1 Þ

i

! P j0 ;

(45)

i¼1

M X n h i X Gkj ðt n ; t i Þ=6 þ Gkj ðt n ; t i  1 Þ=3 f j ðt i  1 Þ j¼1i¼1

þ Δt

M n 1 h X X

i Gkj ðt n ; t i Þ=3 þGkj ðt n ; t i  1 Þ=6 f j ðt i Þ;

(46)

j¼1i¼1

h

i

μkj ðt n Þ ¼ Δt Gkj ðt n ; t n Þ=3 þGkj ðt n ; t n  1 Þ=6 ¼ ¼

Δt 6

Δt 6

Gkj ðt n ; t n  1 Þ

GW ðak  aj ; aj þct n ; ΔtÞ:

(47)

Since in reality separations between different wheels are sufficiently large, thus, according to the property of the Green function, μkj ðt n Þ ¼ 0 if k aj. At t n , Eq. (27) becomes 2=3 R ; wW k ðt n Þ  wk ðt n Þ þzk ðak þct n Þ ¼ C k ðF k ðt n ÞÞ

(48)

and according to Eqs. (34) and (44), it becomes (note: μkj ðt n Þ ¼ 0 if k aj)

αk þ βk f k ðt n Þ  wRk0 ðt n Þ  γ k ðt n Þ  μkk ðt n Þf k ðt n Þ þ zk ðak þ ct n Þ ¼ C k f k ðt n Þ þ P k0

2=3

:

(49)

By letting bk ðt n Þ ¼ αk ðt n Þ wRk0 ðt n Þ  γ k ðt n Þ þ zk ðak þ ct n Þ;

(50)

dk ðt n Þ ¼ βk ðt n Þ  μkk ðt n Þ;

(51)

Eq. (49) becomes

2=3 bk ðt n Þ þ dk ðt n Þf k ðt n Þ ¼ C k f k ðt n Þ þ P k0 ; ðk ¼ 1; 2; :::MÞ: _W w k ðt n Þ,

wW k ðt n Þ,

(52)

wRk ðt n Þ

Now the iteration approach is employed to determine and f k ðt n Þ, where k ¼ 1, 2, …., M and n ¼ 1, 2, …N, according to equations established above. To start the iteration process, all the velocity and dynamic wheel-rail forces are assumed to vanish at t 0 ¼ 0, but the displacement of the rail is set to be (see Eq. (32)) wRk ðt 0 Þ ¼ wRk0 ðt 0 Þ ¼ uk ðt 0 Þ;

(53)

and that of the wheel to be the sum of the rail displacement and the local compression at the wheel-rail contact point (see Eq. (27), i.e. 2=3

R wW k ðt 0 Þ ¼ wk0 ðt 0 Þ þ C k P k0 :

(54)

In other words, the wheel–rail interaction problem is made to be an initial value problem, and the steady-state solution will be achieved with a sufficiently large number of iterations.

5. Results Results for time domain Green functions defined and derived in Section 3 and wheel–rail interaction formulated in Section 4 are produced for two tracks; one being a conventional ballasted railway track, and the other being the track with two rail dampers installed either side of the rail at the mid-span of each sleeper bay (Fig. 1). A set of typical parameters for the track and dampers are given in Section 5.1. Green functions and wheel-rail forces for the first track are presented in Section 5.2, mainly for comparison with results in Ref. [9]. Green functions of the second track are presented in Section 5.3. And finally, wheel-rail forces due to a rail indentation are presented in Section 5.4 for a single wheel and Section 5.5 for four such wheels from two neighbouring bogies.

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

143

Table 1 Parameters for the track. ρ ¼7850 kg/m3 E¼ 2.1  1011 N/m2 G ¼ 0.81  1011 N/m2 A ¼7.69  10  3 m2 I¼ 30.55  10  6 m4 κ¼ 0.4 kP ¼3.5  108 N/m ηP ¼1.37  10  4 ω mS ¼162 kg L ¼ 0.6 m kB ¼ 70  106 N/m ηB ¼ 6.71  10  4 ω

Density of rail material Young's modulus of rail material Shear modulus of rail material Cross-sectional area of the rail Bending moment of inertia of the rail Shear coefficient of the rail cross-section Vertical rail pad stiffness Rail pad loss factor Mass of half a sleeper Sleeper spacing Vertical ballast stiffness Loss factor of ballast

Table 2 Parameters for the rail damper. Tuned frequency (vertical/rotational) Damper length Mass of two dampers Inertia moment of two dampers Vertical stiffness of two damper springs Rotational stiffness of the two dampers Loss factor of dampers

1000 Hz/928 Hz lD ¼ 0.2 m mD ¼ 15 kg JD ¼ 0.058 kg m2 kDw ¼ 5.92  108 N/m kDΨ ¼ 1.97  106 Nm ηD ¼ 3.0  10  4 ω

5.1. Parameters of the track and damper Parameters for half the track structure are listed in Table 1, and those for the rail dampers are listed in Table 2. They are taken from Ref. [16] apart from the damping and the stiffness of the ballast. The viscous damping model in which loss factor is proportional to the equivalent frequency,ω ¼ Ω  βc, is used here to avoid non-causal response [23]. Damping is introduced by using complex stiffness, e.g. for the ballast, the complex stiffness is kB ð1 þ iηB ωÞ. For the ballast, the stiffness is changed to 70  106 N/m from 50  106 N/m. With the addition of the rail dampers, the period of the track structure is still equal to the sleeper spacing, but within each period, there are two discrete supports, one being the railpad/sleeper/ballast system and the other being two rail dampers. For the vertical dynamics of a railway track up to about 3000 Hz, the Timoshenko beam model may be employed to model the rail. The rail displacement vector is defined as q ¼ ðw; ψ ÞT , where w is the vertical displacement (directed downwards) of the rail and ψ is the rotation angle (directed clockwise) of the cross-section due to the bending moment only. See [16] for more details. 5.2. Green function and wheel-rail force for the first track Based on the discussion in Ref. [21], the time domain Green function for rail vertical vibration is calculated with 15 r j r15,  25:12 r β r 25:12 rad/m and  3200 r Ω=2π r3200 Hz. A 2D inverse fast Fourier transform (FFT) algorithm is used on a matrix of order 3200  3200, implying that Δβ ¼ 0:0025  2π rad/m and ΔΩ ¼ 2π  2 rad/s. According to the principle of FFT, the Green function is available between x0 ¼  200 m and 200 m at spacing Δx0 ¼ 0:125 m, and between τ ¼  0:25 s and 0.25 s at spacing Δτ ¼ 1:5625  10  4 s. For a train running at 300 km/h (83.33 m/s), the sleeper passing time is 7:20  10  3 s, which is 46 times Δτ. According to Eq. (20), the response of the rail at any moment t is determined by the force time history fðτÞ and the Green function matrix Gðx0 ; x0 þ ct; t  τÞ with τ r t. Now Gðx0 ; x0 þ ct; t  τÞ is available for 0:25 rt  τ r 0:25 s. This is equivalent to t  0:25 r τ rt þ 0:25:

(55)

In other words, the contribution to the rail response of the force time-history of length 0.25 s before t can be evaluated. For a train running at 300 km/h, it travels about 35 sleeper bays ð83:33  0:25=0:6 ¼ 34:72Þ in 0.25 s.

5.2.1. Comparison with Ref. [9] for Green function Fig. 3 shows the stationary Green function with both the loading and response point at mid-span. It is almost identical to Fig. 9 in Ref. [9], as a result of the fact that, the track parameters used here are actually the same as those in Ref. [9]. The frequency component at the pinned-pinned frequency (about 1070 Hz) is clearly seen in the Green function.

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Displacement (m/(Ns))

144

Displacement (m/(Ns))

Time (τ) (s)

Time (τ) (s) Fig. 3. Stationary Green function for rail vertical vibration with both the loading and response positions at mid-span. Upper plot for τr 0.04 s and lower for 0.04 s r τ r 0.08 s.

5.2.2. Comparison with Ref. [9] for wheel-rail interaction Here, interactions between a single wheel travelling at 60 m/s along the track without any wheel-rail roughness are simulated. The mass of the wheel is 550 kg, its radius is 0.46 m, and half the axle load is 100 kN. At the start of the simulation, the wheel is above a sleeper from which the distance over which the wheel travels is measured. Fig. 4 shows the wheel and rail displacements at the wheel-rail contact point as the wheel travels over the first ten sleeper bays, and Fig. 5 shows the wheel-rail force as the wheel rolls over the sixth to the tenth sleeper bays. The inserted small plots in these two figures are taken from Ref. [9] for comparison. From Fig. 4 it can be seen that, at the early stage of the iterations, no significant transitory behaviour is present and the steady-state solution is achieved after iterations for about two sleeper bays, in contrast to the huge transitory behaviour indicated in Fig. 19 of Ref. [9] for a speed of 24 m/s. The fast convergence is due to the use of the initial displacements of the wheel and rail given in Eqs. (53) and (54). From Fig. 4 it can also be seen that, the pattern of the steady-state displacement curves are quite the same as those shown in the small plot, although the displacement predicted in this paper is a little lower. This difference may be caused by the difference in the treatment of the Duhamel integration. This reason may be also used to explain the difference shown in Fig. 5 in the wheel-rail force between this paper and Ref. [9].

145

Wheel and rail displacements (m)

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Distance (m)

Wheel-rail force (N)

Fig. 4. Wheel (    ) and rail (    ) displacements at the wheel–rail contact point for the wheel travelling at 60 m/s over the first ten sleeper bays. No wheel-rail roughness is present. The inserted small plot is Fig. 20 (d) of Ref. [9].

Distance (m) Fig. 5. Wheel-rail force for the wheel travelling at 60 m/s over the sixth to tenth sleeper bays. No wheel-rail roughness is present. The inserted small plot is Fig. 21 (d) of Ref. [9].

5.2.3. More results for Green function To determine the physical wave speed of the rail as part of the railway track, stationary Green functions at different positions should be consulted. They are shown in Fig. 6 for five response positions: 0 m, 5 m, 10 m, 15 m and 20 m away from the loading point above a sleeper. The distance from the response point to the loading point divided by the time-delay gives the average speed of the wave which is the first to arrive at the observation point. The wave speed (indicated by the slopes of the dashed lines) can be estimated to be around 2000 m/s. This is the speed of the free wave of the rail at the pinned-pinned frequency. The moving Green function, denoted by Gw ðx0 ; x0 þ ct; τÞ, for rail vertical vibration with load speed 100 m/s is shown in Fig. 7 for x0 þ ct ¼ 0:3 m and x0 ¼ 0 m. The loading position is at the mid-span when x0 þ ct ¼ 0:3 m, and x0 ¼ 0 means the response position is the same as the loading position. Effect of the load speed can be seen by comparing Figs. 3 and 7. The moving Green function as function of τ with load speed being 100 m/s and x0 þ ct ¼ 0:3 m is shown in Figs. 8 and 9 for two response points, the first one being 4.25 m (around 7 sleeper bays) ahead the load (i.e. x0 ¼ 4:25, Fig. 8), the second one being behind the load by 4.25 m (x0 ¼  4:25, Fig. 9). It can be seen that response behind the load is stronger than that ahead the load by the same distance, as already pointed out in Section 3.6.

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Displacement (m/(Ns))

146

Time (τ) (s)

Displacement (m/(Ns))

Fig. 6. Stationary Green function for rail vertical vibration with the impulse at above a sleeper. (a)x0 ¼ 0 m; (b)x0 ¼ 5 m; (c) x0 ¼ 10 m; (d) x0 ¼ 15 m; (e) x0 ¼ 20 m.

Time (τ) (s) Fig. 7. Moving Green function for rail vertical vibration with load speed 100 m/s. Both the loading and response positions are at mid-span.

5.3. Green functions of the second track To determine the physical wave speed of the rail as part of the railway track with rail dampers, stationary Green functions should be checked. They are shown in Fig. 10 for four response positions: 5 m, 10 m, 15 m and 20 m away from the loading point above a sleeper. It can be seen that the wave speed is not constant (indicated by changes of the slopes of the dashed lines), but instead, it decreases significantly as the distance between the response point and the loading point increases, indicating that waves of higher propagating speed decay faster than those of lower propagating speed. However, the speed approaches to a constant value (around 500 m/s) as the distance is sufficiently large. The wave speed is much lower than that in the original track. This is caused by the rail dampers which have been tuned to supress vibration waves around the pinned-pinned frequency [16]. The moving Green function as function of τ with load speed being 100 m/s and x0 þ ct ¼ 0:3 m is shown in Figs. 11 and 12 for two response points, 10 m ahead the load (i.e. x0 ¼ 10, Fig. 11), and 10 m behind the load (x0 ¼  10, Fig. 12). It is seen that response behind the load is much stronger than that ahead the load by the same distance. In addition to that, comparison between Figs. 11 and 12 reveals that the oscillating frequency of the response ahead the load is lower than that behind the load.

147

Displacement (m/(Ns))

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Time (τ) (s)

Displacement (m/(Ns))

Fig. 8. Moving Green function for rail vertical vibration with load speed 100 m/s, x0 þ ct ¼ 0:3 m and x0 ¼ 4:25 m.

Time (τ) (s) Fig. 9. Moving Green function for rail vertical vibration at 100 m/s, x0 þ ct ¼ 0:3 m and x0 ¼  4:25 m.

5.4. Wheel-rail force due to a single wheel rolling over a rail indentation Formulations developed in Section 4 is now applied to determine wheel-rail forces due to an indentation situated on the rail head. This rail head indentation can also be a replacement of a wheel flat. The indentation is described by the following equation: zðxÞ ¼ 

  e 2πx 1  cos ; ð0 rx r lÞ 2 l

(56)

where e ¼ 0.35 mm stands for the depth of the indentation and l ¼ 60 mm stands for its length. Wheel-rail forces are predicted for a single wheel from a high-speed train travelling at 60 m/s and 100 m/s along the tracks. Half the axle load is 62.72 kN, and the mass and radius of the wheel are, respectively, 890 kg and 0.42 m. As demonstrated in Section 5.2.2, the steady-state response due to an idea wheel moves along a smooth rail is achieved after iterations for about two sleeper bays. Thus simulations in this section are performed for five sleeper bays, and the indentation is just located at the middle of the third sleeper bay. As the wheel rolls over the indentation, wheel-rail force changes dramatically, and therefore the time step for iteration must be sufficiently small. For a given wheel speed, it is suggested that there should be more than 50 time steps within the indentation-passing duration.

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Displacement (m/Ns)

148

Time (τ) (s)

Displacement (m/Ns)

Fig. 10. Stationary Green function for rail vertical vibration with the impulse at above a sleeper. (a) x0 ¼5 m; (b) x0 ¼ 10 m; (c) x0 ¼ 15 m; (d) x0 ¼20 m.

Time (τ) (s)

Displacement (m/Ns)

Fig. 11. Moving Green function for rail vertical vibration with load speed 100 m/s, x0 þ ct ¼ 0:3 m and x0 ¼ 10 m.

Time (τ) (s) Fig. 12. Moving Green function for rail vertical vibration with load speed 100 m/s, x0 þ ct ¼ 0:3 m and x0 ¼  10 m.

149

Wheel-rail force (N)

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Wheel position (m)

Wheel and rail displacements (m)

Wheel position (m)

Wheel position (m)

Wheel position (m)

Fig. 13. Wheel-rail force and displacement position-histories due to the wheel moving along the track without rail dampers. (a) Wheel-rail force at 60 m/s; (b) Wheel-rail force at 100 m/s; (c) Wheel (solid line) and rail (dashed line) displacements at 60 m/s; Thick line stands for rail indentation; (d) Wheel (solid line) and rail (dashed line) displacements at 100 m/s.

For the track without rail dampers, results are shown in Fig. 13. It can be seen that loss of wheel-rail contact occurs even at 60 m/s. The distance of loss of contact is 30 mm at 60 m/s and 33 mm at 100 m/s. The maximum impact wheel-rail force changes from 230 kN at 60 m/s to 276 kN at 100 m/s, an increase of 20 percent. Fluctuations of the wheel-rail force at the pinned-pinned frequency of the rail are clearly shown in the plots. Before the impact, fluctuations in the wheel-rail force and displacements at the wheel-rail contact point are just caused by the varying dynamic stiffness of the rail within a sleeper bay (the so-called parametric excitation [8]). It can be seen that the wheel-rail force fluctuates more at 100 m/s than at 60 m/s, but this is not true for the displacements at the wheel-rail contact point. For the track with rail dampers, results are shown in Fig. 14. It can be seen that double losses of wheel-rail contact occur at 60 m/s. The distances of loss of contact is 32 mm and 8 mm. There is only one loss of contact at 100 m/s, and the distance of loss of contact is 36 mm. Interestingly, the maximum impact wheel-rail force changes from 289 kN at 60 m/s to 270 kN at 100 m/s, a small reduction. Fluctuations of the wheel-rail force at the pinned-pinned frequency of the rail are clearly supressed by the rail dmpers. Thus, by installing the rail dampers, the maximum wheel-rail impact force grows from 230 kN to 289 kN at 60 m/s, an increase of 26 percent. The increase in wheel-rail impact force is caused by the increased mass of the track with the dampers. However at 100 m/s, the maximum wheel-rail impact force is not significantly affected by the rail dampers.

X. Sheng et al. / Journal of Sound and Vibration 377 (2016) 133–154

Wheel-rail force (N)

150

Wheel position (m)

Wheel and rail displacements (m)

Wheel position (m)

Wheel position (m)

Wheel position (m)

Fig. 14. Wheel-rail force and displacement position-histories due to the wheel moving along the track with rail dampers. (a) Wheel-rail force at 60 m/s; (b) Wheel-rail force at 100 m/s; (c) Wheel (solid line) and rail (dashed line) displacements at 60 m/s. Thick line stands for rail indentation; (d) Wheel (solid line) and rail (dashed line) displacements at 100 m/s.

5.5. Wheel-rail forces caused by four wheels moving along the rail The usefulness is demonstrated in this section of formulations in Section 4 in dealing with interactions between multiple wheels and the track. First of all, calculation is performed for four wheels moving at 100 m/s over the rail indentation defined in Eq. (56). No rail damper is present. The wheels are the same as that in Section 5.4. The initial positions of the first to the fourth wheels, according to the dimensions of two neighbouring bogies of the high-speed train, are 0 m,  2.5 m,  7.624 m and  10.124 m. At t ¼ 0, the indentation is located ahead of the first wheel by 1.5 m (i.e. at the mid-span of the third sleeper bay). At the moment, the bogies and car bodies are excluded, leaving the wheels being coupled via the rail only. Wheel-rail forces are evaluated for a time duration in which the wheels move 15 m, or 25 sleeper bays. In that time duration, all of the four wheels will have a chance to impact the indentation. Fig. 15 shows the wheel-rail forces as function of wheel position. It can be seen that, as a wheel impacts the indentation, the wheel-rail force does not change significantly if compared with the case of a single wheel (Fig. 13 (b)). However, the impact causes other wheels to experience a small impact. It can be seen that, if Wheel A is ahead of Wheel B, then the small impact experienced by Wheel B as Wheel A rolls over the indentation is greater than what is other around. This is the effect of the wheel speed. Secondly, calculation is performed for the same four wheels moving at 100 m/s along the rail with a corrugation defined as z(x)¼Acos(2πx/λ), where A¼0.01 mm is the amplitude, and λ ¼0.1 m is the wavelength. At 100 m/s, the excitation

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Due to

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1st wheel

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Due to 1st wheel Due to 2nd wheel Wheel position (m)

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Fig. 15. Wheel-rail forces due to four wheels travelling at 100 m/s along the rail having an indentation. No rail damper is present. (a) the first wheel; (b) the second wheel; (c) the third wheel; (d) the fourth wheel.

frequency of the corrugation is 1000 Hz, close to the pinned-pinned frequency of the track without rail damper. Wheel-rail forces are evaluated for a time duration in which the wheels move 3 m, or 5 sleeper bays. Fig. 16 shows the wheel-rail force of the first wheel as function of wheel position. It is seen that the steady-state solution, that is a period function with the period being a sleeper bay, is achieved after iterations for less than one sleeper bay. In other words, the calculation method detailed in Section 4 is quite effective, even though multiple wheels are present interacting with each other. Fig. 17 shows wheel-rail forces as the four wheels move at 100 m/s over the same sleeper bay. Due to the low decay rate of rail vibration around the pinned-pinned frequency [16], interactions between the wheels are rather significant, causing the wheel-rail force position-histories of the four wheels to be different from each other. On the other hand, interactions between the wheels can be significantly weakened by installing rail dampers onto the rail, since the rail damper can largely increase the decay rate of rail vibration [16]. As a result, the wheel-rail force position-histories of the four wheels are almost identical to each other, as shown in Fig. 18.

6. Conclusions In this paper, the time domain moving Green function of a railway track as an infinitely long periodic structure is defined, derived and discussed. The moving Green function is defined as the Fourier transform of the response of the rail due to a

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Fig. 16. Wheel-rail force of the first wheel due to four wheels moving at 100 m/s over a rail corrugation with amplitude of 0.01 mm and wavelength 0.1 m. No rail damper is present.

Wheel position (m) Fig. 17. Wheel-rail forces due to four wheels moving at 100 m/s over the same sleeper bay. The rail has a corrugation with amplitude of 0.01 mm and wavelength 0.1 m. No rail damper is present. ───, the first wheel, ─ ─ ─, the second wheel, ─  ─, the third wheel,      , the last wheel.

moving harmonic load, from the load frequency domain to the time domain. The response of the rail due to a moving harmonic load is calculated using the Fourier transform-based method. For the Green function, the following can be summarised: (1) The time domain moving Green function matrix,Gðx0 ; x0 þ ct; τÞ, can be produced by calculating a two-dimensional inverse Fourier transform with respect to wavenumber in the track direction and the frequency of the moving load. (2) The time domain moving Green function matrix is a real matrix, and satisfies Gðx0 ; x0 þ ct; τÞ ¼ 0 for τ r0. (3) The time domain moving Green function matrix,Gðx0 ; x0 þ ct; τÞ, is a periodic function of x0 þ ct with the period being the same as the period of the track structure. Its Fourier series form is given in this paper. (4) A relationship is established between the moving Green function and the conventional stationary Green function of the track as a linear and time-invariant structure. The moving Green function matrix, Gj:k:S, can be obtained by replacing x0 with x0 þcτ, and x0 with x0 þ cðt  τÞ, in the stationary Green function matrix, G ðx0 ; x0 ; τÞ. (5) If x0 a 0, then there is τ0 such that Gðx0 ; x0 þct; τÞ ¼ 0 for 0 r τ r τ0 . τ0 is termed the time-delay of the moving Green function matrix Gðx0 ; x0 þct; τÞ at x0 and t.

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Wheel position (m) Fig. 18. Wheel-rail forces due to four wheels moving at 100 m/s over the same sleeper bay. The rail has a corrugation with amplitude of 0.01 mm and wavelength 0.1 m. Rail damper is present. ───, the first wheel, ─ ─ ─, the second wheel, ─  ─, the third wheel,      , the last wheel.

(6) The time-delay of a moving Green function matrix with the response point being behind the loading point is shorter than that of the moving Green function matrix with the response point being ahead the loading point by the same distance. (7) The magnitude of a moving Green function with the response point being behind the loading point is much stronger than that of the moving Green function with the response point being ahead the loading point by the same distance. The oscillating frequency of the response ahead the load is lower than that behind the load. The time-domain moving Green function is applied in this paper to deal with wheel-rail interactions via the Duhamel integral equation. Since the track is an infinitely long structure, the integration should be carried out from a sufficient long past moment, rather just from zero. It is found that doing so can significantly speed up the achievement of the steady-state response. It is demonstrated that wheel-rail impact force due to a discrete irregularity can be significantly increased by installation of rail dampers for certain wheel speeds. However, the rail damper can significantly weaken interactions between multiple wheels by increasing the decay rate of rail vibration.

Conflict of interest statement We, the authors of this paper, certify that we have no affiliation with, or involvement in, any organisation or entity with any financial interest, or nonfinancial interest in the subject matter or materials discussed in this manuscript.

Acknowledgements The authors acknowledge the support to this work from the National Natural Science Foundation of China (U1434201), the China Railway (2015Z003-B) and Scientific Research Foundation of State Key Laboratory of Traction (2015TPL_T08).

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