Generalization of the Fourier transform-based method for calculating the response of a periodic railway track subject to a moving harmonic load

The 2.5D finite element presentation is employed here to describe the vibration of the rail. A unique discretization is made for every cross-section of the rail, and nodal lines parallel to the x-axis along the track direction are formed by nodes in a cross-section and corresponding counterparts in other cross-sections. The displacements of the n nodes on the x cross-section are denoted by a vector having 3n elements,where u, v and w are displacement components in the longitudinal (x-), lateral (y-) and vertical (z-) directions. According to the 2.5D FEM [18, 29], the differential equation of motion of the rail is given bywhere M, K ₀ and K ₂ are 3n × 3n symmetric matrices, and K ₁ is an anti-symmetric matrix; f(x, t) denotes the nodal force vector, in units N/m, consisting of two parts, one being the externally applied loads, and other being those provided by the supports.

The externally applied loads on the rail are assumed to be harmonic with radian frequency Ω and moving in the x-direction at speed c. At t = 0, the loads are applied at the x ₀ cross-section. The corresponding nodal force vector is given bywhere \({\text{i}}=\sqrt{-1},\) δ(·) is the delta-function, and p ₀ denotes the amplitude vector of the loads. In case of a moving constant load such as an axle load, Ω = 0.

It is assumed that the track structure is periodic in the x-direction with period equal to L, where L may be equal to, or greater than, the sleeper spacing L ₀. In other words, at every length L in the x-direction, the track repeats all the details found in the interval [0, L], which is termed the 0th bay. The track consists of an infinite number of identical bays of length L, and the jth bay is located from x = jL to x = (j + 1)L, where j = −∞,…,−1, 0, 1, 2,…,+∞. Within each, say the 0th bay, there are a number, S, of supports (including attachments such as rail dampers) having arbitrary degrees of freedom. The sth support in the 0th bay is located at x = x _s , where 0 ≤ x _s  < L. The sth support in the jth bay is located at x = jL + x _s . The supports may produce not only point forces to part of the nodes of the main structure, but also torques. The point forces produced by the sth support in the jth bay are denoted by a force vector f _js (t) which is applied at x = jL + x _s and assumed to contain N _s components. A torque applied by the support can be one of the following: that in the cross-sectional (yz) plane (see Fig. 1), that in a longitudinally vertical plane (parallel with the xz plane) and that in a horizontal plane (parallel with the xy plane). A torque in the cross-sectional plane can be replaced by two vertical (or two lateral) point forces and therefore they are not treated as a torque. Remaining torques are denoted by a torque vector m _js (t) consisting M _s components. This torque vector may be represented by two force vectors, m _js (t)/Δx and −m _js (t)/Δx, applied, respectively, at two cross-sections separated by a distance Δx: x = jL + x _s  + Δx and x = jL + x _s ; see Fig. 1 for illustration. In Fig. 1, a torque, M _y , is applied in the vertical plane containing the kth nodal line. This torque is replaced by two forces F _z and \(F_{z}^{'} ,\) where \(F_{z} = - \frac{{M_{y} }}{\Updelta {x}}\) and \(F_{z}^{'} = \frac{{M_{y} }}{\Updelta {x}}.\) Another torque, M _z , is applied in the horizontal plane containing the lth nodal line. This torque is replaced by two forces F _y and \(F_{y}^{'} ,\) where \(F_{y} = \frac{{M_{z} }}{\Updelta {x}}\) and \(F_{z}^{'} = - \frac{{M_{z} }}{\Updelta {x}}.\)

The nodal force vector provided to the rail by all the supports is given bywhere U _s is a matrix of order 3n × N _s , N _s  ≤ 3n, with elements being either 1, −1, or 0 and such that \({\varvec{U}}_{s}^{\text{T}} {\varvec{U}}_{s}\) is a unit matrix of order N _s  × N _s . U _s describes the connectivity between the sth support and the rail. As an element of U _s , U _s (i, k) = 1 if the kth element of f _js (t) acts at the ith degree of freedom of the rail and in the same direction. U _s (i, k) = −1 if the kth element of f _js (t) acts at the ith degree of freedom of the rail but in the opposite direction. U _s (i, k) = 0 if otherwise.

V _s is similar to U _s but is of order 3n × M _s , where M _s  ≤ 3n. Again \({\varvec{V}}_{s}^{\text{T}} {\varvec{V}}_{s}\) is a unit matrix of order M _s  × M _s . If the equivalent force at the x = jL + x _s  + Δx cross-section of the kth element of m _js (t) acts at the ith degree of freedom of the rail and in the same direction, then V _s (i, k) = 1. If the equivalent force at the x = jL + x _s  + Δx cross-section of the kth element of m _js (t) acts at the ith degree of freedom of the rail but in the opposite direction, then V _s (i, k) = −1. Otherwise V _s (i, k) = 0.

U _s and V _s may be termed connectivity matrices of the sth support.

The dynamics of the sth support is described by the receptance matrix of the support observed at the degrees of freedom corresponding to f _js (t) and m _js (t), and this matrix is denoted by H _s (ω), where ω is angular frequency. This is a symmetric matrix of order (N _s  + M _s ) × (N _s  + M _s ). It may be decomposed into four sub-matrices as below:with the upper left sub-matrix being of order N _s  × N _s , the upper right sub-matrix being of order N _s  × M _s , the lower left sub-matrix being of order M _s  × N _s , and the lower right sub-matrix being of order M _s  × M _s .

Substitution of Eqs. (3) and (4) into Eq. (2) yields the differential equation of the track structure:

Equation (8) is identical to Eq. (8) in [15] and the solution of Eq. (8) for q _e (x, t) must be the same as that presented in that reference:where, by denoting the 3n × 3n matrix Q _e (x) is determined by the following inverse Fourier transform:where β is the wavenumber in the x-direction in units rad/m. Since K ₁ is an anti-symmetric matrix, according to Eq. (12), D(β, ω) is a Hermitian matrix; i.e., it is identical to its conjugate transpose.

It can be seen that Eq. (9) is different from Eq. (9) in [15] due to the extra supports in a bay. However, the procedure to find q _c (x, t) is similar to that presented in [15] and details are given in the Appendix for readers’ convenience. The key point is that since the track is periodic in the track direction and the moving loads are of the same frequency Ω, the force vector of a support in a bay is identical to that of the corresponding support in another bay apart from a time lag, i.e., Equations (14) and (15) have been termed the periodic conditions in the literature. q _c (x, t) is given by (see the Appendix),where ω is given by Eq. (11), and where where δ _rs is the Dirac-delta.

Equation (16) combined with Eq. (10) gives the total response of the rail. If observation is made from a reference frame moving with the loads, then the displacements of the structure are given by Eqs. (10) and (16) by setting x = x′ + x ₀ + ct, i.e.,where x′ is the coordinate relative to the moving frame of reference. As in the case of stationary harmonic loads, the matrix in Eq. (22) before the moving load vector is also termed receptance matrix. It can be seen that the receptance matrix of the rail is not temporally constant, but instead, it is a periodic function of time t with period equal to L/c. This periodic matrix is given byIt is in the Fourier series form, that iswith the constant term (i.e., the term with j = 0) being given byand the jth term being given by

It can be seen from Eqs. (25) and (26) that each term is expressed as an inverse Fourier transform of a particular matrix in the wavenumber (β-) domain. An inverse Fourier transform may be performed using the FFT technique. According to FFT, responses within \(|x^{\prime} | \le x_{\hbox{max} }^{\prime}\) will be available, where, \(x_{\hbox{max} }^{\prime} = \frac{2\uppi }{2\Updelta {\beta} },\) with Δβ being the wavenumber resolution. For example, if Δβ = 2π × 0.0025 rad/m, then \(x_{\hbox{max} }^{\prime} = 200\) m. In other words, rail responses within a total length of 400 m will be available. This makes it comfortable to deal with interactions between a whole high-speed train and a railway track.

It is also worthy of pointing out that if the dynamic stiffness of all the supports and dampers are evenly distributed along the track, then the track becomes a continuously supported and damped structure. In this situation, all the terms in Eq. (24) vanish except for that with j = 0. In other words, Eq. (24) becomes \({\varvec{Q}}(x^{\prime} ,\;t) = {{\tilde{\varvec{Q}}}}_{0} (x^{\prime} );\) i.e., the rail receptance matrix is time-invariant.

The spectra of the support force vector in the 0th bay are given in Eq. (74). It can be shown that the support force time-history is given by

Displacements of all or part of the degrees of freedom of all the supports in the 0th bay is denoted by a vector g ₀(t). The spectrum of it is denoted by \({\hat{\varvec{{g}}}}_{0} (f),\) where f is spectral frequency. A receptance matrix, denoted by G(2πf), may be defined for the supports such thatwhere \({\hat{\varvec{{{f}}}}}_{0} (f)\) and \({\hat{\varvec{{{m}}}}}_{0} (f)\) are defined in Eq. (63), and the minus sign indicates that forces and moments are opposite in direction to associated displacement vectors. It can be shown thatIn Eqs. (27) and (29), ω is given by Eq. (11). The terms in the bracket are Fourier transforms. These transforms are from the wavenumber (β-) domain to the space (x = x ₀ + ct) domain. Equations (27) and (29) indicate that the support force and displacement are oscillating at frequency Ω but their amplitudes decay as the loads move away along the track.

Support forces in the jth bay can be worked out according to Eqs. (14) and (15),and displacements of the supports are given by

Characteristic free vibration at frequency ω of a periodic structure satisfieswhere the non-dimensional quantity \(\bar{\beta }L\) is termed propagation constant at the given frequency [1]. As shown in [15], \(\bar{\beta }\) is the root of the following equationwhere since ω is independent of β [see Eq. (11) with the load speed being vanishing], matrix in Eq. (20) has been denoted alternatively by A(β, ω).

According to Eq. (21), A(β, ω) is a periodic function of β with period equal to 2π/L. Thus, if \(\bar{\beta }\) is a root of Eq. (33), then \(\bar{\beta } \pm \tfrac{j2\uppi }{L}\;(j = \pm 1,\; \pm 2, \ldots )\) are roots of Eq. (33) as well.

A set of typical parameters for the track structure are listed in Table 1. These parameters are for half the railway structure and correspond to a track with concrete sleepers and moderately stiff rail pads. The rail damper is constructed with a metal block on a layer of elastomeric material, and allowed to have translational vibration in the vertical direction as well as rotational vibration in the longitudinally vertical (i.e., x–z) plane. Parameters used for the dampers are listed in Table 2, similar to those used in [26].

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 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 3,000 Hz, the Timoshenko beam model can be employed to model the rail. According to the Timoshenko beam theory, the differential equation for the rail subject to a unit vertical moving harmonic load is given by 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; F _j1(t) is the vertical force applied on the rail by the jth sleeper and F _j2(t) the force applied by the two dampers in the jth bay; M _j1(t) is the torque exerted on the rail in the longitudinally vertical plane by the jth sleeper and M _j2(t) the torque from the two dampers in the jth bay; and finally, S = 2, x ₁ = 0 and x ₂ = L/2. Comparing Eqs. (34) and (35) with Eq. (6), it follows thatThat the connectivity matrix V ₁ = V ₂ = 0 is the result of the Timoshenko beam model of the rail in which the rotation angle of the cross-section due to the bending moment only is chosen to be one of the degree of freedom.

The receptance matrix of the support including a railpad, a sleeper and the ballast, is given bywhere k _P and k _B are complex stiffness of the railpad and ballast, m _S is half the sleeper mass and b _S = 0.25 m is the width of the sleeper. Sleepers are assumed to be rigid and vibrate in the vertical direction only.

The receptance matrix of the two dampers is given byand the meanings of symbols appearing in Eq. (37) are defined in Table 2. Each rail damper has two natural frequencies, one for vertical vibration and the other for rotational vibration, as shown in Table 2. These frequencies are all close to the first pinned–pinned frequency of the original track, which is around 1,070 Hz [19].

Figure 3 shows the driving point receptances of the supports at the rail/support connecting points. At the natural frequency of vertical vibration, the corresponding receptance of a rail damper is the minimum (zero if there is no damping; Fig. 3, dashed line), providing effective constrains to the rail at the mid-span. It is expected that the dampers will damp the pinned–pinned vibration of the rail. Since in the pinned–pinned vibration mode, rail rotation is least at mid-span, the rotational resonance of the rail damper will have an insignificant effect. For frequencies well below the first pinned–pinned frequency, the receptances of the rail dampers are much higher than those of the sleepers (Fig. 3, solid line). It thus can be reasoned that, at those frequencies, the addition of the dampers has an insignificant effect. However, for frequencies well above the first pinned–pinned frequency, the receptances of the rail dampers are still smaller than those of the sleepers. It thus can be reasoned that, at those frequencies, the addition of the dampers will have some effect.

Propagation constants (or wavenumber) of the track/damper system can be produced by solving Eq. (33). Alternatively, a contour plot of 1/|det(A(β, ω))| on the frequency–wavenumber plane, as shown in Fig. 7, can also be employed to show propagation wavenumbers for given frequencies, which are displayed as yellow curves. These yellow curves are also called dispersion curves of the periodic structure. A point on a dispersion curve defines a wavenumber and a frequency. If there is no damping in the track, the periodically supported rail allows free vibration at that frequency to propagate, without attenuation, along the rail at that wavenumber. For comparison, propagation wavenumbers of the original track are also shown here on the left (from [19]). It can be seen that the presence of the dampers generates more bounding frequencies [19], but the most important effect of the dampers is that they convert Pass Band 2 of the original track into two stop bands, and pushes the lower bounding frequency of Pass Band 3 to a much higher value, thus significantly increasing the total width of stop frequency bands.

When the load is moving, the displacement amplitude of the rail at the loading point is not, due to the discrete supports, constant but instead a periodic function of time. The variation of the displacement amplitude of the loading point with time t varying over [0, L/c] is equivalent to the variation due to the initial loading position, x ₀, varying over [0, L] at t = 0.

The vertical displacement amplitude of a sleeper to a unit vertical force applied at the top of the railpad is given by 1/(k _B – m _S ω ²), and that due to a torque in the longitudinally vertical plane is zero. The vertical displacement amplitude of the mass of a damper to a unit vertical force at its bottom is given by −1/(m _D ω ²), and that due to a torque in the longitudinally vertical plane is zero. Thus, the matrix, G(ω), in Eqs. (29) and (31) for calculating the vertical displacements of the sleeper and the damper mass is given by

Figure 8b shows rail receptance at the loading point to a stationary load applied above a sleeper (in solid line) and at the mid-span (in dashed line; i.e., above a damper). For comparison, rail receptance of the original track is also shown in the figure on the left (Fig. 8a).

It can be seen that the pinned–pinned frequency at 1,070 Hz (indicated by letter A) of the original track disappears, as expected, thanks to the dampers. However, the installation of the dampers generates two new peaks (indicated by letters B and C). Peak B corresponds to the bounding frequency B shown in Fig. 7b. At this frequency, sleepers behave like pins and the point at the mid-way (that is the position of the damper) between two neighboring sleepers has the maximum response across the span defined by these two sleepers. In other words, the pinned–pinned frequency of the original track is shifted down to a much lower value by the dampers. Peak C, at about 1,250 Hz, corresponds to the bounding frequency C shown in Fig. 7b. At this frequency, dampers behave like pins and the point at the mid-way (that is the position of the sleeper) between two neighboring dampers has the maximum response. Thus this is a pinned–pinned frequency created by the dampers. The effect of these two pinned–pinned frequencies on rail roughness growth is unknown at the moment, and to be investigated in the future.

Rail receptance at the loading point to a load initially being above a sleeper (in solid line) and at the mid-span (in dashed line) is shown in Fig. 9, with the left plot for a stationary load and the right plot for a load moving at 100 m/s. It can be seen that Peak B is flattened by the load speed, and Peak C is split into two sub-peaks.

Figure 10 shows receptance (at the initial moment) of a sleeper to a load initially being above the sleeper (in solid line) and at the mid-span (in dashed line) next to the sleeper, with the left plot for a stationary load and the right one for a load moving at 100 m/s. It can be seen that the effect of the initial loading point within the associated sleeper bay on the response of the sleeper is quite small and this effect becomes even smaller as the load moves fast.

Figure 11 shows receptance (at the initial moment) of a damper to a load initially being above the left neighboring sleeper (in solid line) and at the mid-span (i.e., above this damper; in dashed line), with the left plot for a stationary load and the right one for a load moving at 100 m/s. Peak B is due to the pinned–pinned vibration of the rail: in this vibration mode, the mid-span of the rail, and the damper attached here, have peak responses. Load speed splits this peak into two sub-peaks.

Comparison between the rail, sleeper, and damper is shown in Fig. 12 for receptance magnitude and in Fig. 13 for phase angle. It can be seen from Fig. 12 that, for frequencies higher than 500 Hz, rail vibration is largely absorbed by the railpad, with a small fraction transmitted to the sleeper. The phase angles corresponding to the peaks at around 500 Hz are close to −90°. Between 450 and 1,300 Hz, the damper vibrates more strongly than the rail, but not in phase. The effect of damper vibration on total noise radiation from the track is to be investigated in the future.

As a harmonic load moves along the rail, vibration wave are generated in the rail ahead and behind the load. Due to the load speed, the wave ahead the load is different from that behind the load, not only in amplitude, but also in wavelength and decay rate. The wave behind the load exhibits, more or less, higher amplitude, longer wavelength and less decay rate than the wave ahead the load.

Figure 14 shows the waves generated in the rail by a unit load of 2,000 Hz moving at 100 m/s along the rail. Due to damping in the track, vibration attenuates with distance from the load. However, significant vibration is still observed even 20 m away from the load.

Figure 15 shows the displacement of a damper generated by the same load as a function of distance between the damper and the load. Initially the load is above the left sleeper of the damper, i.e., the initial distance between the damper and the load is −0.3 m. The damper’s response increases as the load approaches the damper, and then decreases as the load moves away. However, the decay is not purely exponentially. Amplitude modulation at the sleeper passing frequency is evident in Fig. 15: the response of the damper is a little smaller when the load is at a sleeper than those when the load is just before and after the sleeper.

The rate of vibration decay along the rail may be worked out by plotting the vibration level against distance from the load, as shown in Fig. 16. Vibration level is defined as 20log(|w|). Two straight lines can be seen in the figure. The slope of the left straight line is the decay rate, in dB/m, of vibration waves behind the load, and that of the right straight line is the decay rate of vibration waves ahead the load. Decay rates determined by this way is different from those numerically computed from the so-called dispersion equation [or propagation constant equation, i.e., Eq. (33)], since the former includes the effect of the load speed, but the latter does not.

Figure 17 shows the decay rate of the rail vibration wave ahead a load moving at 100 m/s. The curve in solid line is for the track with dampers, and that in dashed line is for the original track. Load frequencies range from 50 to 2,000 Hz at a step of 50 Hz.

It can be seen that, for frequencies higher than about 500 Hz, the decay rate of the original track is quite small. Even for frequencies within the stop band (from 1,070 to 1,300 Hz, see [19]) between the second and the third pass bands (see Fig. 7), the decay rate still remains small (less than 2 dB/m). This is because that the stop band is too narrow to avoid the effect of the load speed: propagating waves are still generated by the load moving at 100 m/s.

However, with the dampers, decay rates for frequencies above 500 Hz are significantly increased. This is caused by the combined effect of wider stop bands (see Fig. 7b) and extra damping in the dampers.

Thus, decay rate for a stationary load may not be sufficient to assess the acoustic behavior of a track; it is important to consider the effect of train speed.