Analytical evaluation of track response in the vertical direction due to a moving load

Abstract

In the literature, typical analytical track response models are composed of beams (which represent the rail) on viscoelastic or elastic foundations. The load is usually considered as a single concentrated force (constant or varying in time) moving with constant speed. Concentrated or distributed loads or multilayer track models have rarely been considered. One can find some interesting results concerning analysis of distributed loads and multilayer track structures that include both analytical and numerical approaches. However, there is a noticeable lack of sufficient comparison between track responses under concentrated or distributed load and between one and multilayer track models. One of the unique features of the present paper is a comparison of data obtained for a series of concentrated and distributed loads, which takes into account a wide range of track parameters and train speeds. One of the fundamental questions associated with the multilayer track model is the level of coupling between the rail and the vibrations of the sleepers. In this paper, it is proved that sleepers are weakly coupled with the rail if the track is without significant imperfections, and the steady-state response is analyzed for this case. In other words, sleeper vibrations do not influence the rail vibrations significantly. Therefore the track is analyzed by means of a two-stage model. The first step of this model determines rail vibration under a moving load, and then the sleeper vibration is calculated from previously obtained kinematic excitation. The model is verified by comparison of the obtained results with experimental data. Techniques based on Fourier series are applied to the solution of the steady-state track response. Another important problem associated with track response under moving loads arises from the analysis of the effect of longitudinal forces in rails on vertical displacement. It is shown that, in the case of the steady-state response, longitudinal forces do not influence rail displacements significantly and this observation remains correct for a wide range of track parameters and train speeds. The paper also analyzes the legitimacy of the statement that additional rail deflection between sleepers, compared to the continuous rail support, can be considered as a track imperfection.

On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

“…In the moving coordinate system ( = ; = − V ), for the load constant in time, (19) and expression (20) lead to the ordinary differential equation:…”

“…(1) Analysis of Timoshenko beam under moving constant and varying loads (presented, e.g., in [6][7][8][9]) (2) Analysis of a beam on elastic half-space [10,11] (3) Response of beam on nonlinear foundation (e.g., [8,[12][13][14]) (4) Dynamic response of beam on random foundation; see [15][16][17] (5) Dynamic response of track as multilayered structure (see [18,19], analytical approach; [20][21][22], numerical approach); (6) Analysis of set of distributed moving forces, described by Heaviside functions (e.g., [7]), rectangular function [9,19], cosine square formula [8,12,13], or Gauss function [19] (7) Effect of axial force on dynamic response [19,23] (8) Analysis of set of forces varying harmonically and associated with track imperfections including the phase of sine function for particular axles (numerically [20][21][22]) and analytical approach [13,19] 2 Shock and Vibration…”

“…Applications of Fourier series to obtain the solution of railway track response are practically limited to the case of bridges vibrations so far [4,22]. An example of use of the Fourier series in bounded interval to study the railway track response is presented in [19]. It is shown that solutions obtained in this manner can approximate the track response with very high accuracy depending on a number of Fourier series coefficients and the assumed length of interval.…”

“…(2) In the case when damping properties of rail foundation can be neglected, due to the load being distributed on very small spans and the distance between wheels being relatively large (practically more than about 5-6 m), dynamic factor for maximum rail displacements ds can be described by simple formula (based on [19,23]):…”

“…Currently, these problems can be numerically solved. The following track and load parameters are assumed in further study [19] Figure 1 presents the dynamic factor due to rail foundation damping for two options: higher values describe dynamic factor for viscous properties of fasteners r = 100 ro and sleeper foundation properties p = 10 po ; lower values describe dynamic factor for the reference values ro and po . As can be observed, these relations have various curvatures and show sensitivity related to the effect of damping on dynamic factor.…”