Rolling stock vibrodynamic impact model on railway pipelines

. The present paper presents the rolling stock vibrodynamic impact model on railway pipelines, obtained on the basis of experimental studies conducted under field conditions at a railway station. The experimental research program provided for the determination of the effect of vibration-dynamic effects of rolling stock on the working condition of pipes and butt joints by conducting vibration-measuring work on the investigated section of the railway pipeline during the passage of various series of locomotives. The proposed modeling method makes it possible to obtain a correlation function of the oscillatory process of a railway pipeline, on the basis of which a spectral density is constructed to identify the amplitude-frequency range at which a stable resonance region occurs, leading


Introduction
Increase in traffic flows on railways in modern conditions has led to a significant increase in the destructive effects of rolling stock on railway pipelines, which negatively affected their reliability [1]. This fact is confirmed by the analysis of statistical data on pipeline damage at a number of railway stations. As per the data obtained, the predominant number of damages (up to 70%) occurs on pipes laid in the immediate vicinity of station tracks [2].
A number of scientific papers are devoted to the issues of influence of various types of impacts (dynamic, seismic, vibration) on the condition of underground pipelines [3][4][5][6]. It is necessary to point out that dynamic, seismic, vibration effects on the condition of underground pipelines are similar in their destructive ability and, consequently, scientific papers devoted to this topic can be used to analyze the existing problem.
The article [3] reveals the effect of seismic wave propagation on underground pipelines. There has been worked out a model aimed at estimating the axial deformation of the pipe body and the relative displacement of pipe joints under seismic influences. Meanwhile, the maximum deformation of the pipe body and the displacement of the joint relative to the pipe inevitably causes destruction of a pipeline. The study of the influence of seismic impacts on the wall thickness and diameter of the underground pipeline was carried out in [4]. The outcomes of the study demonstrated that the peak axial deformation of the pipeline increases along with decrease in the thickness and diameter of the pipeline being exposed to seismic waves.
The article [5] presents the results of a study of the effect of rail fasteners on the vibrodynamic effect of rolling stock transmitted to the ground of the roadway and then to underground pipelines. Changes in the amplitude values of the soils were recorded with an increase in the speed of passing trains. Similar actions occur with respect to underground pipelines, while the amplitude values of the pipe body and the butt joint are higher than the amplitudes of the soil. This fact is explained by the fact that the soils are a scattering natural environment, and the underground pipeline is a concentrated element laid in the ground. On the other side of the ground, underground pipelines are mainly subjected to force, vibration and corrosion effects. Vibration actions are the decisive cause of fatigue destruction of the metal of the pipe and butt joints, and also affect the nature of the force interaction of the surrounding soil with the underground pipeline [6].
The consequences of the influence of vibrodynamic influences on the working condition of pipelines are inextricably linked with their reliability. In [7], the reliability of pipelines using continuous Markov chains was evaluated. The proposed model made it possible to objectively assess the reliability of pipelines for the purpose of long-term forecasting and ensuring their operable condition for the entire period of operation.
In order to study the effect of the vibration-dynamic effects of rolling stock on the working condition of pipelines, full-scale experimental studies were carried out at the railway station. Studies have demonstrated that the frequency of pipeline failures is really influenced by the vibrodynamic effects of railway transport [8]. Meanwhile, dynamic loads from railway transport transmitted through the ground cause various types of damage to pipelines.
The experimental research program provided for the determination of the effect of vibration-dynamic effects of rolling stock on the condition of the pipe and butt joint by conducting vibration-measuring work on the test section of the existing pipeline laid along the railway track.
The carried out experimental studies allowed to: -to quantify the degree of influence of the vibrodynamic load on the condition of the pipeline under study and the conditions under which this influence manifests itself to the greatest extent; -to identify the main frequency-amplitude characteristics of the oscillatory process of the pipeline using the theory of vibrations and vibrometry, as well as mathematical statistics; -to establish the amplitude-frequency range of the spectral density at which a stable resonance region occurs, leading to the destruction of the pipeline.

Materials and methods
The methodology of experimental studies of the pipeline section laid on the territory of the railway station was developed on the basis of the current train schedule. In compliance with the experimental research program, measurements were made of the magnitude of vibration accelerations of the pipeline section under study during the passage of various series of locomotives along the station track.
For this purpose, VI6-6TN vibration measuring equipment was used, complete with DU-5S primary acceleration converters mounted on a pipe and butt joint for measuring the magnitude of vibration accelerations in three directions: vertical, horizontal and axial. The received signals from the vibration sensors during the passage of the rolling stock were transmitted to the vibration measuring equipment and then recorded by the NO10M oscilloscope.
The processing of records of vibrodynamic processes was carried out in accordance with the basic provisions of the theory of vibrations and vibrometry. Table 1 demonstrates the values of vibration accelerations of the pipe and the butt joint of the pipeline section under study when passing along the station track of various series of locomotives. As per Table 1, the highest values of vibration accelerations of the pipeline section under study were obtained during the movement of the TEM-2 shunting locomotive. The results obtained are explained by the fact that the TEM-2 shunting locomotive has unsprung bogies, all other locomotives have sprung bogies. It is the "hard running" of the TEM-2 locomotive that causes large values of vibration accelerations of the pipeline section under study. It should also be outlined that the values of vibration accelerations of the butt joint are almost three times higher than the values of vibration accelerations of the pipe, since the joint is much heavier than the pipe.
In order to investigate effect of the vibration dynamic effects of rolling stock on the condition of pipelines, it is necessary to have data not only on the values of vibration accelerations, but also on other parameters of the oscillatory process, the determination of which is one of the main tasks of this study. These parameters include the frequency and amplitude of the oscillatory process of the pipeline.
The oscillatory process of the pipeline is a stationary (in a broad sense) process, since it is caused by a number of random constantly changing factors.
The mathematical model of the oscillatory process is stationary random processes [9][10][11][12]. Characteristics of stationary random processes: correlation function - The correlation function of a stationary random process is defined as . The correlation function of the oscillatory process shows the presence of a connection between any points of the process.
In the measurement experiment, their point estimates are used as characteristics of stationary random processes. The mathematical expectation is In the work, measurements of the ordinates of the oscillatory process were carried out at  For comparative analysis, a normalized correlation function is used: Where is the The spectral density describes the distribution of amplitudes in the form of a series expansion in harmonics with certain oscillation frequencies.
The spectral density of the oscillatory process of the pipeline is obtained by numerical integration in formula (3) using samples of oscillation frequencies and is expressed by the sum of the following form: Tthe measurement time of the oscillatory process.
In this case, the normalized spectral density has the form: the influence of the variance R (0) is excluded here. Table 2 shows the values of frequencies and amplitudes of the oscillatory process obtained as a result of experimental studies on the section of the pipeline laid on the territory of the railway station during the passage of various series of locomotives.

Outcomes
As a result of the experimental studies carried out on the pipeline section laid along the station track, during the passage of the rolling stock, an exponentially damped oscillatory process was obtained. The normalized correlation function of the oscillatory process of the pipeline section under study during the passage of the wheelset of the TEM-2 locomotive bogie is shown in Fig. 1. The correlation function has such a form when a short-term disturbance (shock) affects the structure. In this case, it is a blow from the wheelset of the TEM-2 locomotive bogie.

As a result of the analysis of numerical values of arrays
and expressions for the approximation of correlation functions, a function of the following type was selected: As for formula (5), the parameter  characterizes the attenuation intensity of the oscillatory process, and the parameter characterizes the  average oscillation frequency: For the normalized correlation function, based on (1), the expression follows: Since the oscillatory process of the investigated section of the pipeline during the passage of the wheelset of the locomotive trolley is harmonic in nature, the displacement frequencies coincide with the acceleration frequencies. Figure 2 shows a graph of the normalized spectral density of the oscillatory process of the pipeline section under study during the passage of the wheelset of the TEM-2 locomotive bogie.
For harmonic oscillations, the acceleration amplitude and the displacement amplitude are related by the following expressions: Where Y is the acceleration amplitude, g; And is the displacement amplitude, micrometrs;  is the oscillation frequency, Hz.
The resulting displacement amplitude of the pipeline section under study during the passage of various series of locomotives is determined by the expression: are the squares of the displacement amplitudes of the pipeline section under study in the vertical, horizontal and axial directions, respectively.

Analysis of outcomes
The outcomes of experimental studies conducted on the territory of the railway station confirmed the influence of the vibration-dynamic effects of rolling stock on the working condition of pipelines.
The values of vibration accelerations of the investigated section of the pipeline when passing along the station track of various series of locomotives (table 1) are obtained. A comparative analysis of the data obtained showed that the values of the vibration accelerations of the butt joint are almost three times higher than the vibration accelerations of the pipe. This fact explains the predominant amount of damage on the butt joints of railway pipelines. This is consistent with the statistical data of accidents on railway pipelines, which indicate that more than 50% of all types of damage occur at butt joints. The destruction of the butt joint occurs when the filler bulges out of the joint under vibrodynamic influences from the rolling stock. As a result, water leaks in the joint, and sometimes the joint itself is destroyed. Damage such as fractures and cracks are formed on the pipes during vibrodynamic impacts from the rolling stock.
Also, the change in the magnitude of vibration accelerations of the pipeline section under study, depending on the series of locomotives (table 1), has been detected. The maximum values of vibration accelerations are fixed during the passage of the TEM-2 shunting locomotive. The explanation of the obtained values of vibration accelerations is that the bogie of this locomotive series has the largest weight of the unsprung part and the maximum stiffness of the spring suspension compared to bogies of other locomotive series.
The calculations carried out showed that during the passage of the TEM-2 shunting locomotive, the maximum range of displacement amplitudes of the investigated section of the pipeline was obtained in the range from 100 to 240 micrometrs (table 2). The amplitudes of displacement of the investigated section of the pipeline of the identified range are the cause of most cases of damage on railway pipelines.
The main probabilistic characteristics of the oscillatory process of the pipeline section under studythe normalized correlation function (Fig.1) and the normalized spectral density ( fig.2), which are directly related to the amplitude-frequency spectrum of the oscillatory process of the pipeline section. The frequency range in the range from 80 to 120 Hz has been determined, in which stable resonance regions may appear, leading to damage to pipelines, while the resonant frequency value Hz res 100 =  .

Conclusion
The paper demonstrates a method for modeling the effect of vibrodynamic effects of rolling stock on the working condition of railway pipelines. The proposed method is based on the spectral analysis of stationary random processes with a continuous spectrum. The method made it possible to obtain an explicit form of the normalized correlation function and the normalized spectral density of the oscillatory process of the pipeline to identify a stable amplitude-frequency resonance region, which is the cause of its destruction.
A further direction of research into the vibro-dynamic effects of rolling stock on railway pipelines is to obtain an explicit view of the characteristics of the oscillatory process of the pipeline when implementing bogies of various series of locomotives. This will allow us to identify the ranges of amplitude-frequency spectra at which stable resonance regions arise. The identification of stable resonance regions will reduce the impact of vibration-dynamic effects of rolling stock on the destruction of pipelines.