Estimation of the Horizontal Curvature of the Railway Track Axis with the Use of a Moving Chord Based on Geodetic Measurements | Journal of Surveying Engineering | Vol 148, No 4

In order to verify the method of the moving chord in the operated railway track, measurements of Cartesian coordinates of the track axis were carried out using a satellite antenna in a static manner. These measurements covered two geometric layouts with clearly different values of the radius of the horizontal arc and the nature of the curvature. The distances between the measurement points were approximately 5 m.The First Geometric LayoutThe obtained set of xi, yi track axis coordinates for the first geometric layout made it possible to draw an appropriate graph, which is shown in Fig. 3. The estimated maximum measurement error was ±25  mm. The ordinate of the plot at the first measurement point (for x=0) is y=92.820  m.The basis for identifying the track axis in the horizontal plane is the estimation of its horizontal curvature along its length. Therefore, to create the possibility of further analysis, go to the linear system [i.e., determine the distances (the variable L expressed in m) of individual measurement points from the selected starting point i0]. In the considered case, it was assumed that the starting point (i.e., L=0) has Cartesian coordinates x0=211.644  m, y0=149.530  m.The distance between two consecutive measuring points is (20) Δli/i+1=(xi+1−xi)2+(yi+1−yi)2The linear coordinate Li [i.e., the distance from the point (x0,y0)] is determined from (21) Eq. (21) uses the coordinates of all n points located on straight and arc segments.Using this aforementioned procedure (after taking the chord length lc=20  m), the horizontal curvature of the geometric layout shown in Fig. 3 was determined. The obtained graph of the curvature k(L) is shown in Fig. 4. In this case, the curvature is positive, which corresponds to the presence of a downward convexity in the curvilinear segment.As you can see, the obtained k(L) plot consists of two types of elements: •Sections oscillating around a horizontal course that decribe a curvature of a fixed value (zero on straight sections of the track and nonzero on circular arcs).•Segments oscillating around a linear wave (i.e., straight lines inclined to the L axis) that describe the variable curvature appearing on the transition curves.On the basis of this graph, it is possible to determine the value of the radius of the circular arc and the lenghts of the transition curves, as well as the location of the characteristic points (lying at the connections of straight sections with the transition curves and transition curves with a circular arc).It is assumed that the curvature of the track in straight sections is equal to zero, and the disturbances occurring there in the curvature diagram are the result of the existing deformations and measurement error. The arithmetic mean kCA¯ is determined from the selected range of curvature values that undoubtedly belong to the circular arc; its reciprocal is given by the value of the radius (22) The value obtained by Eq. (22) should be properly rounded (to full meters) and then used in the further calculation procedure. In the case under consideration, the mean value kCA¯=0.0008332428  rad/m was obtained, with standard deviation σk=0.0000719681  rad/m. Using Eq. (22), it corresponds to the value of the radius R=1,200.13045  m.There are transition curves on both sides of the circular arc. At the end of the left-hand straight section in Fig. 3 is the beginning of the TC1 transition curve (i.e., BTC1 point), and at the beginning of the right-hand line is the beginning of the TC2 transition curve (i.e., BTC2 point). The ends of both curves (i.e., the ETC1 and ETC2 points) mark the beginning and the end of the circular arc, respectively.In order to determine the linear coordinates of the mentioned characteristic points, it is necessary to determine the lines of the least squares describing the regions of the k(L) plot with variable curvature values. Fig. 5 shows an enlarged fragment of the curvature plot in Fig. 4 including the TC1 transition curve.For the selected range of curvature values belonging to the TC1 transition curve (i.e., connected to the straight line on the left side in Fig. 3), we obtain (23) For the beginning of this curve (BTC1 point), the value of curvature k=0; hence, its linear coordinate is (24) And for the end of the curve (ETC1 point), the value of the curvature k=kCA¯; therefore, its linear coordinate is equal to (25) The length of the transition curve connected with the straight line on the left in Fig. 3 result directly from the values of the determined coordinates LBTC1 and LETC1(26) lTC1=LETC1−LBTC1=kCA¯b1The further calculation procedure will take place in the x, y coordinate system; therefore, the appropriate Cartesian coordinates of the designated characteristic points should be determined. For a linear coordinate LBTC1, find such an interval of measurement points ⟨i,i+1⟩ that LBTC1∈⟨Li,Li+1⟩. The abscissa xBTC1 and the ordinate yBTC1 can be now determined from the following formulas: (27) xBTC1=xi+xi+1−xiLi+1−Li(LBTC1−Li)(28) yBTC1=yi+yi+1−yiLi+1−Li(LBTC1−Li)Similarly, the abscissa xETC1 of the end of the TC1 transition curve as well as the corresponding ordinate yETC1 are determined.The curvature equation for the TC1 transition curve shown in Fig. 5 is as follows:k=0.00000428498 L−0.0008361 where L is expressed in m and k in rad/m.The linear coordinates of the beginning of the curve [based on Eq. (24)] and its end [based on Eq. (25)] are: LBTC1=195.134  m, LETC1=389.591  m. Thus, the length of the transition curve lTC1=194.457  m determined from Eq. (26).Among the measurement points, the linear coordinate LBTC1∈⟨194.998;200.024⟩  m. This corresponds to the Cartesian coordinates x∈⟨400.000;404.830⟩  m and y∈⟨200.000;201.294⟩  m. On this basis—using Eqs. (27) and (28)—the Cartesian coordinates of the BTC1 point can be determined as xBTC1=400.131  m, yBTC1=200.035  m.The linear coordinate of the end of the curve LETC1∈⟨385.008;389.996⟩  m. This corresponds to the Cartesian coordinates x∈⟨582.190;586.905⟩  m and y∈⟨253.747;255.410⟩  m. On this basis—using Eqs. (27) and (28)—it is possible to determine the Cartesian coordinates of the ETC1 point as xETC1=586.522  m, yETC1=255.275  m.Fig. 6 shows an enlarged fragment of the curvature plot in Fig. 4 including the TC2 transition curve connecting the circular arc with the right-hand straight section in Fig. 3.For the selected range of curvature values belonging to the TC2 transition curve (i.e., connected to the straight section on the right in Fig. 3), we obtain (29) For the beginning of this curve (BTC2 point), the value of curvature k=0; hence, its linear coordinate is (30) And for the end of the curve (ETC2 point), the value of curvature k=kCA¯; therefore, its linear coordinate is equal to (31) The length of the transition curve connected with the straight section on the right in Fig. 3 results directly from the values of the determined coordinates LBTC2 and LETC2(32) lTC2=LBTC2−LETC2=−kCA¯b2The further calculation procedure will take place in the x, y orthogonal coordinate system; therefore, the appropriate Cartesian coordinates of the designated characteristic points should be determined. This is done in a similar way to the TC1 transition curve.The determined equation of curvature for the transition curve shown in Fig. 6 is as follows: k=−0.00000415252L+0.006857The linear coordinates of the beginning of the curve [based on Eq. (30)] and its end [based on Eq. (31)] are as follows: LBTC2=1,651.309  m, LETC2=1,450.649  m. Thus, the length of the curve [determined from Eq. (32)] is lTC2=200.660  m.Among the measurement points, the linear coordinate LBTC2∈⟨1,650.007;1,655.014⟩ m. This corresponds to the Cartesian coordinates x∈⟨1,371.532;1,372.827⟩ m and y∈⟨1,170.375;1,175.204⟩ m. On this basis, it is possible to determine the Cartesian coordinates of the BTC2 point as xBTC2=1,371.869  m, yBTC2=1,171.631  m.The linear coordinate of the end of the curve LETC2∈⟨1,450.033;1,455.009⟩ m. This corresponds to the Cartesian coordinates x∈⟨1,314.310;1,315.998⟩ m and y∈⟨978.802;983.508⟩ m. Based on this, the Cartesian coordinates of the ETC2 point can be determined as xETC2=1,314.519  m, yETC2=979.385  m.The list of the determined geometrical parameters of the measured layout is presented in Table 1.Table 1. List of the determined parameters of the first geometric layoutTable 1. List of the determined parameters of the first geometric layoutLayout elementRadius, R (m)Length, lTC (m)Coordinate, L (m)Abscissa, x (m)Ordinate, y (m)TC1 transition curve—194.457———The beginning of TC1——195.134400.131200.035The end of TC1——389.591586.522255.275Circular arc1,200.130————TC2 transition curve—200.660———The end of TC2——1,450.6491,314.519979.385The beginning of TC2——1,651.3091,371.8691,171.631The numerical data in Table 1 fully identify the measured geometrical layout, which is shown in Fig. 3. They were determined by estmating the horizontal curvature made by the moving chord method. The new method of curvature estimation has proved to be useful here.The Second Geometric LayoutThe obtained set of track axis coordinates for the second geometric layout made it possible to draw a graph y(x), which is shown in Fig. 7. The estimated maximum measurement error was ±20  mm. The ordinate of the plot at the first measurement point (for x=0) is y=–12.504  m.As before, the estimation of the curvature of the track axis in the horizontal plane was carried out in a linear system. In the considered case, it was assumed that the starting point (i.e., L=0) has Cartesian coordinates x0=6.470  m, y0=11.644  m. Assuming the chord length lc=20  m, the horizontal curvature of the geometric layout shown in Fig. 7 was determined. The obtained graph of the curvature k(L) is shown in Fig. 8. In this case, the curvature is negative, which corresponds to the appearance of a convexity directed upward in the curvilinear segmant.On the basis of this graph, it is possible to determine the value of the radius of the circular arc by Eq. (22). In the case under consideration, the mean value kCA¯=−0.0020866355  rad/m was obtained, with standard deviation σk=0.0000540623  rad/m. This corresponds to the radius value R=479,2403885  m.At the end of the left-hand straight section in Fig. 7 is the BTC1 point, and at the beginning of the right-hand straight section is the BTC2 point. The ends of both transition curves (i.e., the ETC1 and ETC2 points) mark the beginning and the end of the circular arc, respectively. Fig. 9 shows an enlarged fragment of the curvature plot, in which the region covering the TC1 transition curve (with variable curvature values) is described by the least squares line.For the curvature values belonging to the TC1 transition curve (i.e., connected to the straight section on the left in Fig. 7), the following equation was obtained: k=−0.0000171975L+0.003332.The linear coordinates of the beginning of the curve [based on Eq. (24)] and its end [based on Eq. (25)] are: LBTC1=193.726  m, LETC1=315.060  m. Thus, the length of the transition curve lTC1=121.334  m determined from Eq. (26).Among the measurement points, the linear coordinate LBTC1∈⟨190.015;195.006⟩ m. This corresponds to the Cartesian coordinates x∈⟨55.646;56.940⟩ m and y∈⟨195.170;200.000⟩ m. On this basis—using Eqs. (27) and (28)—the Cartesian coordinates of the BTC1 point can be determined as xBTC1=56.608  m, yBTC1=198.761  m.The linear coordinate of the end of the curve LETC1∈⟨315.016;320.000⟩ m. This corresponds to the Cartesian coordinates x∈⟨92.774;94.684⟩ m and y∈⟨314.438;319.058⟩ m. On this basis, the Cartesian coordinates of the ETC1 point are xETC1=92.791  m, yETC1=314.478  m.Fig. 10 shows an enlarged fragment of the curvature plot in Fig. 8 including the TC2 transition curve connecting the circular arc with the right-hand straight section in Fig. 7.For the marked range of curvature values belonging to the TC2 transition curve, we obtain k=0.0000165356L–0,013574.The linear coordinates of the beginning of the curve [based on Eq. (30)] and its end [based on Eq. (31)] are as follows: LBTC2=820.894  m, LETC2=694.703  m. Thus, the length of the transition curve is lTC2=126.191  m, determined from Eq. (32).Among the measurement points, the linear coordinate LBTC2∈⟨820.015;825.012⟩  m. This corresponds to the Cartesian coordinates x∈⟨463.288;468.117⟩ m and y∈⟨611.759;613.054⟩ m. On this basis the Cartesian coordinates of the BTC2 point are: xBTC2=464.137  m, yBTC2=611.986  m.The linear coordinate of the end of the curve LETC2∈⟨689.993;695.000⟩  m. This corresponds to the Cartesian coordinates x∈⟨344.588;349.178⟩  m and y∈⟨572.979;574.963⟩  m. The Cartesian coordinates of the ETC2 point are: xETC2=348.906  m, yETC2=574.845  m.The list of the determined geometrical parameters of the second measured layout is presented in Table 2.Table 2. List of the determined parameters of the second geometric layoutTable 2. List of the determined parameters of the second geometric layoutLayout elementRadius, R (m)Length, lTC (m)Coordinate, L (m)Abscissa, x (m)Ordinate, y (m)TC1 transition curve—121.334———The beginning of TC1——193.72656.608198.761The end of TC1——315.06092.791314.478Circular arc479.240————TC2 transition curve—126.191———The end of TC2——694.703348.90757.845The beginning of TC2——820.894464.137611.986The numerical data in Table 2 fully identify the measured geometrical layout, which is shown in Fig. 7.Summary of Verification Carried OutThe identification of both exemplary geometric layouts was carried out by estimating the horizontal curvature made by the moving chord method. The new method of curvature estimation has proved to be fully useful here. The obtained diagrams of horizontal curvature for the measured railway track (Figs. 4–6 and 8–10) clearly differ from the diagrams for model layouts presented in Koc (2020, 2021a, b). They are oscillatory in nature, which results from the track deformation and measurement error. However, this did not prevent the basic geometrical parameters of the measured layout from being estimated.It can be assumed that in the corresponding model solution, there is a circular arc with a radius of 1,000 m in the first geometrical layout, and in the second layout, an arc with a radius of 480 m. This means that there is a much lower value of curvature in the first layout than in the second one. Therefore, the graph of curvature in Fig. 4 can be a bit misleading and the oscillation overexposed compared to Fig. 8 because these drawings have a different vertical scale. The estimation of the radius in the second geometry is more precise. For this case, the standard deviation is σk=5,406E-5  rad/m, while for the first geometry it is σk=7,197E-5  rad/m.The verified method made it possible to determine the location of individual geometric elements along the length of the entire layout (Tables 1 and 2). At the same time, a certain degree of differentiation in the length of the transition curves was demonstrated. In the first geometric layout, lTC1=194,457  m and lTC2=200,660  m were obtained, and in the second one, lTC1=121,334  m and lTC2=126,191  m. This probably follows from the assumed (simmetrical) model solution; as it should be assumed, there should be, respectively, 200 m and 120 m here.



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