Journal of Marine Science and Technology, Vol. 21, No. 2, pp. 191-197 (2013) 191 DOI: 10.6119/JMST-012-0403-1
A NUMERICAL STUDY OF CUBIC PARABOLAS ON RAILWAY TRANSITION CURVES
Tsung-I Shen1, Che-Hao Chang1, Kuan-Yung Chang1, and Cho-Chien Lu2
Key words: railway, transition curve, route alignment, cubic parabola. The consistency and continuity of track alignment design are essential in improving the travel speed and efficiency of transportation. The quality of alignment design affects the ABSTRACT arrangement of stations and facilities along the track line. This paper mainly focuses on the design of transition Alignment comprises straight lines, circular arcs, and transi- curves of the cubic parabola type in track alignment design. tion curves [6, 7, 18]. A good alignment design utilizes smooth Equations for the transition curves are first derived from the shapes. Prior to the use of computers in engineering (before theory of cubic parabolas using calculus techniques. They 1975), alignment design mainly involved adopting circular are then analyzed using numerical analysis methods. The arcs for ease of construction, and due to the limited develop- proposed formulation is evaluated by comparison of its ment of information technology and computers. However, calculated results with data in 684 actual cases of transition circular curves can no longer meet the requirement of pas- curves. The accuracy of the proposed method is verified in senger comfort in modern rail systems. Therefore, transition this study by comparing the estimated results with collected curves must be placed between two circular curves and where data from actual engineering projects, and the applicability straight lines meet circular curves. According to relevant of the new method to the engineering practice of track align- railway alignment regulations and research [2, 5, 11, 13], the ment design is justified. The chainage and coordinates of the majority of transition curves used in rail engineering are de- control points as well as other curve points and the tangential signed based on the cubic parabola. A cubic parabola is a angle can be easily calculated using the transition curve nonlinear function curve, and its calculation is difficult and equations. cumbersome for coordinates of any given point on the curve using a geometric mathematical method. The coordinates that include the N coordinate and the E coordinate are a datum I. INTRODUCTION coordinate system. Track alignment design serves as an im- Upon satisfying the basic requirements of safety, stability, portant reference for the rest of the construction; therefore, and comfort, rail systems seek to optimize speed and effi- engineering practice demands the ability to calculate coordi- ciency [16, 19]. Railway alignment design is important be- nates of any point on the alignment. The present study at- cause it dictates the spatial arrangement of the railways and tempts to use numerical methods to acquire numerical solu- thus the locations of civil infrastructure and mechanical and tions of cubic parabola type transition curves. electrical facilities. Alignment design code [11, 12, 15] is the Numerical analysis is a method of obtaining approximate basic standard for railway alignment design in Taiwan. solutions for a particular problem using mathematical algo- Trains in operation are exposed to the risk of derailment rithms [3, 4, 14]. Due to the complexity of most engineering resulting from severe vibration induced by centrifugal forces problems, analytical solutions often cannot be obtained. Nu- if the following three situations occur: (1) the train enters a merical analysis is therefore necessary to reach an approxi- curve from a straight line; (2) the train travels from a curve to mation. Newton’s method is a typical numerical algorithm another curve of different radius; (3) the train leaves a curve used to find the approximate roots of nonlinear functions. and enters a straight line. These sudden changes can also Newton’s method involves finding the roots of the Taylor cause discomfort to passengers. series expansion of a function using an iterative process, in order to estimate its actual roots. The main concept is to start with an initial guess for a root of the function, then substitute Paper submitted 12/15/11; revised 03/06/12; accepted 04/03/12. Author for this root into the function. The resulting value is again used correspondence: Cho-Chien Lu (e-mail: [email protected]). as a root and substituted into the function. Eventually, the 1 Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan, R.O.C. actual root is reached after many iterations. The current study 2 Department of Civil Engineering, National Kaohsiung University of Applied utilizes Newton’s numerical method to analyze the cubic pa- Sciences, Kaohsiung, Taiwan, R.O.C. rabola type of transition curves, and proposes a numerical
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railway horizontal alignment: (1) Clothoid; (2) Half-Sine; and IP (3) Cubic Parabola. Railway regulations and alignment stan- 404K + 647.84116 E2 = 178309.75153 dards in Taiwan state that transition curves must be designed E1 = 176357.20381 N2 = 2504691.52877 N1 = 2504586.81972 as cubic parabolas. The current study presents basic theories SC R = 405M CS of the cubic parabola and goes on to discuss its application in
94°34′ 14.3″ engineering practice. LS1 = 90M LS2 = 90M TS ST
62°34′ 35.2″ III. NUMERICAL MODELS OF TRANSITION CURVES Fig. 1. A typical curve of route alignment. The proposed numerical model of transition curves in- volves computing engineering quantities at any point on the curve based on a geometric mathematical model. These quan- solution strategy for nonlinear equations describing such tran- tities include chainage, the N-coordinate, the E-coordinate, sition curves [8, 13]. Following the discussion of basic theo- and the tangential angle. The N-E coordinate system, intro- ries of transition curves, curve alignment formulas are derived. duced previously, is the datum coordinate system in engi- The algorithm and the procedure of the derivation are sum- neering. The tangential angle is defined as the angle between marized. Actual alignment designs of some major domestic the tangent line at a given point and the tangent line of the transport engineering projects are collected. These cases are route (the line connecting the TS control point and IP control analyzed using the proposed formulations, and the results are point). A geometric mathematical model can distinctly ex- compared with the actual designs to evaluate the correctness press the geometric relationship between the radius of curva- and applicability of the proposed equations. ture R and the arc length in the track alignment. However, it is not easy to directly calculate the coordinates of a random II. THE ROLE OF RAILWAY TRANSITION point on the curve. To use the design outcome from the geo- CURVES metric mathematical model, engineers must set up measuring equipment at the starting point of the curve, refer to the IP A transition curve is a segment with a gradually changing point as the backsight point, then set out positions of other radius connecting a straight rail section with a curved section. points on the curve according to geometric relationships in- It has three functions: (1) ease the transition and prevent acci- volving deflection angle, chord length, etc. It is difficult to dents caused by sharp changes in direction; (2) reduce shak- verify results obtained using this approach, and systematic ing of vehicles in operation; (3) make the transition smoother, errors often accumulate. A total station is a measuring in- more stable, and safer. strument widely used in modern route surveying. Modern The radius of a transition curve changes with its length [1]. total stations contain programs for coordinate-based numerical A typical route alignment is shown in Fig. 1. There are four methods to aid in setting-out and surveying operations. In en- control points on this alignment: (1) Tangent to Spiral, TS; gineering practice, chainage, N- and E-coordinates, and tan- (2) Spiral to Curve, SC; (3) Curve to Spiral, CS; and (4) Spiral gential angle at every chainage point must be prepared prior to to Tangent, ST [6, 7, 18]. The radius of curvature of the surveying. In light of this, geometric numerical models can straight section can be considered infinite, gradually changing no longer meet the design requirements of curved routes. It to the radius of the circular curve through the transition curve. is therefore important to develop new mathematical models In addition to transition curves, to ensure traffic safety and that can describe geometric relationships and process engi- provide comfort to passengers, superelevation will also be neering information. introduced into the alignment design to help offset centrifugal The purpose of this study is to develop a numerical model forces developed from the tilting plane. It is well known for transition curves, which allows direct calculation of coor- nowadays that simultaneous introduction of the transition dinates and other engineering quantities at any given point on curve and superelevation ramp is the only effective solution the curve. We start from the curvature of transition curves, and for straight track–circular curve railway connections; this has then cover (in order) cubic parabolic equations, length of been applied in practice for years. Furthermore, it is also transition curves, horizontal and vertical distances of transi- known that both should be defined with the same function and tion curves, and the use of Newton’s method to solve for possess the same length [9]. length of the curve. Route alignment can be divided into horizontal alignment and vertical alignment. In track engineering, straight lines are 1. Curvature of Transition Curves favored in horizontal alignment. However, when constrained Assume that the purpose of a transition curve in between by topography and the surrounding environment, plane curves a straight section and a circular arc section is to provide a are required to change the direction of the route. There are linear change of curvature. A typical transition curve is shown three types of transition curves that are commonly used in in Fig. 2. In this study, “r” denotes the radius of curvature of
T.-I Shen et al.: A Numerical Study of Cubic Parabolas on Railway Transition Curves 193
y = f(x) k
straight line transition curve circular curve
r = R k2 k circular curve x
k transition curve 1 x x straight line SC x TS SC Lx TS Fig. 2. A typical introduction curve. Fig. 3. Curvature change of the transition curve.
the transition curve. The radius of curvature at the intersec- 3. Equation of the Arc Length of the Transition Curve tion of the transition curve and the tangent is infinitely large In railway engineering design, the radius of curvature of ∞ (R = ), and the radius of curvature at the intersection of the the circular curve as well as the start and end point of the transition curve and the circular curve is the same as that of tangent can be treated as constants. The curvature of the the circular curve itself (r = R). Fig. 3 shows the curvature transition curve, which connects the tangent and the circular change of the transition curve. In this figure, K1 represents curve, increases linearly. Therefore, the arc length of the the curvature of the tangent, which is expressed mathemati- transition curve must be determined first before solving for cally as (1a); K2 represents the curvature of the circular curve, other details of the curve. In general calculus textbooks, the and the mathematical expression is given as (1c); and Kx is arc length can be calculated as: the curvature of the transition curve. The curvature at a point that is of distance x away from the start point (TS) can be dy 2 calculated from (1b). Ldx=+∫ 1 (5) x dx =∞= k1 (1/ ) 0 (1a) Integrating Eq. (3) gives the expression for dy/dx, and = x substituting this into Eq. (5) gives Eq. (6), which allows the kx (1b) RLx calculation of arc length of cubic parabola transition curves.
1 1 = 4 k2 (1c) x 2 R Ldx=+∫ 1 (6) x 4RL22 2. Equations of Cubic Parabola Transition Curves The relationship between the curvature and the y coordi- An analytical equation cannot be easily obtained by inte- nate of the curve can be derived by integrating the curvature grating Eq. (6). This study utilizes McLaughlin’s binomial expression. In general, calculus textbooks, the curvature (k) is theorem to find the approximate solution of Eq. (6). given as (2). Assuming the transition curve is gentle enough, In order to expand Eq. (6) to a McLaughlin binomial series, (dy/dx)2 is then close to zero, and the curvature becomes we take “a” as that given in Eq. (7) and substitute it into (6) to Eq. (3). arrive at Eq. (8).
4 dydx22/ = x k = (2) a 22 (7) x 3 4RL 1/+ ()dy dx 2 2 1 fa()=+ (1 a )2 (8) 2 ==dy x kx (3) dx2 RL McLaughlin’s binomial series can be tabulated as shown in Table 1. Twice integrating Eq. (3) gives Eq. (4), which is the equa- Finally, is expanded to Eq. (9): tion of the cubic parabola transition curve: f ()n (0) 1 1 1 5 fa()=+ f (0) an =+−+− 1 a a23 a a 4 x3 ∑ y = (4) n!2816128 6RL (9)
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Table 1. McLaughlin’s Binomial Series. (12a) can be obtained by re-arranging Eq. (11). Eq. (4) for the f ()n (0) y coordinate (y) has been derived previously. The equations n order fa()n () f ()n (0) for the x coordinate and y coordinate are summarized in Eqs. n! 1 (12a) and (12b). original fa()=+ (1 a )2 1 - 1 2 1 − 1 1 11xx44 1st. order (1+ a ) 2 a xL=+/1 − 2 2 2 x 22 22 1044RL 72 RL 11 − 3 1 1 2nd. order −+(1a ) 2 − − a2 22 4 8 34 15xx44 5 +− − 22 22 (12a) 311 2 3 1 3 3rd. order −−(1 +a ) + a 20844RL 2176 RL 222 8 16
− 7 5311 15 5 4 4th. order −−−(1 +a ) 2 − − a x3 2222 16 128 y = (12b) 6RL
5. Solving the Arc Length Equation Using Newtown’s Eq. (10) can be obtained by substituting “a” in Eq. (7) into Method Eq. (9): The equation for the x coordinate (12a) contains 4 variables,
2 which makes it difficult to solve. As a result, a numerical 11xx44 fx()=+ 1 − method is used in this study. First, we express dy/dx as K 2844RL22 RL 22 (13a). The square of K can then be expressed as (13b). Dif-
34ferentiating dy/dx gives us Eq. (13c). Substituting (13) into 15xx44 Eq. (11), we obtain Eq. (14), and substituting (13) into Eq. (2) +− (10) 1644RL22 128 RL 22 gives Eq. (15).
dy x2 Integrating Eq. (10) with respect to x gives the equation for K == (13a) the arc length of the transition curves: dx2 RL
4 2 x L = fxd()x K = (13b) x ∫ 4RL2 2 48 dy2 x =+11xx − = ∫ 1 2 (13c) 28(4RL22 ) (4 RL 222 ) dx RL
15xx12 16 +− 1124 1 6 5 8 223 224 dx Lx=+1 K − K + K − K (14) 16(4RL ) 128 (4 RL ) x 10 72 208 2176 11xx59 =+ − 22 x 22 222 1/dydx xRL / xRL / 10(4RL ) 72 (4 RL ) == = 333 (15) R 222 13 17 2222 + 15xx 1/+ ()dy dx 1/2+ ()xRL ()1 K +− 208(4RL223 ) 2176 (4 RL 224 )
2 Multiplying (14) by (15) gives: 11xx44 =+x 1 − 22 22 1044RL 72 RL L xRL2 /1115 x =+−+−1 KK24 K 6 K 8 (16) R 3 10 72 208 2176 34 2 2 44 (1+ K ) 15xx +− (11) 20844RL22 2176 RL 22 Multiplying both sides by 1/2:
4. Equations for x Coordinate and y Coordinate of the 2 L xRL/2 1 1 1 5 Transition Curve x =+−+−1 K 24KK 6 K 8(17) 2R 3 10 72 208 2176 The equation for the x coordinate (x) of the transition curve ()1+ K 2 2
T.-I Shen et al.: A Numerical Study of Cubic Parabolas on Railway Transition Curves 195
Substituting K into the equation above: IV. CASE STUDY: THE DESIGN OF RAILWAY TRANSITION CURVES IN THE TAIWAN L K 11 1 5 RAILWAY RECONSTRUCTION BUREAU x =+−+−1 KK24 K 6 K 8 2R 3 10 72 208 2176 ()1+ K 2 2 The Railway Reconstruction Bureau (R.R.B.) is the main executive organization for major railway construction in (18) Taiwan. This study gathered actual cases of basic railway route alignment from the R.R.B. as the basis for analysis. After expanding Eq. (18) and rearranging, we obtain the Transition curves in these alignments are all formulated as nonlinear Eq. (19) cubic parabolas [11, 12, 15]. In Taiwan, in order to ensure traffic safety, the R.R.B. re- 2925RK97−+ 6120 RK 17680 RK 5 − 127296 RK 3 views design drawings of track route alignment. Alignment designs by consulting firms may differ due to the usage of −++=12729460RK 636480 L (1 K 23/2 ) 0 (19) different application software. This review mechanism can x ensure the quality of design of the curve parameters. This study collected data on transition curves from 684 R The in Eq. (19) is the radius of curvature of the circular actual R.R.B. engineering projects. Each transition curve can L curve. x is calculated according to the standard from three be described by four control points, including (1) Tangent to R L factors: superelevation, cant deficiency, and speed. and x Spiral, TS; (2) Spiral to Curve, SC; (3) Curve to Spiral, CS; K can both be treated as known variables, leaving as the un- and (4) Spiral to Tangent, ST [7, 18]. Each point on the curve known for which the equation needs to be solved. This study has its chainage, coordinates (including the N coordinate and K uses Newtown’s method to solve for in the nonlinear equa- the E coordinate), and tangential angle. We compare data col- K tion. Eq. (20) is set as the objective function. The value at lected on transition curves with those calculated by the method K K K K the iteration step of n , n-1, is the solution. The values 1, proposed herein. K2, and K3 are calculated according to Eqs. (21a), (21b), and (21c), respectively. 1. Application of the Numerical Models The reasons for developing equations for the cubic parab- 97 5 3 f( K )=−+ 2925 RK 6120 RK 17680 RK − 127296 RK ola transition curves are (1) to provide the client with a tool to verify a consulting firm’s design product, and (2) to provide −++23/2 12729460RK 636480 Lx (1 K ) (20) site engineers with a convenient way of obtaining coordi- nates of any given point on the curve. The client examines two fK() aspects of the design product: (1) values of various key pa- KK=− (21) 1 fK'( ) rameters on the curve drawings and (2) overall route align- ment and coordinate details. fK() The outcome of this research leads to rapid and accurate KK=− 1 21fK'( ) calculation of coordinates for any given point on the cubic 1 (21b) parabola transition curve. The calculation procedure is sum- marized in the following five steps: fK() KK=− 2 (21c) 32 ∆ fK'(2 ) Step 1: Obtain the tangential angle ( ) of the curve. This is the angle between the two tangent lines at the start and end of the alignment. The radius (R) of the circular Rearranging Eq. (14) leads to Eq. (22a). K in (22a) is com- curve is then decided based on the topography and puted using Newtown’s method, while L is calculated ac- x surrounding environment. cording to the standard. The x coordinate and y coordinate, (x) Step 2: The total length of the transition curve, L , is evalu- and (y), of the transition curve can be solved by substituting s ated. The length is taken as the maximum of three L into (22a) and (22b) [3, 4, 8, 14]. x criteria specified in rail system alignment codes or standards. The three factors involved in the determi- L nation of the curve length are design speed (V), su- x = x (22a) 24 6 8 KK K5 K perelevation (C), and cant deficiency (Cd). 1+−+ − 10 72 208 2176 Step 3: K is computed using a numerical method, following which x of this point can be calculated. In order to use the transition curve equation proposed in this paper, x3 y = (22b) the radius of the circular curve (R) and total length of 6RLx the transition curve (Ls) must be determined first.
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Both parameters are treated as given in the Eq. (20). Table 2. Differences in curve details between calculated The x coordinate and the y coordinate, (x) and (y), are results and actual data. the unknowns that must be solved. Differences in curve details Step 4: Calculate the x coordinate and y coordinate, (x) and Chainage ∆E (mm) ∆N (mm) ∆θ (") (y), from the equations of the transition curve. The difference (cm) x value is determined first, then y is calculated [10]. Mean (µ) 0.01 0.10 -0.10 0.00 Step 5: This final step is to transform the coordinate (x, y) Standard 0.50 2.18 4.51 1.08 from the x–y coordinate system to the N–E co- deviation (σ) ordinate system. This relates two 2–D Cartesian co- ordinate systems through a rotation followed by a translation. The rotation is defined by one rotation mean difference and standard deviation in the horizontal co- angle (α) and the translation is defined by two origin ordinates are -0.01 mm and 4.51 mm, respectively. This shift parameters (xo, yo). The transform equation is shows the high precision of the proposed method and veri- shown as Eqs. (23a) and (23b). fies its application in the engineering practice of alignment design [17]. =−+αα() Excos( ) y sin xo (23a) V. CONCLUSION Nx=+ sin(αα ) y cos() + y (23b) o The theory behind the cubic parabola-based transition curves discussed in this paper is not new; however, clear ex- 2. Verification of the Numerical Models planations of related concepts are indeed rare in textbooks and In our research, equations for cubic parabola transition published literature. The main design concept of the curved curves are written into a computer program. Error analysis is route proposed here is based on coordinates, and has two performed on all 684 transition curves collected from actual advantages: (1) it is convenient to survey using modern in- cases. For each transition curve, four control points are taken struments; and (2) measured results are more easily verified. as the baseline for comparison. They are (1) Tangent to Spiral, McLaughlin’s Binomial Series was first used to construct the TS; (2) Spiral to Curve, SC; (3) Curve to Spiral, CS; (4) Spiral equation of the transition curve’s arc length. Newton’s method to Tangent, ST. Each of the four points contains (1) chainage was subsequently applied to solve for K in the equations con- (S), (2) N coordinate (N), (3) E coordinate (E), and (4) tan- taining radius of curvature R, arc length of transition curve Lx, gential angle (θ). There are 10944 comparisons analyzed in and K. The arc length of the transition curve is set equal to the total. total length of the curve (Lx = Ls), and Lx and R are constants. Each comparison includes the following: (1) difference in In engineering practice, the radius of circular arcs (R) and the chainage (∆S); (2) difference in the N coordinate (∆N); (3) total length of the curve (Ls) are regulated in the route align- difference in the E coordinate (∆E); and (4) difference in the ment design code. From these two quantities, the value of K deflection angle (∆θ). The difference in chainage is in units of can be calculated. As a result, K and R can be seen as con- cm, the differences in the coordinates are in mm, and the dif- stants when solving for the x and y coordinates. The x coor- ference in the deflection angle is in seconds. dinate changes with the arc length, Lx, while the y coordinate Table 2 presents the results of the comparison, from which varies with both x and Lx. it can be deduced that the proposed equations of cubic parab- K represents the gradient (dy/dx) at a given point on the ola transition curves and the developed computer program transition curve. The tangential angle can be obtained by can be applied to route alignment design in rail engineering. taking the arc tangent in accordance with the K value at the The mean difference of the tangential angle (∆θ) is 0.00", and point of interest. The tangential angle can be used in survey- the standard deviation is 1.08". This demonstrates that the ing at the TS control point. curve design generated by the proposed equations in this study The proposed transition curve equations can completely is in agreement with the traditional alignment design using a describe the features of the curve and calculate coordinates non-coordinate method. The mean difference in the chainage of any point on the curve, including pegs. Railway align- is 0.01 cm, and the standard deviation is 0.50 cm. Results ment personnel can conveniently and rapidly review and show consistency with the traditional method of alignment verify design drawings and details provided by consulting design by setting out pegs on the curve. This means the total firms. route length calculated from the new method does not differ Although the research outcomes can be directly applied to from the traditional method. Furthermore, the proposed co- railway alignment practice, the following recommendations ordinate method can provide better and more accurate control are offered to facilitate computerization in this area: over costs. Differences in the vertical coordinates between the proposed method and actual data have a mean difference 1. A standard protocol of operation should be in place to regu- of 0.01 mm and a standard deviation of 2.18 mm, while the late procedures of route alignment design. This can prevent
T.-I Shen et al.: A Numerical Study of Cubic Parabolas on Railway Transition Curves 197
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