Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 679817, 13 pages http://dx.doi.org/10.1155/2014/679817
Research Article Analytical Method of Modelling the Geometric System of Communication Route
Wladyslaw Koc
Faculty of Civil and Environmental Engineering, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland
Correspondence should be addressed to Wladyslaw Koc; [email protected]
Received 10 October 2013; Accepted 14 April 2014; Published 11 May 2014
Academic Editor: Youqing Wang
Copyright © 2014 Wladyslaw Koc. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper presents a new analytical approach to modelling thecurvature of a communication route by making use of differential equations. The method makes it possible to identify both linear and nonlinear curvature. It enables us to join curves of thesame or opposite signs of curvature. Solutions of problems for linear change of curvature and selected variants of nonlinear curvature in polynomial and trigonometric form were analyzed. A comparison of determined horizontal transition curves was made and examples of negotiating these curves into a geometric system were given.
1. Introduction railway routes. However, it is easy to note that there is a distinct disproportion in taking interest in this problem. A A rapid progress in computer calculation technique algo- search for new solutions is going on this sphere on vehicular rithmsbasedoncurrentknowledgeandinspiredbyfurther roads (e.g., [4–12]). As regards the railway lines, the situation prospecting is also taken into account in the field of geometri- is entirely different. Publications on new transition curves are cal shaping of communication routes. In the sphere of shaping relatively rare and in the majority were published long ago the communication routes, vehicular roads and railway lines, (e.g., [13–17]). the development of satellite measuring technique GNSS [1, 2] The application of transition curves is aimed at ensuring a also plays a crucial role. A particular significance here will be continuous change of the unbalanced side acceleration bet- attached to the question of vehicular dynamics. Therefore the ween intervals of motor road (or railway route) of diversi- modelling of curvature will obtain a key role. fied curvature in an advantageous way for the interaction This subject also includes the design of elements of the dynamics within the road-vehicle system. Such a requirement route of diversified curvature3 [ ]. The elaboration of a method relates to all sorts of transition curves. Under this situation it for a precise determination of the coordinate route points might appear that there is one particular algorithm to create seems to be the most appropriate area of action. To perform them, common to all the analyzed curves. In fact most of the thisactivityuseismadeoftoolsnecessarytoprovideana- solutions that have been known so far appear independently lyticalsolutions,thatis,themostadvantageousforpractical and bear various names (sometimes coming from the name application. The junction of elements of a vehicular route or of their author). A general knowledge of the determination of railway line of diversified curvature should ensure a continu- transition curves equations would make it possible to com- ouschangeoftheunbalancedsideacceleration,inanadvan- pare various forms of curves with each other and to make an tageous way for interaction dynamics within the road-vehicle assessment of their practical application. system. This is strictly connected with the proper shaping of Twenty years ago this issue had already been sufficiently the curvature. explained with respect mainly to railway routes [16]. Atten- The most developed investigation branch concerning this tion was then concentrated on working out a technique of problem has been for years the study of transition curves identification of unbalanced accelerations occurring on vari- connecting straight lengths of route with circular arcs. The ous types of transition curves. It was based on a comparative problem of transition curves is related to vehicular roads and analysis of some selected transition curves with the use of 2 Mathematical Problems in Engineering a dynamic model. In the method acceleration was a factor that y excitedthetransversevibrationsofthecarriage[18]. The main ΔΘ conclusion resulting from the considerations was to prove M1 the relation existing between the response of the system and ΔΘ the class of the excitation function. The dynamic interactions Δl Δy were smaller (i.e., more advantageous) if the class of the M function was higher. It turned out that evidently the largest K acceleration values were noted on the cubical parabola (class Δx 0 of function C ). With respect to Bloss curve and a cosine 1 curve (class C )theyaresignificantlysmaller,whereason Θ Θ1 2 sine curve (class C )theyarethesmallest. x After years the mentioned identification method of unbalanced accelerations [16] provided a source of inspira- Figure 1: Schematic diagram for the explanation of the notion of tionfortheelaborationofanewtechniqueformodellingthe curvature. geometric system of the communication route based on join- ing two points of the route with diversified curvature. 𝑛 theobtainedfunction𝑘(𝑙) is of class C in the range ⟨0, 𝑙𝑘⟩, where 𝑛=min(𝑛1,𝑛2). 2. Analytical Method of The application of the method under consideration makes Modelling the Curvature it possible to combine various geometric elements, for exam- ple, a straight with a circular arc, and also, after introducing an A measure of the route bending is the ratio of the angle which adequate sign of curvature, two circular arcs of a consistent determines the direction of the vehicle’slongitudinal axis after 𝐾 run or converse arcs. covering a certain arc (Figure 1). The curvature of curve at On determining the curvature function 𝑘(𝑙) afundamen- point 𝑀 is called the boundary which is aimed at by the rela- ΔΘ 𝐾 tal task is to find the coordinates that correspond to a curve in tion of acute angle between tangents to curve at points the rectangular coordinate system 𝑥, 𝑦.Asolutionforsucha 𝑀 and 𝑀1,tothelengthofarcΔ𝑙 when point 𝑀1 tends along 𝐾 𝑀 system is required by the satellite measuring technique GNSS curve to point [19–21], which becomes an aid in the determination of the ΔΘ 𝑑Θ coordinate points in a uniform, 3D-system of coordinates 𝑘= lim = . (1) Δ𝑙 → 0 Δ𝑙 𝑑𝑙 WGS 84 (the world geodetic system 1984). Next the measured ellipsoidal coordinates (GPS) are transformed to Gauss- Iftheoperationprocedurebytheuseofrectangularcoor- Kruger (𝑋, 𝑌) conformal coordinates [22]. An assumption is dinates 𝑥, 𝑦 is to be continued, it is necessary to take into con- made that the beginning of system 𝑥, 𝑦 is at the terminal point sideration the sign of the curvature. As illustrated in Figure 1 oftheinputcurve(withcurvature𝑘1), while the axis of angle ΔΘ = Θ1 −Θ>0; the curvature is assumed here that abscissae is tangent to this curve at this point. it has a positive value which, as can be seen, accompanies the The equation of the sought after connection can be curves with downward convexities. A negative value of cur- written in parametric form vature corresponds to angle ΔΘ < 0 and appears on curves of an upward convexity. 𝑥 (𝑙) = ∫ cos Θ (𝑙) 𝑑𝑙, Making a generalization of the identification method (4) applied to unbalanced accelerations occurring on various 𝑦 (𝑙) = ∫ Θ (𝑙) 𝑑𝑙. types of transition curves [16], it is possible to search for cur- sin vature function 𝑘(𝑙) among the solutions of the differential equation Parameter 𝑙 is the position of a given point along the curve length. Function Θ(𝑙) is determined on the basis of the (𝑚) (𝑚−1) 𝑘 (𝑙) =𝑓[𝑙, 𝑘, 𝑘 ,...,𝑘 ] , (2) formula with conditions for the transition curve at the outset (for 𝑙= Θ (𝑙) = ∫ 𝑘 (𝑙) 𝑑𝑙. (5) 0) and at the end (for 𝑙=𝑙𝑘) The presented method of modelling the curvature has a 𝑘 𝑖= 0 𝑘(𝑖) (0+)={ 1 for universal character. It can be applied to both vehicular roads 0 for 𝑖= 1,2,...,𝑛1, and railway routes. In the case of railway routes there arises an (3) additional possibility, for modelling in a similar way, the (𝑗) − 𝑘2 for 𝑗=0 superelevation ramp, understood here as a determined differ- 𝑘 (𝑙𝑘 )={ 0 for 𝑗=1,2,...,𝑛2, ence of heights of the rail. The principle of operation is similar to curvature modelling. The only differences are that 𝑘1 has to where 𝑘1 and 𝑘2 indicate the curvature magnitudes on both be replaced by the value of superelevation ℎ1 at the beginning endsofthecurve(takingnoteofanadequatesign).Theorder of the superelevation ramp and 𝑘2 has to be replaced by the of the differential equation2 ( )is𝑚=𝑛1 +𝑛2 +2,and value of superelevation ℎ2 at the end of the superelevation Mathematical Problems in Engineering 3 ramp. The ramp length, of course, corresponds to the length As can be seen the acceleration increment is here a con- of the transition curve. Using the same procedure it is possible stant value. However, from condition (10)itfollowsthat to find the unbalanced acceleration 𝑎(𝑙). 𝜓 V 𝑙 ≥ 𝑎 −𝑎 . 𝑘 𝜓 2 1 (15) 3. Transition Curve of Linear Curvature dop
3.1. Finding Out the Curvature Equation. Linear change of The superelevation ramp equation in the case under con- 𝑙 curvature along a defined length 𝑘 is obtained by assuming sideration is similar to (8)and(12) two principal conditions 1 𝑘(0+)=𝑘, 1 ℎ (𝑙) =ℎ1 + (ℎ2 −ℎ1)𝑙. (16) 𝑙𝑘 − (6) 𝑘(𝑙𝑘 )=𝑘2, The speed of lifting the wheel 𝑓(𝑙) on the superelevation and a differential equation ramp (assuming that the travelling speed of the train V = 𝑘 (𝑙) =0. (7) const.) is determined by the equation After the determination of constants the solution of the 𝑑 V 𝑓 (𝑙) = V ℎ (𝑙) = (ℎ −ℎ ). differential problem (6), (7)isasfollows: 2 1 (17) 𝑑𝑙 𝑙𝑘 1 𝑘 (𝑙) =𝑘 + (𝑘 −𝑘 )𝑙. 1 2 1 (8) 𝑓(𝑙) 𝑙𝑘 Thus, also is here a constant value, but from condition (11)itfollowsthat Function Θ(𝑙) is obtained from formula (5). In the case under consideration 𝑓 V 𝑙 ≥ ℎ −ℎ . 1 𝑘 𝑓 2 1 (18) 2 dop Θ (𝑙) =𝑘1𝑙+ (𝑘2 −𝑘1) 𝑙 . (9) 2𝑙𝑘 The assumed length of the transition curve must fulfill the 3.2. The Determination of the Transition Curve Length (for condition Railway Lines). With regard to rail tracks the transition curve must satisfy two kinematic conditions 𝜓 𝑓 𝑙𝑘 ≥ max (𝑙𝑘 ,𝑙𝑘 ) . (19) 𝜓 ≤𝜓 max dop (10) 3.3. Coordinates of Transition Curve Connecting Uniform Cur- 𝑓 ≤𝑓 , max dop (11) vatures. The determination of 𝑥(𝑙) and 𝑦(𝑙) by using (4)will need the expansion of integrands into Maclaurin series. After where 𝜓max is the maximum value of acceleration increment 3 𝜓 completing the whole procedure the following parametric on transition curve in m/s , dop is permissible value of accel- 3 equations are obtained: eration increment in m/s , 𝑓max is the maximum speed of 𝑓 lifting the wheel on superelevation ramp in mm/s, and dop is permissible value of speed of lifting the wheel in mm/s. 𝑥 (𝑙) = ∫ cos Θ (𝑙) 𝑑𝑙 The unbalanced acceleration 𝑎(𝑙) occurring on the tran- sition curve results from speed V of the train, magnitude of 𝑘2 𝑘 =𝑙− 1 𝑙3 − 1 (𝑘 −𝑘 )𝑙4 curvature, and ordinates of the superelevation ramp; its 6 8𝑙 2 1 course is similar to 𝑘(𝑙) and ℎ(𝑙).Analogouslywith(8) 𝑘 4 3 1 𝑘1 1 2 5 𝑘1 6 𝑎 𝑙 =𝑎 + (𝑎 −𝑎)𝑙, +[ − (𝑘2 −𝑘1) ]𝑙 + (𝑘2 −𝑘1)𝑙 ( ) 1 2 1 (12) 120 40𝑙2 72𝑙 𝑙𝑘 𝑘 𝑘 6 2 where 𝑎1 and 𝑎2 are accelerations at ends of the transition 𝑘 𝑘 2 −[ 1 − 1 (𝑘 −𝑘 ) ]𝑙7 curve 5040 2 2 1 112𝑙𝑘 2 ℎ 2 ℎ V 1 V 2 5 𝑎1 = 𝑘1 −𝑔 ,𝑎2 = 𝑘2 −𝑔 (13) 𝑘 𝑘 3 rad 𝑠 rad 𝑠 −[ 1 (𝑘 −𝑘 )− 1 (𝑘 −𝑘 ) ]𝑙8 1920𝑙 2 1 3 2 1 𝑘 384𝑙𝑘 (ℎ1 and ℎ2 are ordinates of the superelevation ramp at its 8 4 ends). 𝑘 𝑘 2 V +[ 1 − 1 (𝑘 −𝑘 ) On the assumption that the speed of the train =const., 362880 1728𝑙2 2 1 the formula for acceleration increment 𝜓(𝑙) is as follows: 𝑘 1 𝑑 V + (𝑘 −𝑘 )4]𝑙9 ⋅⋅⋅ 𝜓 (𝑙) = V 𝑎 (𝑙) = (𝑎2 −𝑎1) . (14) 4 2 1 𝑑𝑙 𝑙𝑘 3456𝑙𝑘 4 Mathematical Problems in Engineering
0.015 𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 0.01 3 2 0.005 𝑘1 2 1 3 𝑘1 4 𝑘1 5 = 𝑙 + (𝑘2 −𝑘1)𝑙 − 𝑙 − (𝑘2 −𝑘1)𝑙 0 2 6𝑙𝑘 24 20𝑙𝑘 (rad) 0 20406080
Q −0.005 𝑘5 𝑘 l (m) +[ 1 − 1 (𝑘 −𝑘 )2]𝑙6 −0.01 720 2 2 1 48𝑙𝑘 −0.015 −0.02 𝑘4 1 +[ 1 (𝑘 −𝑘 )− (𝑘 −𝑘 )3]𝑙7 336𝑙 2 1 3 2 1 Figure 2: Diagram of angle Θ(𝑙) defined by9 ( )forcircular 𝑘 336𝑙𝑘 arcs connection of curvatures 𝑘1 = 1/1200 rad/m and 𝑘2 = −1/700 rad/m by means of transition curve of length 𝑙𝑘 =60m. 𝑘7 𝑘3 −[ 1 − 1 (𝑘 −𝑘 )2]𝑙8 40320 2 2 1 384𝑙𝑘 cos Θ(𝑙) and sin Θ(𝑙) should be expanded to Taylor series. 𝑘6 𝑘2 After integration of the equations take the following form: −[ 1 (𝑘 −𝑘 )− 1 (𝑘 −𝑘 )3]𝑙9 +⋅⋅⋅. 2 1 3 2 1 12960𝑙𝑘 864𝑙 𝑘 𝑥 (𝑙) = ∫ Θ (𝑙) 𝑑𝑙 (20) cos
2 1 𝑘1 =𝑥(𝑙0)+cos (− 𝑙𝑘)(𝑙−𝑙0) Equations (20)for𝑘1 =0describethecurveintheform 2 𝑘2 −𝑘1 of clothoid used to connect a straight with a circular arc. 2 1 1 𝑘1 3 −[ (𝑘2 −𝑘1) sin (− 𝑙𝑘)](𝑙−𝑙0) 6𝑙𝑘 2 𝑘2 −𝑘1 3.4. Coordinates of Transition Curve Connecting Inverse Cur- vatures. At inverse arcs (i.e., with diversified curvature signs) 1 1 𝑘2 −[ (𝑘 −𝑘 )2 (− 1 𝑙 )](𝑙−𝑙 )5 there arises the problem related to the determined coordi- 2 2 1 cos 𝑘 0 40𝑙 2 𝑘2 −𝑘1 nates 𝑥(𝑙) and 𝑦(𝑙), due to the course of function Θ(𝑙).For 𝑘 𝑘 = 𝑘 Θ(𝑙) sign 1 sign 2 function is a monotonic function, 1 1 𝑘2 𝑘 ≠ 𝑘 Θ(𝑙) +[ (𝑘 −𝑘 )3 (− 1 𝑙 )] (𝑙 − 𝑙 )7 whereas for sign 1 sign 2 in the diagram of function 3 2 1 sin 2 𝑘 −𝑘 𝑘 0 336𝑙𝑘 2 1 there appears an extremum at point 𝑙0,where +⋅⋅⋅
𝑙0 Θ (𝑙0) =𝑘(𝑙0) =𝑘1 + (𝑘2 −𝑘1) =0. (21) 𝑙𝑘 (23)
𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 Values 𝑙0 and Θ(𝑙0) follow from the relations: 1 𝑘2 =𝑦(𝑙)+ (− 1 𝑙 )(𝑙−𝑙 ) 0 sin 2 𝑘 −𝑘 𝑘 0 𝑘 2 1 𝑙 =− 1 𝑙 0 𝑘 2 𝑘2 −𝑘1 1 1 𝑘 +[ (𝑘 −𝑘 ) (− 1 𝑙 )](𝑙−𝑙 )3 (22) 6𝑙 2 1 cos 2 𝑘 −𝑘 𝑘 0 𝑘2 𝑘 2 1 Θ(𝑙 )=− 1 𝑙 . 0 2(𝑘 −𝑘 ) 𝑘 1 1 𝑘2 2 1 −[ (𝑘 −𝑘 )2 (− 1 𝑙 )] (𝑙 − 𝑙 )5 2 2 1 sin 2 𝑘 −𝑘 𝑘 0 40𝑙𝑘 2 1
2 Figure 2 illustrates a scheme of angle Θ(𝑙) for the con- 1 3 1 𝑘1 7 −[ (𝑘2 −𝑘1) cos (− 𝑙𝑘)] (𝑙 −0 𝑙 ) nection of inverse circular arcs of radii 𝑅1 = 1200 mand 3 2 𝑘 −𝑘 336𝑙𝑘 2 1 𝑅2 = 700 m. The length of the transition curve 𝑙𝑘 =60mwas determined for speed value V =90km/h using the procedure 1 1 𝑘2 +[ (𝑘 −𝑘 )4 (− 1 𝑙 )](𝑙−𝑙 )9 as described at Section 3.2 (the assumed superelevation 4 2 1 sin 2 𝑘 −𝑘 𝑘 0 3456𝑙𝑘 2 1 values were ℎ1 =20mm and ℎ2 =45mm). In this situation the parametric equations of transition +⋅⋅⋅ . curve (20)areeffectivefor𝑙∈⟨0,𝑙0⟩;for𝑙∈(𝑙0,𝑙𝑘⟩ functions (24) Mathematical Problems in Engineering 5
On completing the parametric equations (20)and(23) 0.4 along with (24)acorrectsolutionofthetaskforthecase of diversified curvature signs is obtainedFigure ( 3). From a 0.3 practical point of view the determination of the magnitude of 0.2 the tangent slope angle at the end of the transition curve is (m) very important. It amounts to y 1 0.1 Θ (𝑙 ) = (𝑘 +𝑘 ) 𝑙 . 𝑘 2 1 2 𝑘 (25) 0 AknowledgeofΘ(𝑙𝑘) makes it possible to add the tran- 020406080 sition curve to the other one having curvature 𝑘2 at its initial x (m) point (satisfying the tangency condition of both the curves). Figure 3: Diagram of horizontal ordinates 𝑦(𝑥) defined by20 ( ), (23), and (24) for the joint of circular arcs of curvatures 𝑘1 = 4. Transition Curve of Polynomial Curvature 1/1200 rad/m and 𝑘2 = −1/700 rad/m with transition curve of length 𝑙𝑘 =60m (using different horizontal and vertical scales). From the point of view of vehicles’ dynamics, the transition curveoflinearchangeofcurvatureisnotthemostadvan- tageous solution [23]. A definitely better one is nonlinear solution whose characteristics are interesting enough to be The adopted length of the transition curve must satisfy compared with the Bezier curves preferable lately in literature condition (19). [4, 7]. The differential equation (2)allowstoappointanunlim- 4.2. Coordinates of Transition Curve Connecting Uniform Cur- ited number of curves with nonlinear curvature change. In vatures. On expanding functions cos Θ(𝑙) and sin Θ(𝑙) into the case of geometric layout of communication routes it will Maclaurin series and after integration, the following paramet- be appropriate to consider the transition curves of poly- ric equations are acquired: nomial and trigonometric curvature. In fact such forms of curves are used in the standard ways to connect a straight line with a circular arc. 𝑥 (𝑙) = ∫ cos Θ (𝑙) 𝑑𝑙
𝑘2 𝑘4 𝑘 4.1. Formulation of the Curvature Equation. To formulate the =𝑙− 1 𝑙3 +[ 1 − 1 (𝑘 −𝑘 )] 𝑙5 6 120 2 2 1 curvature equation in the polynomial form, let us assume the 5𝑙𝑘 following boundary conditions: 𝑘 𝑘(0+)=𝑘,𝑘(𝑙−)=𝑘 +[ 1 (𝑘 −𝑘 )] 𝑙6 1 𝑘 2 12𝑙3 2 1 (26) 𝑘 𝑘 (0+)=0, 𝑘 (𝑙−)=0 𝑘 𝑘6 𝑘3 1 −[ 1 − 1 (𝑘 −𝑘 )+ (𝑘 −𝑘 )2]𝑙7 and the differential equation 5040 42𝑙2 2 1 4 2 1 𝑘 14𝑙𝑘 (4) 𝑘 (𝑙) =0. (27) 𝑘3 1 −[ 1 (𝑘 −𝑘 )− (𝑘 −𝑘 )2]𝑙8 96𝑙3 2 1 16𝑙5 2 1 After solving the differential problem (26), (27)the 𝑘 𝑘 following formula for curvature 𝑘(𝑙) is obtained: 𝑘8 𝑘5 𝑘2 +[ 1 − 1 (𝑘 −𝑘 )+ 1 (𝑘 −𝑘 )2 2 2 1 4 2 1 3 2 2 3 362880 1080𝑙 36𝑙 𝑘 (𝑙) =𝑘 + (𝑘 −𝑘 ) 𝑙 − (𝑘 −𝑘 ) 𝑙 . 𝑘 𝑘 1 𝑙2 2 1 3 2 1 (28) 𝑘 𝑙𝑘 1 2 Θ(𝑙) + (𝑘 −𝑘 ) ]𝑙9 +⋅⋅⋅ The form of function determined on the basis of (5) 72𝑙6 2 1 and (28)isasfollows: 𝑘 1 1 Θ (𝑙) =𝑘𝑙+ (𝑘 −𝑘 ) 𝑙3 − (𝑘 −𝑘 ) 𝑙4. 𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 1 𝑙2 2 1 3 2 1 (29) 𝑘 2𝑙𝑘 𝑘 𝑘3 1 On rail routes the length of the transition curve is = 1 𝑙2 −[ 1 − (𝑘 −𝑘 )] 𝑙4 2 24 2 2 1 determined like at Section 3.2 for solution of linear curvature. 4𝑙𝑘 The following conditions should be fulfilled here: 1 𝑘5 𝑘2 𝜓 3 V 5 1 1 6 𝑙 ≥ 𝑎 −𝑎 , − (𝑘2 −𝑘1)𝑙 +[ − (𝑘2 −𝑘1)] 𝑙 𝑘 2 1 3 720 2 2 𝜓 10𝑙 12𝑙𝑘 dop 𝑘 (30) 2 𝑓 3 V 𝑘 𝑙 ≥ ℎ −ℎ . + 1 (𝑘 −𝑘 )𝑙7 𝑘 2 𝑓 2 1 3 2 1 dop 28𝑙𝑘 6 Mathematical Problems in Engineering
0.001 series; after integration the following parametric equations of the transition curve are obtained: 0.0005 𝑥 (𝑙) = ∫ Θ (𝑙) 𝑑𝑙 0 cos 0 20406080100 −0.0005 l (m) =𝑥(𝑙0)+cos Θ(𝑙0)(𝑙−𝑙0) (rad/m) k −0.001 1 − Θ(𝑙 )Θ (𝑙 )(𝑙−𝑙 )3 6 sin 0 0 0 −0.0015 1 − Θ(𝑙 )Θ (𝑙 )(𝑙−𝑙 )4 −0.002 24 sin 0 0 0
Figure 4: Diagram of curvature 𝑘(𝑙) described by (28)forthe 1 2 𝑘 = 1/1200 −{ cos Θ(𝑙0)[Θ (𝑙0)] connection of circular arcs of curvatures 1 rad/m and 40 𝑘2 = −1/700 rad/m with the use of transition curve of length 𝑙𝑘 = 1 90 m. + Θ(𝑙 )Θ(4) (𝑙 )} (𝑙 − 𝑙 )5 120 sin 0 0 0
1 6 𝑘7 𝑘4 𝑘 − cos Θ(𝑙0)Θ (𝑙0)Θ (𝑙0)(𝑙−𝑙0) −[ 1 − 1 (𝑘 −𝑘 )+ 1 (𝑘 −𝑘 )2]𝑙8 72 40320 192𝑙2 2 1 16𝑙4 2 1 𝑘 𝑘 1 3 +{ sin Θ(𝑙0)[Θ (𝑙0)] 4 336 𝑘 2 𝑘 +[ 1 (𝑘 −𝑘 ) − 1 (𝑘 −𝑘 )] 𝑙9 +⋅⋅⋅. 1 18𝑙5 2 1 432𝑙3 2 1 − Θ(𝑙 )Θ (𝑙 )Θ(4) (𝑙 ) 𝑘 𝑘 336 cos 0 0 0 (31) 1 2 − Θ(𝑙 )[Θ (𝑙 )] }(𝑙−𝑙 )7 504 cos 0 0 0 Equations (31)for𝑘1 =0describe the so-called Bloss curvewhichcanbeusedtojoinastraighttoacirculararc. 1 2 +{ Θ(𝑙 )[Θ (𝑙 )] Θ (𝑙 ) 384 sin 0 0 0 4.3. Coordinates of Transition Curve Connecting Inverse Cur- 1 − Θ(𝑙 )Θ (𝑙 )Θ(4) (𝑙 )} (𝑙 − 𝑙 )8 vatures. When the arcs are inverse (i.e., the curvature signs 1152 cos 0 0 0 0 are different, Figure 4) there occurs a problem relating to the determined coordinates 𝑥(𝑙) and 𝑦(𝑙) which results from 1 4 +{ cos Θ(𝑙0)[Θ (𝑙0)] function Θ(𝑙).Forsign𝑘1 = sign 𝑘2 function Θ(𝑙) is a mono- 3456 𝑘 ≠ 𝑘 tonic function, while for sign 1 sign 2 in the diagram of 1 2 Θ(𝑙) 𝑙 + Θ(𝑙 )Θ (𝑙 )[Θ (𝑙 )] function there appears extremum at point 0,where 1296 sin 0 0 0
2 3 2 2 3 1 (4) Θ (𝑙 )=𝑘(𝑙)=𝑘 + (𝑘 −𝑘 )𝑙 − (𝑘 −𝑘 )𝑙 =0. + sin Θ(𝑙0)[Θ (𝑙0)] Θ (𝑙0) 0 0 1 𝑙2 2 1 0 3 2 1 0 1728 𝑘 𝑙𝑘 (32) 1 2 9 − Θ(𝑙 )[Θ(4) (𝑙 )] }(𝑙−𝑙 ) +⋅⋅⋅ 10368 cos 0 0 0 Parametric equations of transition curve (31)arevalidfor 𝑙∈⟨0,𝑙0⟩. The determination of value 𝑙0 makes it necessary to 𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 solve (32) which is cubic. The sought-after value 𝑙0 ∈⟨0,𝑙𝑘⟩, satisfying conditions of the problem is defined by the formula =𝑦(𝑙0)+sin Θ(𝑙0)(𝑙−𝑙0)
1 𝜑 𝜋 1 3 𝑙 =[ − ( + )] 𝑙 , + [cos Θ(𝑙0)Θ (𝑙0)] (𝑙 −0 𝑙 ) 0 2 cos 3 3 𝑘 (33) 6 1 + [ Θ(𝑙 )Θ (𝑙 )](𝑙−𝑙 )4 where angle 𝜑 follows from the relation 24 cos 0 0 0
1 2 𝑘 +𝑘 −{ Θ(𝑙 )[Θ (𝑙 )] 𝜑= 1 2 . 40 sin 0 0 cos 𝑘 −𝑘 (34) 2 1 1 − Θ(𝑙 )Θ(4) (𝑙 )} (𝑙 − 𝑙 )5 120 cos 0 0 0 Under such circumstances in order to determine para- metric equations of the transition curve for 𝑙∈(𝑙0,𝑙𝑘⟩,func- 1 − [ Θ(𝑙 )Θ (𝑙 )Θ (𝑙 )] (𝑙 − 𝑙 )6 tions cos Θ(𝑙) and sin Θ(𝑙) should be expanded into Taylor 72 sin 0 0 0 0 Mathematical Problems in Engineering 7
1 3 −{ cos Θ(𝑙0)[Θ (𝑙0)] As soon as the differential problem (26), (37)hasbeen 336 solved, a formula for curvature, is obtained 1 (4) + sin Θ(𝑙0)Θ (𝑙0)Θ (𝑙0) 336 1 1 𝜋 𝑘 (𝑙) = (𝑘1 +𝑘2)− (𝑘2 −𝑘1) cos 𝑙. 1 2 2 2 𝑙 (38) + Θ(𝑙 )[Θ (𝑙 )] }(𝑙−𝑙 )7 𝑘 504 sin 0 0 0
1 2 Θ(𝑙) −{ cos Θ(𝑙0)[Θ (𝑙0)] Θ (𝑙0) The form of function determined by the use of (5) 384 and (38)isasfollows: 1 (4) 8 + sin Θ(𝑙0)Θ (𝑙0)Θ (𝑙0)} (𝑙 −0 𝑙 ) 1152 1 1 𝑙 𝜋 Θ (𝑙) = (𝑘 +𝑘 )𝑙− (𝑘 −𝑘 ) 𝑘 𝑙. 1 4 1 2 2 1 sin (39) +{ Θ(𝑙 )[Θ (𝑙 )] 2 2 𝜋 𝑙𝑘 3456 sin 0 0
1 2 − Θ(𝑙 )Θ (𝑙 )[Θ (𝑙 )] The length of the transition curve on railway routes 1296 cos 0 0 0 follows from conditions: 1 2 − Θ(𝑙 )[Θ (𝑙 )] Θ(4) (𝑙 ) 1728 cos 0 0 0 𝜓 𝜋 V 𝑙𝑘 ≥ 𝑎2 −𝑎1 , 1 2 9 2 𝜓 − Θ(𝑙 )[Θ(4) (𝑙 )] }(𝑙−𝑙 ) +⋅⋅⋅ , dop 10368 sin 0 0 0 (40) 𝑓 𝜋 V 𝑙 ≥ ℎ −ℎ . (35) 𝑘 2 𝑓 2 1 dop where The assumed length of the curve must satisfy the condi- 1 1 tion (19). Θ(𝑙 )=𝑘𝑙 + (𝑘 −𝑘 )𝑙3 − (𝑘 −𝑘 )𝑙4 0 1 0 𝑙2 2 1 0 3 2 1 0 𝑘 2𝑙𝑘 5.2. Ordinates of Transition Curve Connecting Uniform Curva- 6 6 2 Θ (𝑙 )= (𝑘 −𝑘 )𝑙 − (𝑘 −𝑘 )𝑙 tures. After the expansion of functions cos Θ(𝑙) and sin Θ(𝑙) 0 𝑙2 2 1 0 𝑙3 2 1 0 𝑘 𝑘 into Maclaurin series to be next integrated the following para- (36) 6 12 metric equations are obtained: Θ (𝑙 )= (𝑘 −𝑘 )− (𝑘 −𝑘 )𝑙 0 𝑙2 2 1 3 2 1 0 𝑘 𝑙𝑘 12 Θ(4) (𝑙 )=− (𝑘 −𝑘 ). 𝑥 (𝑙) = ∫ cos Θ (𝑙) 𝑑𝑙 0 3 2 1 𝑙𝑘 𝑘2 𝑘4 1 𝜋2 =𝑙− 1 𝑙3 +[ 1 − 𝑘 (𝑘 −𝑘 )] 𝑙5 6 120 60 𝑙2 1 2 1 Making use of parametric equations (31)andalso(35)one 𝑘 can obtain a correct solution of the problem with diversified 𝑘6 1 𝜋2 signs of curvature. −[ 1 − 𝑘3 (𝑘 −𝑘 ) 5040 504 2 1 2 1 Value of the tangent slope at the end of the transition 𝑙𝑘 curvemakesitpossibletoconnecttoitacurveofcurvature𝑘2 1 𝜋4 1 𝜋4 satisfying simultaneously the tangency condition of both the − 𝑘 (𝑘 −𝑘 )− (𝑘 −𝑘 )2]𝑙7 1680 4 1 2 1 2016 4 2 1 curves.Theabovevalueisthesameasinthecaseoflinear 𝑙𝑘 𝑙𝑘 curvature. It is defined by (25). 𝑘8 1 𝜋2 +[ 1 − 𝑘5 (𝑘 −𝑘 ) 362880 12960 𝑙2 1 2 1 5. Transition Curve of Curvature in 𝑘 1 𝜋4 Trigonometric Form − 𝑘2(𝑘 −𝑘 )2 5184 𝑙4 1 2 1 5.1. Formulation of Curvature Equation. As before boundary 𝑘 conditions (26) and another differential equation are 1 𝜋4 1 𝜋6 − 𝑘3 (𝑘 −𝑘 )− 𝑘 (𝑘 −𝑘 ) assumed. 12960 𝑙4 1 2 1 90720 6 1 2 1 𝑘 𝑙𝑘 𝜋2 1 𝜋6 𝑘(4) (𝑙) + 𝑘 (𝑙) =0. + (𝑘 −𝑘 )2]𝑙9 +⋅⋅⋅ 2 (37) 25920 6 2 1 𝑙𝑘 𝑙𝑘 8 Mathematical Problems in Engineering
𝑙∈(𝑙,𝑙 ⟩ Θ(𝑙) Θ(𝑙) 𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 For 0 𝑘 functions cos and sin should be expanded into Taylor series. The integration provides the following equations: 1 1 1 𝜋2 = 𝑘 𝑙2 −[ 𝑘3 − (𝑘 −𝑘 )] 𝑙4 2 1 24 1 48 𝑙2 2 1 𝑘 𝑥 (𝑙) = ∫ cos Θ (𝑙) 𝑑𝑙 1 1 𝜋2 =𝑥(𝑙)+ Θ(𝑙 )(𝑙−𝑙 ) +[ 𝑘5 − 𝑘2 (𝑘 −𝑘 ) 0 cos 0 0 720 1 144 1 𝑙2 2 1 𝑘 1 3 − sin Θ(𝑙0)Θ (𝑙0)(𝑙−𝑙0) 4 6 1 𝜋 6 − (𝑘 −𝑘 )] 𝑙 1 4 4 2 1 1440 𝑙 − sin Θ(𝑙0)Θ (𝑙0)⋅(𝑙−𝑙0) 𝑘 24 1 2 1 1 𝜋2 −{ Θ(𝑙 )[Θ (𝑙 )] −[ 𝑘7 − 𝑘4 (𝑘 −𝑘 ) 40 cos 0 0 40320 1 2304 1 2 2 1 𝑙𝑘 1 (4) 5 + sin Θ(𝑙0)Θ (𝑙0)} (𝑙 −0 𝑙 ) 4 4 120 1 𝜋 2 1 𝜋 + 𝑘 (𝑘 −𝑘 ) − 𝑘2 (𝑘 −𝑘 ) 1 2304 𝑙4 1 2 1 3840 𝑙4 1 2 1 −{ Θ(𝑙 )Θ (𝑙 )Θ (𝑙 ) 𝑘 𝑘 72 cos 0 0 0 6 1 𝜋 8 1 (5) 6 + (𝑘 −𝑘 )] 𝑙 +⋅⋅⋅. + sin Θ(𝑙0)Θ (𝑙0)} (𝑙 −0 𝑙 ) 80640 6 2 1 720 𝑙𝑘 (41) 1 3 +{ Θ(𝑙 )[Θ (𝑙 )] 336 sin 0 0
1 (4) 𝑘 =0 − Θ(𝑙 )Θ (𝑙 )Θ (𝑙 ) Equations (41)for 1 describe the curve known as 336 cos 0 0 0 cosinusoid which can be used to connect a straight with a 1 2 circular arc. − Θ(𝑙 )[Θ (𝑙 )] 504 cos 0 0 1 − Θ(𝑙 )Θ(6) (𝑙 )} (𝑙 − 𝑙 )7 5.3. Transition Curve Coordinates Connecting Inverse Curva- 5040 sin 0 0 0 tures. Near inverse arcs in the diagram of function Θ(𝑙) there 1 2 appears the extremum at point 𝑙0,where +{ Θ(𝑙 )[Θ (𝑙 )] Θ (𝑙 ) 384 sin 0 0 0
1 (5) − cos Θ(𝑙0)Θ (𝑙) Θ (𝑙) 1 1 𝜋 1920 Θ (𝑙0)=𝑘(𝑙0)= (𝑘1 +𝑘2)− (𝑘2 −𝑘1) cos 𝑙0 =0. 2 2 𝑙𝑘 1 − Θ(𝑙 )Θ (𝑙 )Θ(4) (𝑙 ) (42) 1152 cos 0 0 0 1 − Θ(𝑙 )Θ(7) (𝑙 )}(𝑙−𝑙 )8 40320 sin 0 0 0 The transition curve parametric equations (41)areeffec- 1 4 tive for 𝑙∈⟨0,𝑙0⟩. To determine value 𝑙0 one must solve (42). +{ cos Θ(𝑙0)[Θ (𝑙0)] The equation can be written in the form 3456 1 2 + Θ(𝑙 )Θ (𝑙 )[Θ (𝑙 )] 1296 sin 0 0 0 𝜋 𝑘1 +𝑘2 𝑙 = . 1 2 cos 𝑙 0 𝑘 −𝑘 (43) + Θ(𝑙 )[Θ (𝑙 )] Θ(4) (𝑙 ) 𝑘 2 1 1728 sin 0 0 0 1 − Θ(𝑙 )Θ (𝑙 )Θ(6) (𝑙 ) 12960 cos 0 0 0 It is easy to prove that for sign 𝑘1 ≠ sign 𝑘2 it is necessary to satisfy the condition |𝑘1+𝑘2|<|𝑘2−𝑘1|,thus|(𝑘1+𝑘2)/(𝑘2− 1 (5) − cos Θ(𝑙0)Θ (𝑙0)Θ (𝑙0) 𝑘1)| < 1. The formula for length 𝑙0 is as follows: 6480 1 2 − Θ(𝑙 )[Θ(4) (𝑙 )] 10368 cos 0 0 1 𝑘 +𝑘 𝑙 = ( 1 2 )𝑙 . 1 0 𝜋 arccos𝑘 −𝑘 𝑘 (44) − Θ(𝑙 )Θ(8) (𝑙 )}(𝑙−𝑙 )9 +⋅⋅⋅ 2 1 362880 sin 0 0 0 Mathematical Problems in Engineering 9
𝑦 (𝑙) = ∫ sin Θ (𝑙) 𝑑𝑙 where 1 𝑙 Θ(𝑙 )= (𝑘 +𝑘 )𝑙 ± 𝑘 √−𝑘 𝑘 =𝑦(𝑙0)+sin Θ(𝑙0)(𝑙−𝑙0) 0 2 1 2 0 𝜋 1 2
1 3 𝜋 + cos Θ(𝑙0)Θ (𝑙0)(𝑙−𝑙0) Θ (𝑙0 )=∓ √−𝑘1𝑘2 6 𝑙𝑘 1 4 𝜋 + Θ(𝑙 )Θ (𝑙 )⋅(𝑙−𝑙 ) Θ (𝑙 )=∓ √−𝑘 𝑘 24 cos 0 0 0 0 1 2 𝑙𝑘 1 2 −{ Θ(𝑙 )[Θ (𝑙 )] 1 𝜋2 40 sin 0 0 Θ (𝑙 )= (𝑘 +𝑘 ) 0 2 2 1 2 𝑙𝑘 1 (4) 5 − cos Θ(𝑙0)Θ (𝑙0)}(𝑙−𝑙0) 3 120 (4) 𝜋 Θ (𝑙0)=± √−𝑘1𝑘2 1 𝑙3 (46) −{ Θ(𝑙 )𝑄 (𝑙 )Θ (𝑙 ) 𝑘 72 sin 0 0 0 1 𝜋4 Θ(5) (𝑙 )=− (𝑘 +𝑘 ) 1 (5) 6 0 2 4 1 2 − Θ(𝑙 )Θ (𝑙 )}(𝑙−𝑙 ) 𝑙𝑘 720 cos 0 0 0 𝜋5 1 3 (6) Θ (𝑙0)=∓ √−𝑘1𝑘2 −{ cos Θ(𝑙0)[Θ (𝑙0)] 𝑙5 336 𝑘 6 1 (4) (7) 1 𝜋 + sin Θ(𝑙0)Θ (𝑙0)Θ (𝑙0) Θ (𝑙 )= (𝑘 +𝑘 ) 336 0 2 6 1 2 𝑙𝑘 1 2 + Θ(𝑙 )[Θ (𝑙 )] 𝜋7 504 sin 0 0 Θ(8) (𝑙 )=± √−𝑘 𝑘 . 0 7 1 2 𝑙𝑘 1 (6) 7 − cos Θ(𝑙0)Θ (𝑙0)} (𝑙 −0 𝑙 ) 5040 In expressions with double signs ± and ∓,theupperone 𝑘 >0𝑘 <0 1 2 corresponds to the case 1 , 2 , while the lower one −{ Θ(𝑙 )[Θ (𝑙 )] Θ (𝑙 ) 𝑘 <0 𝑘 >0 384 cos 0 0 0 refers to 1 , 2 . Thevalueoftangentslopeattheendofthetransition 1 (5) curvemakesitpossibletoconnecttoitacurveofcurvature𝑘2 + sin Θ(𝑙0)Θ (𝑙0)Θ (𝑙0) 1920 satisfying the tangency condition of both the curves. The value is the same as in the previous cases. It is determined 1 (4) + Θ(𝑙 )Θ (𝑙 )Θ (𝑙 ) by (25). 1152 sin 0 0 0 1 − Θ(𝑙 )Θ(7) (𝑙 )} (𝑙 − 𝑙 )8 6. Comparison of Horizontal Ordinates of 40320 cos 0 0 0 Determined Transition Curves 1 4 +{ Θ(𝑙 )[Θ (𝑙 )] 3456 sin 0 0 6.1. The Shaping of Horizontal Ordinates of Comparable Curves. If the analysis is carried out for railway routes, then 1 2 the necessity of meeting kinematic conditions (10)and(11) − cos Θ(𝑙0)Θ (𝑙0)[Θ (𝑙0)] 1296 will require a diversification of the lengths of the compared 1 2 with each other curves (using the same speed of trains). − Θ(𝑙 )[Θ (𝑙 )] Θ(4) (𝑙 ) 1728 cos 0 0 0 Moreover, it is necessary to individually consider cases of connecting homogeneous curvatures (sign 𝑘1 = sign 𝑘2)and 1 (6) inverse arcs (sign 𝑘1 ≠ sign 𝑘2). − sin 𝑄(𝑙0)𝑄 (𝑙0)𝑄 (𝑙0) 12960 Itisassumedthatatjointsofconsistentarcswithradii 𝑅 = 1200 𝑅 = 700 1 1 mand 2 m an attempt is made to attain a − 𝑄(𝑙 )𝑄 (𝑙 )𝑄(5) (𝑙 ) V = 110 6480 sin 0 0 0 speed km/h. Hence it is necessary to employ on arcs superelevation of ℎ1 =80mm and ℎ2 = 115 mm, respectively. 1 (4) 2 The retention of the same kinematic parameters can provide − sin Θ(𝑙0)[Θ (𝑙0)] 10368 diversified lengths of the transition curve:
1 9 + Θ(𝑙 )Θ(8) (𝑙 )} (𝑙 − 𝑙 ) +⋅⋅⋅ , (i) for a curve of linear curvature 𝑙𝑘 =40m, 362880 cos 0 0 0 (ii) for a curve of polynomial curvature 𝑙𝑘 =60m, (45) (iii) for a curve of trigonometric curvature 𝑙𝑘 = 62, 832 m. 10 Mathematical Problems in Engineering
The above length relations result directly from the equa- 2.5 tions given at Sections 3.2, 4.1,and5.1. 2 Figure 5 gives diagrams of horizontal ordinates for com- 1.5 pared transition curves of positive and negative curvature 1 values. As seen the curvature ordinates of linear curvature 0.5 evidently deviate from the ordinates of other curves. In turn, 0 (m) curves of polynomial and trigonometric curvature are in y −0.5 0 10203040506070 principle consistent. A difference can only be noted in the −1 x (m) final area. And so, the class of the function describing the −1.5 curvature plays here a significant part: with respect to a curve 0 −2 of linear curvature it is a function of class C , while regarding −2.5 1 the other two curves it is a function of class C . At the joint of inverse arcs of radii 𝑅1 = 1200 mand𝑅2 = Figure 5: Diagrams of horizontal ordinates of curves joining 700 m an attempt is made to reach the speed of V =90km/h. consistent arcs for positive and negative values of curvature (using Hence it follows that it is necessary to apply superelevation different horizontal and vertical scales): curve of linear curvature: line in violet colour, curve of polynomial curvature in green colour, along arcs attaining ℎ1 =20mm and ℎ2 =45mm, respec- and curve of trigonometric curvature in red colour. tively. By the use the same kinematic parameters it is possible to obtain the following lengths of the transition curve: 1.5 (i) for a curve of linear curvature 𝑙𝑘 =60m, 1 (ii) for a curve of polynomial curvature 𝑙𝑘 =90m, 0.5 (iii) for a curve of trigonometric curvature 𝑙𝑘 = 94, 248 m. 0 Figure 6 presents diagrams of horizontal ordinates of (m) y 0 20 40 60 80 100 compared transition curves for positive and negative values −0.5 x (m) of curvature. Ascanbeseen,likeinthecaseofconsistentarcs,thecurve −1 ordinates of linear curvature deviate significantly from the −1.5 ordinates of the remaining curves, but the shape of these curves also differs one from the other. The class of function Figure 6: Diagrams of horizontal ordinates of curves connecting describing the curvature is still important. However, in the inverse arcs for positive and negative values of curvature (using range of the same class various geometric solutions are different horizontal and vertical scales): the curve of linear curvature possible. is in violet color, the curve of polynomial curvature is in green color, and the curve of trigonometric curvature is in red color. 6.2. Negotiation of Transition Curves in a Geometric System. The transition curves do not appear alone in a geometric (ii) for 𝑘2 <0 systembutmustbeconnectedwithsomearcsintheneigh- borhood. If the input arc is circular, then its equation is as 𝑦 (𝑥) =𝑦 + √𝑅2 −(𝑥 −𝑥)2 follows: 𝑆 2 𝑆 𝑠 1 (i) for 𝑘1 >0 𝐾 (50) 𝑥𝑆 =𝑥𝐾 + 𝑅2,𝑦𝑆 =𝑦𝐾 − 𝑅2, √1+𝑠2 √1+𝑠2 √ 2 2 𝐾 𝐾 𝑦 (𝑥) =𝑅1 − 𝑅1 −𝑥 , (47)
where 𝑥>𝑥𝐾. (ii) for 𝑘1 <0 In (49)and(50) 𝑥𝐾 denotes the abscissae of the transition √ 2 2 𝑦 (𝑥) =−(𝑅1 − 𝑅1 −𝑥 ), (48) curve end, 𝑦𝐾 istheordinateofthetransitioncurveend,while 𝑠𝐾 isthevalueoftangentattheendofthetransitioncurve. where 𝑥<0. Values for numerical examples under consideration are given in Table 1. However, if the output arc is also a circular arc, then it is Figures 7 and 8 illustrate the geometric systems made up expressed in the following equation: oftwocirculararcsofconsistentcurvature,connectedwith various types of transition curves. As seen the application of a (i) for 𝑘2 >0 given type of transition curve leads to an adequate position of 𝑦 (𝑥) =𝑦 − √𝑅2 −(𝑥 −𝑥)2 theoutputcirculararc.Thepositionofthisarcbytheuseofa 𝑆 2 𝑆 linear transition curve clearly deviates from such cases where
𝑠𝐾 1 (49) advantage is taken of curves with polynomial and trigono- 𝑥𝑆 =𝑥𝐾 − 𝑅2,𝑦𝑆 =𝑦𝐾 + 𝑅2, metricalcurvature.Withregardtothelatteramutualposition √1+𝑠2 √1+𝑠2 𝐾 𝐾 of the circular arc varies insignificantly. Mathematical Problems in Engineering 11
Table1:Comparisonofvalues𝑥𝐾, 𝑦𝐾,and𝑠𝐾 for numerical examples analyzed.
Curve 𝑙𝑘 [m] Θ(𝑙𝑘) [rad] 𝑠𝐾 𝑥𝐾 [m] 𝑦𝐾 [m]
Case 1: 𝑘1 = 1/1200 rad/m and 𝑘2 = 1/700 rad/m Curve C0 40 0,045238 0,045269 39,988 0,825 Curve C1 (p) 60 0,067857 0,067961 59,963 1,821 Curve C1 (t) 62,832 0,071060 0,071180 62,796 1,940
Case 2: 𝑘1 = 1/1200 rad/m and 𝑘2 = −1/700 rad/m Curve C0 60 −0,017857 −0,017859 59,999 0,143 Curve C1 (p) 90 −0,026786 −0,026792 89,991 0,627 Curve C1 (t) 94,248 −0,028050 −0,028057 94,238 0,714
Case 3: 𝑘1 = −1/1200 rad/m and 𝑘2 = 1/700 rad/m Curve C0 60 0,01786 0,017859 59,999 −0,143 Curve C1 (p) 90 0,026786 0,026792 89,991 −0,627 Curve C1 (t) 94,248 0,028050 0,028057 94,238 −0,714
Case 4: 𝑘1 = −1/1200 rad/m and 𝑘2 = −1/700 rad/m Curve C0 40 −0,045238 −0,045269 39,988 −0,825 Curve C1 (p) 60 −0,067857 −0,067961 59,969 −1,821 Curve C1 (t) 62,832 −0,071060 −0,071180 62,976 −1,940
6 1.5
5 1
4 0.5
3 0 (m) y
−50 (m) 0 50 100 150
2 y −0.5 x (m) 1 −1
0 −1.5 −40 −20 0 20 40 60 80 100 120 −2 x (m) Figure 9: Examples of geometric systems consisting of two inverse Figure 7: Examples of geometric systems consisting of two circular circular arcs (brown line), connected with different types of transi- arcs with consistent curvature (brown line) connected with different tion curves, for positive values of input curvature (using different transition curves, for positive values of curvature (using different horizontal and vertical scales): curve of linear curvature in violet horizontal and vertical scales): curve of linear curvature in violet line, curve of polynomial curvature in green, and curve of trigono- line, curve of polynomial curvature in green, and curve of trigono- metric curvature in red. metric curvature in red.
0 −40 −20 0 20 40 60 80 100 120 Figures 9 and 10 illustrate geometric systems made up of −1 x (m) two inverse circular arcs connected with various types of −2 transition curves. Here the layout of the output circular arc is much more −3 diversified. The position of the arc obtained by the application (m) of the transition curve of linear curvature diverts by several y −4 dozen meters from situations where curves of polynomial and −5 trigonometric curvature have been used. The difference for −6 curves of polynomial and trigonometric curvature attains, in the numerical example analyzed, an order of several meters. −7 The presented examples are taken, in the very nature of things, at random. They reveal wide possibilities for the pre- Figure 8: Examples of geometric systems consisting of two circular sented analytical method in modelling geometric systems. Its arcs with consistent curvature (brown line) connected with different application makes it possible to generate diversified layout of transition curves, for negative values of curvature (using different the communication routes adapted to the field conditions and horizontal and vertical scales): curve of linear curvature in violet other circumstances. Simultaneously the method ensures line, curve of polynomial curvature in green, and curve of trigono- correctness of the obtained solution in view of the kinematic metric curvature in red. parameters magnitudes in force. 12 Mathematical Problems in Engineering
2 References 1.5 [1] W. Koc and C. Specht, “Application of the Polish active GNSS 1 geodetic network for surveying and design of the railroad,” in Proceedings of the 1st International Conference on Road and Rail 0.5 Infrastructure (CETRA ’10),pp.757–762,May2010. (m)
y 0 [2] P.Lipar, M. Lakner, T. Maher, and M. Zura,ˇ “Estimation of road −50 0 50 100 150 centerline curvature from raw GPS data,” Baltic Journal of Road −0.5 x (m) and Bridge Engineering,vol.6,no.3,pp.163–168,2011. −1 [3] L. Tasci and N. Kuloglu, “Investigation of a new transition −1.5 curve,” TheBalticJournalofRoadandBridgeEngineering,vol.6, no. 1, pp. 23–29, 2011. Figure 10: Examples of geometric systems consisting of two [4] H.-H. Cai and G.-J. Wang, “A new method in highway route inverse circular arcs (brown line), connected with different types design: joining circular arcs by a single C-Bezier´ curve with of transition curves, for negative values of input curvature (using shape parameter,” Journal of Zhejiang University: Science A,vol. different horizontal and vertical scales): curve of linear curvature 10, no. 4, pp. 562–569, 2009. in violet line, curve of polynomial curvature in green, and curve of [5] S. Dimulyo, Z. Habib, and M. 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[20] B. Hoffmann-Wellenhof, H. Lichtenegger, and E. Wasle, GNSS Global Navigation Satellite Systems—GPS, GLONASS, Galileo & More,Springer,Wien,Austria,2008. [21] P. Misra and P. Enge, Global Positioning System: Signals, Mea- surements, and Performance, Ganga-Jamuna Press, Lincoln, Mass, USA, 2012. [22] C. Specht, “Accuracy and coverage of the modernized Polish Maritime differential GPS system,” Advances in Space Research, vol.47,no.2,pp.221–228,2011. [23] W. Koc and K. Palikowska, “Dynamic properties evaluation of the selected methods of joining route segments with different curvature,” Technika Transportu Szynowego,vol.9,pp.1785– 1807, 2012 (Polish). Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014
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