LOGISTICS Identification of Transition Curves...

Identification of Transition Curves in Vehicular Roads and Railways

Wladyslaw Koc Gdansk University of Technology, Poland

In the paper attention is focused on the necessity to systematize the procedure for determining the shape of transition curves used in vehicular roads and railway routes. There has been presented a universal method of identifying curvature in transition curves by using differential equations. Curvature equations for such known forms of transition curves as clothoid, quartic parabola, the Bloss curve, cosinusoid and sinusoid, have been worked out and by the use these equations it was possible to determine some appropriate Cartesian coordinates. In addition some approximate solutions obtained in consequence of making certain simplifying assumptions orientated mainly towards railway routes, have been provided. Notice has been taken of limitations occurring in the application of smooth transition curves in railway systems, which can be caused by very small values of the horizontal ordinates in the initial region. This problem has provided an inspiration for finding a new family of the so-called parametric transition curves, being more advantageous not only over the clothoid but also over cubic parabola as far as dynamics is concerned. Keywords: vehicular roads, railway routes, transition curves, identification method, a new solution.

1. INTRODUCTION solutions that have been known so far, appear independently and use various names (sometimes The problem of transition curves applies to both originating from the name of their author). roads and railways. However, it could be seen that A knowledge of a general method of determining there is a clear disproportion in the interest of this transition curve equations would make it possible problem. In case of roads there still can be to compare different forms of curves and to observed an effort of searching for new solutions prepare an assessment of their usefulness for in the area (e.g. [1, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, practical application. 16, 23, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 42, As a matter of fact, this problem was to a large 43]). In the case of railroads, the situation is extent explained 20 years ago, mainly with respect completely different − the investigations of new to railway lines [24]. It was then that the method of transition curves are not as numerous, and, what is identifying unbalanced accelerations, occurring on more, the works were worked out relatively long various types of transition curves, was worked out. time ago (e.g. [2, 9, 10, 15, 17, 18, 19, 22, 24, 26, The method was based on a comparative analysis 28, 35, 38, 39, 40, 41]). of some selected transition curves provided with It is obvious that the use of transition curves is a dynamic model. Acceleration was in it a factor aimed at ensuring a continuous change of an exciting transverse vibrations of the vehicle [20]. unbalanced side acceleration between vehicular The basic conclusion resulting from the road lengths (or railway sections) of considerations given to the subject was to indicate diversified curvature, in a way that is advantageous the existing relation between the response of the for the dynamics of the road – vehicle system system and the class of the exciting function. The interactions. Such a requirement concerns all types dynamic effects were smaller (that is more of transition curves. In this situation it might seem advantageous), if the class of the function was that there is one defined algorithm of their higher. It appeared that the largest acceleration formation, which is common for the whole family values were definitely connected with a third-order of curves under consideration. However, all the 31 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

parabola (function class C0), whereas in the case of by using differential equations. It describes the the Bloss curve and the cosinusoid (class C1) they way to find some solutions satisfying an arbitrary were significantly smaller. The smallest values number of assumed conditions. The solutions can were noted on sinusoid (class C2). be of entirely different form with respect to the The transition curves are also defined in given conditions. a different way. As regards vehicular roads they 2.2. DETERMINATION OF CARTESIAN are often defined by function of angle θ(l) which is responsible for changing the direction of the COORDINATES longitudinal axis of a vehicle after it has travelled Making use of the transition curves in field a length of a certain arc. The necessitates the determination of their coordinates is, traditionally dominated by the use of curvature in terms of the Cartesian system x, y. In order to k(x) with a rectangular system of coordinates. The this one should first determine function θ(l) acceleration values over the length of the transition curve result from the curvature distribution. As can ( ) = ( ) (3) be expected it is the curvature distribution that should form the basis for the identification of the and next the transition𝛩𝛩 𝑙𝑙 curve∫ 𝑘𝑘 𝑙𝑙equation𝑑𝑑𝑑𝑑 expressed in transition curves. In general it can be linear or a parametric form: nonlinear. ( ) = cos ( ) (4) 2. METHODOLOGY FOR IDENTIFYING TRANSITION CURVES 𝑥𝑥(𝑙𝑙) = ∫sin 𝛩𝛩( 𝑙𝑙) 𝑑𝑑𝑑𝑑 (5) 2.1. DETERMINATION OF THE 𝑦𝑦 𝑙𝑙 ∫ 𝛩𝛩 𝑙𝑙 𝑑𝑑𝑑𝑑 CURVATURE EQUATIO Here it is necessary to explain that the determination of x(l) and y(l) by using equations Let us make an attempt at generalizing the (4) and (5) will require expansion of the method presented in the paper [24] with a view to integrands, in a general way, into Taylor applying it also to typical vehicular roads. In the (or Maclaurin) series [21]. curvature distribution k(l) one can make use of a similar procedure. The curvature should be 2.3. A SIMPLIFIED METHOD OF described by a function of an appropriate class to DETERMINING THE TRANSITION produce a lesser (i.e. more advantageous) dynamic CURVE EQUATION effect. Function k(l) should be sought among the On railway routes, as well as vehicular roads of differential equation solutions fast traffic, where note is taken of great circular arc radii and relatively long transition curves, use is ( )( ) = , ( ), ( ), … , ( 1)( ) (1) made of a commonly simplified technique for determining the transition curve equation which 𝑚𝑚 ′ 𝑚𝑚− with 𝑘𝑘conditions𝑙𝑙 𝑓𝑓at�𝑙𝑙 f𝑘𝑘irst𝑙𝑙 (for𝑘𝑘 𝑙𝑙 l = 0)𝑘𝑘 and then𝑙𝑙 � (for consequently provides us with this formula in the form of explicit function y(x). The simplification of l = lk) of the transition curve the procedure is based on the assumption that the ( ) + modeled curvature k(l) is related to its projection (0 ) = 0 for i = 0, 1, 2, ..., n1 on axis x, that is, l = x, and lk = xk . As a result of 𝑖𝑖 𝑘𝑘 1 such assumptions we obtain an initial equation for for j = 0 curvature k0(x). The determination of function y(x) ( )( ) = (2) in an accurate way by analytical approach is 𝑅𝑅 𝑗𝑗 − 0 for j = 1, 2, ..., n2 impossible, for the reason that it would require to 𝑘𝑘 𝑙𝑙𝑘𝑘 solve the differential equation The order of the differential equation (3) is ( ) m = n1 + n2 + 2, and the obtained function k(l) is n 0( ) = 3 a function of class C within the interval 0, , ′′ 2 {1 +𝑦𝑦[ (𝑥𝑥 )] }2 where = min( 1, 2). 𝑘𝑘 𝑥𝑥 〈 𝑙𝑙𝑘𝑘 〉 ′ The presented mathematical notation is an Therefore k (x) is also 𝑦𝑦traditionally𝑥𝑥 treated as 𝑛𝑛 𝑛𝑛 𝑛𝑛 0 identification of the shape of the transition curves an initial curvature, being an approximation of 32 LOGISTICS INFRASTRUCTURE Identification of Transition Curves...

target curvature k(x). The transition from k0(x) to Having expanded function cos Θ(l) and sin k(x) takes place in such a way that k0(x) is Θ(l) into Maclaurin series [21], on the basis of assumed as equation of the second derivative of equations (4) and (5) we have function y(x) being sought; thus, 1 5 ( ) = cos ( ) = 2 2 + ( ) = ( ) 40 0 (6) 1 1 9 13 + … ′′ 3456𝑥𝑥 𝑙𝑙 4 4∫ 𝛩𝛩599040𝑙𝑙 𝑑𝑑 𝑑𝑑 6 6𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 (11) Then, the equation𝑦𝑦 𝑥𝑥 is integrated𝑘𝑘 𝑥𝑥 twice, which 𝑘𝑘 𝑘𝑘 gives y'(x) and y(x); conditions y(0) = 0 and 𝑅𝑅 𝑙𝑙 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑙𝑙 1 1 ( ) = sin ( ) = 3 7 + y'(0) = 0 are taken into account. 6 336 3 3 1 11 Curvature k(x) of the transition curve obtained … 𝑘𝑘 4224𝑦𝑦 𝑙𝑙 0 5 ∫5 𝛩𝛩 𝑙𝑙 𝑑𝑑𝑑𝑑 𝑅𝑅 𝑙𝑙 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 (12) differs, of course, from the initial curvature k0(x).

The difference depends on the tangent slope value 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − y'(x). With regard to the transition curves used for The simplified form of clothoid in terms of the railway purposes (if advantage is taken of a system x, y system is obtained on accepting the following of coordinates where the outset of the curve is assumptions: abscissa x = l, the final point abscissa = , the initial curvature 0( ) = tangent to the x-axis) the value of y'(x) along the 1 length is small, therefore the difference between . 𝑥𝑥𝑘𝑘 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑥𝑥 curvatures k0(x) and k(x) is, in practice, 𝑘𝑘 insignificant. 𝑅𝑅 𝑙𝑙 𝑥𝑥 1 ( ) = 3 (13) 6

3. ANALYSIS OF SOME KNOWN FORMS 𝑦𝑦 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 OF TRANSITION CURVES The simplified clothoid carries its own name, the cubic parabola, and it has been used for years 3.1. CLOTHOID as the basic type of transition curve in railway The clothoid is a curve proposed in 1874 by engineering. It does not mean at all that the French physicist Marie Cornu, in connection with solution is most advantageous. his research in the field of optics (diffraction of light). The following basic boundary conditions 3.2. QUARTIC PARABOLA Now the number of conditions has been + (0 ) = 0 increased. At the same time the conditions are (7) being differentiated for the first and the second half 1 𝑘𝑘( ) = of the transition curve. An appropriate differential − equation is used. 𝑘𝑘 and the differential equation𝑘𝑘 𝑙𝑙 𝑅𝑅 ( ) = 0 (14) ( ) = 0 (8) ′′′ 𝑘𝑘 𝑙𝑙 ′′ • For the first half of the transition curve (i.e. are used. After determining𝑘𝑘 𝑙𝑙 the constants, the 0, for 2 ) we have the conditions: 𝑘𝑘 solution of the differential problem (7), (8) is as 𝑙𝑙 follows: 𝑙𝑙 ∈ 〈 〉(0+) = (0+) = 0

1 ′ (15) ( ) = (9) 1 1 𝑘𝑘 =𝑘𝑘 2 2 − 𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝑘𝑘 Thus, we are dealing here with a linear change Solution of differential𝑘𝑘 � 𝑙𝑙 � problem𝑅𝑅 of (14), (15) is of curvature. From expressions (3) and (9), by the as follows: use of integration, it is possible to find angle θ . 2 2 1 ( ) = 2 (16) ( ) = 2 2 (10) 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑅𝑅2𝑙𝑙 𝑙𝑙 𝛩𝛩 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 ( ) = 3 3 2 (17)

𝛩𝛩 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 33 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

2 7 1 2 4 ( ) = cos ( ) = 2 4 + sin + cos + … 63 8 2 12 12 2 12 2 2 13 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 8 … 𝑘𝑘 (18) (25) 3159𝑥𝑥 𝑙𝑙 4 ∫ 𝛩𝛩 𝑙𝑙 𝑑𝑑𝑑𝑑 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 � 𝑅𝑅� 𝑅𝑅 𝑙𝑙𝑘𝑘 � 𝑅𝑅�� �𝑙𝑙 − �

𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 1 4 2 10 The approximate solution is obtained for ( ) = sin ( ) = + 1 4 2 6 2 405 3 6 ( ) = + 2 0 2 ; the following 1 16 … 2 14580𝑦𝑦 𝑙𝑙 5 ∫10 𝛩𝛩 𝑙𝑙 𝑑𝑑 𝑑𝑑 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 (19) 𝑅𝑅 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 = = 𝑘𝑘conditions𝑥𝑥 − are valid:𝑥𝑥 − 2 𝑥𝑥 12 , 2 96 . 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − ′ 𝑙𝑙 𝑙𝑙 𝑙𝑙 𝑙𝑙 A simplified form of the quartic parabola is 𝑅𝑅 𝑅𝑅 2 𝑦𝑦 � 1� 2 𝑦𝑦 � � 2 2 ( ) = + 2 + 3 obtained for 0( ) = 2 . 48 6 2 3 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 1 4 2 𝑘𝑘 (26) 𝑘𝑘 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 𝑦𝑦6 𝑥𝑥 − 𝑅𝑅 𝑅𝑅 𝑥𝑥 − 𝑅𝑅 𝑥𝑥 𝑅𝑅 𝑙𝑙 𝑥𝑥 − 1 4 ( ) = 2 (20) 6 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 Over the whole length of the curve there occurs 𝑦𝑦 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 • For the second half of the transition curve a nonlinear change of the curvature. A similar situation will take place with respect to other (i.e. for , ) the conditions are as 2 smooth transition curves under consideration for 𝑙𝑙𝑘𝑘 follows: which the number of the boundary conditions 𝑙𝑙 ∈ 〈 𝑙𝑙𝑘𝑘 〉 n1 = n2 (for n1 ≠ n2 the curvature distribution 1 1 + = becomes asymmetrical) . 2 2

𝑘𝑘 𝑘𝑘 � 𝑙𝑙 � 1 𝑅𝑅 3.3. BLOSS CURVE ( ) = (21) − In 1936, German engineer A. E. Bloss proposed 𝑘𝑘 𝑘𝑘 (𝑙𝑙 ) =𝑅𝑅0 a transition for railways, in which as a curvature he used a simple polynomial of 3rd ′ − 𝑘𝑘 𝑙𝑙𝑘𝑘 degree in relation to the length of the arc. In Now we have vehicular roads this curve is very often called Göldner curve. Further on the number of 1 4 2 conditions is increased ( ) = + 2 (22) 2 (0+) = (0+) = 0 𝑅𝑅 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑙𝑙 −1 2 𝑙𝑙 − 𝑙𝑙2 ( ) = + 2 3 (23) ′ 6 3 2 1 𝑙𝑙𝑘𝑘 𝑘𝑘( ) = 𝑘𝑘 (27)

𝛩𝛩 𝑙𝑙 𝑅𝑅 − 𝑅𝑅 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − After using the equations (18) and (19) for 𝑘𝑘 𝑘𝑘 (𝑙𝑙 ) =𝑅𝑅0 estimating the values 2 and 2 , we obtain 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 ′ − 𝑘𝑘 𝑙𝑙𝑘𝑘 𝑥𝑥 � � 𝑦𝑦 � � and use is made of differential equation

(4)( ) = 0 (28)

The solution of problem𝑘𝑘 𝑙𝑙 (27), (28) provides the curvature equation

3 2 ( ) = 2 3 (29) (24) 2 3

𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 1 3 1 4 ( ) = sin ( ) = + sin ( ) = 2 3 (30) 2 12 2 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 1 2 1 + cos sin 𝑅𝑅 𝛩𝛩 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝑦𝑦2 𝑙𝑙 4 ∫ 𝛩𝛩12𝑙𝑙 𝑑𝑑𝑑𝑑 𝑦𝑦2 � � 24 2 � �12� 𝑙𝑙 − 1 7 1 8 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 ( ) = cos ( ) = 4 + 5 1 3 1 14 2 16 2 cos𝑅𝑅 𝑅𝑅 𝑅𝑅cos 𝑅𝑅 + 1 3 � 12 � � �𝑙𝑙 2− � −192� 3 �12 � − 9 + … 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝑥𝑥72𝑙𝑙 2 6 ∫ 𝛩𝛩 𝑙𝑙 𝑑𝑑 𝑑𝑑 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙(31)− 𝑅𝑅 𝑙𝑙𝑘𝑘 � 𝑅𝑅�� �𝑙𝑙 − � − � 𝑅𝑅 � 𝑅𝑅� 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 34 LOGISTICS INFRASTRUCTURE Identification of Transition Curves...

1 1 + + + ( ) = sin ( ) = 4 5 (0 ) = (0 ) = (0 ) = 0 4 2 10 3 1 1 ′ ′′ 10 11 1 + …𝑘𝑘 𝑘𝑘 (32) 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑦𝑦60 𝑙𝑙 3 6 ∫ 𝛩𝛩44 𝑙𝑙 3 𝑑𝑑7𝑑𝑑 𝑅𝑅 𝑙𝑙 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑙𝑙 − ( ) = (40) − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑘𝑘 The simplified form of Bloss curve, using the 𝑘𝑘 𝑙𝑙( ) =𝑅𝑅 ( ) = 0 x, y system, is obtained on the assumption ′ − ′′ − 3 2 ( ) = 2 3 Conditions𝑘𝑘 𝑙𝑙𝑘𝑘(40) 𝑘𝑘determine𝑙𝑙𝑘𝑘 the order of the 0 2 3 . differential equation. They are expressed in the

𝑘𝑘 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 1 4 1 5 following form: ( ) = 2 3 (33) 4 10 4 2 𝑦𝑦 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 (6)( ) + (4) = 0 3.4. COSINUSOID 2 (41) 𝜋𝜋

Satisfying conditions (27) the curvature is 𝑘𝑘 𝑙𝑙 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑙𝑙 identified by means of another differential Having solved the differential problem (40), equation. (41) it is possible to obtain a formula for the curvature equation: 2 (4) ( ) + 2 ( ) = 0 (34) 𝜋𝜋 ′′ 1 1 2 𝑙𝑙𝑘𝑘 ( ) = sin 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 2 (42) The curvature equation is as follows 𝜋𝜋 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙 𝑙𝑙 −1 𝜋𝜋 𝑅𝑅 � 𝑙𝑙 𝑙𝑙� 2 ( ) = + 2 + cos 1 1 4 2 2 4 2 (43) ( ) = cos 𝑘𝑘 𝑘𝑘 2 2 (35) 𝑙𝑙 𝑙𝑙 𝜋𝜋 𝜋𝜋 𝑘𝑘 𝑘𝑘 𝛩𝛩 𝑙𝑙 − 𝜋𝜋 𝑅𝑅 𝑅𝑅 𝑙𝑙 𝑙𝑙 𝜋𝜋 4 𝑅𝑅 � 𝑙𝑙 𝑙𝑙� 𝑅𝑅 𝑅𝑅 𝑙𝑙𝑘𝑘 9 𝑘𝑘 𝑙𝑙 1 − � 𝑙𝑙� ( ) = cos ( ) = 6 + … (44) ( ) = sin 648 2 2 2 (36) 𝜋𝜋 𝑘𝑘 𝑙𝑙 𝜋𝜋 𝑘𝑘 𝑥𝑥 𝑙𝑙 ∫ 𝛩𝛩 𝑙𝑙 𝑑𝑑𝑑𝑑 𝑙𝑙 −2 𝑅𝑅 𝑙𝑙 𝑙𝑙 4 𝑘𝑘 𝛩𝛩 𝑙𝑙 𝑅𝑅 𝑙𝑙 − 𝜋𝜋 𝑅𝑅 4 �𝑙𝑙 𝑙𝑙� ( ) = sin ( ) = 5 7 ( ) = cos ( ) = 7 + 30 3 315 5 2016 2 4 6 𝜋𝜋 𝜋𝜋 9 6 𝜋𝜋 + … 𝑘𝑘 𝑘𝑘 9 𝑦𝑦 𝑙𝑙 ∫7 𝛩𝛩 𝑙𝑙 𝑑𝑑 𝑑𝑑 𝑅𝑅 𝑙𝑙 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑙𝑙 (45)− 𝑘𝑘 5670 𝑥𝑥 𝑙𝑙 2∫6 𝛩𝛩 …𝑙𝑙 𝑑𝑑 𝑑𝑑 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑙𝑙 (37) 𝜋𝜋 25920 𝜋𝜋 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑅𝑅 𝑙𝑙 𝑙𝑙 − 2 4 A simplified form of the sinusoid is obtained on 4 6 ( ) = sin ( ) = 2 4 + making the assumptions x = l , = and 48 1440 1 1 2 6 𝜋𝜋 𝜋𝜋 ( ) = sin 8 0 2 . 𝑘𝑘 𝑘𝑘 𝑦𝑦 𝑙𝑙 ∫6 𝛩𝛩 …𝑙𝑙 𝑑𝑑 𝑑𝑑 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙 𝑘𝑘 𝑙𝑙 (38) 𝜋𝜋 𝑥𝑥 𝑙𝑙 80640 𝜋𝜋 𝑅𝑅 𝑙𝑙𝑘𝑘 𝜋𝜋 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑥𝑥 − 1 � 𝑥𝑥� 2 2 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − ( ) = + 3 + sin A simplified form of the cosine curve is secured 4 2 6 8 3 (46) 1 𝑙𝑙𝑘𝑘 𝑙𝑙𝑘𝑘 𝜋𝜋 on the assumption that ( ) = 0 2 𝑦𝑦 𝑥𝑥 − 𝜋𝜋 𝑅𝑅 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑥𝑥 𝜋𝜋 𝑅𝑅 � 𝑙𝑙𝑘𝑘 𝑥𝑥� 1 4. LIMITATIONS NOTED IN THE USE OF cos . 2 𝑘𝑘 𝑥𝑥 𝑅𝑅 − SMOOTH TRANSITION CURVES IN 𝜋𝜋

𝑅𝑅 𝑙𝑙𝑘𝑘 RAILWAY ROUTES � 𝑥𝑥� 1 2 ( ) = 2 + cos 1 4 2 2 (39) A comparison of transition curves related to 𝑙𝑙𝑘𝑘 𝜋𝜋 railway routes requires that certain assumptions 𝑦𝑦 𝑥𝑥 𝑅𝑅 𝑥𝑥 𝜋𝜋 𝑅𝑅 � �𝑙𝑙𝑘𝑘 𝑥𝑥� − � 3.5. SINE CURVE should be made with respect to the permissible values of kinematic parameters being in force, In addition to the assumptions made earlier acceleration growth ψper and the speed of lifting more conditions are set out. the railway rolling stock wheel at the

superelevation ramp fper . The assumption of equal

values of ψper and fper calls for the elongation of particular smooth transition curves in relation to the cubic parabola (by the use of an appropriate coefficient A). 35 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

Fig. 1. The shaping of the horizontal ordinates along Fig. 2. The shaping of horizontal ordinates in the outset the length of the cubic parabola and the smooth region of transition curves under consideration. transition curves. Table 1. Selected values of horizontal ordinates Thus it is possible to make a comparison of the y [mm]. horizontal ordinates (Fig. 1). In the given case the Transition x = 5 m x = 10 m x = 15 m x = 20 m curve in the form of cubic parabola is a reference curve adopted for the following geometric system: Cubic 0.37202 2.97619 10.04464 2.80952 • maximal speed of trains v = 100 km/h, parabola Quartic • radius of circular arc R = 700 m, 0.00581 0.09301 0.47084 1.48810 parabola • value of superelevation along arc h0 = 80 mm, Bloss curve 0.01524 0.23975 1.19280 3.70370 • length of transition curve (with a rectilinear Cosinusoid 0.01162 0.18562 0.93728 2.95150 Sinusoid 0.00036 0.01143 0.08642 0.36183 superelevation ramp) lk = 80 m.

Let us, now, take a closer look at the shaping of The lengths of the smooth transition curves in the horizontal ordinates of the transition curves of Fig. 1 are as follows: for sinusoid and quartic Fig. 1 along the length of the first 20 m. This is parabola (A = 2) – lk = 160 m, for cosinusoid illustrated in Fig. 2. Table 1 provides particular (A = π/2) – lk = 125,664 m, and for Bloss curve numerical values. When analyzing the data of (A = 1,5) – lk = 120 m. Table 1, it is surprising that the usefulness of The theoretical analyses performer and the smooth transition curves causes such big doubts. experimental works carried out unambiguously The horizontal ordinates of the outset region are indicate a lesser (that is, more advantageous) very small with respect to these curves. They are dynamic interactions while travelling on smooth many times smaller than the ordinates of the cubic transition curves. As already mentioned parabola. The Bloss curve takes relatively the best a dominant role here is played by the class of place among the smooth curves. The most function describing the curvature. However, in advantageous sinusoid with respect to dynamics, in spite of their indisputable advantages the range of fact, seems to be impossible to be carried out, in application of smooth transition curves In railway a given case its ordinates along the distance of the tracks under exploitation is very limited. The basic first 20 m do not reach even 1 mm. reason for the existing skepticism about this The presented considerations relating to the question seems to be the very low value of the railway routes can also be referred to vehicular horizontal ordinates In the initial region of the roads. It all leads to the conclusion that the major curves analyzed. It is often difficult to set them out cause of the difficulties encountered, lies in correctly in the field and in practice, leads to excessive smoothing of the curvature near the shortcuts of the transition curve made in original smooth transition curve. In order to take comparison with the brief foredesign. preventive measures, it is necessary to resign from the zeroing condition of the curvature derivative at the outset point, and assume a certain numerical value instead, smaller, however, than it occurs in the case of clothoid, or a cubic parabola. In this 36 LOGISTICS INFRASTRUCTURE Identification of Transition Curves... way an idea to find a new family, called parametric The largest values of the kinematic parameters 3 2 transition curves, has emerged [18]. = are noted at point 0 3 (2 ) , where also the − 𝐶𝐶 curvature derivative is the maximum 5. PARAMETRIC TRANSITION CURVE 𝑙𝑙 −𝐶𝐶 𝑙𝑙𝑘𝑘

(3 2 )2 1 Advantage is taken of differential equation (27) max ( ) = ( ) = + 0 3 (2 ) (52) with the following conditions (where constant − 𝐶𝐶 ′ ′ C > 0): −𝐶𝐶 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑙𝑙 𝑘𝑘 𝑙𝑙 �𝐶𝐶 (3 2 )2 � Expression = + determines 1 3 (2 ) + − 𝐶𝐶 (0 ) = 0 ( ) = the relation between the length of the parametric 𝐴𝐴 �𝐶𝐶 −𝐶𝐶 � − (47) transition curve and the cubic parabola (for which 𝑘𝑘 𝑘𝑘 𝑙𝑙𝑘𝑘 𝑅𝑅 (0+) = ( ) = 0 in the calculation formulae A = 1). This coefficient

′ 𝐶𝐶 ′ − assumes the smallest value for C = 1 and amounts 4 𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑘𝑘 𝑙𝑙𝑘𝑘 = = 1.3333 The solution of the differential problem (27), to 3 . (47) leads to the determination of the curvature A simplified form of the parametric curve is equation of a new transition curve which on obtained𝐴𝐴 on the assumption that x = l , = , 3 2 2 account of the occurring parameter C, will be ( ) = + 2 3 0 2 3 . 𝑘𝑘 𝑘𝑘 referred to as parametric curve. The equation has 𝐶𝐶 − 𝐶𝐶 −𝐶𝐶 𝑙𝑙 𝑥𝑥 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 the form 𝑘𝑘 𝑥𝑥 𝑥𝑥 𝑥𝑥3 2− 𝑥𝑥 2 ( ) = 3 + 4 5 6 12 2 20 3 (53) 3 2 2 2 3 𝐶𝐶 − 𝐶𝐶 −𝐶𝐶 ( ) = + 2 3 (48) 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅 𝑙𝑙𝑘𝑘 𝐶𝐶 − 𝐶𝐶 −𝐶𝐶 𝑦𝑦 𝑥𝑥 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 The parametric transition curve was determined 𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 Function (48) describing the curvature is because of the limitations indicated at an earlier a function of class C0. However, due to the fact point regarding the use of smooth transition curves that the transition near the region of the circular arc on railway routes. Therefore it will be legitimate to is mild, and grows milder in the initial area (than in consider its properties while working on the the case of the clothoid and the cubic parabola) the application of the simplified solution and the obtained curve can still be included among smooth utilization of equation (53). transition curves. Moreover, the curve, in view of dynamics, is more advantageous than both the clothoid and the cubic parabola. From equation (48) it follows that

3 2 2 ( ) = 2 + 3 4 2 3 2 4 3 (49) 𝐶𝐶 − 𝐶𝐶 −𝐶𝐶

𝛩𝛩 𝑙𝑙 𝑅𝑅𝑙𝑙𝑘𝑘 𝑙𝑙 𝑅𝑅𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅𝑙𝑙𝑘𝑘 𝑙𝑙 The parametric transition curve equations are as follows:

2 ( ) = cos ( ) = 5 40 2 2 𝐶𝐶2 (3 2 ) 6 9 (2 ) 4(3 2 ) 7 + 𝑅𝑅 𝑙𝑙𝑘𝑘+ Fig. 3. The shaping of the horizontal ordinates along the 𝑥𝑥36𝑙𝑙 2 3 ∫ 𝛩𝛩 𝑙𝑙 504𝑑𝑑𝑑𝑑 2𝑙𝑙 −4 𝑙𝑙 − 𝐶𝐶 − 𝐶𝐶 𝐶𝐶 −𝐶𝐶 − − 𝐶𝐶 length of the parametric transition curve (C = 0,25, (3 2 )(2 ) 8 𝑅𝑅 𝑙𝑙𝑘𝑘 + … 𝑅𝑅 𝑙𝑙𝑘𝑘 C= 0,5, C=0,75, and C = 1,0), the cubic parabola, and 96 2 𝑙𝑙5 𝑙𝑙 (50) − 𝐶𝐶 −𝐶𝐶 the Bloss curve. 𝑘𝑘 𝑅𝑅 𝑙𝑙 𝑙𝑙 3 2 ( ) = sin ( ) = 3 + 4 6 12 2 Fig. 3 shows the shaping of the horizontal 3 𝐶𝐶2 − 𝐶𝐶 2 5 7 (3 2 ) 8 ordinates of the obtained parametric curve for 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑅𝑅+𝑙𝑙𝑘𝑘 … (51) 20𝑦𝑦 𝑙𝑙 3 ∫ 336𝛩𝛩 𝑙𝑙3 3𝑑𝑑𝑑𝑑 192 𝑙𝑙 3 4 𝑙𝑙 − 0, 1 in the background of appropriate −𝐶𝐶 𝐶𝐶 𝐶𝐶 − 𝐶𝐶 diagrams for the cubic parabola and the Bloss 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 − 𝑅𝑅 𝑙𝑙𝑘𝑘 𝑙𝑙 𝐶𝐶curve.∈ 〈 As〉 can be seen in this figure, for 0, 1 ordinates y(x) differ evidently along the whole length of the transition curve. The 𝐶𝐶 ∈ 〈 〉 37 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

parametric curve C = 1 has horizontal ordinates which are an approximation to the cubic parabolic ordinates. The ordinates of the remaining curves are fund between the cubic parabolic ordinates and the Bloss curve (i.e. curve C = 0). For C 1 ordinates y(x) are similar along the entire length, and also share similarity with ordinates of≥ the cubic parabola (particular curves are different only in length). All these factors create a situation where the use of transition curves which have C 1, would not be advantageous; they make the process of entering the circular arc easier. However,≥ the initial region is still affected Fig. 4. The shaping of the horizontal ordinates of some by violent disturbances in the curvature, which are selected parametric curves (and the cubic parabola, and characteristic of the cubic parabola. the Bloss curve) in the outset region. Thus, solutions that may prove useful for applications in practice should be sought among In the case analyzed it might appear that the parametric curves of (0, 1). The selection best solution would be to apply parametric curve criterion should, of course, be based on values of C = 0,5. It indicates quite clearly the horizontal the horizontal ordinates 𝐶𝐶in ∈the outset region. Fig. 4 ordinates in the initial region (though twice smaller illustrates the shaping of the horizontal ordinates in than the cubic parabola). In addition it is the outset region of some selected parametric characterized by relatively mild transfer from curves. a straight to a transition curve; the curvature The choice of the form of the parametric curve derivative at the initial point presents only 36 % of (i.e. the acceptance of the required value of the value which appears on the cubic parabola. parameter C) will depend on a particular geometric situation. The problem of fundamental significance 6. CONCLUSIONS will become the value analysis of the horizontal • The requirements imposed upon transition ordinates of the outset region. Table 2 illustrates curves relating both to vehicular roads and the magnitudes of these ordinates along the length railway routes are clearly defined and for this of the first 20 m for various values of parameter C reason there should be one common algorithm (for the numerical data used in the paper). The to create them. Meanwhile all the solutions ordinates are much greater than those for the that have been known so far, are still used smooth transition curves given in Table 1. independently and bear various names. Table 2. Selected values of horizontal ordinates y [mm] for various parametric curves.

Curve A lk [m] x = 5 m x = 10 m x = 15 m x = 20 m C = 0.1 1.47544 118.035 0.03991 0.43272 1.82934 5.17768 C = 0.2 1.45185 116.148 0.06533 0.63122 2.48293 6.68835 C = 0.3 1.42941 114.353 0.09148 0.83501 3.15259 8.23309 C = 0.4 1.40833 112.667 0.11831 1.04364 3.83676 9.80796 C = 0.5 1.38889 111.111 0.14574 1.25648 4.53310 11.40720 C = 0.6 1.37143 109.714 0.17365 1.47253 5.23825 13.02267 C = 0.7 1.35641 108.513 0.20187 1.69041 5.94746 14.64299 C = 0.8 1.34444 107.556 0.23016 1.90814 6.65403 16.25237 C = 0.9 1.33636 106.909 0.25816 2.12293 7.34869 17.82905 C = 1.0 1.33333 106.667 0.28537 2.33089 8.01849 19.34291

Moreover, the transition curves are also defined in different ways, with respect to vehicular roads they are often denoted by angle function θ(l) which is responsible for changing the direction of the longitudinal axis 38 LOGISTICS INFRASTRUCTURE Identification of Transition Curves...

of the vehicle after travelling along a certain region of entering the circular arc, and is arc, whereas in railway routes the major disturbance at the starting point (though functions are related to the use of curvature smaller than in the case of the clothoid, Or the k(x) with a system of rectangular coordinates. cubic parabola as well). The acceptance of an • As can be expected the basis for the appropriate value of parameter (0, 1) identification of the transition curves should depends on a particular geometric situation be the curvature distribution along their length and occurs as a result of the value analysis𝐶𝐶 ∈ of deciding about the occurring unbalanced horizontal ordinates in the outset region of the accelerations. The dynamic interactions are transition curve. smaller (i.e. more advantageous), if the function class describing curvature is higher. REFERENCES The paper presents a method identifying [1] Arslan A., Tari E., Ziatdinov R., Nabiyev R.: curvature on transition curves by differential Transition curve modeling with kinematical equations. The method makes a reference to properties: research on log‒aesthetic curves. the approach used in [24] related to Computer-Aided Design and Applications 2014, identification of accelerations. The curvature 10, 11(5), 509-517. equations have been determined for some [2] Baluch H.: Optimization of track geometrical known forms of transition curves, such as, the systems. WK&L, Warsaw, Poland, 1983 clothoid, the quartic parabola, the Bloss curve, (in Polish). the cosinusoid and the sinusoid. Taking [3] Baykal O., Tari E., Coskun Z., Sahin M.: New advantage of these equations the Cartesian transition curve joining two straight lines. Journal of Transportation Engineering 1997, 123 (5), 337– coordinates were found. Approximate 347. solutions were also provided after some [4] Bolton K.: Biarc curves. Computer-Aided Design simplified Assumption had been made, 1975, 7 (2), 89–92. orientated to a large extent towards railway [5] Bosurgi G., D’Andrea A.: A polynomial routes. parametric curve (PPC‒CURVE) for the design of • Smooth transition curves, i.e. curves of horizontal geometry of highways. Computer-Aided nonlinear distribution of curvature, have been Civil and Infrastructure Engineering 2012, 4, known for a long time and possess a number 27(4), 303-312. of unquestionable advantages. First of all they [6] Cai H., Wang G.: A new method in highway route are characterized by minor values of dynamic design: joining circular arcs by a single C-B´ezier curve with shape parameter. Journal of Zhejiang interactions. The range of their applications in University − Science A, 2009, 10, 562–569. railway routes has been limited so far. [7] Dimulyo S., Habib Z., Sakai M.: Fair cubic Unfortunately, the curve have one main transition between two circles with one circle drawback, namely, a very small value of the inside or tangent to the other. Numerical horizontal ordinates in the outset region, in Algorithms 2009, 51, 461–476. practice often impossible for execution and [8] Farin G.: Class a B´ezier curves. Computer Aided hard to be maintained. Geometric Design 2006, 23 (7), 573–581. • The basic reason for the difficulties occurring [9] Freimann E.: Transition curves with changing in some known forms of smooth transition curvature in the reverse curve. Eisenbahningenieur 1973, No. 5 (in German). curves is connected with curvature being too [10] Grabowski R. J.: Smooth curvilinear transitions much mitigated in the outset region. Thus it is along vehicular roads and railway lines. Zeszyty necessary to find a new form of the transition Naukowe AGH 1984, No. 82, Cracow, Poland curve, and abandon the condition of zeroing (in Polish). the curvature derivative at the outset point. [11] Habib Z., Sakai M.: G2 cubic transition between For this purpose use has been made of the two circles with shape control. International curvature identification method described in Journal of Computer Mathematics 2003, 80 (8), 959–967. the paper, contributing simultaneously to the 2 formation of a family of parametric transition [12] Habib Z., Sakai M.: G Pythagorean hodograph curves. quintic transition between two circles with shape control. Computer Aided Geometric Design 2007, • The parametric transition curve recommended 24, 252-266. for practical application is characterized by a mild proceeding of the curvature In the

39 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

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Wladyslaw Koc Gdansk University of Technology, Poland [email protected]

41 Identification of Transition Curves... Logistics and Transport No 4(28)/2015

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