Optimum Railway Transition Curves—Method of the Assessment and Results

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Article Optimum Railway Transition Curves—Method of the Assessment and Results

Krzysztof Zboinski and Piotr Woznica *

Faculty of Transport, Warsaw University of Technology, 75 Koszykowa Street, 00-662 Warsaw, Poland; [email protected] * Correspondence: [email protected]

Abstract: This article discusses the optimization of railway transition curves, through the application of polynomials of 9th and 11th degrees. In this work, the authors use a 2-axle rail vehicle model combined with mathematically understood optimization methods. This model is used to simulate rail vehicle movement negotiating both a transition curve and circular arc. Passenger comfort is applied as the criterion to assess which transition is actually is the best one. The 4-axle vehicle was also used to verify the results obtained using the 2-axle vehicle. Our results show that the traditionally used in a railway engineering transition—3rd degree parabola—which is not always the optimum curve. This fact is especially valid for the longest curves, with lengths greater than 150 m. For such cases, the transition curves similar to standard curves of 9th and 11th degrees is the optimum ones. This result is conﬁrmed by the use of the 4-axle vehicle.

Keywords: railway transition curve; optimization; simulation

1. Introduction Citation: Zboinski, K.; Woznica, P. Today, transport negatively influences both the environment and human health. It is Optimum Railway Transition also a big energy consumer. Due to this fact, engineers tend to project sustainable transport Curves—Method of the Assessment systems with a smaller negative impact on health and ecosystems. In railway engineering, and Results. Energies 2021, 14, 3995. the infrastructure, train speed, and train dynamics influence train energy consumption. https://doi.org/10.3390/en14133995 If we take into account infrastructure 3D shape, it has a big impact on dynamics, and dynamics of train strongly influences journey comfort. Academic Editor: João Pombo The main aim of this study is the optimization and the shape assessment of the railway transition curves. The authors of the current article have used two advanced Received: 5 June 2021 rail vehicle models, using the full dynamics of the vehicles, to find new better shapes of Accepted: 30 June 2021 Published: 2 July 2021 railway transitions, not used in the engineering practice. The dynamics of a full vehicle- track system, which means the complete description of the vehicle interactions with the

Publisher’s Note: MDPI stays neutral railway track, is still not very popular. The idea to represent the whole rail vehicle with with regard to jurisdictional claims in a mathematical point today is not sufﬁcient. The simplicity of the rail model makes that published maps and institutional afﬁl- the full examination of vehicle dynamics is impossible. The advanced vehicle model gives iations. the chance of better assignment of dynamical properties of railway transitions to freight or passenger vehicles. The mentioned approach will include criteria for the assessment of transitions other than the comfort of the passengers. For the cargo trains, the main criterion can be the wear in wheel-rail contact. The authors of [1,2] have shown that using tools, which are characteristic for the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. railway vehicle dynamics in the shape assessment of the railway transition curves, can give This article is an open access article many advantages. It may provide new information not available applying the classical distributed under the terms and approach treating all elements of the system as the mathematical point. The results of the conditions of the Creative Commons simulations obtained, using such a method, are not always the same as the results obtained Attribution (CC BY) license (https:// using the classical approach. The mentioned works have concluded that the simulation creativecommons.org/licenses/by/ methods (used by the advanced vehicle models) can be at least a complement to the 4.0/). traditional method. This relates particularly to new high-speed railway lines, where types

Energies 2021, 14, 3995. https://doi.org/10.3390/en14133995 https://www.mdpi.com/journal/energies Energies 2021, 14, 3995 2 of 14

of rail vehicles operating are specified. Simulation (numerical) studies can find a certain justification, since there are today many software packages of AGEM type (automatic generation of the equations of motion) for railway vehicles. The programs of the lead author for the fast construction of mathematical models of these vehicles are also included. Up-to-date, in road engineering, the clothoid [3] is the curve most frequently used by engineers. The characteristic feature of this curve is the fact that in this curve, a linear change of its curvature k is a function of the curve length l exists. So, a relatively gradual increase in the lateral acceleration of the rail vehicle can be expected. The Equations for x and y coordinates in a function of l are as follows: ! l4 l8 x(l) = l 1 − + − ... , (1) 40C2 3456C4 ! l3 l4 l8 y(l) = 1 − + − ... . (2) 6C 56C2 7040C4 In Equations (1) and (2), C is the constant being the product of circular arc radius and the total curve length. In railway engineering, the engineers use the simplification of the clothoid. This simplification is the 3rd degree (cubic) parabola. In such an approach, simplification x = l is applied, so the Equation (2) has the following form:

x3 y(x) = . (3) 6C

The curve (3) has the linear curvature, because the simpliﬁcation k = d2y/dx2 is also applied. If we use the exact formulae for the curvature [3], mentioned linearity of the curvature is not to be a fact at the end of the curve.

2. Literature Survey Today, we may observe a large number of works, which deal with searching for new shapes of transition curves. It has a strong connection, especially in the context of new high-speed train lines construction. In the light of high-speed trains is also visible [4–8]. If we study literature related to railway transition curves formation, we state that a clear division of work is visible. Four groups of the works can be defined, and some works can be presented here as the typical works from each group [9–12]. The first group is represented by Tari and Baykal. The authors of [9] defined the so-called lateral change of acceleration (LCA) of railway vehicles while negotiating the transition curve as the key criterion for evaluating journey comfort in the railway transition curve. The continuity in time of function of lateral change of acceleration must be absolutely provided. They searched only for the transition curves, which satisfied this condition. Moreover, Hasslinger [13] imposed this condition on LCA. The second group is represented mainly by the authors of [10]. It is arising from the fact that the advanced model of a railway vehicle (simulation software) was used to assess the dynamical properties of the most popular transition curves used in railway engineering. The authors of [10] took some of the most important dynamical characteristics like lateral and vertical acceleration of vehicle body, wheel/rail lateral and vertical forces, and derailment coefficient to compare the shapes of the transition curves. The same idea can also be found in the work by authors of [14–26]. The third group is the group, where the properties of transition curves are studied, treated the curve as the mathematical object with all its mathematical properties. In such a group of works, authors mostly try to find the curves better than the 3rd degree parabola (the clothoid) [11,27–32]. From the perspective of the impact of the infrastructure shape on passenger comfort is the work [12] by Kufver. Due to this fact, he used the famous European standard elaborated Energies 2021, 14, 3995 3 of 14

by CEN [33]. In mentioned standard, the percentage PCT of both seating and standing passengers with discomfort feelings was described by the proper Equation. The values of PCT had a form of the 2nd degree parabola in the function of the curve length, and it was mathematically possible to ﬁnd such a value of the length of the curve, for which the percentage had a minimum value. Analyzing the literature, one can state that the works where the full dynamics of the vehicle-track system and optimization methods are used jointly are not popular. The classical approach to track-vehicle interactions is still applied [34]. A relatively simple vehicle model (or sometimes just simply a point) and the maximum value of unbalanced lateral acceleration and its change, which should not be exceeded as the demand is still rare. The approach in which during the transition curve shape optimization the advanced rail vehicle model of the vehicle-track system and mathematically understood optimization methods are used is still a certain novelty.

3. Method As the transition curve, the polynomial of 9th and 11th degrees was applied. High degrees of polynomials were chosen, due to the ﬂexibility of shape modeling. Odd degrees were used, taking into account the fact that such curves have only one so-called standard curve. General Equations for curve, curve curvature, cant, and inclination of superelevation ramp used is:

n n−1 n−2 n−3 4 3 ! 1 A l A − l A − l A − l A l A l y = n + n 1 + n 2 + n 3 + ..... + 4 + 3 , (4) R n−2 n−3 n−4 n−5 2 l l0 l0 l0 l0 l0 0 " # d2y 1 A ln−2 A ln−3 A l = = ( − ) n + ( − )( − ) n−1 + + · 3 k 2 n n 1 n−2 n 1 n 2 n−3 ..... 3 2 , (5) dl R l0 l0 l0 " # A ln−2 A ln−3 A l = ( − ) n + ( − )( − ) n−1 + + · 3 h H n n 1 n−2 n 1 n 2 n−3 ..... 3 2 , (6) l0 l0 l0 " dh A ln−3 A ln−4 A = = ( − )( − ) n + ( − )( − )( − ) n−1 + + · · 3 i H n n 1 n 2 n−2 n 1 n 2 n 3 n−3 ..... 3 2 1 , (7) dl l0 l0 l0 where: y—transition curve offset [m], n—degree of the polynomial, R—curve radius [m], l0—total curve length [m], l—curve current length [m], k—curvature [1/m], h—superelevation ramp (cant) [m], i—inclination of the superelevation ramp [-]. For each characteristic, fundamental geometrical demands in the extreme points of the curve were used [34]. For example, for curvature function mentioned demand is: - The value of curvature in the beginning point of the curve—0, - The value of curvature in the last point of the curve—1/R. In this work, one criterion for transition curve assessment is used. This a normalized integral of the lateral acceleration of the vehicle body along the route. Using the Equation, we have: L Z C −1 QF1 = LC | ÿb| dl (8) 0 where: QF1—quality function, Energies 2021, 14, x FOR PEER REVIEW 4 of 15

In this work, one criterion for transition curve assessment is used. This a normalized integral of the lateral acceleration of the vehicle body along the route. Using the Equation, we have:

LC QF = L−1 y dl 1 C  b (8) 0

Energies 2021, 14, 3995 where: 4 of 14 QF1—quality function, Lc—total length of the route [m], 2 yb —lateral acceleration of vehicle body [m/s ], Lc—total length of the route [m], L—current curve length [m]. ÿ —lateral acceleration of vehicle body [m/s2], b This criterion is a fundamental criterion in railway dynamics, since vehicle L—current curve length [m]. dynamicsThis criterion improvement is a fundamental is guaranteed criterion after inthe railway optimization. dynamics, since vehicle dynamics improvementThe length is guaranteed of the curve after was the calculated, optimization. using the traditional method, known in engineeringThe length practice of the [34]. curve In wasthis calculated,work, two using standard the curves traditional of 9th method, and 11th known degrees in engineeringwere used as practice the initial [34]. ones. In this These work, curves two standardare: curves of 9th and 11th degrees were used as the initial ones. These curves1  are:5 l 9 5 l 8 l7 7 l6 

y9 = − + − 2 +  (9)  9 7 8 6 7 5 6 ! 4  R1  185 l lo 5 l4 lo l l7o l 6 lo  = − + − + y9 7 6 2 5 4 (9) R 18 lo 4 lo lo 6 lo  11 10 9 8 7  1 7 l 7 l 15 l 15 l l y11 =  − + − + 3!  (10) 1R 711l11l9 7 2l10l8 15 2l9 l7 15 l28 l6 l7 l5  y =  0− 0+ −0 +03 0  (10) 11 R 11 l9 2 l8 2 l7 2 l6 l5 In this work, two models of rail0 vehicles0 were0 applied.0 The0 first model has a 2-axle structure.In this In work, many two earlier models works of rail[26], vehicles it was called were applied.a “2-axle The(freight) ﬁrst modelwagon has”. Like a 2-axle every structure.wagon of In this many type, earlier it has works a body [26 ],connected it was called with a “2-axletwo wheelsets (freight) with wagon”. spring Like-damping every wagonelements. of this The type, structure it has of a body the model connected and its with parameters two wheelsets correspond with spring-damping to a typical real elements.wagon. This The structure model is of of the key model importance and its parameters here, because correspond in this to work a typical, it was real wagon. used to Thisinvestigate modelis the of key dynamical importance properties here, because and optimize in this work, the shape it was of used the to transition investigate curves. the dynamicalThe nominal properties model and of optimize this vehicle the shape is shown of the transitionin Figure curves. 1c. The The vehicle nominal model model is ofsupplemented this vehicle is withshown the in Figuremodel 1 ofc. The the vehicle track, as model presented is supplemented in Figure with 1a,b. the The model entire oftrack the- track,vehicle as system presented is discussed in Figure in1a,b. detail The [26]. entire The track-vehicle model parameters system of is this discussed system inare detailalso presented [26]. The modelby the parametersauthors of [26]. of this system are also presented by the authors of [26].

FigureFigure 1. 1.2-axle 2-axle vehicle vehicle model: model: ( a(a)) Track Track vertically, vertically, (b ()b track) track laterally, laterally, (c )(c vehicle.) vehicle.

TheThe second second modelmodel isis MK-111MK-111 [[26].26]. This car car is is a a British British passenger passenger car. car. It It was was used used in inthis this work work to to verify verify the the results results of of optimization of the the shape shape of of transition transition curves curves made made withwith the the previous previous model. model. TheThe nominalnominal modelmodel of of this this vehicle vehicle is is shown shown in in Figure Figure2. 2. The The parametersparameters of of a a bogie bogie for for this this wagon wagon were were adopted adopted by by the the authors authors of of [ 26[26].]. The The remaining remaining model parameters were adopted on the basis of the obtained data at British Rail Research, Derby. The model itself consists of seven rigid elements. The track model included in the vehicle was identical to that for a 2-axle car. EnergiesEnergies 2021 2021, 14, 14, x, FORx FOR PEER PEER REVIEW REVIEW 5 5of of 15 15

Energies 2021, 14, 3995 modelmodel parameters parameters were were adopted adopted on on the the basis basis of of the the obtained obtained data data at at British British 5 Rail of Rail 14 Research,Research, Derby. Derby. The The model model itself itself consists consists of of seven seven rigid rigid elements. elements. The The track track model model incincludedluded in in the the vehicle vehicle was was identical identical to to that that for for a a2 -2axle-axle car. car.

FigureFigureFigure 2. 2. 2.44-axle- 4axle-axle vehicle vehicle vehicle model. model. model.

TheTheThe approach approachapproach to toto the the the shape shape shape modeling modeling modeling was was was widely widely widely defined deﬁned defined [25]. [ 25[25].]. Dynamics Dynamics Dynamics of of of relative relative relative motionmotionmotion is is is used used in in in this this this method. method. method. The The The deﬁnition d efinition definition of vehicleof of vehicle vehicle dynamics dynamics dynamics is relative is is relative relative to the to track- to the the trackbasedtrack-based-based moving moving moving reference reference reference frame frame frame (reference (reference (reference frame frame frame 01 xyz0 10xyz1xyz at at Figure at Figure Figure3). 3). 3). The The The ideaidea idea appliedapplied applied ininin this thisthis work workwork relied reliedrelied on onon the thethe passage passagepassage of ofof rail railrail model modelmodel through throughthrough the thethe route routeroute (track) (track)(track) consisted consistedconsisted in: in:in: SStraighttraightStraight track, track, track, transition transition curve,curve, curve,and and and circular circular circular arc. arc. arc. The The The geometrical geometrical geometrical track track track model model model was was generallywas gen- gen- erallyanalyzederally analyzed analyzed in three in in three dimensions. three dimensions. dimensions.

FigureFigureFigure 3. 3. 3.MovingMoving Moving coordinate coordinate coordinate reference reference reference frame frame frame 0 01 01xyz.xyz.1xyz.

ItIt Itis is is also also also worth worthy worthy yof of notice notice thatthat that inin in thethe the currentcurrent current work, work work no, no, no track track track deformation—both deformation deformation——bothboth lateral lat- lat- eralanderal and vertical—causedand vertical vertical——causedcaused by by forces by forces forces in in the in the wheel/trackthe wheel/track wheel/track contact contact contact is is takenistaken taken intointo into account.account. account. The The The shape of the track is fixed and oriented in relation to the track center-line. The work has shapeshape of of the the track track is is fixed fixed and and oriented oriented in in relation relation to to the the track track cent centerer-line.-line. The The work work has has a theoretical character, and the analysis of track displacements is rather characteristic of a atheoretical theoretical character character, and, and the the analysis analysis of of track track displacements displacements is is rather rather characteristic characteristic of of works connected with railway maintenance. worksworks connected connected with with railway railway maintenance. maintenance. The scheme of the software is presented in Figure4. There are two iteration loops. The first one is the integration loop. It stopped when the assumed length of the route is reached by the vehicle model. The second loop is the optimization process loop. This loop stopped when the limit assumed value of iteration is reached unless the optimum solution is not found earlier. In practice, 200 iterations were enough to find optimum solutions. The Energies 2021, 14, x FOR PEER REVIEW 6 of 15

The scheme of the software is presented in Figure 4. There are two iteration loops. Energies 2021, 14, 3995 The first one is the integration loop. It stopped when the assumed length of the route6 of 14 is reached by the vehicle model. The second loop is the optimization process loop. This loop stopped when the limit assumed value of iteration is reached unless the optimum solution is not found earlier. In practice, 200 iterations were enough to find optimum typicalsolutions. time The of calculationtypical time was of calculation not greater thanwas not 1 h forgreater each than optimization, 1 h for each using optimiza Intel Coretion, Duo 2 GB processor. using Intel Core Duo 2 GB processor.

Figure 4. The scheme of software. Figure 4. The scheme of software.

4.4. ResultsResults ofof thethe OptimizationOptimization 4.1.4.1. ResultsResults ofof thethe OptimizationOptimization forfor 2-Axle2-Axle VehicleVehicle ThisThis sectiomsectiom presentspresents thethe generalgeneral resultsresults ofof thethe optimizationoptimization ofof railwayrailway transitiontransition curvescurves performed performed by by the the authors authors of the of currentthe current work. work. In the In optimization the optimization made, the made, routes the ofroutes the vehicle, of the asvehicle, mentioned, as mentioned, were always were composed always composed of straight of track, straight transition track, curve, transition and circularcurve, and arc. circular Lengths arc. of straight Lengths track of straight were always track were the same always and the equal same to 50and m. equal Similarly, to 50 them. lengthsSimilarly, of the lengths circular of arc the were circular also thearc samewere andalso equalthe same to 100 and m. equal to 100 m. In general, the optimization results are always comprised of: Optimum transition curves and their curvatures represented by the optimum polynomial coefﬁcients, dynamics of the rail vehicle, mainly represented by displacements and acceleration of vehicle body and the wheelsets, and creepages in wheel-rail contact. Energies 2021, 14, 3995 7 of 14

In Table1, the authors presented parameters assumed in optimization for two different degrees of the polynomial—9th one and 11th one. These parameters are: The radius of circular arc R, the cant H, the length l0 of the curve, and vehicle velocity v. Admissible unbalanced acceleration alim on the track level was calculated for particular R, H, and velocity v, and ranged from −0.3 m/s2 (small velocities) to 0.6 m/s2.

Table 1. Assumed parameters for 9th and 11th degrees.

Order Number (Degree) Radius R, Cant H Length l0, Velocity v 1 (9th) 600 m, 150 mm 118.35 m, 20.20 m/s 2 (9th) 600 m, 150 mm 142.15 m, 24.26 m/s 3 (9th) 600 m, 150 mm 162.36 m, 27.71 m/s 4 (9th) 600 m, 150 mm 180.46 m, 30.79 m/s 5 (9th) 1200 m, 150 mm 167.40 m, 28.57 m/s 6 (9th) 1200 m, 75 mm 71.07 m, 24.26 m/s 7 (9th) 1200 m, 70 mm 82.41 m, 30.14 m/s 8 (9th) 1200 m, 75 mm 105.98 m, 36.17 m/s 9 (9th) 2000 m, 115 mm 134.99 m, 30.05 m/s 10 (9th) 2000 m, 45 mm 42.64 m, 24.26 m/s 11 (9th) 2000 m, 50 mm 60.29 m, 30.87 m/s 12 (9th) 2000 m, 45 mm 60.60 m, 34.47 m/s 13 (11th) 600 m, 150 mm 133.10 m, 20.20 m/s 14 (11th) 600 m, 150 mm 159.86 m, 24.26 m/s 15 (11th) 600 m, 150 mm 182.58 m, 27.71 m/s 16 (11th) 600 m, 150 mm 202.94 m, 30.79 m/s 17 (11th) 1200 m, 150 mm 188.25 m, 28.57 m/s 18 (11th) 1200 m, 75 mm 79.93 m, 24.26 m/s 19 (11th) 1200 m, 70 mm 82.41 m, 30.14 m/s 20 (11th) 1200 m, 75 mm 119.18 m, 36.17 m/s 21 (11th) 2000 m, 115 mm 151.80 m, 30.05 m/s 22 (11th) 2000 m, 45 mm 47.95 m, 24.26 m/s 23 (11th) 2000 m, 50 mm 67.80 m, 30.87 m/s 24 (11th) 2000 m, 45 mm 68.15 m, 34.47 m/s

Generally, five typical shapes of the curves were obtained in this work. Their curvatures had one of the five possible shapes: 1 The shape of the standard curve—type 1, 2 The shape with the inflection point in the middle part of the curve—type 2, 3 The linear shape as for the 3rd degree parabola—type 3, 4 The convex shape on the route of the transition curve ([0, l0])—type 4, 5 The concave shape on the route of the transition curve ([0, l0])—type 5. Figure5 presents four types of the curvatures—type no. 1, 2, 4, and 5. In Table2, the authors presented results for all 24 cases from Table1, as represented by characteristic features of curvatures of the optimum transition curves shapes. Four cases from this table are taken as the representative ones to present the results of the optimization. For these cases, the single curve radius R and superelevation H for circular arc was assumed. Their values were R = 600 m and H = 0.15 m, respectively. In all 24 analyzed cases from Table2, the betterment of rail vehicle dynamics represented by smaller values of quality function was observed. New optimum transition curves improved all key dynamical characteristics of the vehicle, first of all—lateral displacement and acceleration of vehicle body. This fact can be crucial for journey passenger comfort. Energies 2021, 14, 3995 8 of 14 Energies 2021, 14, x FOR PEER REVIEW 8 of 15

(a) (b)

Figure 5. Types of curvatures: (a)) 1 and 2; ((b)) 4 and 5.

TableIn 2. TableThe results 2, the of authors the optimization presented for results 9th and for 11th all degrees. 24 cases from Table 1, as represented by characteristic features of curvatures of the optimum transition curves shapes. Four Order Number (degree) Radius R, Cant H No. of Type of Curvature cases from this table are taken as the representative ones to present the results of the optimization. For1 (9th) these cases, the single 600 curve m, 150 radius mm R and superelevation 2 H for circular 2 (9th) 600 m, 150 mm 2 arc was assumed.3 (9th) Their values were R 600 = 600 m, 150m and mm H = 0.15 m, respectively. 2 4 (9th) 600 m, 150 mm 1 Table 2. The results5 (9th) of the optimization for 1200 9th m, and 150 11 mmth degrees. 2 6 (9th) 1200 m, 75 mm 3 Order Number7 (9th) (degree) Radius 1200 m, R, 70 Cant mm H No. of Type 3of Curvature 18 (9th) 600 1200 m, m, 150 75 mmmm 42 29 ( (9th)9th) 600 2000 m, m, 150 115 mm 12 103 (9 (9th)th) 600 2000 m, m, 150 45 mmmm 32 11 (9th) 2000 m, 50 mm 3 4 (9th) 600 m, 150 mm 1 12 (9th) 2000 m, 45 mm 3 135 (9 (11th)th) 1200 600 m, 150150 mm mm 2 146 (9 (11th)th) 1200 600 m,m, 150 75 mmmm 23 157 (9 (11th)th) 1200 600 m,m, 150 70 mmmm 53 16 (11th) 600 m, 150 mm 1 8 (9th) 1200 m, 75 mm 4 17 (11th) 1200 m, 150 mm 3 189 (9 (11th)th) 2000 1200 m, m, 115 75 mm mm 31 1910 (11th)(9th) 2000 1200 m, 7045 mmmm 3 2011 (11th)(9th) 2000 1200 m, 7550 mmmm 43 2112 (11th)(9th) 2000 2000 m,m, 11545 mm 3 22 (11th) 2000 m, 45 mm 3 1323 (11 (11th)th) 600 2000 m, m, 150 50 mmmm 32 1424 ( (11th)11th) 600 2000 m, m, 150 45 mmmm 32 15 (11th) 600 m, 150 mm 5 Table316 presents (11th) the values of optimum600 m, 150 coefﬁcients mm of the polynomial1 (Equation (1)) with accuracy17 (11 toth 5) decimal places for1200 cases m, no. 150 2, mm 4, 14, and 16 from Table23. In the rest of the work, it18 will (11th be) named the curves1200 (2), (4),m, 75 (14), mm and (16). 3 Graphical19 (11 resultsth) are limited to1200 the m, case 70 no.mm 2 from Table2. Figure3 6a shows a comparison20 of (11 transitionth) curves (curve1200 offsets m, y75)—initial mm one (black line), optimum4 one (red line), and the21 (11 3rdth degree) parabola (dashed2000 m, line), 115 whereas mm Figure6b shows their3 curvatures. It can be seen22 (11 thatth both) the offset, and2000 the curvature m, 45 mm of the optimum curve places3 it between the standard23 curve(11th) and the 3rd degree2000 parabola. m, 50 mm It is also visible that the curvature3 of the optimum curve24 (11 isth something) between2000 the m, standard 45 mm curve and 3rd degree parabola.3 Figure7 shows the comparison of the lateral displacement and acceleration of the center of mass of the car body for the initial curve, the optimum curve, and also the 3rd

Energies 2021, 14, 3995 9 of 14

degree parabola. Both the displacement graph (Figure7a) and the acceleration graph (Figure7b) clearly show that the optimum curve is better than both the standard curve, and the 3rd degree parabola. Notably, in this context, interesting is Figure7b for accelerations.

Table 3. Optimum polynomial coefﬁcients.

Degree (no. from Table1) Ai Value of Ai

9th (2) A9 −0.09942 9th (2) A8 0,44678 9th (2) A7 −0.71504 9th (2) A6 0.41706 9th (2) A5 0.00002 9th (2) A4 0.00638 9th (2) A3 0.09687 9th (4) A9 −0.22052 9th (4) A8 0.99234 9th (4) A7 −1.58774 9th (4) A6 0.92617 9th (4) A5 0 9th (4) A4 0 9th (4) A3 0.03439 11th (14) A11 0.16689 11th (14) A10 −0.91748 11th (14) A9 1.96613 11th (14) A8 −1.96607 11th (14) A7 0.78582 11th (14) A6 −0.00059 11th (14) A5 0.00090 11th (14) A4 0.01193 11th (14) A3 0.15142 11th (16) A11 0.52781 11th (16) A10 −2.90298 11th (16) A9 6.22068 11th (16) A8 −6.22068 11th (16) A7 2.48827 11th (16) A6 0 11th (16) A5 0.00013 11th (16) A4 0.00163 Energies 2021, 14, x FOR PEER REVIEW 10 of 15 11th (16) A3 0.02473

(a) (b)

FigureFigure 6. 6.Transitions Transitions curves: curves: (a) CurveCurve offsets; offsets; (b (b) )curvatures. curvatures.

Figure 7 shows the comparison of the lateral displacement and acceleration of the center of mass of the car body for the initial curve, the optimum curve, and also the 3rd degree parabola. Both the displacement graph (Figure 7a) and the acceleration graph (Figure 7b) clearly show that the optimum curve is better than both the standard curve, and the 3rd degree parabola. No

tably, in this context, interesting is Figure 7b for accelerations.

(a) (b) Figure 7. Dynamical characteristics; (a) Vehicle body lateral displacements; (b) vehicle body lateral accelerations.

The ratio of the objective function values (the optimum curve/3rd degree parabola) is 0.59 = (0.0018622 [m/s2]/0.0031123 [m/s2]). The ratio of the optimum objective function value to that one for the 9th degree standard curve is 0.35 = (0.0018622 [m/s2]/0.0051947 [m/s2]). Figure 8 presents the comparison of other graphical results—here, the lateral acceleration of the car body around the x-axis for the initial (standard) curve of 9th degree

Energies 2021, 14, x FOR PEER REVIEW 10 of 15

(a) (b) Figure 6. Transitions curves: (a) Curve offsets; (b) curvatures.

Figure 7 shows the comparison of the lateral displacement and acceleration of the center of mass of the car body for the initial curve, the optimum curve, and also the 3rd Energies 2021, 14, 3995 degree parabola. Both the displacement graph (Figure 7a) and the acceleration 10 graph of 14 (Figure 7b) clearly show that the optimum curve is better than both the standard curve, and the 3rd degree parabola. No

tably, in this context, interesting is Figure 7b for accelerations.

(a) (b)

Figure 7. DynamicalDynamical characteristics characteristics;; ( a)) V Vehicleehicle body lateral displacements; (b)) vehiclevehicle bodybody laterallateral accelerations.accelerations.

The ratio ofof thethe objectiveobjective function function values values (the (the optimum optimum curve/3rd curve/3rd degree degree parabola) parabola) is is0.59 0.59 = (0.0018622= (0.0018622 [m/s [m/s2]/0.00311232]/0.0031123 [m/s [m/s2]).2]). The The ratio ratio of the of optimumthe optimum objective objective function function value Energies 2021, 14, x FOR PEER REVIEWto that one for the 9th degree standard curve is 0.35 = (0.0018622 [m/s2]/0.00519472 [m/s112 ]).of 15 value to that one for the 9th degree standard curve is 0.35 = (0.0018622 [m/s ]/0.0051947 [m/s2Figure]). 8 presents the comparison of other graphical results—here, the lateral acceleration ofFigure the car 8 presents body around the comparison the x-axis for of theother initial graphical (standard) results curve—here of 9th, the degree lateral and accel- the erationoptimumand the of optimum curve.the car The body curve. betterment around The betterment the of railx-axis vehicle of for rail the dynamics vehicle initial dynamics (standard) is visible is in curvevisible comparison of in 9thcomparison degree to the standardto the standard curve of curve 9th degree. of 9th degree.

FigureFigure 8. 8.Dynamical Dynamical characteristics-vehicle characteristics-vehicle body body angular angular accelerations accelerations around aroundx-axis. x-axis.

4.2.4.2. Veriﬁcation Verification of of Results Results of of the the Optimization Optimization with with the the Use Use of of the the 4-Axle 4-Axle Vehicle Vehicle TheThe purpose purpose of of this this sub-section sub-section is tois showto show that that the optimumthe optimum transition transition curves curves 9th and 9th 11thand degrees 11th degrees presented presented in Section in 4.1Section maintain 4.1 maintain their positive their properties, positive properties, when a passenger when a vehiclepassenger (4-axle vehicle passenger (4-axle car) passenger is negotiating car) is them.negotiating Recall them. thatthe Recall shape that optimization the shape opti- of themization transition of curves the transition was optimized curves was with optimized a 2-axle vehicle with with a 2-axle a much vehicle smaller with number a much smaller number of degrees of freedom. It was dictated by shorter calculation times, and thus, a practical possibility of solving the task. The analysis of the usefulness of the curves obtained for the needs of the previous section, and example, defined by coefficients (2), (4), (14), and (16) from Table 1 was performed with the ULYSSES program. The model of the four-axle MK-111 wagon, presented earlier in Section 2, was used here. The two variants of the second stage of the wagon suspension, i.e., the soft and the stiff variant, presented by the authors of [25] were used. The aim of the simulations performed concerns the transition curves connecting with the circular arc. The parameters of arc were, as mentioned, R = 600 m and H = 0.15 m. The simulation data were identical to those used for determining the curves (2), (4), (14), and (16). The results concerned the vehicle’s travel along three curves: 1. The 3rd degree parabola equal in length to the standard transition curve of 9th or 11th degree; 2. The optimum curve for v = 24.26 m/s (respectively (2) for the 9th degree or (14) for the 11th degree) or the optimum curve for v = 30.79 m/s (respectively (4) for the 9th degree or (16) for the 11th degree); 3. The standard curve for the 9th or 11th degree. Eight cases were examined in the current work. The first two related to the movement of a vehicle with soft suspension along optimum curves of the 9th degree and different velocities, respectively v = 24.26 and 30.79 m/s. The next two cases concerned the movement of the vehicle with soft suspension along the 11th degree optimum curves with the same two speeds—v = 24.26 and 30.79 m/s. The next pair differed from the first one only in the use of a vehicle model with a stiff suspension. The last pair differed from the second pair again only in the use of the model with a stiff suspension. Two cases among the 8 analyzed (for the curves (2) and (14)) are presented in the figures in the further part of this section. The test results were limited to the presentation of the lateral

Energies 2021, 14, 3995 11 of 14

of degrees of freedom. It was dictated by shorter calculation times, and thus, a practical possibility of solving the task. The analysis of the usefulness of the curves obtained for the needs of the previous section, and example, defined by coefficients (2), (4), (14), and (16) from Table1 was performed with the ULYSSES program. The model of the four-axle MK-111 wagon, presented earlier in Section2, was used here. The two variants of the second stage of the wagon suspension, i.e., the soft and the stiff variant, presented by the authors of [25] were used. The aim of the simulations performed concerns the transition curves connecting with the circular arc. The parameters of arc were, as mentioned, R = 600 m and H = 0.15 m. The simulation data were identical to those used for determining the curves (2), (4), (14), and (16). The results concerned the vehicle’s travel along three curves: 1 The 3rd degree parabola equal in length to the standard transition curve of 9th or 11th degree; 2 The optimum curve for v = 24.26 m/s (respectively (2) for the 9th degree or (14) for the 11th degree) or the optimum curve for v = 30.79 m/s (respectively (4) for the 9th degree or (16) for the 11th degree); 3 The standard curve for the 9th or 11th degree. Eight cases were examined in the current work. The first two related to the movement of a vehicle with soft suspension along optimum curves of the 9th degree and different velocities, respectively v = 24.26 and 30.79 m/s. The next two cases concerned the movement of the vehicle with soft suspension along the 11th degree optimum curves with the same two speeds—v = 24.26 and 30.79 m/s. The next pair differed from the first one only in the Energies 2021, 14, x FOR PEER REVIEW 12 of 15 use of a vehicle model with a stiff suspension. The last pair differed from the second pair again only in the use of the model with a stiff suspension. Two cases among the 8 analyzed (for the curves (2) and (14)) are presented in the figures in the further part of this section. Thedisplacements test results were of the limited center to of the mass presentation of the body of the of lateral the tested displacements model. Each of the of center the pre- of masssented of thefigures body contained of the tested the results model. for Each theof three the presentedcurves, thus figures, defined. contained the results for theIn three the case curves, of the thus, vehicle defined. with soft suspension and velocity v = 24.26 m/s, the opti- mumIn curves the case (2) of theand vehicle (14) showed with soft features suspension better and than velocity the 3rdv =degree 24.26 m/s,parabola the optimum unequiv- curvesocally (2)for andthe considered (14) showed length features of the better parabola, than the equal 3rd to degree the standard parabola polynomial unequivocally curve forof the9th consideredand 11th degree. length This of the was parabola, proved equalby smoother to the standard transitions polynomial between curvethem and of 9th the andadjacent 11th degree. segments This (straight was proved track by and smoother circular transitions arc) in Figuresbetween 9 them and and 10, the and adjacent a lower segmentsmaximum (straight value for track the and curves circular (2) arc)and in(14) Figures than 9for and the 10 3rd, and degree a lower parabola maximum in Figure value s for the curves (2) and (14) than for the 3rd degree parabola in Figures 11 and 12. 11 and 12.

FigureFigure 9. 9.Lateral Lateral displacements displacements for for 9th 9th degree degree (soft (soft suspension, suspension,v v= = 24.26 24.26 m/s m/s22).).

Figure 10. Lateral displacements for 11th degree (soft suspension v = 24.26 m/s2).

Energies 2021, 14, x FOR PEER REVIEW 12 of 15

displacements of the center of mass of the body of the tested model. Each of the presented figures contained the results for the three curves, thus, defined. In the case of the vehicle with soft suspension and velocity v = 24.26 m/s, the optimum curves (2) and (14) showed features better than the 3rd degree parabola unequivocally for the considered length of the parabola, equal to the standard polynomial curve of 9th and 11th degree. This was proved by smoother transitions between them and the adjacent segments (straight track and circular arc) in Figures 9 and 10, and a lower maximum value for the curves (2) and (14) than for the 3rd degree parabola in Figures 11 and 12.

Energies 2021, 14, 3995 12 of 14

Figure 9. Lateral displacements for 9th degree (soft suspension, v = 24.26 m/s2).

Energies 2021, 14, x FOR PEER REVIEW 13 of 15

2 FigureFigure 10. 10.Lateral Lateral displacements displacements for for 11th 11th degree degree (soft (soft suspension suspensionv =v = 24.26 24.26 m/s m/s2).).

FigureFigure 11. 11.Maximum Maximum displacements displacements from from Figure Figure9 under9 under magniﬁcation. magnification.

FigureFigure 12. 12.Maximum Maximum displacements displacements from from Figure Figure 10 10 under under magniﬁcation. magnification.

InIn the the six six rest rest cases, cases, four four times times the the optimum optimum curves curves had had better better properties properties than than 3rd 3rd degreedegree parabola, parabola, which which was was represented represented by by the the smaller smaller maximum maximum values values of of the the lateral lateral 2 displacementdisplacement of of the the 4-axle 4-axle wagon. wagon. It It was was especially especially clear clear for for the the velocity velocity of of 24.26 24.26 m/s m/s2..

5. Discussion The presented method, using the 2-axle vehicle model and mathematically understood optimization, shows that it is possible to optimize the shape of the transition curve for a circular arc and cant. It is crucial from the perspective of designing railway infrastructure to reduce the negative impact of this infrastructure on the vehicle, and consequently, on the passenger. The work showed that the new shapes of the transition curves obtained for a given circular arc result in milder dynamics represented mainly by displacements and accelerations of the vehicle body. The results obtained and presented in the current article constitute a certain novelty in the field of optimization and the properties assessment of railway polynomial transition curves. Obtained curves of 9th and 11th degrees had significantly better properties than what is known from the literature, as well as standard curves of those degrees and the 3rd degree parabola—the curve traditionally used. It is not disturbed by the fact that different circular arc radii generally resulted in different shapes of the curves. It has also been shown that these curves maintain their characteristics also for the 4-axle car model, which corresponds to a real passenger car. This con-

Energies 2021, 14, 3995 13 of 14

5. Discussion The presented method, using the 2-axle vehicle model and mathematically understood optimization, shows that it is possible to optimize the shape of the transition curve for a circular arc and cant. It is crucial from the perspective of designing railway infrastructure to reduce the negative impact of this infrastructure on the vehicle, and consequently, on the passenger. The work showed that the new shapes of the transition curves obtained for a given circular arc result in milder dynamics represented mainly by displacements and accelerations of the vehicle body. The results obtained and presented in the current article constitute a certain novelty in the field of optimization and the properties assessment of railway polynomial transition curves. Obtained curves of 9th and 11th degrees had significantly better properties than what is known from the literature, as well as standard curves of those degrees and the 3rd degree parabola—the curve traditionally used. It is not disturbed by the fact that different circular arc radii generally resulted in different shapes of the curves. It has also been shown that these curves maintain their characteristics also for the 4-axle car model, which corresponds to a real passenger car. This confirms that the author’s efforts made sense. It also shows the correctness of the approach used in this work. The results obtained during the research explain, why polynomial transition curves have not been used. The authors mean their small popularity. The curves of lower degrees are analyzed and used, and they do not satisfy the advanced boundary conditions (e.g., inclination function of bell-shaped). In the case of higher degrees, curves, more often analyzed than used, are expected to satisfy the advanced boundary conditions. Such an orientation to the lower degrees of the polynomial curves (without fulfilling advanced boundary conditions) can probably be treated as a mistake. The obtained new shapes indicate the need for further study of the transition curves and set new directions for research. Note that although the equations of the obtained optimum curves have been defined, the obtained results can be viewed more broadly. Well, the obtained shapes indicate that it is worth taking a closer look at the transition curves connecting the features of the 3rd degree parabola and the well-known polynomial curves of higher degree, which we called the standard polynomial curves in this work. Note that such curves do not necessarily have to be polynomials. Perhaps it would be advantageous to use curve types that are even more flexible from the perspective of shape taking than polynomials. In this context, cubic splines (3rd degree spline curves) come to mind here. The obtained results and the demonstration of the effectiveness of the approach used in this work allow the author to believe that the work is a good starting point for adopting new rules in the future and for constructing new methods of shaping transition curves. Methods, in which the core would be a combination of vehicle motion simulation methods and mathematical optimization methods. Efforts to change these principles and methods are likely to continue as long as no progress is made in this area. In this context, the author recognizes the contribution of this work to the discussion on this issue.

Author Contributions: Both authors (K.Z. and P.W.) equally contributed to the simulation, optimization, data analysis, and writing. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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