applied sciences

Article Analysis of the Loading on an Articulated of Circular Pipes Loaded with Tank

Oleksij Fomin 1 , Juraj Gerlici 2 , Alyona Lovska 3 and Kateryna Kravchenko 2,*

1 Department of Cars and Carriage Facilities, State University of Infrastructure and Technologies, Kyrylivska 9, 04071 Kyiv, Ukraine; [email protected] 2 Faculty of Mechanical Engineering, Department of Transport and Handling , Univerzitna 1, University of Zilina, 010 26 Zilina, Slovakia; [email protected] 3 Department of Wagon Engineering and Product Quality, Ukrainian State University of Railway Transport, Feuerbach Square 7, 61050 Kharkiv, Ukraine; [email protected] * Correspondence: [email protected]; Tel.: +421-944-100-382

Abstract: This research is concerned with the use of double walls filled with foam aluminum for an to decrease loading during operational modes. The research presents the strength calculation for the frame of a wagon with a consideration of the engineering solutions proposed. It was found that the maximum equivalent stresses appeared in the bottom section of the center sill behind the back support; they amounted to about 290 MPa and did not exceed the allowable values. The maximum displacements were in the middle parts of the main longitudinal beams of a section, and they amounted to 8.8 mm. The research also presents the strength calculation for a weld joint in the maximum loaded zones of the frame of a wagon and reports the results of the modal analysis of the frame of the improved wagon. It was found that the oscillation frequencies   did not exceed the allowable values. The results of the research may be useful for those who are concerned about designing innovative units and improving the operational efficiency of Citation: Fomin, O.; Gerlici, J.; Lovska, A.; Kravchenko, K. Analysis railway transport. of the Loading on an Articulated Flat Wagon of Circular Pipes Loaded with Keywords: flat wagon with double sidewalls; bearing structure; dynamic loading; strength; trans- Tank Containers. Appl. Sci. 2021, 11, port mechanics 5510. https://doi.org/10.3390/ app11125510

Academic Editor: Huajiang Ouyang 1. Introduction Higher efficiency of rail transportation within the international transportation network Received: 10 May 2021 can be ensured by the introduction of innovative wagon structures. As is known, one of the Accepted: 10 June 2021 most popular wagon types for international transportation is the flat wagon, particularly Published: 14 June 2021 the articulated flat wagon. A special feature of such wagons is their two-section carrying structure that rests on three . Publisher’s Note: MDPI stays neutral In operation, such wagons bear considerable longitudinal loads, which harm both the with regard to jurisdictional claims in strength of the carrying structure and traffic safety. published maps and institutional affil- Therefore, a decrease in the dynamic loads on articulated flat wagons is of primary iations. importance; it can be achieved by implementing innovative engineering solutions. These will provide the fatigue strength of the carrying structure, decrease repair and maintenance costs, ensure higher security of the transported freight (e.g., by tank containers), security of detachable bodies, traffic safety, etc. Copyright: © 2021 by the authors. The issue of how freight displacements impact the motion characteristics of a flat Licensee MDPI, Basel, Switzerland. wagon is studied in [1]. The authors obtained dependencies of the basic dynamic character- This article is an open access article istics of traffic speed. distributed under the terms and Study [2] presents research into the loading and strength of a long-base car. The re- conditions of the Creative Commons Attribution (CC BY) license (https:// search confirmed the efficiency of engineering solutions implemented in car design. creativecommons.org/licenses/by/ It should be noted that the design of the carrying structure of the car did not include 4.0/). any measures aimed at decreasing dynamic loads due to cyclic stresses.

Appl. Sci. 2021, 11, 5510. https://doi.org/10.3390/app11125510 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 5510 2 of 12

The carrying structure of a flat wagon is analyzed in [3]. The carrying structure of a car intended for intermodal transportation is analyzed in [4]. A special feature of the carrying structure of such a car is a loading/unloading rotating platform. However, the authors did not consider the longitudinal dynamic loads on the carrying structure of the car under operational modes. The dynamic loads on the carrying structure of a car are determined in [5]. The authors studied the application of different types of bogies for railcars and analyzed how the technical characteristics impact the dynamic characteristics of the car. Theoretical and experimental research into the loads on the carrying structure of a freight wagon is demonstrated in [6]. The technique proposed by the authors proved to be efficient, thus confirming that the authors followed the right line in their research. Nevertheless, they did not consider any measures to decrease the dynamic loads on the carrying structure of railcars. Studies [7,8] propose and describe the realization of a technique aimed at decreasing loads on a railcar under operational modes. However, the authors did not study how to decrease the dynamic loads on the carrying structure of a long-base railcar. The analysis of the impact strength of is considered in publication [9]. The au- thor formed a mathematical model that allows one to calculate the acceleration of a on impact. At the same time, no solutions have been proposed that are aimed at ensuring the strength of the vehicle in operation. The unified concept of the impact strength of a railcar body is presented in publica- tion [10]. In the simulation, the assumption was made that the body is absolutely rigid. At the same time, the work did not address the issues of improving the railcar body to ensure its safety. An experimental study on the dynamics of rolling stock is carried out in publica- tion [11]. The authors proposed solutions, the implementation of which will improve the safety of rolling stock. Article [12] defines the main indicators of the dynamics and strength of a platform car in operation. The features of the experimental tests to which the car was subjected are highlighted. It is important to say that the authors of these works have not proposed solutions aimed at reducing the load on the rolling stock in operation. Analysis of the literature [1–12] demonstrates that the issue of determination of the loads on the carrying structure of a flat wagon of circular pipes loaded with tank containers has not been studied yet. Therefore, it requires appropriate research in the field. The purpose of this study was to investigate the loading of a flat wagon of circular pipes loaded with tank containers using mathematical modeling. To achieve the objective, the following tasks were set: - To build a design diagram of a flat wagon for determination of the longitudinal loads on the carrying structure; - To simulate the dynamic loads on the carrying structure of a flat wagon with a standard automatic coupler; - To simulate the dynamic loads on the carrying structure of a flat wagon with a draft gear construct; - To determine the basic strength characteristics of the carrying structure of a flat wagon.

2. Presentation of the Basic Material of the Article The authors proposed the application of circular pipes in the frame of a flat wagon to reduce material consumption [13]. To determine the optimal parameters of the pipes, a strength calculation of the supporting structure of the prototype railcar flat-car model 13-401 was carried out. This type of railcar was chosen as the base railcar because it had several upgrades to improve the efficiency of its operation. For example, it had setting folding or stationary fitting stops on the main longitudinal beams of the frame to be able to transport containers, including in international traffic. Appl.Appl. Sci.Sci. 20202121,, 1111,, 55105510 33 ofof 1212

Appl. Sci. 2021, 11, 5510 3 of 12 foldingfolding oror stationarystationary fittingfitting stopsstops onon thethe mainmain longitudinallongitudinal beamsbeams ofof thethe frameframe toto bebe ableable toto transporttransport containers,containers, includingincluding inin internationalinternational traffic.traffic. BasedBased onon thethe performedperformed calculationscalculations andand obtainedobtained valuesvalues ofof thethe maximummaximum equivalentequivalent stresses,stresses,Based thethe on strengthstrength the performed reservesreserves calculations ofof thethe componentscomponents and obtained ofof thethe valuessupportingsupporting of the structurestructure maximum ofof equivalentthethe flatflat carcar werestresses,were determined.determined. the strength TakingTaking reserves thisthis intointo of the account,account, components accordingaccording of the toto the supportingthe rangerange ofof structure pipes,pipes, thethe of mostmost the flatop-op- car were determined. Taking this into account, according to the range of pipes, the most timaltimal onesones fromfrom thethe viewpointviewpoint ofof thethe minimumminimum materialmaterial consumptionconsumption ofof pipespipes werewere de-de- optimal ones from the viewpoint of the minimum material consumption of pipes were termined.termined. TheThe proposedproposed solutionssolutions werewere confirmedconfirmed byby complexcomplex calculationscalculations ofof dynamics,dynamics, determined. The proposed solutions were confirmed by complex calculations of dynamics, strength,strength, andand fatiguefatigue strength,strength, asas wellwell asas aa modalmodal analysisanalysis ofof thethe loadload--bearingbearing ststructureructure ofof strength, and fatigue strength, as well as a modal analysis of the load-bearing structure of thethe platformplatform carcar underunder operationaloperational loadsloads [14].[14]. the platform car under operational loads [14]. FigFigureure 11 demonstratesdemonstrates anan articulatedarticulated flatflat wagonwagon builtbuilt accordingaccording toto thethe designdesign pro-pro- Figure1 demonstrates an articulated flat wagon built according to the design pro- posedposed [14].[14]. posed [14].

Figure 1. The articulated flat wagon of circular pipes. FigureFigure 1.1. The articulated flatflat wagonwagon ofof circularcircular pipes.pipes.

TheThe dynamic dynamic loadsloads onon the the frame frame of of the the flat flatflat wago wagonwagonn loaded loaded withwith tanktank containers containers underunder operationaloperational modes modes (tensile (tensile jerk) jerk) were were determineddetermined usingusing mathematicalmathematical modelingmodeling basedbased onon thethe mathematical mathematical model model designed designed by by Prof.Prof. G.I.G.I. BogomazovBogomazov [[15].[15].15]. The The designdesign diagramdiagram ofof thethe flat flatflat wagon wagon is is givengiven inin FigureFigure2 2.2..

Figure 2. The design diagram of the flat wagon. FigureFigure 2.2. TheThe designdesign diagramdiagram ofof thethe flatflat wagon.wagon.

The researchresearch included included a loada load of 2.5of 2.5 MN МN on theon frontthe front stops stops of an of automatic an automatic coupler coupler [16,17]. A 1CCThe tank research included was taken a load as of the 2.5 prototype. МN on the front stops of an automatic coupler [16,17].[16,17]. AA 1CC1CC tanktank containercontainer waswas takentaken asas thethe prototype.prototype. The tank containers were taken as masses attached to the flatflat wagon frames. Al Alll tank containersThe tank on thecontainers flat wagon were had taken an equalas masses amount attached of liquid to the freight. flat wagon frames. All tank containers on the flat wagon had an equal amount of liquid freight. containersThe motion on the equations flat wagon are had an equal amount of liquid freight. TheThe motionmotion equationsequations areare 2 0 .. .. 2 M · x + M · h · ϕ = S −2 S − R ; (1) FRI MFR I  x FR MI  h FRI  SL ∑ S  i R ; II MFR IIII x FR  M FR  h  FR  S L  S i  R II ; (1) FRIIII FR FR FR Li1i=1 i II (1) i1 ...... ϕ ϕ IFRI · FRI + MFRI · h · xFRI − g · FRI · MFRI · h =  . .   . .  (2) = l · FFR sign∆1 − sign∆2 + l k1 · ∆1 − k2 · ∆2 ; Appl. Sci. 2021, 11, 5510 4 of 12

..  . .  MFR · zFR = k1 · ∆1 + k2 · ∆2 − FFR sign∆1 − sign∆2 ; (3) ! ! k k k .. I .. I .. I I mi + ∑ mij · xi + mi · zci + ∑ mij · cij · ϕi − ∑ mij · lij · ξij = Si ; (4) j=1 j=1 j=1 ! ! k k 2 .. I .. I Iθi + ∑ mij · cij · ϕi + mi · zci + ∑ mij · cij · xi + j=1 j=1 ! (5) k .. I k I + ∑ mij · cij · lij · ξij − g · mi · zci + ∑ mij · cij · ϕi = 0; j=1 j=1

k ! ..I m + m · z = i ∑ ij FRI 0; (6) j=1

.. I .. I .. I .. I Iij · ξij − mij · lij · xij − mij · cij · lij · ϕi + g · mij · lij · ξij = 0; (7)

2 .. .. 0 · + · · ϕ = − + M FRII xFRII MFRII h FRII ∑ Si RI; (8) i=1 .. .. ϕ ϕ IFRII · FRII + MFRII · h · xFRII − g · FRII · MFRII · h =  . .   . .  (9) = l · FFR sign∆1 − sign∆2 + l k1 · ∆1 − k2 · ∆2 ;

..  . .  MFLI · zFLI = k1 · ∆1 + k2 · ∆2 − FFR sign∆1 − sign∆2 ; (10) ! ! k k k .. II .. II .. II II mi + ∑ mij · xi + mi · zci + ∑ mij · cij · ϕi − ∑ mij · lij · ξij = Si ; (11) j=1 j=1 j=1

k ! II − g · mi · zci + ∑ mij · cij · ϕi = 0; (12) j=1

k ! .. m + m · z = i ∑ ij FLII 0; (13) j=1

.. II .. II .. II .. II Iij · ξij − mij · lij · xij − mij · cij · lij · ϕi + g · mij · lij · ξij = 0; (14) M0 = M + · m + n·I i = z − l · ϕ i = z + l · ϕ FLi FLi 2 B r2 ; ∆1 FLi FLi ; ∆2 FLi FLi ;  0 = · − Si f f r sign xFLi xij

MFLi is the mass of the i-th carrying structure of a flat wagon. IFRi is the inertia moment of the i-th flat wagon relative to the longitudinal axle. SL is the longitudinal impact force to a coupler. f f r is the amplitude value of the dry friction force. mB is the mass. I is the inertia moment of a wheelset. r is the radius of a wheel with moderate wear. n is the number of bogie axles. l is the half-base of the i-th section of a flat wagon. FFR is the absolute value of the dry friction force in a spring group. k1, k2 are the rigidities of springs of a spring suspension of bogies of a flat wagon. k is the number of oscillation modes of the liquid freight. mi is the mass of a body equivalent to the i-th with part of a liquid freight, which does not move relative to the tank. mij is the mass of the j-th pendulum in the i-th tank container. zci is the height of the center of weight of a tank container. cij is the distance between the plane zi = 0 and the fixation point of the j-th pendulum in i-th tank container. lij is the length of the j-th pendulum. Iθ is the reduced inertia moment of the i-th tank container and the liquid freight, which does not move relative to the tank. Iij is the inertia moment of a pendulum. x, y, z are the coordinates corresponding to the longitudinal, angular (about the longitudinal axle), and vertical displacements of a flat wagon, respectively. xi, ϕi are the coordinates corresponding to the longitudinal and Appl. Sci. 2021, 11, 5510 5 of 12

FFR is the absolute value of the dry friction force in a spring group. kk12, are the rigidi- ties of springs of a spring suspension of bogies of a flat wagon. k is the number of oscil-

lation modes of the liquid freight. mi is the mass of a body equivalent to the і-th tank

container with part of a liquid freight, which does not move relative to the tank. mij is

the mass of the j-th pendulum in the і-th tank container. zci is the height of the center of

weight of a tank container. cij is the distance between the plane zi  0 and the fixation

point of the j-th pendulum in i-th tank container. lij is the length of the j-th pendulum.

I  is the reduced inertia moment of the і-th tank container and the liquid freight, which

does not move relative to the tank. I ij is the inertia moment of a pendulum. xyz,, are the coordinates corresponding to the longitudinal, angular (about the longitudinal axle),

and vertical displacements of a flat wagon, respectively. xii,  are the coordinates cor- responding to the longitudinal and angular displacements about the longitudinal axle of  Appl. Sci. 2021, 11, 5510 a tank container. ij is the vertical deviation of the j-th pendulum.5 of 12 The motion of the liquid freight was described using mathematical pendulums [15]. The hydrodynamic characteristics of the liquid freight were determined by the tech- ξ niqueangular presented displacements in [18]. about A the boundary longitudinal value axle of problem a tank container. was made,ij is the which vertical is equivalent to the deviation of the j-th pendulum. problemThe motion of finding of the liquidthe minimum freight was described of a functional. using mathematical The Ritz pendulums method [15 was]. used as a calcula- tion method.The hydrodynamic The authors characteristics used petrol of the liquid as the freight liquid were freight. determined Based by the on tech- the calculations con- ductednique presented for the in case [18]. Aof boundary the maximum value problem possible was made, loading which of is equivalentthe tank to shell the according to [19], problem of finding the minimum of a functional. The Ritz method was used as a calculation2 themethod. authors The authors obtained used petrolthe following as the liquid values: freight. Based mtij on the6.8 calculations and Iij conducted250 t m . The calculation for the case of the maximum possible loading of the tank shell according to [19], the authors included that the tank was filled to 95%. 2 obtained the following values: mij ≈ 6.8t and Iij = 250t · m . The calculation included that the tankThe was equations filled to 95%. were solved with the Runge–Kutta method in MathCad software [20– 23]. TheThe equationsstarting were conditions solved with were the Runge–Kutta made equal method to in zero MathCad [24–software28]. The [20 –results23]. of the research showedThe starting that conditions acceleration were made to equal the tofirst zero section [24–28]. Theof the results flat of thewagon research from showed force application was that acceleration to the first section of the flat wagon from force application was 27.7 m/s2, 2 27.7and itm/s² was, 24.4and m/s it 2wasto the 24.4 second m/s section to the (Figure second3). section (Figure 3).

(a) (b) FigureFigure 3.3.The The accelerations accelerations to the carryingto the carrying structure of structure the flat wagon: of the (a) First flat section wagon from: (a force) First section from force applicationapplication. (.b ()b second) second section section from force from application. force application. The study also addressed the use of a draft gear construct as an alternative to a stan- dardThe automatic study coupler also addressed to decrease the the dynamic use of loadsa draft on gear the frame construct of the flat as wagonan alternative to a stand- ard(Figure automatic4 )[13]. Thecoupler concept to design decrease includes the adynami typical CA-3c loads automatic on the coupler frame body of the flat wagon (Fig- (1), which interacts with the retainer bar (5) through the connecting wedge (2). The retainer urebar 4) has [13]. a spherical The concept surface for design interaction includes with the a shanktypical of theCA automatic-3 automatic coupler coupler and body (1), which interactsthe possibility with of the carrying retainer out movements bar (5) through of the body the in theconnecting horizontal planewedge when (2 the). The retainer bar has acar spherical enters the surface curved sections for interaction of the . with The design the shank also includes of the pistons automatic (3), in which coupler and the possi- throttle valves (4) are located. This concept can be applied to wagons with a center beam bilityconsisting of carrying of two pipes. out It ismovements also possible toof use the this body concept in onthe wagons, horizontal the center plane beam when the car enters theof whichcurved is represented sections by of a singlethe track. tube. In The this case,design the concept also inclu uses onedes piston. pistons (3), in which throttle Appl. Sci. 2021, 11, 5510 6 of 12

valves (4) are located. This concept can be applied to wagons with a center beam consist- ing of two pipes. It is also possible to use this concept on wagons, the center beam of which Appl. Sci. 2021, 11, 5510 is represented by a single tube. In this case, the concept uses one piston. 6 of 12

valves (4) are located. This concept can be applied to wagons with a center beam consist-

Appl. Sci. 2021, 11, 5510 ing of two pipes. It is also possible to use this concept on wagons,6 of 12 the center beam of which is represented by a single tube. In this case, the concept uses one piston.

Figure 4. The draft gear construct of an automatic coupler: 1, body; 2, wedge; 3, piston; 4, throttle valves; 5, retainer bar.

The end parts of the center sill of the flat wagon were filled with damping material with the viscous resistance coefficient β. The draft gear construct was equipped with a Figurepiston 4. The with draft two gear constructthrottle of anvalves automatic (input coupler: and 1, body; output) 2, wedge; to 3, piston;transform 4, throttle the kinetic impact energy valves;Figure 5, retainer4. The bar. draft gear construct of an automatic coupler: 1, body; 2, wedge; 3, piston; 4, throttle Sa to the dissipation energy. The longitudinal loading was transferred from the automatic valves; 5, retainer bar. couplerThe end body parts ofto the the center double sill of- thedisk flat wagonpistons were of filled the withdraft damping gear materialconstruct through the retainer with the viscous resistance coefficient β. The draft gear construct was equipped with a pistonbar usingThe with twoend a throttlefork. parts valvesWhen of the (input the center andpistons output) sill movedof to transformthe flat toward thewagon kinetic the impactwere center energyfilled plate with of dampingthe wagon material (cavity SwithA)a to, the the dissipation input viscous valves energy. resistance wer The longitudinale open coefficient while loading the wasβ .output The transferred draft valves from gear the were automaticconstruct closed. was During equipped the backward with a couplerdirection body of to thethe double-disk pistons pistons(jerk, ofcompression) the draft gear construct, the output through thevalves retainer were open and the input pistonbar using with a fork. two When throttle the pistons valves moved (input toward theand center output) plate of to the transform wagon (cavity the kinetic impact energy valves were closed. The viscous solution flew to cavity B from the coupler body. The op- A),Sa to the the input dissipation valves were open energy. while the The output longitudinal valves were closed. loading During was the backward transferred from the automatic directioneration of diagram the pistons of (jerk, the compression), draft gear the construct output valves is were given open in and Figure the input 5. The viscous resistance valvescoupler were body closed. to The the viscous double solution-disk flew pistons to cavity of B fromthe thedraft coupler gear body. construct The through the retainer baroperationcoefficient using diagram a formed fork. of the When draftby the gear the constructdraft pistons gear is given moved construct in Figure toward5 .was The viscous theover center resistance70 kNs plate/m. Theof the optimal wagon value (cavity of A)coefficientthe, theviscous input formed resistance valves by the draft wer gearcoefficiente constructopen while was was over thedetermined 70 output kNs/m. The valves using optimal were mathematical value closed. of the During modeling the backward of the dy- viscousnamic resistance loading coefficient of the was car determined’s load-bearing using mathematical structure modeling under of theoperational dynamic loads. directionloading of the of car’s the load-bearing pistons structure(jerk, compression) under operational, loads.the output valves were open and the input valves were closed. The viscous solution flew to cavity B from the coupler body. The op- eration diagram of the draft gear construct is given in Figure 5. The viscous resistance coefficient formed by the draft gear construct was over 70 kNs/m. The optimal value of the viscous resistance coefficient was determined using mathematical modeling of the dy- namic loading of the car’s load-bearing structure under operational loads.

FigureFigure 5. The 5. The operational operational diagram of diagram the draft gear of construct.the draft gear construct. It is important to say that the use of the proposed concept contributes to the anti- corrosionIt propertiesis important of the railcarto say cantilever that the parts. use This of concept the ofproposed a harness device concept also contributes to the anti- allows one to reduce the cost of manufacturing the railcar by eliminating the need to use harnesscorrosion devices properties with absorbing of devices. the railcar cantilever parts. This concept of a harness device also allowsThe motionone to equations reduce of the design cost modelof manufacturing have the form the railcar by eliminating the need to use harness devices with absorbing devices. Figure 5. The operational diagram of the draft gear construct. The motion equations of the design model have the form

It is important to say that the use of the proposed2 concept contributes to the anti- M  x  M  h S S  R  x ; (15) corrosion properties of theFR railcarIIIII FR cantilever FR FRparts. L This concept i II of FLa harness device also i1 allows one to reduce the cost of manufacturing the railcar by eliminating the need to use harness devices with absorbing devices. The motion equations of the design model have the form

2 M  x  M  h S S  R  x ; FRIIIII FR FR FR L i II FL (15) i1 Appl. Sci. 2021, 11, 5510 7 of 12

2 .. .. . 0 · + · · ϕ = − − − · M FRI xFRI MFRI h FRI SL ∑ Si RII β xFLI ; (15) i=1 ...... ϕ ϕ IFRI · FRI + MFRI · h · xFRI − g · FRI · MFRI · h =  . .   . .  (16) = l · FFR sign∆1 − sign∆2 + l k1 · ∆1 − k2 · ∆2 ;

..  . .  MFR · zFR = k1 · ∆1 + k2 · ∆2 − FFR sign∆1 − sign∆2 ; (17) ! ! k k k .. I .. I .. I I mi + ∑ mij · xi + mi · zci + ∑ mij · cij · ϕi − ∑ mij · lij · ξij = Si ; (18) j=1 j=1 j=1 ! ! k k 2 .. I .. I Iθi + ∑ mij · cij · ϕi + mi · zci + ∑ mij · cij · xi + j=1 j=1 ! (19) k .. I k I + ∑ mij · cij · lij · ξij − g · mi · zci + ∑ mij · cij · ϕi = 0; j=1 j=1

k ! ..I m + m · z = i ∑ ij FRI 0; (20) j=1

.. I .. I .. I .. I Iij · ξij − mij · lij · xij − mij · cij · lij · ϕi + g · mij · lij · ξij = 0; (21)

2 .. .. 0 · + · · ϕ = − + M FRII xFRII MFRII h FRII ∑ Si RI; (22) i=1 .. .. ϕ ϕ IFRII · FRII + MFRII · h · xFRII − g · FRII · MFRII · h =  . .   . .  (23) = l · FFR sign∆1 − sign∆2 + l k1 · ∆1 − k2 · ∆2 ;

..  . .  MFLI · zFLI = k1 · ∆1 + k2 · ∆2 − FFR sign∆1 − sign∆2 ; (24) ! ! k k k .. II .. II .. II II mi + ∑ mij · xi + mi · zci + ∑ mij · cij · ϕi − ∑ mij · lij · ξij = Si ; (25) j=1 j=1 j=1

k ! II − g · mi · zci + ∑ mij · cij · ϕi = 0; (26) j=1

k ! .. m + m · z = i ∑ ij FLII 0; (27) j=1

.. II .. II .. II .. II Iij · ξij − mij · lij · xij − mij · cij · lij · ϕi + g · mij · lij · ξij = 0; (28) M0 = M + · m + n·I i = z − l · ϕ i = z + l · ϕ FLi FLi 2 B r2 ; ∆1 FLi FLi ; ∆2 FLi FLi ;  0 = · − Si f f r sign xFLi xij where viscous resistance coefficient β is formed by the draft gear construct. The results of the calculation are given in Figure6. Appl. Sci. 2021, 11, 5510 8 of 12

Appl. Sci. 2021, 11, 5510 8 of 12 Appl. Sci. 2021, 11, 5510 8 of 12

(a) (b)

Figure 6. The accelerations(a) to the carrying structure of the flat wagon(b) : (a) First section from force application. (b) second section from force application. FigureFigure 6. 6. TheThe accelerations accelerations to to the the carrying carrying structure structure of of the the flat flat wagon wagon:: (a ()a )F Firstirst section section from from force force application. (b) second section from force application. application.The results (b) second of the section research from demonstrate force application. that acceleration to the first section of the flat wagon from force application was about 25.0 m/s² and about 22.0 m/s2 to the second sec- TheThe results results of of the the research research demonstrate demonstrate that that acceleration acceleration to to the the first first sec sectiontion of of the the flat flat tion. wagonwagon from from force force application application was was about about 25.0 25.0 m/s² m/s and2 and about about 22.0 22.0 m/s2 m/s to the2 to second the second sec- The proposed solution will decrease the loading of the frame of a wagon by 10% in tion.section. comparison to that for a standard automatic coupler. TheThe proposed proposed solution solution will will decrease decrease the the loading loading of of the the frame frame of of a awagon wagon by by 10% 10% in in The basic strength characteristics of the carrying structure of an articulated flat comparisoncomparison to to that that for for a astandard standard automatic automatic coupler. coupler. wagon with a draft gear construct were determined through the strength calculation using TheThe basic basic strengthstrength characteristics characteristics of of the the carrying carrying structure structure of an of articulated an articulated flat wagon flat the FEM in SolidWorks Simulation (CosmosWorks) software. wagonwith a with draft a geardraft constructgear construct were were determined determined through through the strengththe strength calculation calculation using using the Isoparametric tetrahedrons were used as finite elements. The number of elements in theFEM FEM in SolidWorksin SolidWorks Simulation Simulation (CosmosWorks) (CosmosWorks) software. software. the mesh was 1,311,530, and the number of nodes was 418,692. The maximum element IsoparametricIsoparametric tetrahedrons tetrahedrons were were used used a ass finite finite elements. elements. The The number number of of elements elements in in size was 20 mm, the minimum element size was 4 mm, and the maximum element side thethe mesh mesh was was 1 1,311,530,,311,530, and thethe numbernumber of of nodes nodes was was 418,692. 418,692. The The maximum maximum element element size ratio was 2117.6. The percentage of elements with a side ratio of less than three or more sizewas was 20 mm,20 mm, the the minimum minimum element element size size was was 4 mm, 4 mm, and and the maximumthe maximum element element side side ratio than ten was 74.9 and 0.719, respectively. ratiowas was 2117.6. 2117.6 The. percentageThe percentage of elements of elements with awith side a ratio side of ratio less of than less three than or three more or than more ten The design diagram of the carrying structure of the flat wagon section included the thanwas ten 74.9 was and 74.9 0.719, and respectively. 0.719, respectively. action of the vertical static loading P , conditioned by the gross weight of the tank con- TheThe design design diagram diagram of ofthe the carrying carrying structure structure of the of theflat flatwagon wagon section section included included the

tainersthe action and of the the longitudinal vertical static loading loading Pl Ptoν, the conditioned front stops by of the the grosscoupler weight, which of amounts the tank action of the vertical static loading P , conditioned by the gross weight of the tank con- tocontainers 2.5 МN ( andFigure the 7). longitudinal That is, the loading calculationPl to thewas front carried stops out of for the the coupler, case of which such amountsan oper- tainers and the longitudinal loading Pl to the front stops of the coupler, which amounts atingto 2.5 mode MN (Figureas “jerk7.”). The That vertical is, the loading calculation was applied was carried to the outcarrying for the structure case of as such remote an to 2.5 МN (Figure 7). That is, the calculation was carried out for the case of such an oper- loadingoperating, which mode included as “jerk.” the The height vertical of the loading weight was center applied of the to tank the carryingcontainer. structure The stress as ating mode as “jerk.” The vertical loading was applied to the carrying structure as remote stateremote of loading,the supporting which includedstructure theof the height flat car, of the as weightwell as centerthe displacements of the tank container. in it, are pre- The loading, which included the height of the weight center of the tank container. The stress sentedstress state in Figures of the supporting8 and 9, respectively structure.of the flat car, as well as the displacements in it, are statepresented of the insupporting Figures8 andstructure9, respectively. of the flat car, as well as the displacements in it, are pre- sented in Figures 8 and 9, respectively.

Figure 7. TheThe design design diagram diagram of the carrying structure of the flat flat wagon section. Figure 7. The design diagram of the carrying structure of the flat wagon section. Appl. Sci. 2021, 11, 5510 9 of 12

Appl.Appl.Appl. Sci.Sci. 202120202121,,, 11 1111,,, 551055105510 999 ofof 121212

Figure 8. The stress state of the carrying structure of the flat wagon section. FigureFigure 8.8. TheThe stress stress state state of of thethe carryingcarrying structurestructure ofof thethe flatflatflat wagonwagon section.section.section.

Figure 9. The displacements in the units of the carrying structure of the flat wagon section. FigureFigure 9.9. TheThe displacements displacements in in the the unitsunits ofof thethe carryingcarrying structurestructure ofof thethe flatflatflat wagonwagonwagon section.section.section. The maximum equivalent stresses were in the interaction area between the center sill The maximum equivalent stresses were in the interaction area between the center sill and bodyTheThe maximummaximum bolster, and equivalentequivalent they amounted stressesstresses weretowere about inin thethe 290 interactioninteraction МPа. Thus, areaarea they betweenbetween did not thethe exceed centercenter the sillsill and body bolster, and they amounted to about 290 MPa. Thus, they did not exceed the allowableandand bodybody valuesbolsterbolster ,,[16,17]. andand theythey The amountedamounted maximum toto displacements aboutabout 290290 МPа.МPа. were Thus,Thus, in theythethey middle diddid notnot parts exceedexceed of thethe allowable values [16,17]. The maximum displacements were in the middle parts of the mainallowableallowable longitudinal valuesvalues [16,17]. [16,17].section ThebeamsThe maximummaximum, and they displacementsdisplacements amounted to 8.8werewere mm. inin The thethe usemiddlemiddle of the partsparts draft ofof gear thethe main longitudinal section beams, and they amounted to 8.8 mm. The use of the draft gear constructmainmain longitudinallongitudinal will decrease sectionsection the beams beamsmaximum,, andand equivalent theythey amountedamounted stresses toto 8.8 8.8by mm.mm.12% TheinThe the useuse carrying ofof thethe draft draftstructure geargear construct will decrease the maximum equivalent stresses by 12% in the carrying structure ofconstructconstruct the articulated willwill decreasedecrease flat wagon thethe maximummaximum, and by 12.6%equivalentequivalent in the stressesstresses displacements byby 12%12% in inin thethe comparison carryingcarrying structurestructure to those of the articulated flat wagon, and by 12.6% in the displacements in comparison to those valuesofof thethe articulatedarticulatedin the case offlatflat a wagontypicalwagon, , automatic andand byby 12.6%12.6% coupler inin thethe (Figure displacementsdisplacements 10). inin comparisoncomparison toto thosethose values in the case of a typical automatic coupler (Figure 10). valuesvalues inin thethe casecase ofof aa typicaltypical automaticautomatic couplercoupler ((FigureFigure 10).10).

Figure 10. The comparative study of the simulated strength. FigureFigure 10.10. TheThe comparativecomparative studystudy ofof thethe simulatedsimulated strength.strength. Appl. Sci. 2021, 11, 5510 10 of 12

3. Discussion of the Results Obtained The authors determined the dynamic loads on an articulated flat wagon of circular pipes loaded with tank containers. We also determined the accelerations to the carrying structure of the flat wagon with a typical automatic coupler (Figure3). The dynamic loads on the carrying structure of a flat wagon can be decreased by the application of a draft gear construct (Figure4). This will decrease the loads on the frame of a flat wagon by 10% and the maximum equivalent loads by 12% (Figure8) in comparison to the loads when applying a standard automatic coupler (Figure6). It should be noted that the mathematical modeling did not include angular displace- ments of the coupling unit. Moreover, the strength modeling of one section of the frame of the flat wagon took into account that a section rigidly interacted with the bogies, i.e., the gaps between the center plate and plate bowl were disregarded. These restrictions will be included in the authors’ further research in the field. In addition, the loading on the carrying structure of an articulated flat wagon requires experimental tests.

4. Conclusions 1. This study addressed the building of a design diagram of a flat wagon for the deter- mination of longitudinal loads on a carrying structure loaded with four 1CC tank containers. The research was performed in the plane coordinates. The loading of the supporting structure during the “jerk” was taken into account. The calculation included a force of 2.5 MN to the front stops of an automatic coupler. 2. The authors simulated the dynamic loads on the carrying structure of a flat wagon with a standard automatic coupler. The research included the fact that 95% of the tank was filled with liquid freight. The differential equations were solved in MathCad software. The results of the research showed that accelerations to the first section of the flat wagon from force application were 27.7 m/s2, and they were 24.4 m/s2 to the second section. 3. The study investigated the modeling of the dynamic loads on the carrying structure of a flat wagon with a draft gear construct. It was found that acceleration to the first section of the flat wagon from force application was 25.0 m/s2 and about 22.0 m/s2 to the second section. The solution proposed can decrease the dynamic loads on the carrying structure of a flat wagon by 10% in comparison to those generated by the use of a standard automatic coupler. 4. The authors determined the basic strength characteristics of the carrying structure of a flat wagon with the finite element method. It was found that the maximum equivalent stresses were in the interaction area between the center sill and body bolster, and they amounted to about 290 MPa. The maximum displacements were in the middle parts of the main longitudinal beams of a section, and they amounted to 8.8 mm. It was found that the use of the draft gear construct decreased the maximum equiv- alent stresses by 12% in the carrying structure of a flat wagon, and by 12.6% in the displacements in comparison to those generated by the use of a standard automatic coupler.

Author Contributions: Conceptualization, O.F. and A.L.; methodology, O.F. and A.L.; software, A.L.; validation, A.L. and J.G.; investigation, O.F., A.L. and K.K.; resources, O.F., A.L. and K.K.; writing—original draft preparation, O.F., A.L. and K.K.; writing—review and editing J.G.; visualiza- tion, O.F., A.L. and K.K.; supervision, J.G. All authors have read and agreed to the published version of the manuscript. Funding: This publication was issued thanks to support from the Cultural and Educational Grant Agency of the Ministry of Education of the Slovak Republic in the project, “Implementation of modern methods of computer and experimental analysis of properties of vehicle components in the education of future vehicle designers” (Project No. KEGA 036ŽU-4/2021). The authors also gratefully acknowledge funding from specific research into “Innovative principles for creating Appl. Sci. 2021, 11, 5510 11 of 12

resource-saving structures of railroad cars based on the refined dynamic loads and functionally adaptive flash-concepts,” which is funded from the state budget of Ukraine in 2020. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

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