materials

Article Dynamics and Strength of Circular Tube Open Wagons with Aluminum Foam Filled Center Sills

Oleksij Fomin 1 , Mykola Gorbunov 2, Alyona Lovska 3, Juraj Gerlici 4 and Kateryna Kravchenko 4,*

1 Department of Cars and Carriage Facilities, State University of Infrastructure and Technologies, Kyrylivska Str., 9, 04071 Kyiv, Ukraine; [email protected] 2 Department of Railway, Automobile Transport and Handling Machines, Institute of Transport and Logistics, Volodymyr Dahl East Ukrainian National University, Central Avenue 59a, 93400 Sewerodonetsk, Ukraine; [email protected] 3 Department of Wagon Engineering and Product Quality, Ukrainian State University of Railway Transport, Feuerbach Square 7, 61050 Kharkov, Ukraine; [email protected] 4 Department of Transport and Handling Machines, Faculty of Mechanical Engineering, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia; [email protected] * Correspondence: [email protected]; Tel.: +421-944-100-382

Abstract: The study deals with an application of aluminum foam as an energy-absorbing material for the carrying structure of a rail car. The material is particularly recommended for circular tube carrying structures. The authors conducted mathematical modeling of dynamic loads on the carrying structure of an that faces shunting impacts with consideration of the center sill filled with aluminum foam. It was established that the maximum accelerations on the carrying structure of an open wagon were 35.7 m/s2, which was 3.5% lower in comparison with those for a circular

 tube structure without a filler. The results obtained were proved by computer modeling. The  strength of the carrying structure of an open wagon was also calculated. It was established that

Citation: Fomin, O.; Gorbunov, M.; aluminum foam applied as a filler for the center sill decreased the maximum equivalent stresses in Lovska, A.; Gerlici, J.; Kravchenko, K. the carrying structure of an open wagon by about 5% and displacements by 12% in comparison with Dynamics and Strength of Circular those involving the circular tube carrying structure of an open wagon without a filler. The natural Tube Open Wagons with Aluminum frequencies and the oscillation modes of the carrying structure of an open wagon were defined. The Foam Filled Center Sills. Materials designed models of the dynamic loading of the carrying structure of an open wagon were verified 2021, 14, 1915. https://doi.org/ with an F-test. 10.3390/ma14081915 Keywords: circular tube open wagons; carrying structure; aluminum foam; dynamics; strength Academic Editor: Thomas Fiedler

Received: 15 March 2021 Accepted: 9 April 2021 1. Introduction Published: 12 April 2021 The leading position in the global transportation market can be maintained in the rail

Publisher’s Note: MDPI stays neutral industry by introducing new innovative engineering solutions in terms of the design at with regard to jurisdictional claims in wagon manufacturing facilities. The carrying structure of a rail car is one of the most loaded published maps and institutional affil- assembly units regarding operation and material intensity. Therefore, improvements and iations. optimization solutions for the carrying structure of a rail car require particular attention. This requires not only optimal characteristics of the carrying structure in terms of material intensity, but also the application of non-typical materials that enable us to decrease the tare weight and comply, at the same time, with the required strength characteristics. Besides, the issue of lower operational dynamic loading enables us to reduce the damage Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. in the carrying structure, satisfy the requirements for dynamics and strength within the This article is an open access article normative values, extend the life cycle of cars, etc. This can be reached by applying energy- distributed under the terms and absorbing materials in the carrying structures of rail cars. It should be mentioned that the conditions of the Creative Commons aluminum foam used as a structural material for a long time in the mechanical engineering Attribution (CC BY) license (https:// all over the world has proved its efficiency [1–5]. Article [6] (p. 11) suggests that it is creativecommons.org/licenses/by/ advantageous to use the studied foam for the development of energy absorber elements 4.0/). to improve vehicle crashworthiness during low-speed impact. Metal foams belong to a

Materials 2021, 14, 1915. https://doi.org/10.3390/ma14081915 https://www.mdpi.com/journal/materials Materials 2021, 14, 1915 2 of 12

particular class of metals with an interesting mix of mechanical and physical properties. Characterized by low density and high strength, metal foams show a stiffness/weight ratio five times greater than the bulk material and a large plateau in the compressive stress-strain diagram [7] (p. 55). However, the issue of an application of aluminum foam in the structures of rail cars has not been discussed yet. Therefore, substantiating the application of aluminum foam for carrying the structures of rail cars is an urgent task nowadays.

2. Analysis of Recent Research and Publications Every newly designed or significantly modified support part of a railway must meet strict criteria related to its strength and fatigue life. Research [8,9] presents key changes of the structure of an original bogie frame, conditions, and results strength analyses of unfavorable load cases, for which values of load forces come out from valid standards. Tools for computer simulations are widely used in the field of a rail vehicle design. These means of virtual reality allow performing static analysis of rail vehicle parts and dynamic analysis of a rail vehicle multibody system [10,11]. Article [12] presents the structural design of the freight railway wagon with variable use of loading space with regard to the safe operation and assessment of the properties by the calculation methods of simulation analysis. Based on computer-aided simulation analysis, they optimized the chassis frame of the wagon. This wagon will be able to offer even more capacity and utilize fewer resources and energy than current wagons for intermodal transport. Paper [13] describes the most significant and innovative research and development design solutions and computational procedures for a new structure of a railway tank wagon. A strength assessment of a new structure of railway tank wagon is presented. For validation of the new construction assemblies, they created a substitute simulation model. Peculiarities in the determination of the strength for the carrying structure of a wagon are studied in [14]. It deals with the reasons for defects in components of the wagon body. The authors suggest installing reinforcing elements in the interaction zone between the center sill and the bolster beam. The research into the structural peculiarities of BCNHL wagons is given in [15]. The study outlines some possible ways to improve the technical and economic parameters of wagons for higher operational efficiency. But measures to decrease the dynamic loading on the carrying structure of the wagons are not considered. The structural and element analysis of a BOXN25 open wagon with the finite element method is conducted in [16]. The study deals with the strength calculation and the modal analysis of the wagon structure. The strength of the carrying structure of a 12-757 open wagon with operational wear-outs typical for a 1.5-life cycle is researched in [17]. The study gives an improved dynamic loading on the open wagon in operation and the strength calculation for the carrying structure. However, the authors do not suggest any measures to improve the carrying structure of wagons to ensure strength under operational loading modes. The measures for lower dynamic loading on the carrying structure of an open wagon under ferry transportation are proposed in [18]. The authors present a device that can offer viscous resistance to displacements of the body and be used for fixation of rail cars on the train ferry deck. Its use enables us to decrease the dynamic loading on the body undersea transportation by 30% in comparison with that at a standard fixation pattern. However, the study does not touch on the issue of improvements in the carrying structure of a rail car for lower dynamic loads in operation. Study [19] presents peculiarities to be considered during the optimization of carrying structure components for a rail car with an application of the graph-analytical method. The strength calculation was conducted by the finite element method in LIRA-software (version 9.6, Lira-Soft, Kyiv, Ukraine). Materials 2021, 14, 1915 3 of 12

The optimization of open-type freight wagon bodies with the bearing floor is given in [20]. The study presents an algorithm of combined structural and parametric optimiza- tion for the sidewall and the frame of an open wagon with the bearing floor intended for the axial load of 25 tons. The authors propose optimization techniques for the metal structure of a wagon body. However, the determination of dynamic loads on the carrying structure of a wagon with the proposed implementations is not considered. The structural optimization concept of a wagon body of aluminum sandwich panels is given in [21,22]. The characteristic search function for the optimal combination is determined by the maximum stresses and displacements. However, the study does not present the mechanism of how to decrease the longitudinal dynamic loading on the wagon with consideration of the proposed panels. Aluminum foams are a new class of materials with promise of an improvement of vehicle crashworthiness, combining the properties derived from the cellular structure, in particular the lightness, with the typical behavior of metals [23]. With comparable mechanical properties, it is two-thirds lighter than [1,5]. The use of high-strength aluminum alloys for the load-bearing structures of freight wagon bodies will reduce the tare of the wagon to 50% [24,25]. Also, aluminum alloys have high corrosion resistance [24,25]. Flexural studies on foam-filled thin walled aluminum-extruded sections showed higher resistance to bending (7.5 kN) against empty Al-sections (5.8 kN) [26]. The optimum combination of high sound absorption coefficient and frequency range occurred in a crushed foam with good cell interconnectivity. Research [27] (p. 247) presented the compression tests of specimen aluminum foams. The curves were characterized by a typical initial elastic response, followed by a deforma- tion plateau with a positive slope and finally a transition to densification. The compression strength (at the begging of the plateau with the positive slope) was 5 MPa and the stress variations in the elastic regime were found to be nearly linear.

3. Objective and Tasks of the Article The objective of the article is to reveal peculiarities of the dynamic loading on the carrying structure of a circular tube open wagon with consideration of aluminum foam as filler for the center sill. To solve this goal, the following tasks are defined: mathematical modeling of dynamic load; determination of the strength of the load-bearing structure; computer simulation of dynamic loading; modal analysis; verification of the developed models of dynamic loading.

4. Presentation of the Basic Material of the Study The dynamic loading on a wagon in operation can be reduced with the application of aluminum foam as an energy-absorbing material [6,7]. One of the most rational options for using aluminum foam for the wagon structure is to apply it as a filler for wagon Materials 2021, 14, x FOR PEER REVIEW 4 of 13 components of closed sections, e.g., the circular tube carrying structures of wagons. The authors studied the efficiency in applying aluminum foam in the center sill of an open wagon,dinal impact as force. the mostThe design loaded diagram unit is presented in the carryingin Figure 2. Thestructure aluminum (Figure foam 1). was considered as an elastic body with a stiffness of 100 kN/m.

Figure 1. Carrying structure of the circular tube open wagon with aluminum foam as filler for cen- Figureter sill. 1. Carrying structure of the circular tube open wagon with aluminum foam as filler for center sill.

Figure 2. Design diagram of open wagon under shunting impacts.

 MWWWW x  M   P  c  x ; (1)

 IW W M x W g W M l F FR  sign 1 sign 2 l C 1  C 2 ; (2)

MW z W  C1  C 2  F FR  sign  1  sign  2 ; (3)

where nI WS MW M W 2 m b ; M  M W h ; C1  k 1 1 ; C 2  k 2 2 ; (4) r2

12 zWWWW  l ;.   z  l  (5)

MW —mass of carrying structure of the wagon; IW —inertia moment of wagon relative to the longitudinal axle; P—value of longitudinal impact force to automatic cou-

pler; c —stiffness of the aluminum foam, mb —bogie mass; IWS —inertia moment of a wheelset; r —radius of the mean worn-out wheel; n —number of bogie’s axles; l —half-

base of wagon; FFR —absolute value of dry friction force in spring group; kk12, —

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dinal impact force. The design diagram is presented in Figure 2. The aluminum foam was considered as an elastic body with a stiffness of 100 kN/m.

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The dynamic loads on the open wagon body under shunting impacts, as the largest loading on the carrying structure in operation, were defined by the mathematical model developed by Prof. G.I. Bogomaz. The model was composed for the determination of the accelerations as a component of the dynamic loads on a flat wagon loaded with tank containers under shunting impacts. Therefore, it was elaborated for the determination of the accelerations as a component of the dynamic loading on a wagon under the longitudinal

impact force. The design diagram is presented in Figure2. The aluminum foam was consideredFigure 1. Carrying as an elastic structure body of withthe circular a stiffness tube open of 100 wagon kN/m. with aluminum foam as filler for cen- ter sill.

FigureFigure 2. 2.Design Design diagram diagram of of open open wagon wagon under under shunting shunting impacts. impacts.

.. .. . M0 · x + M0 · ϕ = P − c · x ; (1) W MW x  MW   P  c  xW ; WWWW. . (1) .. 0 .. 0   IW · ϕW + M · xW − g · ϕW · M = l · FFR sign∆1 − sign∆2 + l(C1 − C2); (2)  IW W M x W g W M l F FR  sign 1 sign 2 l C 1  C 2 ; (2) ..  . .  MW · zW = C1 + C2 − FFR sign∆1 − sign∆2 ; (3) MW z W  C1  C 2  F FR  sign  1  sign  2 ; (3) where where 0 n · IWS 0 M W = MW + 2 · mb + ; M = MW · h; C1 = k1 · ∆1; C2 = k2 · ∆2; (4) r2 nI M M 2 mWS ; M  M h ; C  k ; C  k ; W W b2 W 1 1 1 2 2 2 (4) ∆1 = zW −rl · ϕW; ∆2 = zW + l · ϕW. (5) M —mass of carrying structure of the wagon; I —inertia moment of wagon rela- W 12 zWWWW  l ;.   zW  l  (5) tive to the longitudinal axle; p—value of longitudinal impact force to automatic coupler; c—stiffnessMW — ofmass thealuminum of carrying foam, structuremb—bogie of the mass; wagon;IWS—inertia IW —inertia moment moment of a wheelset; of wagon r—radiusrelative to of the longitudinal mean worn-out axle; wheel; P—valuen—number of longitudinal of bogie’s impact axles; lforce—half-base to automatic of wagon; cou- FFR—absolute value of dry friction force in spring group; k1, k2—rigidities of springs in pler; c —stiffness of the aluminum foam, mb —bogie mass; IWS —inertia moment of a bogie’s suspension; x , ϕ , z —coordinates corresponding to the longitudinal, angular wheelset; r —radiusW of theW meanW worn-out wheel; n —number of bogie’s axles; l —half- around the longitudinal, and the vertical displacements of the wagon, respectively. base of wagon; F —absolute value of dry friction force in spring group; kk, — The input parametersFR of the model are the technical characteristics of a wagon,12 and the value of longitudinal impact force to the automatic coupler. The calculation was made in the case when the longitudinal loading on the carrying structure of an open wagon was P = 3.5 MN [26,28]. The mathematical model was solved in MathCad software (version 15.1, PTC, Boston, MA, USA) [29–31]. The initial displacements and the speeds were taken to be equal to zero. The results of the calculation are given in Figure3. Materials 2021, 14, x FOR PEER REVIEW 5 of 13

rigidities of springs in bogie’s suspension; xzWWW,, —coordinates corresponding to the longitudinal, angular around the longitudinal, and the vertical displacements of the wagon, respectively. The input parameters of the model are the technical characteristics of a wagon, and the value of longitudinal impact force to the automatic coupler. The calculation was made in the case when the longitudinal loading on the carrying structure of an open wagon was Р = 3.5 MN [26,28]. The mathematical model was solved in MathCad soft- Materials 2021, 14, 1915 ware(version 15.1, PTC, Boston, MA, USA) [29–31]. The initial displacements and5 of the 12 speeds were taken to be equal to zero. The results of the calculation are given in Figure 3.

FigureFigure 3.3. AccelerationsAccelerations onon carryingcarrying structurestructure ofof thethe openopen wagonwagon underunder shuntingshunting impacts.impacts.

TheThe maximum accelerations accelerations on on the the carrying carrying structure structure of ofan anopen open wagon wagon were were 35.7 35.7m/s2, m/s which2, whichwas 3.5% was lower 3.5% in lower comparison in comparison with those with for the those tube for structure the tube without structure fill- withouter. filler. TheThe finite finite element element method method was was used used for for the the determination determination of the of strength the strength of an openof an wagonopen wagon under under shunting shunting impacts impacts [32,33 ].[32 The,33]. center The center sill was sill calculated was calculated with awith cylindrical a cylin- elementdrical eleme installednt installed inside, inside, with the with properties the properties being similarbeing similar to those to ofthose aluminum of aluminum foam: elasticfoam: elastic modulus—5.3 modulus×—1053.3MPa, × 103 Poisson’sMPa, Poisson ratio—0.3;’s ratio mass—0.3; density—800 mass density kg/m—8003 ;kg ultimate/m3; ul- tensiletimate strength—5.0tensile strength·MPa;—5.0∙ yieldMPa stress—1.05; yield stress·MPa.—1.05∙MPa. TheThe optimal optimal number number of elements of elements in a grid in a was grid determined was determined with the with graph-analytical the graph- method.analytical Spatial method. isoparametric Spatial isoparametric tetrahedrons tetrahedrons were used aswere the finiteused as elements; the finite the elements; optimal numberthe optimal of elements number was of defined elements with was the defined graph-analytical with the graph method.-analytical The number method. of units The innumber the grid of was units 224,220, in the andgrid the was number 224,220, of elementsand the number was 723,496. of elements The maximum was 723 size,496. of The an elementmaximum equaled size of 70 an mm, element the minimum equaled size70 mm, was the 14 mm.minimum The minimal size was number 14 mm. of The elements mini- inmal a circlenumber was of 14; elements the size in gain a circle ratio was of elements14; the size in thegain grid ratio was of elements 1.8. The maximumin the grid sidewas ratio1.8. The was maximum 707.55, the sidepercentage ratio was of elements 707.55, the with percentage a side ratio of lower elements than 3 with was a 33.2, side while ratio percentagelower than of3 was elements 33.2, withwhile a percentage side ratio more of elements than 10 waswith 12.8. a side ratio more than 10 was 12.8.The design diagram considered that the carrying structure of an open wagon was st underThe vertical design static diagram loading consideredPV , longitudinal that the force carryingPl on structure the rear support of an open of an wagon automatic was coupler, and also under pressurest from the bulk freight Pb on the side and end walls Materials 2021, 14, x FOR PEER REVIEWunder vertical static loading PV , longitudinal force Pl on the rear support of an auto-6 of 13 (Figure4 ). The lateral pressure from the bulk freight was calculated by the method given inmatic [34]. coupler, Mineral carbonand also was under taken pressure as bulk freight. from the bulk freight Pb on the side and end walls (Figure 4). The lateral pressure from the bulk freight was calculated by the method given in [34]. Mineral carbon was taken as bulk freight.

FigureFigure 4.4. DesignDesign diagramdiagram ofof thethe openopen wagon.wagon.

SteelSteel 09G2S09G2S was taken taken as as the the material material for for the the carrying carrying structure structure of ofa wagon a wagon with with the thevalue value of the of thetensile tensile strength strength σ = σ490= 490MPa MPa and and yield yield stress stress σy =σ 345y = 345MPa. MPa. The Themodel model was wasfixed fixed in the in areas the areas resting resting upon upon the . the bogies. The Theresults results of the of calculation the calculation are given are given in Fig- in Figureure 5. 5The. The research research was was carried carried out out for for the the elastic elastic stage. stage.

(a)

(b)

Figure 5. Stressed state of carrying structure of open wagon (a) top view; (b) bottom view.

The maximum equivalent stresses emerged in the bottom part of the center sill be- hind the rear draft lug and accounted for about 320 МPа. The maximum displacements

Materials 2021, 14, x FOR PEER REVIEW 6 of 13

Figure 4. Design diagram of the open wagon.

Materials 2021, 14, 1915 Steel 09G2S was taken as the material for the carrying structure of a wagon with the 6 of 12 value of the tensile strength σ = 490 MPa and yield stress σy = 345 MPa. The model was fixed in the areas resting upon the bogies. The results of the calculation are given in Fig- ure 5. The research was carried out for the elastic stage.

(a)

(b)

FigureFigure 5. 5. StressedStressed state of state carrying of carryingstructure of structureopen wagon of(a) top open view; wagon (b) bottom (a )view. top view; (b) bottom view.

The maximum equivalent stresses emerged in the bottom part of the center sill be- Materials 2021, 14, x FOR PEER REVIEWhind Thethe rear maximum draft lug and equivalent accounted for stresses about 320 emerged МPа. The maximum in the bottom displacements part7 of 13 of the center sill behind the rear draft lug and accounted for about 320 MPa. The maximum displacements were fixed in the middle part of the center sill and were 9.5 mm (Figure6). The maximum

deformationswere fixed in the were middle 3.64 part ×of the10 −center3 (Figure sill and7 ).were 9.5 mm (Figure 6). The maxi- mum deformations were 3.64 × 10−3 (Figure 7).

(a)

(b)

FigureFigure 6. 6. DisplacementsDisplacements in assembly in assemblyunits of carrying units structure of carrying of open wagon structure (a) top view; of ( openb) wagon (a) top view; (b) bottom view. bottom view.

(a)

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were fixed in the middle part of the center sill and were 9.5 mm (Figure 6). The maxi- mum deformations were 3.64 × 10−3 (Figure 7).

(a)

(b) Materials 2021, 14, 1915 7 of 12 Figure 6. Displacements in assembly units of carrying structure of open wagon (a) top view; (b) bottom view.

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Materials 2021, 14, x FOR PEER REVIEW 8 of 13

(a)

(b) (b)

FigureFigureFigure 7. 7. Deformations Deformations in carrying in in carrying carrying structure structure structure of open of of wagon open open ( wagona wagon) top view; ( a(a)) top( btop) bottom view; view; (view. b(b)) bottom bottom view. view.

ItItIt was waswas establishedestablished established that that that foam foam foam aluminum aluminum aluminum applied applied appliedas the as filler the as filler forthe the forfiller center the for center sill the de- sillcenter decreased sill de- thecreasedcreased maximum the the maximum maximum equivalent equivalent equivalent stresses stresses in stresses the in the carrying carryingin the structure carrying structure ofofstructu an open openre wagonof wagon an openby by aboutwagon 5% by about 5% and displacements by 12% in comparison with those of the circular tube carry- and displacements by 12% in comparison with those of the circular tube carrying structure abouting structure 5% and of andisplacements open wagon without by 12% the in filler comparis (Figure 8)on. with those of the circular tube carry- ofing an structure open wagon of an without open wagon the filler without (Figure the8). filler (Figure 8).

Figure 8. Stress state of the carrying structure of an open wagon without filler.

The mass of the carrying structure of an open wagon increased by 2.6% in compari- son with that of the filler-free structure (at aluminum foam density 300 kg/m3). FigureFigure 8. 8.Stress Stress state state of of the the carrying carrying structure structure of of an an open open wagon wagon without without filler. filler. The acceleration distribution fields relative to the carrying structure of an open wagon for the designed diagram (Figure 4) were determined by computer modeling. The mass of the carrying structure of an open wagon increased by 2.6% in comparison AluminumThe mass foam ofwas the modeled carrying as an structure elastic element of an ofopen the stiffnesswagon 100increased kN/m. This by 2.6%stiff- in compari- 3 withsonness withthatvalue ofthat is thetaken of filler-free the into filler consideration structure-free structure, since (at aluminum it (at is withaluminum this foam value foam density that densitythe 300 effectiveness kg/m 300 kg/m). of 3). the proposedTheThe acceleration acceleration solution is distribution traced. distribution The results fields fields of relative the relativecalculation to the to carryingare the given carrying in structure Figure structure 9.of an open of an wagon open forwagon the designed for the designed diagram (Figure diagram4) were (Figure determined 4) were determinedby computer by modeling. computer Aluminum modeling. foamAluminum was modeled foam was as anmodeled elastic elementas an elastic of the element stiffness of 100the kN/m.stiffness This 100stiffness kN/m. This value stiff- is ness value is taken into consideration, since it is with this value that the effectiveness of the proposed solution is traced. The results of the calculation are given in Figure 9.

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Materials 2021, 14, x FOR PEER REVIEWtaken into consideration, since it is with this value that the effectiveness9 of 13 of the proposed solution is traced. The results of the calculation are given in Figure9.

(a)

(b)

FigureFigure 9. AccelerationAcceleration distribution distribution fields relative fields to relative carrying tostructure carrying of circular structure tube open of circular wagon tube open wagon (a()a) top top view; (b ()b bottom) bottom view. view.

The maximum accelerations emerging in the mid-span of the center sill were 2 30 m/sThe2. The maximum maximum accelerations accelerations in emerging the side walls in the were mid-span about 20 of m/s the2; they center were sill were 30 m/s . 2 Theconcentrated maximum in their accelerations middle sections. in the The side least walls acceleration were aboutvalue was 20 m/sin the ;end they parts were concentrated inof their the carrying middle structure sections. of an The open least wagon. acceleration value was in the end parts of the carrying structureThe natural of an oscillation open wagon. frequencies of the carrying structure of an open wagon were determinedThe natural by a modal oscillation analysis in frequencies CosmosWorks of software the carrying [35,36]. The structure results of of the an cal- open wagon were culation are given in Table 1. determinedSome oscillation by a modal modes analysisin the carrying in CosmosWorks structure of an open software wagon [ are35 , given36]. The in results of the calculationFigure 10. are given in Table1.

Table 1. Natural oscillation frequencies in carrying structure of the open wagon.

Oscillation Mode Frequency, Hz Period, s 1 18.6 0.051 2 19.5 0.041 3 30.9 0.038 4 32.3 0.034 5 33.9 0.033 6 40.3 0.032 7 44.0 0.031 8 48.3 0.030

9 53.3 0.027 10 55.2 0.025

Some oscillation modes in the carrying structure of an open wagon are given in Figure 10. Materials 2021, 14, x FOR PEER REVIEW 10 of 13

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Figure 10. Oscillation modes in carrying structure of open wagon (a) 1st natural frequency; (b) 2nd natural frequency; (c) 3rd natural frequency; (d) 4th natural frequency.

Table 1. Natural oscillation frequencies in carrying structure of the open wagon.

Oscillation Mode Frequency, Hz Period, s 1 18.6 0.051 2 19.5 0.041 3 30.9 0.038 4 32.3 0.034 5 33.9 0.033 6 40.3 0.032 7 44.0 0.031 8 48.3 0.030

9 53.3 0.027 FigureFigure 10. 10.Oscillation Oscillation modes modes in in carrying carrying structure structure of of open open wagon wagon ( a(a)) 1st 1st natural natural frequency; frequency; ( b(b)) 2nd 2nd 10 55.2 0.025 naturalnatural frequency; frequency; ( c()c) 3rd 3rd natural natural frequency; frequency; ( d(d)) 4th 4th natural natural frequency. frequency.

TableTheThe 1. Natural calculationcalculation oscillation conductedconducted frequencies demonstratesdemonstrates in carrying that thatstructure thethe naturalnatural of the open frequenciesfrequencies wagon. areare withinwithin thethe admissibleadmissible values values [ 26[26,28,28,37],37], thus, thus they they exceed exceed 8 Hz. 8 Hz. In this In case,this case, the discrepancy the discrepancy between be- Oscillation Mode Frequency, Hz Period, s thetween first the natural first natural vibration vibration frequency frequency of the carrying of the carrying structure structure of the open of wagon,the open obtained wagon, byobtained mathematical by mathematical1 modeling model and computering and computer simulation,18.6 simulation, is about 3%. is about 3%.0.051 TheThe designeddesigned2 dynamicdynamic modelsmodels werewere checked19.5checked forfor adequacyadequacy withwith anan0.041 F-testF-test [[3388––4040].]. TheThe inputinput parameterparameter3 ofof thethe modelmodel waswas thethe impactimpact30.9 forceforce toto thethe automaticautomatic0.038 coupler,coupler, andand thethe outputoutput waswas the the4 accelerations accelerations on on the the carrying carrying32.3 structure structure of anof openan open wagon. wagon. The0.034 The results results of the of calculationthe calculation are given5are given in Figure in Figu 11re. 11. 33.9 0.033 6 40.3 0.032 7 44.0 0.031 8 48.3 0.030 9 53.3 0.027 10 55.2 0.025

The calculation conducted demonstrates that the natural frequencies are within the admissible values [26,28,37], thus they exceed 8 Hz. In this case, the discrepancy be- tween the first natural vibration frequency of the carrying structure of the open wagon, obtained by mathematical modeling and computer simulation, is about 3%. The designed dynamic models were checked for adequacy with an F-test [38–40]. The input parameter of the model was the impact force to the automatic coupler, and the output was the accelerations on the carrying structure of an open wagon. The results of the calculation are given in Figure 11. FigureFigure 11.11. AccelerationsAccelerations onon carryingcarrying structurestructure ofof thethe openopen wagonwagon atat shuntingshunting impacts.impacts.

2 2 The dispersion of adequacy was Sad = 20.5 and the error mean square SS = 14.8. Thus, the design value of the criterion was Fp = 1.38, which was lower than its tabular value (Ft = 3.07). It implies that the hypothesis on adequacy is not rejected.

5. Conclusions 1. Mathematical modeling of dynamic loads on the carrying structure of an open wagon was conducted with consideration of aluminum foam as the filler for the center sill. The aluminum foam was considered as an elastic body of the stiffness of 100 kN/m. The calculation was made in the case when the longitudinal loading on the carrying structure of an open wagon was P = 3.5 MN. The mathematical model was solved in MathCad software. The maximum accelerations on the carrying structure of an open wagon were 35.7 m/s2, which was 3.5% lower in comparison with those of the tubular structure without the filler. Figure 11. Accelerations on carrying structure of the open wagon at shunting impacts.

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2. The strength of the carrying structure of an open wagon with consideration of alu- minum foam as the filler for the center sill was defined by the finite element method in CosmosWorks software with SolidWorks software for the graphics. The maximum equivalent stresses emerging in the top section of the center sill behind the rear draft lugs accounted for about 320 MPa. The maximum displacements fixed in the middle part of the center sill were 9.5 mm. The maximum deformations were 3.64 × 10−3. It is established that foam aluminum applied as the filler for the center sill decreases the maximum equivalent stresses in the carrying structure of an open wagon by about 5%, and displacements by 12% in comparison with those of the circular tube carrying structure of an open wagon without filler. The mass of the carrying structure of an open wagon increases by 2.6% in comparison with that of the filler-free structure (at aluminum foam density 300 kg/m3). 3. Computer modeling of the dynamic loads on the carrying structure of an open wagon was conducted with consideration of aluminum foam as filler for the center sill. The maximum accelerations emerging in the middle section of the center sill were 30 m/s2. The maximum accelerations in the side walls were concentrated in the middle sections and accounted for about 20 m/s2. The minimum accelerations were in the end parts of the carrying structure of an open wagon. 4. Modal analysis of the carrying structure of an open wagon with consideration of aluminum foam as filler for the center sill was conducted. It determined the natural frequencies in the carrying structure of a flat wagon. It was established that the values of the natural oscillation frequencies do not fall outside the range of the admissible values. 5. Mathematical models of the dynamic loading of the carrying structure of an open wagon with consideration of aluminum foam as filler for the center sill were verified. 2 A F-test was used as a design criterion. Thus, the adequacy dispersion was Sad = 20.5 2 and the error mean square was SS = 14.8. Also, the design criterion value was Fp = 1.38, which was lower than its tabular value (Ft = 3.07). As a result, the hypothesis on adequacy was not rejected. The research will encourage engineers to design innovative units and to improve the operational efficiency.

Author Contributions: Conceptualization, O.F., M.G., and A.L.; methodology, O.F., M.G., and A.L.; software, O.F. and 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, M.G. and J.G.; visualization, O.F., A.L., and K.K.; supervision, M.G., J.G. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Slovak Research and Development Agency of the Ministry of Education, Science, Research and Sport of the Slovak Republic through the Educational Grant Agency of the Ministry of Education of the Slovak Republic in the project No. VEGA 1/0558/18: Research of the interaction of a braked railway wheelset and in simulated operational conditions of a vehicle running in a track on the test bench. 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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