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1 Te International Journal of TRANSPORT &LOGISTICS Medzinárodný časopis DOPRAVA A LOGISTIKA ISSN 45-7X GEOMETRY OF OVAL STRAND CREATED OF n +n +n WIRES Eva Stanová Civil engineering facult, Tecnical Universit of Košice, Vsokoškolská 4, 4 Košice e- mail: eva.stanova@tuke.sk Abstract: Te paper deals wit te matematical geometric modelling of te oval wire strand created of n + n + n wires. Te matematical representation of te wire aes is in form of parametric equations wit variable input parameters. Te number and diameter of te core wires are te initial data. Te parametric equations of te aes of individual wires are derived based on tese data and te number of wires in te laer. Te equations are implemented in Pro/Engineer Wildfire V5 software for creating te geometrical model of te strand. Ke words: wire rope, strand of a rope, oval strand, geometrical model INTRODUCTION Steel ropes are used in man engineering applications. Multiplicit of application requires a variousness of teir structures. Design of structure affects te mecanical properties of te ropes terefore determining optimal values of teir geometrical parameters plas an important role. Computer-aided design is powerful tool in tis process. It allows design a geometrical construction of te rope, determine te geometric parameters and verif teir relevance b creating a geometrical model [, ]. In te geometrical modelling of wire strands and ropes ke part plas matematical epression of te wires. Matematical equations allow create geometrical models of te strands using te CAD sstem [3]. Te ropes of circular cross-section constitute one group of wire ropes. Tese can be formed b strands of various sapes. Te oval strands are one tpe of tem. In tis paper, te matematical epression of oval strand constructed of n + n + n wires is derived and teir implementation to te software Pro/ENGINEER Wildfire is used to construct a geometrical model [4]. GEOMETRICAL CONSTRUCTION OF THE OVAL STRAND Te considered strand is made of two laers of circular wires elicall laid around a core [5]. Te core consists of n wires aving a diameter δ. First laer is created b n wires wit diameter δ and second laer is created b n wires wit diameter δ (Fig..
2 E. Stanová Geometr of oval strand created of n +n +n wires T&L wire of te nd laer wire of te st laer wire of te core Figure Cross- section of te oval strand of n + n + n tpe Tere is te gap between te wires. Te wires of bot laers ave te rigt- and pitc and te winding angle isα. 3 MATHEMATICAL EXPRESSION OF THE WIRE AXES Te wires elicall la around a straigt core. Te surface generated b te wire can be formed b translation of te circle wose center is on te wire ais and te circle lies at te normal plane of tis curve. Terefore, sufficient is to derive a matematical epression of te wire aes curves. 3. Te wire of te first laer Let te rigt-and Cartesian coordinate sstem ( O,, z ; be placed so tat te z ais is identical wit te ais o S of te strand and te ais is perpendicular to te line OX (Fig.. ω os o U Y T V U O T V X S γ O V S U Y T Figure Cross-section of te st laer Figure 3 Wire ais in te st laer Te curve of te wire ais consists of straigt line segments and eli segments (Fig. 3. Let point S be located on te curve wic will be epressed. Te part ST of te curve is a line segment, te part TU is a part of clindrical eli. We will derive te equations of bot parts
3 E. Stanová Geometr of oval strand created of n +n +n wires T&L using te angle of rotation around te z ais as a parameter ψ. We mark γ te angle between te ais and te line OS. Te size of tem is given b formula: γ = arctan ( n δ + δ. ( Ten te parametric equations of te line segment ST ave te form: δ + δ l ( ψ =, ( + δ l ψ = tan( ψ γ, (3 ( + + δ tan( ψ γ z ψ = n l, tanα (4 were ψ ;γ. Te part TU of te wire ais curve is clindrical eli. Its ais o passing troug te point Y (Fig. 3 is parallel to te z-ais. Using te transformation of te coordinate sstem we obtain te equations of te eli part: + δ ψ = cos( ψ γ, (5 ( + + δ sin( ψ γ ψ = n, (6 + δ( ψ γ + ( n z ψ =, (7 tanα were ψ γ ; γ + π. Tese two segments are repeated in te curve. Te are onl rotated about te angle κ = kπ around te z ais and translate about te eigt + δ π + ( n =. (8 tan α 3. Te wire of te second laer Using te previous metod, we obtain a matematical epression of te wire ais curve of second laer. For te angle between te ais and te line OS is valid relationsip γ = arctan ( n δ + δ + δ Te line segment of te curve is epressed b te equations. (9 δ + δ + δ l ( ψ =, ( 3
4 E. Stanová Geometr of oval strand created of n +n +n wires T&L ( ψ ( ψ + δ + δ ( ψ = tan, ( l γ ( n + + δ + δ tan( ψ γ z l =, ( tan α were ψ ;γ. Te eli segment of te curve can be epressed b te forme ( ψ ( ψ ( ψ + δ + δ ( ψ =, (3 cos γ ( n + + δ + δ sin( ψ γ =, (4 ( n + + δ + δ ( ψ γ tan =, (5 z α for ψ γ ; γ + π. Te translation of te segment in te curve is ( n + + δ + δ tan π ( ψ =. (6 α 3.3 Oter wires of te laer Wires of one laer are te same surfaces. Tis fact can be used. Eac wire in te laer is given b te previous one sifted b a particular size w in te aial direction. Te size is depended on te one pitc lengt and te number n of wires in te laer. It can be calculated from te relationsip: w =. (7 n So, te curve of an wire ais of te laer can be epressed b parametric equations ( ψ s ( ψ cosκ s ( ψ sin κ ( ψ s ( ψ sin κ + s ( ψ cosκ z ( ψ zs ( ψ + k + i w =, (8 =, (9 =, ( in wic for te first laer we use te equations ( - (4, ψ ;γ for line segment ( s = l and te equations (5 (7, ψ γ ; γ + π for elical segment ( s =. For te second laer we use te equations (9 ( for s = l and (3 (5 for s =. 4
5 E. Stanová Geometr of oval strand created of n +n +n wires T&L 4 MODELLING OF THE WIRES IN THE LAYER Based on te matematical epression it is possible to construct te geometrical model of te wires and subsequentl of te strand. To illustrate tis possibilit te models of te strands wit 4++3 wires and 4++4 wires are constructed. In tis case te parametric equations are implemented in Pro/ENGINEER Wildfire software for te geometric modelling. Let us assume, tat te diameter of te core wires, te gap between tem and te winding angles of bot laers are given. Te strands selected to illustrate differ onl b te number of wires in te second laer. Diameters and gaps for te laers must be calculated. It is possible on te base of te derived equations. Basic geometrical parameters of te strands are listed in Table. Table Basic parameters of te strand of tpe 4++3 and tpe 4++4 Strand of wires Core First laer Second laer Number of te wires n Winding angle α ( -,,, Diameter δ of te wire (mm,8,446,94,749 Gap between wires (mm,,87,76,6 Geometrical model of te strand wit 4++3 wires is sown in Figure 4. Figure 4 Geometrical model of te strand of 4++3 wires Geometrical model of te strand of 4++4 wires compared to te previous canges onl in te second laer, wic as a different number of wires (Fig. 5. 5
6 E. Stanová Geometr of oval strand created of n +n +n wires T&L Figure 5 Geometrical model of te strand of 4++4 wires 5 CONCLUSION In order to create te geometrical model of te oval strand, te parametric equations ave been developed and implemented in Pro/ENGINEER Wildfire V5 software for te modelling. Developed equations allow us to create te model of oval strand wit two laers and te core consisting of n wires. Based on te geometric parameters tat define te core, we can to determine necessar parameters for an given number of wires in te laer. Te strands selected to illustrate differed b te number of wires in te second laer. Te parametric equations wit concrete input parameters were implemented in te said software and geometrical models of te wires and consequentl two strands were created. Te described metod of te creation of geometric model of oval strand can be used to eplicit computer modelling and analsis of wire strands and ropes. Acknowledgements Tis work was supported b VEGA /3/ Teoretická a eperimentálna analýza adaptívnc lanovýc a tensegrit sústav pri statickom a dnamickom namáaní s uvažovaním účinkov vetra a seizmicit. 6
7 E. Stanová Geometr of oval strand created of n +n +n wires T&L References [] USABIAGA, H., PAGALDAY, J. M.: Analtical procedure for modeling recursivel and wire b wire stranded ropes subected to traction and torsion loads. International Journal of Solids and Structures 45 (8. [] STANOVÁ, E., FEDORKO, G., FABIAN, M., KMEŤ, S.: Computer modelling of wire strands and ropes Part I: Teor and computer implementation. In: Advances in Engineering Software, Volume 4, Issue 6 (, Imprint: Elsevier Ltd., ISSN , pp [3] FABIAN, M., SPIŠÁK, E.: Navrování a výroba s pomocí CA.. tecnologií. Brno : CCB, 9, 398 p. ISBN [4] FEDORKO, G., MOLNÁR, V., MADÁČ, K.: Základ aplikácie Pro/Engineer v tecnicke konštrukcii. Košice: vdavateľstvo Fakult BERG, Tecnická univerzita v Košiciac, 8, s. 87, ISBN [5] ttp://nengli.en.alibaba.com/productsowimg/ /oval_strand_steel_wire_rope.tml#insearc 7
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