E. Stanová. Key words: wire rope, strand of a rope, oval strand, geometrical model

Transcription

1 Te International Journal of TRANSPORT & LOGISTICS Medzinárodný čaopi DOPRAVA A LOGISTIKA ISSN 45-7X GEOMETRY OF OVAL STRAND CREATED OF n n n WIRES E. Stanová Abtract: Te paper deal wit te matematical geometric modelling of te oval wire trand created of wire. Te matematical repreentation of te wire axe i in form of parametric equation wit variable input parameter. Te number and diameter of te core wire are te initial data. Te parametric equation of te axe of individual wire are derived baed on tee data and te number of wire in te layer. Te equation are implemented in Pro/Engineer Wildfire V5 oftware for creating te geometrical model of te trand. Key word: wire rope, trand of a rope, oval trand, geometrical model INTRODUCTION Steel rope are ued in many engineering application. Multiplicity of application require a varioune of teir tructure. Deign of tructure affect te mecanical propertie of te rope terefore determining optimal value of teir geometrical parameter play an important role. Computer-aided deign i powerful tool in ti proce. It allow deign a geometrical contruction of te rope, determine te geometric parameter and verify teir relevance by creating a geometrical model [, ]. In te geometrical modelling of wire trand and rope key part play matematical expreion of te wire. Matematical equation allow create geometrical model of te trand uing te CAD ytem [3]. Te rope of circular cro-ection contitute one group of wire rope. Tee can be formed by trand of variou ape. Te oval trand are one type of tem. In ti paper, te matematical expreion of oval trand contructed of n n n wire i derived and teir implementation to te oftware Pro/ENGINEER Wildfire i ued to contruct a geometrical model [4]. GEOMETRICAL CONSTRUCTION OF THE OVAL STRAND Te conidered trand i made of two layer of circulary wire elically laid around a core [5]. Te core conit of n wire aving a diameter. Firt layer i created by n wire wit diameter and econd layer i created by n wire wit diameter (Fig. ). wire of te nd layer wire of te t layer wire of te core

2 Figure Cro- ection of te oval trand of n n n type Tere i te gap between te wire. Te wire of bot layer ave te rigt- and pitc and te winding angle i. 3 MATHEMATICAL EXPRESSION OF THE WIRE AXES Te wire elically lay around a traigt core. Te urface generated by te wire can be formed by tranlation of te circle woe center i on te wire axi and te circle lie at te normal plane of ti curve. Terefore, ufficient i to derive a matematical expreion of te wire axe curve. 3. Te wire of te firt layer Let te rigt-and Carteian coordinate ytem O ; x, y, z be placed o tat te z axi i identical wit te axi o S of te trand and te x axi i perpendicular to te line OX (Fig. ). y ω o o U Y T V U O x T V X S γ O V S U Y T x y Figure Cro-ection of te t layer Figure 3 Wire axi in te t layer Te curve of te wire axi conit of traigt line egment and elix egment (Fig. 3). Let point S be located on te curve wic will be expreed. Te part ST of te curve i a line egment, te part TU i a part of cylindrical elix. We will derive te equation of bot part uing te angle of rotation around te z axi a a parameter. We mark te angle between te x axi and te line OS. Te ize of tem i given by formula arctan. () Ten te parametric equation of te line egment ST ave te form l x, () l tan y, (3)

3 ;. were tan n tan z l, (4) Te part TU of te wire axi curve i cylindrical elix. It axi o paing troug te point Y (Fig. 3) i paralel to te z-axi. Uing te tranformation of te coordinate ytem we obtain te equation of te elix part: were ;. co x, (5) in y n, (6) z, (7) tan Tee two egment are repeated in te curve. Tey are only rotated about te angle te z axi and tranlate about te eigt k around. (8) tan 3. Te wire of te econd layer Uing te previou metod, we obtain a matematical expreion of te wire axi curve of econd layer. For te angle between te x axi and te line OS i valid relationip arctan Te line egment of te curve i expreed by te equation ψ ;γ. were z l ψ x l ψ ψ. (9), () y l tan ψ γ, () tanψ γ, () tan α Te elix egment of te curve can be expreed by te forme y ψ ψ x co ψ γ, (3) inψ γ, (4)

4 for z ψ ψ γ tan, (5) α ;. Te tranlation of te egment in te curve i ψ tan α. (6) π 3.3 Oter wire of te layer Wire of one layer are te ame urface. Ti fact can be ued. Eac wire in te layer i given by te previou one ifted by a particular ize in te axial direction. Te ize i depended on te one pitc lengt and te number w n of wire in te layer. It can be calculated from te relationip w. (7) n So, te curve of any wire axi of te layer can be expreed by parametric equation x ψ x ψ co κ y ψ in κ y, (8) ψ x ψ in κ y ψ co κ z, (9) ψ z ψ k i w in wic for te firt layer we ue te equation () - (4), te equation (5) (7), te equation (9) () for l, () ; for elical egment ( and (3) (5) for. 4 MODELLING OF THE WIRES IN THE LAYER ; for line egment ( l ) and ). For te econd layer we ue Baed on te matematical expreion it i poible to contruct te geometrical model of te wire and ubequently of te trand. To illutrate ti poibility te model of te trand wit 4++3 wire and 4++4 wire are contructed. In ti cae te parametric equation are implemented in Pro/ENGINEER Wildfire oftware for te geometric modelling. Let u aume, tat te diameter of te core wire, te gap between tem and te winding angle of bot layer are given. Te trand elected to illutrate differ only by te number of wire in te econd layer. Diameter and gap for te layer mut be calculated. It i poible on te bae of te derived equation. Baic geometrical parameter of te trand are lited in Table. Table Baic parameter of te trand of type 4++3 and type 4++4 Strand of wire Core Firt layer Second layer Number of te wire Winding angle Diameter Gap α n ,,, of te wire (mm),8,446,94,749 between wire (mm),,87,76,6 Geometrical model of te trand wit 4++3 wire i own in Figure 4.

5 Figure 4 Geometrical model of te trand of 4++3 wire Geometrical model of te trand of 4++4 wire compared to te previou cange only in te econd layer, wic a a different number of wire (Fig. 5). Figure 5 Geometrical model of te trand of 4++4 wire 5 CONCLUSION In order to create te geometrical model of te oval trand, te parametric equation ave been developed and implemented in Pro/ENGINEER Wildfire V5 oftware for te modelling. Developed equation allow u to create te model of oval trand wit two layer and te core coniting of n wire. Baed on te geometric parameter tat define te core, we can to determine neceary parameter for any given number of wire in te layer.

6 Te trand elected to illutrate differed by te number of wire in te econd layer. Te parametric equation wit concrete input parameter were implemented in te aid oftware and geometrical model of te wire and conequently two trand were created. Te decribed metod of te creation of geometric model of oval trand can be ued to explicit computer modelling and analyi of wire trand and rope. Acknowledgement Ti work wa upported by VEGA /3/ Teoretická a experimentálna analýza adaptívnyc lanovýc a tenegrity útav pri tatickom a dynamickom namáaní uvažovaním účinkov vetra a eizmicity. Reference [] USABIAGA, H., PAGALDAY, J. M.: Analytical procedure for modeling recurively and wire by wire tranded rope ubected to traction and torion load. International Journal of Solid and Structure 45 (8). [] STANOVÁ, E., FEDORKO, G., FABIAN, M., KMEŤ, S.: Computer modelling of wire trand and rope Part I: Teory and computer implementation. In: Advance in Engineering Software, Volume 4, Iue 6 (), Imprint: Elevier Ltd., ISSN , pp [3] FABIAN, M., SPIŠÁK, E.: Navrování a výroba pomocí CA.. tecnologií. Brno : CCB, 9, 398 p. ISBN [4] FEDORKO, G., MOLNÁR, V., MADÁČ, K.: Základy aplikácie Pro/Engineer v tecnicke konštrukcii. Košice: vydavateľtvo Fakulty BERG, Tecnická univerzita v Košiciac, 8,. 87, ISBN [5] ttp://nxengli.en.alibaba.com/productowimg/ /oval_strand_steel_wire_rope.tml#insearc