Permutation Analysis of Track and Column Braiding

34Joumal of Donghua University (Eng. Ed.) Vol. 21, No.2 (2004)Permutation Analysis of Track and Column BraidingLI Yu-ling(李毓陵)', DING Xin(丁辛)', HU Liang-jian(胡良剑)21. Collegeof Textiles; 2. College of Science, Donghua University, Shanghai 200051Abstract: The positions of braiding carrier in track andcarriers and the blank lattices the unocupied ones. The latticescolumn braiding are represented by a diagrammatic braidinginside the bold frame, shown in Fig. 1(a), fom the main partplan and a corresponding lattice-array is defined. A set isof the array termed as the main array, in which the carriersthen formed so that the permutation analysis can becan be moved in both row and c∞olumn direction. Those latticesperformed to represent the movement of carriers in aoutside the frame form the supplement part of the array calledbraiding process. The process of 4 step braiding is analyzedthe side array, in which the carriers can be moved in eitheras an example to describe the application of the proposedrow or column direction depending upon the position of themethod by expressing a bralding cycle as a product ofside array. Above the lttie-array there is a datc arraydisjoint cycles. As a result, a mapping relation between thenamed Z array, indicating the moving pattern of carriers in Ydisjoint cycles and the movement of carriers is deduced.direction.At the left side of the lattice array, Harray showsFollowing the same analysis principles, a process of 8-stepthe moving pattern of carriers in X direction. Starting frombraiding and the corresponding initial state of the latticethe top row, each row in Z-array represents a single movementarray is developed. A successful permutation analysis to theof carriers in column direction. The number indicates theprocess manifests the general suitability of the proposedmoving distance in terms of“steps" and the sign shows themethod.Keywords: track and colunn braiding ，braidingdirection. Take the first row of Z- array for example. The firstelement“+ 1”means the first column of carriers moving oneputtern, latticearray ，permutation, cyclestep downward and the second clement “- 1”means thsecond colum of carriers one step upward. Similarly, startingIntroductionfrom the most outside column in Harray, element “+ 1”means the corresponding row of cariers moving one stepTrack and column braiding is an available 3-Drightward and'. 1”the corresponding row of carriers one stepbraiding methodl~s], which performs braiding process byleftward. By means of the above representation, the braidingmoving braiding carriers along row or column directionprocesses can be described unambiguously.alternatively. Li and DingeJ examine the braiding methodand concentrate their study on the movement of braidingcarriers on the braiding platform, on which the braidingfo.1 |fo.2tracks are located. According to the braiding principle, af.f |fu2study of braiding process is actually a study of the movingpattern of yarn ends on the braiding platform. In their[-1 +1another paper(61， the positions of braiding carriers arafo.2represented by a diagrammatic braiding plan and acorresponding lattice-array is defined. The current studytries to deal with the braiding process with permutation(a) Braiding plan(b) Set Fanalysis in order to seek the fundamentals of the processFig. 1 Mathematical abstraction of 4 - step braidingand the formation of braiding patterns.Mathematically, the braiding plan in Fig. 1(a) can beAbstraction of Track and Column Braidingdefined by a set F, as shown in Fig. 1(b). Each element inProcessset F, fi，represents the corresponding cell in the lattice-array. The array indicating carriers movement,乙and HFig. 1(a) is a braiding plan of 4- step braiding with 2 rowsarray, can be regarded as a map carried on F, denoted by φand 2 columns of carriers. The positions possibly ocupied by(Z, H). For the convenience of mathematical analysis,braiding carriers in the braiding platform are represented by aboth ends of element in X or Y direction are joint togetherlattice- array, in which the filled lattices represent the existing中国煤化impl, Jo! is cnsideredReceived Nov.18, 2003"THCNMHGSupported by Shanghai Key Discipline ProjectCorrespondence should be addrened to LI Yu-lingJoumal of Donghua University (Eng. Ed.) Vol.21, No.2 (2004) 35as a neighbor element of fs.1; fi.o is considered as arepresented by permutation P1 :neighbor element of fi,3, etc. And because the number of(fo,1 f,s fz.1 fs,1 fo.z fi.zlattices is not changed during braiding process, thenp(Z,p1=(f. fz fs1fo.I fs,z fo.2H) is both a onc-to-one and an onto mapping. Hence theprocess of mappingp(Z, H) is actually a permutation.fi.z fs.z f.o fr,s fr.o fs.s\(2)Therefore, the braiding process can be analyzed by meansf1,z f2.z fs.o f1,; fz,o fz,s)of permutation theory to obtain generally purposed resultsBecause the permutation of each column is independent, itfor further studies.can be represented by a cycle. Furtbermore, each cycle isdisjoint, then Equation(2) can be written in a powerMathematical Preliminariesform: .(1) Suppose A is a set of n elements, A = {a,p:= (fo,1 f,1 f2,1 fs1)+laz, . a}. A permutation of A is a mapping from A into. (fo.z f1.2 fz.2 fs,2)-(3)A, which is both one-to-one and onto. In other words, apermutation of A is a one-to-one function from A onto A,The index of power of each cycle indicates thedenoted bycharacteristics of carriers' movement in the correspondingcolumn and is exactly the same as those data in the first rowφ=(u!aau)(1)in Z-array. For the elements that keep in position in thecycle such as fi,o, fi,g, fz.o，f2,a are omitted in Equation(2) The suffix of the elements in the first line of the(3).permutation expression may not be necessary in a sequentialSimilarly the second step ( in X direction ) isrepresented by permutation P:order. For example, h = ( 1and鸭=pz= (f.。 f.n f,z f.a)+1are the same permutation. As a rule in(f2。fz. fz.x fz.s)-'(4)alaicthis paper, the suffix in the first line of permutationAlso，the index of power of each cycle indicates theexpression is arranged in an ascending order.movement of carriers in the corresponding row and(3) The product of two permutations φ and中ismatches the data of the first column in H-array. Thedefined as 内内(a;) =内(内(a,))， which means,elements that keep in position in the cycle are also omittedin Equation(4) .performing中first and then(4) The power of permutation φ is defined recursivelyThe same procedures apply to the third and fourthbyφ=IA,如=中-,n=1,2,3,.,whereIsisthe:stepsidentity mapping.p:= (Jo,I f.，fr.n fs.)-(5) The minimal integer λ satisfying中 = Ia is called(fo.z fL.z fz.2 fs.z)+1(5)the degree of permutation φ.(6) If a permutation mapsan - + a2, az→ag, .ps= (f.,。 f.r f.z fh,s)-am1→am，am→a, it is called an m-cycle, or a cycle withlength m. In a more compact notation, a permutation of(f2,o fz.1 fz.z fr.s)+lm-cycle can be written as (a, az，.. am). Especially, aBy observing Equations(3) ~(6), a brief summation2-cycle is called a transposition.can be made:(7) If two cycles have no common elements, they area) Each step of the braiding process can be represented bycalled disjoint cycles.permutation and written as a product of disjoint cycles.(8) Every permutation of a finite set is a product ofEach cycle corresponds to one step of movement ofdisjoint cycles and the degree of a permutation is the leastcarriers in one column or in one row.common multiple of the lengths of its disjoint cycles.b) When the elements in a cycle are arranged in the ordersame as the lattice-array, i.e. in Y direction from topPermutation Analysis of 4- Step Braidingto bottom and in X direction from left to right, thepermutation can be expressed in a power form wittThe braiding patterns Z and H of Fig.1(a) arents in Z or H array.For4中国煤化工le consits of four stepsin conatively. Therefore, az=[$1 +1]H=[+1 +1]compleu心1 aru;YHCNMHGnabed twg rjuir vaii uuscripermutationStarting from Y direction, the first step of braiding can beP:36Joumal of Donghua University (Eng. Ed.) Vo1.21, No.2 (2004)P=p●P.内.p|f-1. f-.9-1. |1-1.4P-[(fo,1 f.1 f2.1 f2.1)+(Jo.2 h.2 fz.z fs.:)-]fo,1| fo.2| fo,s | fo.4[(f.o f.1 h.: h.s)+(f2,o fz1 fz,z f2.s)-]|5.-11.05.15.25.3hhs | fhn,6[(1o.1 f1.1 f2.1 fs1)-(fo.z h.z fz.z 5.z)*]|fz,-1fz,o| f2.1f2.2| f2.3 I f2,4t f2.s I fz.s[(fo hi.t fi.z h,s) '(fz,o fz.1 f2.z fz.s)#]fs.-1fs.o|fs,1 | fs.z| fs.3jfs4 fs.5| fa.6. -(fo.1 f2.3 fs.z fi.o)(fh.1 f2.1 fz.z fz)(fs.1) (fo.z) (fz,o) (f3)7)f,-1.4.of41 fs.z| f43jf,.f.sfo,6 IFrom Equation(7), it can be seen thatfs.1| fs.z|Js,a| fs,sa) In P, there are 4 elements fs.1. fo.2, fz,o and f.s,thatfo,1|fe,z |fs.s fo,skeep in position after a braiding cycle. According thepermutation theory, these elements should be omitted in(a) SetFthe cycle. Because of their special meaning in braidingprocess, these elements should be listed in Equation(7).Referring to the lattice-array in Fig.1， these fourelements are the blank lattices in the side array. It'rmeans that the unoccupied positions in the braiding plancan be educed by permutation analysis.b) There are two disjoint cycles in permutation P and thelength of each cycle is 4. Hence the degree of P is 4( theleast common multiple of disjoint cycles' length). Itmeans that all the carriers can be divided into twogroups and will return to their original positions after 4braiding cycles. This verifies the result in literaturest,.(b) Lattice-arrayobtained from braiding practices.c) In P，there are 8 elements in the cycles and eachFig.2 8-step braidingelement corresponds to a existing carrier in the braidingSimilar to 4- step braiding, each step of braidingplan. It means that 8 carriers are nceded according toprocess can be expressed in a form of cycle power:Fig. 1(a).Although the method of permutation analysis isp一(61,10o,1 “乐1 0,1)#C_-r,z o.:”压z fa,x)-(f-1.1' 6.1 - sa 0x)+2《J1. Jo. “fs. 后.)-developed from 4 - step braiding process, it can be appliedto the track and column braiding with any other braidingr- (h.patterns. Once the carrier's movements are determined,the arrangement of the braiding plan, positions of carrierin the braiding platform and other braiding parameters canbe obtained by the method.1L-.s fo.s ”原3压s)-*C51.4 0o.4 ”s. fs.4 )#Analysis of 8-Step Braiding《反- 5。- 5s.s 5o)-'<_ {.。 “f,s 5o)+1The result of a braiding cycle can also be expressed byReferring to the method introduced in by Kostar etpermutational. [口，an 8-step braiding pattern is given as follow:P=p.p... "阳「+2-2+2-27- (5-,I 5:压1)(h.x 5; 1.s)Ch.x As h.)Z=-2+2-2+2(h.s压4压)(o1 h2↓ h1 22 ox)8)+2-2+2-2(1-4.3 5e压o h 5ss)L-2 +2 -2 +2」(5..s)h, s)(fs. ,)(s, s)(fr.-1 )(f,o)(h, -1)<4.00H=Some results can be obtained from Equation(8)+1+1-1-1a) There are. 8 disinint cvcles in P. It means that all the-1 -1 +1 +1]中国煤化工groups with differentBased on the carrier's movement pattern, a lattice-arrayth of 3 and other four ofMHwith 4X4 main array, together with the corresponding setCNMH G 15(he last commonF, is show in Fig.2.multiple of 3 and 5). It is to say that 15 braiding cyclesJoumal of Donghua University (Eng. Ed.) Vo1.21, No.2 (2004) 37are needed for all the carriers return to their originalcarriers groups,positions according to 8-step pattern. In other words,- the length of a cycle is the numbers of carriers withinonce the movement of carriers is determined, the groupsa group, andof carriers with individual law of movement as well as一the least common multiple of cycles is the numbers ofthe repeating cycles of movement of each group can bebraiding cycles that all the carriers return to theirobtained.original positions.b) There are 16 elements retain their positions in P, whichare the single element within a bracket. These elementsReferencesare all included in the side array. Hence the braidingplan in Fig. 2(b) can be drawn. It is significant for[1] W. Li, Hanmad and A. E-Shiekh. Structural Analysis ofrealizing this complex pattern that the unoccupied3-D braided Preforms for Composites Part I: Four-stepPreforms. J. Text. Inst., 1990. 81(4); 491- 514positions of braiding platform can be worked out by[2] TD Kostar and T-W Chou. Process Simulation andpermutation.Fabrication of Advanced Multi-step Three dimensionalBraided Preforms. Journal of Materials Science. 1994.Conclusions29(8); 2159- 2167[3] Han Qirui. Pattern of the Bobbin Movement in 3-D Braida. With the braiding plan, track and column braidingProcess for Composite. Journal of Tianjin Institute ofprocess can be clearly described. It is the method thatTextile science and technology. 1994. 13(2): 1 -5permutation analysis relies on.[4] Li Yu-ling and Ding Xin. Some Aspects of 3 - D Continuousb. When assuming that both ends of elements of set Fin XBraiding. See: Dang Jia-li edited. Present Situation andand Y direction are joint together to form a continuousDevelopment for Composite Materials of the Setting-in 2lstloop, each step of a braiding cycle can be expressed byCentury. Changsha: Hunan Science and TechnologyPubisher. 1998: 1019- 1023permutation directly.c. When the permutation expressing each step is written as[5] 道得锟，吴以心,李兴国编著、立体织物与复合材料.上海:中国纺织大学出版社.1998; 62-78a cycle power, the index of eacb cycle power indicates[6] Li Yu_-ling and Ding Xin. Some Basic Concepts Used inthe characteristics of carriers'movement in theTrack and Column Braiding. See: Mao Tian-xian edited.corresponding column or row, and is exactly the same asPresent Situation and Development of Composite Materials.those data in乙or H-array.Hefei: Publisher of China science and Technologyd. The expression of a complete braiding cycle byUniversity. 2000: 339 - 342permutation is to multiply the permutation of single[7] 高绪珏主编.近世代数.沈阳:辽宁人民出版社. 1985: 14-braiding steps according to the processing order.. When permutation of every braiding cycle is written as a[8] Li Tie-fei, Chen Xu-wei and Li Yu-ling. Rational Structureproduct of disjoint cycles,of Braiding Bed for Track- and-column Braiding. Journal ofDong Hua University. 2001. 27(6);14-16- the numbers of disjoint cycles are the numbers of中国煤化工MYHCNMHG