Model of the Feeding Process of Anisotropic Warp Knitted Fabrics

Size: px

Start display at page:

Download "Model of the Feeding Process of Anisotropic Warp Knitted Fabrics"

Rosaline Garrison
5 years ago
Views:

1 Zgnew Mo³ajczy Technca Unvesy of ódÿ Insue of Knng Technoogy and Sucue of Kned oducs u. eomsego ódÿ oand Mode of he Feedng ocess of Ansooc Wa Kned Facs Asac The mode of he feedng ocess of ansooc wa ned facs ncudes a mahemaca mode of a ned fac n whch he suface dsuon of vaae feaues of he ned fac sucue s defned n he fom of numeca maxes he cea of seecng he feedng mehod ae defned on he ass of he exsng echnooges and he defnon of focng he feedng sysem on he ass of he nemacs of oo-fomng eemens and fnay he gaden of engh of he fed and equed heads n sucua eemens of he ned fac. Idenfyng he feedng sysem namcs a consan-engh head feedng fomuaes he assumons of he hysca mode. The mahemaca mode desces me couses of vaaons n namc foces n heads n a sch eea. On he ass of he agohm of cacuaons a numeca smuaon of head feedng was caed ou fo he sucue of a jacquad ned fac deemnng he exeme vaues of foces n he fed wa heads. Key wods: wa nng ansooc wa nng feedng suface dsuon mahemaca mode. Inoducon A chaacesc feaue of ansosucua ned facs s he vaaon n he comonen sches suface sucue. The sches fom a gven comnaon of heads of sucua eemens of dffeen saa confguaons and dffeen enghs. In he ocess of manufacung wa ansomohc ned facs seecng he omum aamees fo feedng he nng zone of wa nng machnes wh heads s a ey faco n ogammng he echnoogca ocess []. The mode of feedng assumes a sysem of acve wa unwndng fom wa eams wh a consan o vaae ogammed oaona seed of he eams. ha he ovefeeds and defcences of heads eween he enghs of fed was and head equemens n sucua eemens of he ned fac have mnmum vaues. n 6 - defnng he focng of he feedng sysem S() whch s he sum of: S'() - focng cononed y dsacemen of nng eemens of he wanng machnes ac dsacemen of heads n he nng zone ye of sch conons of fac ae-u. The focng was deemned emcay y dga anayss of cues of he head movng eween he ac es oe and he needes. ΔS() - he focng defned y vaaon of he head un-n gaden fo he seeced feedng mehod. n s 7 and 8 - a mahemaca mode of he feedng ocess descng he me couses of he ac es oe defecons y() and he vaay of he namc foces n heads fomng a sch eea. The mahemaca mode was fomuaed on he ass of he hysca mode of he feedng ocess esened n he foowng chae. The mode of feedng he manufacued ansooc wa ned fac wh wa heads s esened n he fom of an agohm wh defned successve ses of he mode aa souons (Tae ). n - a mahemaca mode of an ansooc wa ned fac desced n he fom of numeca maxes of A comonen sches []. n - a geomeca-emca mode of he sucua eemens of he fac. n - a defnon of numeca maxes L (V ) of suface dsuon of he head engh n sch eea. n 4 - deemnng he head gaden Δ accodng o one of fou osse mehods of consan-engh head feedng. n 5 - a conon of seecng he feedng mehod accodng o cea deemnng ha he head gaden n he as fomed couse s cose o zeo and Fgue. A hysca mode of a consan-engh feedng sysem. Wam eam Knng zone 58

2 Tae. Agohm of he feedng mode. S e Defnng numeca maxes of -h comonen sches. umeca max of he -h comonen sch A { a } o A { B } fo o R o R j whee R R - hegh and wh of a sch eea. 4 Assgnng an eemen of max A o he vaue of head engh a whee: f(abd ) ABd - dmenson aamees of he fac e - emca coeffcen of he oen mode e Descng numeca maxes of head enghs L of he -h comonen sch L { } Tansfomaon of max L no symmeca max L' Deemnng gadens of head un-n' s Δ fo eemen a Δ ( ) - ( ) z : whee: ( ) - engh of he fed head ( ) z - engh of he equed head accodng o a gven mehod of head feedng: ' A': ( Δ ) A - feedng R R cons ( aveage equemen whn he ms of a sch eea) z R ' B': ( Δ ) B - feedng ( ) R (of a vaae head engh of z n a gou of heads a he hegh of he sch eea) R ' C': ( Δ ) C - feedng ( ) R (of a vaae head engh of z n he fac couse) ''D': ( Δ ) - feedng a vaae head engh n a gou of heads of equa B C o he aveage equemen n couses of a seaaed gou of heads. z Idenfcaon of he Feedng Sysem Dynamcs a Consan-engh Thead Feedng A hysca mode of he feedng sysem n a wa-nng machne a consanengh head feedng n he ocess of manufacung ansooc sucues (Fgue ) was defned fo he foowng assumons: n a quas-fa sysem efeed o wa heads n he wh of he sch eea R n aamees of he sysem desced y sffness he educed mass of he ac es oe eascy and aenuaon of heads. Theads ae eaed as weghess vscoeasc odes n he assumed hysca mode efecs he geomey of he wa head feedng zone n a a gven decon of head movemen on he ac es oe he hysca mode s a sysem sujec o vaae nemac focng S () n he hysca mode aows fo he hyseess of head oads caused y he fcon agans he ac es oe n he ac es oe s eaed as an nfexe eemen eascay suoed n such a way ha has one degee of feedom of movemen Seecng a feedng mehod 'A' 'B' 'C' 'D' accodng o he cea meeng he foowng conons: Δ 0 fo R (fo he as couse of he sch eea) Δ Δ - head ovefeed n eemen a - head defcency n eemen a Defnng he funcon of head focng S() n he feedng ocess S() (S'() Δ S ) fo heads o R S'() -focng cononed y he dsacemen of nng eemens n he wa- nng machne Δ S - Δ a comonen vaae of focng defned y he gaden of head un-n's Deemnng dsocaons of he ac of he feedng sysem namcs es oe y() on he ass of denfcao n ds ds a S( ) a S( ) whee: a a a - coeffcens descng he consucon of he feedng sysem z - coeffcen of he sysem aenuao n - coeffcen of he sysem eascy z mn mn R z y a Deemnng he namc oads n heads ρ z ds S ( ) S ( ) (): ds ( ) y (cos β sn) (cos β sn) μ e Symos used n Fgue : - enson n -h head eween he ac es oe and he nng zone - enson n -h head eween he wa eam and he ac es oe m - on educed o one head mass of he ac es oe s - coeffcen of he ac es oe sffness educed o one head - coeffcen of head eascy - coeffcen of head aenuaon - geomeca aamees of he sysem β - geomeca aamees of he sysem n v - oaona and nea seed of head feedng (n cons. and v cons. fo consan-engh wa unwndng) R - wae eea of he sch descng he nume of heads affecng he ac es oe S - oa nemac focng of he feedng sysem S'() - cycc focng dung fomaon of one couse of fac ΔS - gowh of focng equa o he gaden of head un-ns. 59

3 Mahemaca Mode of he Feedng Sysem The hysca mode of he feedng sysem namcs (Fgue ) can e desced y he equaon of moon whee: e - he quany descng he neeaon eween foces and ;. e μ - he coeffcen of fcon eween he ac es oe and he head movng on he ac es oe ρ - he ange of enccemen of he ac es oe wh head. Tang no accoun he oees of wa heads n he hysca mode as desced y he Kevn-Vog mode foces and can e esened y equaon and (4) whee λ λ - he head eongaon afe and efoe he ac es oe. Susung eaons and (4) o equaon we oan equaon (5). Fom he conon of wa head connuy ' S ( S ΔS ) (6) λ λ y( cos β sn ) we can deemne he quany: R R s y λ λ λ λ e λ ( cos β sn ) S y e (7) acng eaon (7) n equaon (5) we oan equaon (8). Afe ansfomaons and goung of ems eaon (8) can e wen as equaon (9). The ef sde of eaon (9) s he vaue of foce (equaon ) hence y acng he aove eaon n he equaon of moon and eaangng we oan equaon (0) whee: - he coeffcen of he sysem π π aenuaon - he coeffcen of he sysem eascy a a a - coeffcens descng he geomey of he feedng sysem R ( cos β e sn ) λ - he nume of heads movng on he ac es oe n he decon of he wa eam when conon s fufed: ΔS - V. Δ - Δλ - Δλ < 0 fo: ΔS - he gowh of he nemac focng of he sysem Δλ Δλ - gowh of head eongaon n he ams of head and V - he seed of head devey Δ - he se fo dscee cacuaons afe me. - he nume of heads movng on he ac es oe n he decon of needes when conon s fufed: (4) (5) ΔS - V. Δ - Δλ - Δλ > 0 - he nume of heads no movng on he ac es oe when conon s fufed: ΔS - V. Δ - Δλ - Δλ 0 R Deendng on one of he hee defned conons of head movemen on he ac es oe coeffcens a a and a have he vaues: n fo he gou of heads (conon ) cos β e sn a e (4) n fo he gou of heads (conon ) a cos β e (5) n fo he gou of heads (conon ) cos β sn a (6) Coeffcens π and π ae deemned fom eaons (7) and (8). Snce: h - he eave coeffcen of sysem aenuaon z h mr (9) and ω o - he fequency of fee vaon of he non-aenuaed sysem z ω 0 m R (0) equaon (0) has he fom esened n. Accodng o conons and he foces n heads () fo he successve ses of cacuaons ae deemned fom he foowng eaons: n fo conon - he eaon n fo conon - he eaon and n fo conon - he eaon (4). e sn λ e ds λ e e ( cos β sn ) [ S λ y( cos β sn )] ( cos β sn ) ( cos β sn ) e a R y S e e y a ds S ( ) ( ( ) ) S a S Equaons and 0. (8) (9) (0) Reasaon of he Feedng Mode Agohm fo he Sucue of a Jacquad Wa Kned Fac The sujec of anayss s he sucue of a jacquad wa ned fac consuced fom wo comonen sches. The fs comonen sch foms a eeaae se of oen chan oos; he ohe s he wef sch of vaed confguaon and engh of heads n successvey fomed couses. Due o he eeaay of eemens whn s sch eea he chan sch s an somohous oo sucue. The wef sch consss of hee eeaae eemens a - ye '' '' and '' aanged n he 60

4 eea suface accodng o he asc desgn of he fac aen (Fgue a). The aen ecoded n he fom of cooued squaes (c - ed z - geen - whe) conans nfomaon on he suface dsuon of wef heads n one wae (eemen '') o jonng wo (eemen '') o hee waes (eemen '') of he fs comonen sch. An ansosucua sysem of snge eemens of he wef sch sucue was ecoded n he fom of a numeca max A (Fgue ) on he ass of he defned nces of sch codng. The numeca max A ceay defnes he ansooc chaace of he fac sucue and s he ey aamee n denfyng he vaay of he fac feaues as we as he echnoogy of s oducon. The agohm (Tae ) aamees of he fac sucue wee defned accodng o se : couse hegh B0.95 mm wae wh A.80 mm chan sch head damee d ³ 0.4 mm (oyese yan of nea densy. ex) wef head damee d w 0. mm (oyese yan of nea densy 9.4 ex). The head enghs n chan oos and n wefs cacuaed accodng o an oen geomeca mode ae: ³.4 mm w. mm w.9 mm w 7 mm (eave eo ε.9%). By assgnng he vaues of head enghs n eemens (a ) o he eemens a of max A we oan max L. Fo mehod 'A' ha s feedng a consan head engh equa o he aveage equemen z n he sch eea he gadens of he head un-ns Δ wee deemned. The quanes Δ Δw B wee shown n he ned feds of he head un-n max V' (Fgue c). Fo se 6 he agohm Tae comonens of nemac focng ΔS wee deemned whee: ΔS -Δ. Thus fo he successve fou wef heads ΔS exessed n mm: n fo : ΔS -. ΔS -. ΔS -. ΔS ΔS 5.0 ΔS 6. n fo : ΔS 0. ΔS 0. ΔS. ΔS 4. ΔS 5.0 ΔS 6. n fo : ΔS.0 ΔS. ΔS 0.8 ΔS ΔS 5.0 ΔS 6. z z ( cos β sn ) ( a a a ) s a ( cos β sn ) s a a ( cos β sn ) [ ] [ ( cos β sn ) ] [ ] s a S h ω y a 0 R m a S e a) aen dawng max A umeca max of he second comonen sch (mase wef) C B Z R R 6 ) c) Fgue. A mahemaca mode of fac sucue. nd comonen sch nd comonen sch R R 4 6/00 6/00 6/00 6/ /4 6/-0 6/- 6/ / 6/-04 6/- 6/ / 6/-5 6/-096/ /96/-6 6/-0 6/ /- 6/- 6/- 6/ ' MATRIX V ( S ) ( ) S ( cos β sn ) ( cos β sn ) { B } accodng o mehod A" y ( ) ( ) ( ) ( ) S cos β sn y e ( cos β sn ) ( ) S cos β sn y ( cos β sn ) Equaons 7 8 and 4. (7) (8) (4) 6

a) 00 0 0 0 sch segmen fo R.. - Tme 4 0 [s].. - Tme 4 0 [s] ) sch segmen Fgue. a) Focng S() fo he wef head ; ) A me couse of namc foces n head. n fo 4 : ΔS 4 0. ΔS 4 0. ΔS 4 0. ΔS 44 -.0 ΔS 54-4.

5 a) sch segmen fo R.. - Tme 4 0 [s].. - Tme 4 0 [s] ) sch segmen Fgue. a) Focng S() fo he wef head ; ) A me couse of namc foces n head. n fo 4 : ΔS 4 0. ΔS 4 0. ΔS 4 0. ΔS ΔS ΔS Fo cacuaons of he ac es oe dsacemens y() and namc oads of heads () he foowng nu daa was aen (symos as n Fgue ): m.7 g; s 0.5 c/mm; 4680 c; 00 cms; μ0.; 7 mm; 44 mm; 46 o ; β4 o ; nng seed n400 couses/mn; u8/''e. The fna esu n he fom of an exame me couse of foces n he head fo focng S() s esened n Fgues a and. 6 Concusons The mahemaca mode of he feedng ocess fo ansosucua wa ned facs wh defned assumons of he namc sysem of consan-engh head feedng fo he focng efecng vaae aamees of he fac sucue and feaues of he nng ocess on wa-n- ng machnes shoud e hefu n seecng omum conons fo he nng ocess. Ths omsaon s defned y he mng vaues of foces n heads max and mn. Beyond hese vaues esecay fo mn >> max dsuances of he nng ocess occu whch fuhe eads o unfavouae changes n he fac sucue. Refeences. Z. Mo³ajczy Dga Idenfcaon of Yan Tae-U Gaden n Ansooc Wa Kned Facs. Zeszyy auowe W³óenncwo o. 58 6h Inenaona Texe Confeence IMTEX ' Z. Mo³ajczy Mode of Saa Sucue of Ansooc Wa Kned Facs. Fes & Texes n Easen Euoe Vo. 9 o. A/June Receved..00 Revewed

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma