Archer bow arrow behaviour in the vertical plane
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1 t of iongining nd ioni Vo. 8, No., 6 ow ow viou in t vti pn OR ZNEVKYY viv tt ntitut of Pyi Cutu, Kotyuko t., viv, 9, kin Toti nd xpint ut of on t po of, ow nd ow viou in t vti pn pntd. T i of t i to dvop tod of oput iution of tti nd dyni inttion twn t, ow nd ow yt in od to povid y wit pti ondtion. od of n ody i pntd ni yt opod of fw oid odi, wi onntd to ot nd to t gound wit vioti nt. Mni nd tti od of ow nd ow goty in vti pn in d nd dwn itution i invtigtd. n yti, igid, onnttd ti nt nd ti ting t in ftu of t od. Nui ut of oput iution of, ow nd ow inttion pntd in gpi fo, wi k t tod y to u y potn nd o. Ky wod: y, ioni, tti oding, oput iution. ntodution Duing pid otion n ow i in oon otion p wit ow nd n. T in otion in t vti pn i ud y t ow gip i ting yt. iutnouy t ow toug t ting nok point i invovd y t wo yt in dftion out of t vti pn. Ti pid otion u ow dftion in t t pn. Rigoou t-dinion nyi of t p yt i vy opitd nd wit t ot uption i not nti. T in pt of potnti ngy, tod in ow i, i tnfod to kinti ngy of t ongitudin otion of t ow nd t ting oon otion. o ngigi pt of ngy i tnfod nd dft t otion of t ow. toug t ow i in p otion of t wo yt, t po n idid nd dud to two pt yt. T fit on i in t vti pn, nd t ond i in t t pn.
2 64. ZNEVKYY T fit tp in t undtnding of n ow viou in intn iti w td to t xpntion of t pnonon tt fo ot two ntui n known t pdox. T invtigtion of t pdox w id out y n of ig-pd pk potogpy wi owd u to u dit vidn of wt n ow do it v t ow. T pdox i t ft tt n ow do not fy t tgt ong t in pntd y it xi. T fo ting on t ow duing it do not oinid uffiinty wit ti xi. T ting fo i xtd on t ow in t ow pn. n tting poition, t ow do not i in ti pn, nd it xi inin towd it t n ng of fw dg. Evn in t t nok nd d point of t ow do i in t ow pn, t ongitudin ow xi of t ting fo in do not oinid noug wit t ow pn u t initi p of t ow xi i not tigt noug. o, t in of ting fo do not oinid wit t in of o-tion nt of t ow. T d ting pu t ow nok point into t ow pn. Tfo t ow ov fowd nd igty tun ding t ng wit ti pn. T ipu no to t xi of t ow ud y t of t fing fo t ting, w t oun-ik fo of t ting pu t ow duing it tion otion uting in ignifint nding of t ow ft it tnit t ow. t fto ow t ow to undut ound t ow nd nd to foow tigt ou towd it tgt witout tiking t ow nd. Mni nd tti od of t t otion w invtigtd y PKK 6 nd ipovd y KOO 4 nd ZNEVKYY 8, 9. W ttd nyti oding of t yt viou in t vti pn uing pudo-tti, nd dyni ppo. dyni nyi of t tdition yti ow n id out in Jpn 5, ut u ow i vy diffnt fo odn pot ow nd in it dign un pt v not n tkn into ount. n, t no nyti od, wi k t tudi of t, ow nd ow inttion in t vti pn poi nd wi ipov oputtion tod fo djuting t pt of t yt. T i of t w to dvop tod of tti oding nd oput iution of tti nd dyni inttion in t ow ow yt in od to povid y wit pti ondtion. n t pp, t u w d of ioni tod t od of un ody yt of ni oito toti ni nd tti gng qution of t ond od t pinip of d t t Cuy po, tod of ittion nd t Rung Kutt tod oputtion tod MtCD nd Mtti pkt xpint tod ig-pd vido nyi ntu ni oding.
3 ow ow viou in t vti pn 65. ody oding n toti oding, un ody i pntd ni yt opod of fw oid odi, wi onntd to ot wit vioti nt. od tutu i uit tking into onidtion t i of, ody poition nd t t of it inttion wit t uounding. pot ting ow duing t ti of oon otion of ting nd n ow wit fixd nd tying to kp tdy poition of ody. T oto pog of ooting i inititd fo t ik ign to n d. ow intt wit in t point of ontt, i.., t point t wi nd i in ontt wit ow nd figu. u of ot ti of ting nd ow oon otion.. nd tivy dp ontt ong t inn id of t tu u, t point of ontt n ud to pivot point. u of dipnt of t ow nd, w n du t otion to otogon dition, wi uit to di t otion of ow nd in t vti pn ty foow: t ongitudin -xi ong t upp xtity nd t -xi ppndiu to it. ig.. od of t ody: ongitudin dition, ppndiu dition T nxt tp to otin uit od i kinti inkg of t goup wit ow pi E-D-. n t ongitudin dition, t nu of t dg of fdo qu two, i.., t dipnt of t ody tiv to gound ottion otion of t point D tiv to t point E nd t dipnt of t upp xtity ontting ow nd tiv to t ody D. n t ppndiu dition, w d wit on dg of fdo, u tiity of t oud gid D i ignifinty gt tn it tiity tiv to t ody in t vti dition DE.
4 66. ZNEVKYY Tu w n od n ody in t ongitudin dition d on two pti nd two vioti nt. T fit pti wit t od t ody xpt t upp xtity tt i in ontt wit ow nd nd vitu oodint. T fit vioti nt di inttion in t ody gound yt nd i pntd y t offiint of tiffn vioity k. T ond on, i.., n upp xtity of t nd t, vitu oodint u of it oon otion wit nd ontt point. T oponding vioti nt wit t offiint nd k di t inttion twn t upp xtity nd t ining pt of ody. T od of otion in t ppndiu dition inud on pti of t wit vitu oodint nd vioti nt wit t offiint nd k. Tking ount of t ftu givn ov, w n div t xpion pnting t kinti ngy of t ody od figu : potnti ngy: T P diiption funtion: Φ k k k. Pi yo xp divtion wit ti, i.., d dt.. ow nd ow oding T i uption in t oding of nd, i nd n ow i ti otion igid odi in gn vti pn. nd, tii nd ign oud odd on igid ody wo kinti ngy figu i xpd foow:
5 ow ow viou in t vti pn 67 T nd d, w y x x y i n ngu dipnt tiv to t point xy i tngu yt of oodint fixd to t nd. nting t two t xpion into w otin: T nd xc yc, w i x C, y C t oodint of nt of gvity i t ont of inti tiv to t point. ig.. Dyni ti od of ow nd n ow
6 . ZNEVKYY Kinti ngy of i i pntd y: z d z T µ, w i t ngt of i µ i t ditiutd z i t o-odint fixd to i. T upp nd ow i kd wit t oponding uipt nd. nting t xpion fo dipnt in ± ± z o ± ± z into t t qution fo t ngy T w iv t: in in o o o o i T,4 w i t of i i t ont of inti of i tiv to nd of t nd t point i t ditn fo t nd of t nd to t i nt of. T qution fo potnti ngy of i y xpd foow figu : P ϕ, 5 w nd tnd fo t tiffn of i. Expion fo t kinti ngy of ting i dividd into t pt pnting t pt of ting pinnd to t nok point of t i nd t ow: ting T T T T T, 6 w i t of ting i t ngt of t ting n in untung ow. 68
7 ow ow viou in t vti pn 69 T xpion fo t potnti ngy of ting n i: P w f i t ditiutd tiffn of t ting n in tung ow. f, 7 tnd fo t ngt of t ting T xpion fo t kinti ngy of t ow figu i givn y: T z dz P ow z µ, 8 w i t ngt of t ow µ i t ditiutd z i t oodint fixd to it ongitudin xi P i t of t ow d. T inttion twn ow nd ow i did oding t tu od ony toug t ontt t t nok point. n initi poition of t ow tiv to t ow in it in vti pn i dtind y t in tt od n ow d. T in fixd to t nd nd v t iity to tun nd dipp jut t t ont n ow tt to ov. Du to iz nd, t in do not uut ngy, tfo ti inttion wit n ow i not tkn into ount witin t fwok of t od. T ut of ig-pd vido nyi ow tt n ow otion in t vti pn n ud to igid ift 7,. nting nd z into t ngy qution 8 w otin: T ow, 9 w i t of t ow nd tnd fo t ditn fo t ti to t ow nt of. T xpion fo t potnti ngy of t ow i utd wok of inti fo oding t pinip of d t on t vitu ongitudin dipnt << : P ow w i n initi ttitud ng of t ow. g,
8 . ZNEVKYY 4. Dyni viou ft o tti tnfotion in, 7, 9,, w gt t xpion fo t wo kinti nd potnti ngy of t ow ow yt duing ti oon otion twn ting nd ow ooting: 4 y x T C C, ϕ ϕ g f f P, w o in o in 4. Ot two pt of t ting v n tkn into ount wit -inti titi of i pinnd to t nok point. oving t dyni po, w do not onid gvity fo ting on t ow. ut w tk into onidtion ow wigt u it fo ont ting on t ow t vu t ont of inti fo. nting xpion nd into t gng qution of t ond od i i i i i q q P q T q T dt d Φ, w gt yt of diffnti qution of t ond od tiv to gntd oodint,,,,,,,, i q : k k 7
9 ow ow viou in t vti pn 4 x k C 4 y k C C C y x ϕ 4 4 ϕ g g, w i t gntd fo: 4 f f. T initi ondition of t po foow:,, t 7
10 7. ZNEVKYY w t ontnt,,,, t oution of t tti po t nxt tion. Zo vu of divtion opond to t nn of y tniqu ting i toppd nd poition i otion. oding to t ut of uf toyogpy, t oto pog of ooting i inititd fo t ign to n d. Tu it n ifid n opn oop. T ntiiptd n i tizd y n ining tivity of t. ptoi jo ony. Ti u i tivtd to t n ng of tti quiiiu in t t pn. No u ow ignifint ng of tivity. Tfo, no gntd fo oud pnt in t gng qution, wi di t viou of t yt in t vti pn i. T dition of ow otion t vto of t nt of i did y t xpion: tn ζ α ζ, 4 w ζ i t ng of pojtion, nd α i t ng of ttk. n ow v ting jut ft it tion o qu to zo. T yt of qution wit initi ondition pnt t Cuy po fo non-in odiny diffnti qution of t ond od. t i ipoi to otin nyti oution fo t po, tfo w ud t Rung Kutt tod ppid in t pog NDov Mtod Expiit Rung Kutt fo t pkg of Mtti 4. Ti tti od di dyni of t ow nd ow yt in t vti pn o, to pi, ow nd ow oon otion in t in pn of t ow. ndd, t yt pfo D otion, ut it dipnt in t t pn i undd ti tn dipnt in t in pn. T infun of t t otion on t in otion i ngigiy u of ignifinty diffnt vu of dipnt nd t oponding ngy tti viou t u onid t t of t inttion twn n nd ow t t intnt of ting. n odn tniqu of t pot of y, you not quid to od, ut to gp ow nd. T nt of t of ow i otd t t point of t inttion twn t nd nd t nd. T of i, tiiz, igt nd ot igid joint onit t in qut of t wo ow. T of n ow, ting, nd ov nt of i onit out % of t wo of t yt T oponding gvity fo
11 ow ow viou in t vti pn ngigiy opd to t dwing fo. o, t gvity fo v no infun on t ow p nd opntd fo t fo of nd. Tfo, in tti po, w n u ony two xtn fo, wi t on t nd nd on t ting, i.., t fo t t point t pivot point nd nok point, ptivy figu. ig.. of tti od: d ow, tung ow T oponding tti od of t ow in givn itution i pntd y in in in in o o o o o o o o 5 in in ϕ ϕ o o in in f f φ φ tn tn, 7
12 74. ZNEVKYY w nd t fo in ting n dwing fo onto t x of oodint t ngt of n ow. nd t pojtion of t i dwing ditn dtind y T initi ondition fo t pivot point dipnt duing t ow nd nd t nd inttion givn y:. 6 Mtti od of t d ow i pntd y t foowing qution: o o in in in o in ϕ 7 in ϕ f, w i t ting ngt nd i t fo of t ting tting figu. T pt of ow wit d ting kd wit t uipt. T yt of qution 5 nd 6 inud tnndnt nd non-in funtion tt nnot ovd nytiy. Tfo to tudy of t tti po, w d u of nui tod of ittion uing t oput pog ind fo t pkg of Mtd i Pofion 6. Expint oding Two kind of i od popu in ni nyi of t odn pot ow, i.., n ti tip 4 nd igid joind to i y n ti nt. ot of t od v o dvntg nd iittion. fo t nyi of tnion nd dfotion of t i nd t infun of it p on t ow ffiiny, t fit od i tt. f w tudy t tti nd dyni of t ow wo yt, t ond od i o fftiv u of it ipiity nd on piion. Coon pt of t od vitu ngt of i, nt of, ont of inti nd vitu tiffn v n popod y
13 ow ow viou in t vti pn 75 CKMN. T od w ppid to i Engi ow. n od to u t od fo odn pot ow wit uvd i, w v to odify it. ig. 4. fo t tnfotion of t uvd i to t ikn od of t i : i f itution witout ting, d ow, tung ow dig of ow fo v. dw ngt W popo xpint nd nyti tod to dtin t pt of oigin od. yti p of t ow in it in pn i tudid in two poition figu 4: t ow i d wit ting, nd t ow i onvntiony tung. T k in ow vitu pt of ting nd i, wi tnfo t uvd ow to t ikn. T ting i in ontt wit t uvd pt of t i in t d poition twn t point nd T. W u t ngt of t uv T to qu to t ngt of t tigt in T. T oigin O of t Ctin oodint yt i pinnd to t nd in t idd of t ow. O i t xi of yty, xi O i d on t i. ing ip gi qution figu 4 T T T T w iv t two i goti pt of t od:
14 76. ZNEVKYY OT OT OT OT T T T T, 8 w i vitu ngt of t i OT OT OT OT. ing fo, ngti nd goti ignnt figu 4, : E d E ϕ ϕ w otin t fo pt of t od: T ϕ ϕ, T k ϕ, 9 k w E, ptivy t potnti ngy of t i nd t fo of tung ow i tiffn of t vitu ti nt intd of ditiutd tiffn of i T T ϕ i t ng of vitu i t f itution E i.., witout ting k. T o of not k i xpind in t. T T nt typ of pot ow dignd wit noin qu upp nd ow i. o pviou typ w dignd wit i of diffnt p nd tiffn T tod ntiond ov i uit fo dtintion of vitu pt pty fo t upp nd ow i, too pviou tion. 7. Pti ppition t u onid t odn pot ow nd ow d oding to T ntntion y dtion tndd d : WN&WN Ruv ow. T ow onit of Wint Ri 5 nd ong i 7, i.., t ow nd ngt i 65 figu nd t wo ngt of t ow ud twn tip of t i 778. T tndd u of ow yty in t vti pn nd ti i 6 nd t ow fo i 78 N. T ow No wit 5% point w ud. Rtd pt of t ow foow t pviou tion: 5 6 g 7 g 68. kg 68. kg N ϕ.647 ϕ.676 4
15 ow ow viou in t vti pn 77. kg 8 kg x C y C f 55 N 7 g W otin t ow pt in it d poition oving t yt of qution 7: N. T ow pt : 78.4 g 7.6 kg 5. ig. 5. Pt of ow nd inttion: vitu dipnt of t ody oi fo ting on t ody oi fo ting on t ow nd T pt : 6 kg.8 kg. kg.6 N 9. N 9.4 N k 7 kg k 78 kg k 45 kg. oving t tti po 5, w iv t t pt und initi ondition nd 6: T ot oution foow: N 9 N 78 N N.
16 78. ZNEVKYY T dyni of, ow nd ow duing ti oon otion, n initi t igt P 7, i pntd in figu 5, 6, 7. ig. 6. Pt of ow nd ow inttion: ongitudin pd of n ow ongitudin dipnt of n ow ppndiu dipnt of ting nd ow nok point d ongitudin dipnt of t pivot point nd nd nd ngu dipnt of nd T ti of ow nd ow oon otion fo ting up to ow ooting, i.., out.54, w utd oniding t intnt of t xiu ongitudin pd V 6 to opond dity to zo vu of tion, figu 6. oi fo ting twn ow nd nd nd ft ting vi in t ng o to % of it initi tti vu figu 5. fo, ot
17 ow ow viou in t vti pn 79 ti figu 5, i tnfd to t gound vi n ody, wi it dipnt in n figu 5. u of t diffnt t of tti nd dyni n of fo, n ow p on ting fw iit dp, figu 6 opd to it d poition. Dipnt of t pivot point, i.., t point t wi ow nd nd nd in ontt, i out on iit nd ppoxity i qu in t ongitudin 9.4 < <. nd ppndiu dition figu 6d. ig. 7. Pt of ow intn iti: ng of ttk of n ow ttitud ng ngu pd of t ow in t in pn ting nd ow oon otion intn iti i opnid y intniv oition, wi ud y t dtution of t tti n of fo t t intnt of ting. T vn fu y of oition duing t otion. T ut of oput iution k it poi to dtin t ow nd ow djutd pt, wi iniiz t ng of ttk nd ngu pd of n ow tt i tt fo good ot figu 6.
18 8. ZNEVKYY 8. Conuion. T ut of t oding of t ow ow yt ot wit w-known ut of ig-pd vido nyi: t po of oon otion ignifint non-in t.. oi fo ting twn ow nd nd nd ft ting vi in t ng o to % of it initi tti vu. fo, ot ti, i tnfd to t gound vi n ody, wi it dipnt in n.. u of t diffnt t of tti nd dyni n of fo, n ow p on ting fw iit dp opd to it d poition. 4. ting nd ow oon otion intn iti i opnid y intniv oition, wi ud y t dtution of t tti n of fo t t intnt of ting. T fu vn y of oition duing t otion. knowdgnt T uto i gtfu to Pofo. Goof, iug nivity, nd Doto. Go, tuttgt nivity, fo ti utnti p. Rfn EDEMNN-NER J., GOOER., Coodintiv pt of y n ppo uing uf toyogpy, XV yp. ioni in pot, d. y.j. Ri, M.M. Vitn, Po.. Kontnz: VK, 998, GRO., ZNEVKYY., -ow-ow yt djutnt in t vti pn, intifi Poding of t XX ntntion ypoiu of ioni in pot, d. y K.E. Giniki, nividd d Extdu, pin,, CKMN C.N., NGER., KOPTEG P.E., y: t tni id, Rdnd: Ntion id y oition, 947, 48 p. 4 KOO.W., ow-ow inttion in y, Joun of pot in, 998, 6, pp OM., OTK., iution of t p nd dyni of Jpn ow ppition of g dftion toy, t ook of t 4 t nt. Conf. on t Engining of pot, Kyoto, Jpn,, 7. 6 PKK R., Moding nd iution of t optito ow ow yt, unpuid doto dittion, Wzw, W, Wn it Zigt. igpd fi, ZNEVKYY., Dyni of ow ow yt, Joun of utotion nd nfotion in, 999,, pp ZNEVKYY., t dftion of y ow, pot Engining,, Vo. 4, No., pp. 4. ZNEVKYY., T tni pt of t pot ow tuning in kinin, uy in Engi, Tni Nw,,, pp. 4 9.
19 ow ow viou in t vti pn 8 ZNEVKYY., ORKO., Mni nd tti oding of ow nd ow inttion, t of iongining nd ioni,, 4, upp., pp ZNEVKYY., T tod of iution nd nyi of pot y pt in kinin, uy in Engi, doto. dittion, Pyioni ntitut, viv, 996.
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