Modeling and Computer Simulation of Bow Stabilization in the Vertical Plane

Transcription

1 SSN (print) nternationa Journa of Sport Science and Engineering Vo. 0 (008) No. 0, pp Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pane hor Zanevkyy Caiir Puaki Technica niverity, KWFiZ, Macewkiego, Rado 6-600, Poand (Received October 4 007, accepted January 0 008) btract. The ai of the reearch i to deveop a ethod of echanica and atheatica odeing and coputer iuation of dynaic tabiiation of bow rotation in the vertica pane intending to get practica recoendation for the port of archery. Behavior of a fexibe tabiier in the ain pane of the odern port bow deigned in the frae of nternationa rchery Federation i anayed uing a echanica and atheatica ode. The ode i deigned baing on Euer-Bernoui bea and agrange equation of the econd kind. n engineering oriented ethod baed on virtua ode and Rayeigh-Rit procedure i deveoped to tudy natura frequencie of the archer-bow-tabiier yte. The reut of odeing of the archer-bow-arrow yte correate with we-known reut of a high-peed video anayi: the proce of coon otion ha a ignificant non-inear character. Keyword: port archery, bow, dynaic, tabiiation, odeing, iuation.. ntroduction The ain feature of the odern target bow deigned according FT (nternationa rchery Federation) Standard [3] are a rigid centra part naed a rier with a hande and two fexibe ib fixed at the end of the hande (Fig. ). The untrung ib have got recurved horn at the free end and are bent backward off the interna part of a bow. To ake a hot, a bow i ituated verticay with it ain pane. Depite a ipe contruction at firt ight, bow ove urpriingy copicated. t otion i in 3D pace before, during, and after the hot. Dipaceent of the bow rier i ignificanty aer than dipaceent of the arrow, tring dipaceent, and two ib dipaceent in the vertica pane. the yte ove ateray too, but thi oveent i aer in coparion to the oveent in the ain (vertica) pane. To avoid oe part of bow rier otion, odern port bow are uppied with a ong cantiever rod (or a uti-rod packet) ounted in front of the rier directy to a target. dditionay, there are fro one to three hort cantiever rod ounted on the rier in different direction reativey to the ain rod but their roe in bow tabiiation i econdary and their avaiabiity on the rier i optiona. Bow rier oveent before the hot, i.e. before the tring reeae off archer finger, i under the archer contro and directy affect aiing. Moveent during the hot, i.e. during tring and arrow coon otion i party controed by the archer hand hoding a bow rier and party i free. Thi oveent affect aiing to oe extent. Bow oveent after the hot, i.e. after an arrow aunche the tring doe not affect arrow fight but caue the archer to odify a tye to anticipate the oveent. Eion (996) caified thi otion a dipaceent, rotation, and vibration and quaified the character of the according to the three phae entioned above []. To uarie a tabiier yte behavior, he pointed two ain function: to axiie the dynaic tabiiation of the bow, particuary with repect to the bow recoi and to etabih a 'baanced bow' with the force on the bowhand running through the bow ar. The ain part of the hande otion i rotation. The ai of the reearch i to deveop a ethod of echanica and atheatica odeing and coputer iuation of dynaic tabiiation of bow rotation in the vertica pane intending to get practica recoendation for the port of archery. Pubihed by Word cadeic Pre, Word cadeic nion

2 4 hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan Fig. : rcher with a port bow [3].. Modeing of a Stabiier.. Coon pproach to the Mode Deign et conider a copound hinge and a rode echani a a bow chee ode [7]. tabiier i odeed a an eatic rod joined to the hande ike a cantiever bea. ccording to the reut of the video invetigation [], a bow tabiier bend according the ain ode of natura ociation. Energetic ethod of dynaic echanic obtain coprehenive accuracy for echanica engineering cacuation. Therefore, we can deign a echanica and atheatica ode of a bow with tabiier uing a hypothetica function of the ain ode of the bea. ccording to the theore of appied echanic, a preciion of natura frequencie obtained with the energetic ethod i near the preciion of hypothetic function having been ued [4]. Sport archer tretch a bow during the tie of coon otion of a tring and an arrow with the fixed hand trying to keep a teady poe of a body. Body a i ignificanty greater than bow a. Therefore, we can aue the point of contact a iovabe, i.e. a pivot point. nguar dipaceent of the rier (eaured in radian) i uch aer than unite; therefore we can ue a inear ode for it odeing. nertia propertie of the bow are odeed with a oad reativey to the pivot point. So, the ode of the tabiier coud be a one end pinned eatic bea with a point a that ha got a oent of inertia on the ae end. To approbate hypothetic function of tabiier dynaic bend, we conider a range of ratio for bea and joined oad a. f the oent of inertia of the oad i uch ore than the oent of inertia of the bea reativey the pivot point, the probe i near anayi of natura ociation of a cantiever eatic bea. f the other way round, the probe i about the eatic bea with one free end and another pinned end. t the very beginning, we aue the hypothetic function of the ain ode a a function of tatic bend of a cantiever bea oaded by a concentrated force at the free end (Fig. a): F η () = ( 3 ), () 6ε where F i a oading force; ε i ditributed bend tiffne of the bea; i ength of the bea; i ongitudina coordinate; η i tranvere dipaceent. The function () atifie three of four boundary condition, i.e. the geoetrica condition and one of two dynaic condition. The geoetrica condition are ero dipaceent and ero ange of cro-ection turn at the fixed end. Ony one dynaic condition i atified by the function (), i.e. ero oent of the force, but ero cro-ection force at the free end in t E-ai addre: [email protected]. SSci eai for contribution: [email protected]

3 nternationa Journa of Sport Science and Engineering, (008), pp atified. However, thi function i very fruitfu in the probe becaue the error of the ain natura frequency inear 0,73 % [6]. Fig. : Schee ode of a bow tabiier a a cantiever eatic bea (a) and a one-end hinged eatic bea: ode and (b); ode 3, 4, and 5 (c); the ode 6 (d). a hypothetic function of the ain natura ociation ode of one hinged bea (Fig. b) we aue a u of one-haf inuoid wave and a inear function: π η = in, () where А i a function of tie; i an ange of inuoid wave turn. ike () the function () atifie ony three of four boundary condition, i.e. ero dipaceent and ero force oent at the pinned end of the bea and one dynaic of the two boundary condition, i.e. ero force oent at the free end. nother dynaic boundary condition i not atified, i.e. ero cro-ection force at the free end. Depite thi, the function () aow an appreciated preciion of the ain natura frequency becaue the error i near,0 % [0]. Becaue there are no reut on the probe of natura frequencie of a one end pinned bea with a oad in we-known echanica and atheatica pubication, we conider thi probe uing Haiton variation principe: t δ ( T P) dt = 0, t where T = µ η η d t t and 0 = 0 η P = ε d are kinetic and potentia energy 0 correpondingy; µ i ditributed a of the bea; t i tie; i oent of inertia of the oad reativey the hinge axi (ee Fig. b). Pacing the two at expreion of energy in the Haiton functiona, we get correpondent differentia equation: and boundary condition 4 η η µ ε = 0 ; 4 t 3 3 η η η η = 0, η = 0, ε = ; =, = 0, = 0. (3) t Soution of the probe (3) obtained uing Kryov function are root of the deterinant: ch h k v( k 4 ) v( k 4 ) ( k) co( k) h( k) in( k) ( k) in( k) ch( k) co( k) = 0, (4) SSci eai for ubcription: [email protected]

4 6 hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan 3 where 4 ω k = are dienione vaue of natura frequencie; i a of the bea; ω i circuar ε natura frequencie; v = i dienione vaue of the oent of inertia of the oad. Zero oution of the equation (4) correpond with coon rotation of the bea and the oad reativey the hinge axi. v=0, we get k=0; 3,97; 7,069; 0,0; 3,35;, which are the ae a we-known oution for π ( 4i 3) the bea with one hinged end, where i i a nuber of natura frequency [5]. There i no ero 4 oution a v= : k=,875; 4,694; 7,855; 0,996;, which are the ae a we-known oution for the π cantiever bea ( i ). Soution of the ain frequency for different reationhip of a-inertia paraeter are preented on the Fig. 3 (ODE ine). Fig. 3: Main natura frequency v. bow and tabiier a-inertia paraeter: ODE i conidered a an exact oution (eq. 4)... Mode Verion of the Stabiier ing a hypothetica function of the ain natura ode, we can appy agrange equation of the econd kind: d T T P, dt = 0 q i qi qi where q i are generaied coordinate; prefix how a partia derivation in tie, i.e. ( ) t. Mode verion i deigned a a u of () and a inear function: η = 3 (6) η generaied coordinate, there are А and. The ange of the oad turning i =. fter =0 ubtituting of the hypothetica function (6) in the equation of energie and then in (5), we get expreion for the ain natura frequency: 4 / 400 ( k) = / 33 / 35, (7) / 3 v where (/) i a ign of diviion. For the ode verion, we appied the hypothetica function (). ike ha been ade before, we get the (5) SSci eai for contribution: [email protected]

5 nternationa Journa of Sport Science and Engineering, (008), pp η π ange of the oad turn that i preented here a expreion =. Correpondent expreion =0 of the ain frequency i: ( / π v ) 4 4 π π ( k) = / vπ / 3 v. n the ode verion 3, hypothetica function are aued with the expreion: η = ; η =. (9) c Virtua tiffne and a of the bea are ocated at the free end (Fig. c). ing the function (), we 3ε 33 η get: c = ; 3 c =. The ange of the oad turning i =. Correpondent expreion for the 40 =0 ain frequency i: ( / 3 v) 40 ( k) = (0) v Mode verion 4 i different of the ode verion 3 ony with the vaue of virtua a c = that 3 correpond with the vaue of the oent of inertia of the bea reativey the hinge axi. Here, a forua for the ain natura frequency i: 4 ( k) 4 9 = ( / 3 v) v (8). () 33 Mode verion 5 i a cobination of two previou ode verion, i.e. 3 and 4: c = ; 40 4 = 40. Genera virtua a c = correpond with the vaue of the bea oent of 3 inertia. The ain natura frequency i cacuated with a forua: ( 33 / 40) 4 ( k) = 3 / 33 / 40. () / 3 v n the ode verion 6 the bea a i ocated in three point, i.e. at the bea-end 0 = = 6 and in the idde of the bea =. Stiffne i concentrated in two point, i.e. in the idde of the 3 6ε bea and at the free end: c = c = (Fig. d). Thi i equa to the tiffne of the cantiever bea: 3 3ε c =. ing the function (6), we get an expreion for the ain natura frequency: 3 4 ( k) = ( 9 / 6) / 89 / / 576 / 3 v. The reut on the reative accuracy for the a ix ode verion (7), (9)-(3) reativey the ODE oution (4) are grouped in the Tabe. Fro the practica point of view, appreciated reut regard the accuracy of the ain natura frequency i obtained with the ode verion,, and 6. But the bet accuracy in a wide range of a-inertia paraeter of the bea and the oad (,5 < gv < ) verion. Mode verion i appreciated ony for a oad ( < v < 3), we get uing ode g. ing the ode verion 6, we get ediocre eve of accuracy, and ony in a narrow range of reationhip ( g v ) the accuracy i appreciated. The ain natura frequency reut obtained with the ode verion,, and 6 and the ODE oution (4) are preented in a non-dieniona for (Fig. 3). Taking into account a rea reationhip between bow and tabiier a-inertia paraeter (3) SSci eai for ubcription: [email protected]

6 8 hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan (, 5 < g v <, 0) and conidering the reut of cacuation (ee Tabe and Fig. 3), we can chooe the ode verion a the bet one. The aet error of the ain natura frequency reut for thi ode i 0,6 % when the reationhip v=0,0537 (gv=-,7). Tabe : Reated error of the ain frequency for different ode verion of a bow and tabiier yte (%) gv Nuber of a ode verion ,30,0 055,69 959,79-34,6 5,6-5 9,30, 550,0 496,07-34,6 5,7-4 9,8, 65,79 35,44-34,58 5,0-3 8,0,5 07, 90,0-34,5 4,6 -,54 3,4 5,59 5,7-30,66,8-0,7 5,5 3,7-5,30 -,90 3,08 0 0,66,0 0,98-7,40 -,35 4,36 0,7,7 0,76-7,60 0,5 4,5 0,73,80 0,7-7,65 0,69 4,5 3 0,73,8 0,74-7,6 0,74 4,55 3. pprobation of the Mode 3.. Mode of iing n a tatic probe of the bow and tabiier yte [], we can aue ony two externa force, which act to the hande and to the tring, i.e. the force in the point correponding О (the pivot point) and А (nock point). atheatica ode of the bow at the drawn ituation i (Fig. 4): ξ = in inγ ; ξ = in inγ ; η = h co coγ ; c ( ϕ ) = F in( γ ); η = coγ co h ; c ( ϕ ) = F in( γ ); Fξ = F F inγ S F inγ ; Fη = F S coγ F coγ η = f ; F = f ; tgφ = ; tgφ =, S S Fξ ξ where, are ib ength;, are ength of tring branche in the drawn bow ituation; S, S are ength of the branche of a free tring; h, h are virtua ength of the rier, i.e. the ditance fro the pivot point (O) to the point of virtua eatic eeent of ib; c, c are tiffne of the ib; F, F are force of the tring branche; F ξ, F η are projection of the drawn force to the axi of co-ordinate; a i a ength of an arrow (and drawn ditance too); f i a paraeter of tiffne of a tring. The ubdivide and ark correponding the upper and ower ib. Matheatica ode of the braced bow i: co co h h = coγ ; in in = inγ ; B B B S FB in( B γ B ) = c ( B ϕ ); FB in( B γ B ) = c( B ϕ ); F f B B =,(5) S where B i the tring ength in the braced ituation; FB i the force of the tring tretch; S i the ength of a free tring (ee Fig. 4). Subdivide В ean paraeter of a bow with a braced tring. Syte of equation (4) and (5) incude trancendent and non-inear function, o they coud not be oved anayticay. Therefore, conidering the tatic probe, we appied nuerica ethod uing a progra B F B η B ; B B (4) SSci eai for contribution: [email protected]

7 nternationa Journa of Sport Science and Engineering, (008), pp Find fro the oftware package Mathcad 000i Profeiona ( Fig. 4: Static chee ode: braced bow (); drawn bow (). 3.. Bow, Stabiier, and rrow Coon Motion The ain attept to the dynaic probe on bow and arrow yte ha been founded [8, 9]. ing coordinate of the nock point, ib ange, the anguar co-ordinate of the rier, and the anguar co-ordinate of the arrow we coud for expreion of energy (Fig. 5): ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ψ η µ η ξ H a co h r co h r h h d T a 0 ; ( ) ( ) ( ) ( ) ( ) ( )( ) = a a d g d S S f S S f c c P 0 0 ψ η µ ψ µ ξ ϕ ϕ (6) where 3 = i a of a tring pinned to the nock point; i a of a tring; i a of an arrow; a ( ) µ i ditributed a of an arrow haft; i a co-ordinate fixed to the arrow axi; ψ i an attitude ange of an arrow; H i oent of inertia of a rier reativey the pivot point; are a of ib with added a of tring, ( 3 ) pinned to their nock point; are oent of inertia of ib reativey their joint to the rier with addition the ae part of tring a; are ditance of center of a of ib to their joint; i gravity contant.,, r r g SSci eai for ubcription: [email protected]

8 0 hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan Bow and arrow interaction ha been decribed according the ode (6) taking into account ony the interaction at the nock point. The initia poition of the arrow reativey the bow in it ain (vertica) pane i deterined with a ret that hod an arrowhead. ret i fixed to the hande and ha got abiity to turn and to diappear jut an arrow tart oveent. Thank a a ie and a, the ret doe not accuuate ignificant aount of energy, therefore it interaction with an arrow i not taken into account in the frae of the ode. Soving the dynaic probe ike the tatic probe (4) and (5), we do not conider gravity force acted the bow. But we take into conideration an arrow weight becaue it force oent acted the arrow i the ae vaue a the oent of inertia force. Kinetic and potentia energy of the tabiier with the ain natura for (6) according the ode verion i: ε Tt 33 = t t t P = ct, (7) ; ( ) where ct = 3 i bend tiffne of a cantiever tabiier oaded by a tranvere force at the free end. 3 t Pacing the expreion (6) and (7) to the agrange equation (5), we get a yte of differentia equation of the econd power reativey the generated co-ordinate qi ξ, η, ψ,,,, : ( a ) ξ e S es4 = 0; ( a ) η arψ a g e S es3 = 0; ψ ar ( η ξ ψ g) = 0 ; ( ) r h b c ( ϕ ) e ( bs bs) = 0 ( ) r h b c ( ϕ ) e ( b S b S ) 0 ; ( h ) [ ( ) ( ) ] H e = h tt r h b b 3 rh [ b ( ) b ( ) ] 3 4 t 0 t [ S ( b h ) S b ] e [ S ( b h ) S b ] = 0; t ( ) 4c = 0 0 t 35 t, (8) where i a oent of inertia of the arrow reativey it tai; r i a ditance fro the tai to the center of a of the arrow; f ( S ) f ( S ) e = ; e = ; = S S ; = S3 S4 ; S S S = h b η ; S = h b ξ ; S3 = h b3 η ; S4 = h b4 ξ ; b = co( ); b = in( ); b3 = co( ); b4 = in( ). ; SSci eai for contribution: [email protected]

9 nternationa Journa of Sport Science and Engineering, (008), pp 03-4 Fig. 5: Dynaic chee ode (a); chee ode of an arrow (b). The initia condition of the probe are: t = 0, ξ = a; η = η 0; = 0; = 0; = 0; ψ = ψ 0; = 0; ξ = 0; η = 0; = 0; = 0; = 0; ψ = 0 = 0, where contant η 0, 0, 0 are the oution of the tatic probe (4). Zero vaue of derivation correpond the anner of the port archer technique, i.e. a breathing i topped and a poe i otione. Stabiier paraeter pay in the two at differentia equation (8). The yte of equation (8) with initia condition (9) repreent a Cauchy probe for the ordinary differentia equation of the econd power. t i ipoibe to get anaytica oution for the probe, therefore we ued Runge-Kutta ethod appied in the progra NDSove fro the package Matheatica 5. ( 4. Coputer Siuation et conider a odern port bow with paraeter: = 48 c; = 95,3 g; = = 63,44 kgc ; r = r =,7 c; c = c = 078 Nc; ϕ = -0,06; ϕ = 0,06; h h = 43,4 c; = 70 kgc ; S = 80 c; S = 90 c; f = 900 N; = 7 g; = 70 c; = H a = 5 g; = 55,5 kgc ; r = 4,3 c; t=,486 ; t =0,304 kg; c t =70 N/. Fro the tatic probe (4), we have got η 0 = 4 ; 0 = 0,86; 0 = 0,694. The ret point ituate at the coordinate η P0 = 34. The reut of oution of the probe for the paraeter above are preented in the graph (Fig. 6, 7). n arrow aunche a tring nock point a their coon ongitudina acceeration becoe ero. t the = = a (9) SSci eai for ubcription: [email protected]

10 hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan intant an arrow ha the axia ongitudina peed. The tie of bow and arrow coon otion i about 5,8. The graph decribe a proce of bow tabiiation in the vertica pane during the bow and the arrow ove together. Bow rier anguar otion cockwie i party copenating by tabiier bend countercockwie. onotone character of thee oveent tetifie a beow reonance regie of the proce. Bow and arrow coon otion ha an extreey non-inear character. During the otion, the acceeration of the ongitudina dipaceent ξ decreae fro approxiatey 640 g to 70 g, and then increae again up to 50 g (ee Fig. 6 a).; in the finihing phae it decreae to ero. Siutaneouy, the ongitudina projection of the arrow veocity increae (non-inear again) up to 5.8 / (ee Fig. 6 b). t the tie, the rier turn on a a ange = rad (ee Fig. 6 c), i.e. it ove forward with it upper part. Thi i proved correct by high peed video fi []. String and arrow coon otion (interna baitic) i accopanied with intenive ociation, which are caued by detruction of the tatic baance of force at the point of tring reeae. There are even fu cyce of ociation during the otion (ee Fig. 7). The eve and the character of dynaic tabiiation are to a certain degree depended on tabiier bea tiffne. Reut of the cacuation experient on the iue are preented in Tabe. We can notice that tabiier bea tiffne increae caue bow rier turn increae. Coparing the bow without a tabiier (= 0,009 rad) and the bow with aboutey oid tabiier ( = 0,006 rad), we can deterine a difference of a bow rier turn about 65 %. Tabe : Kineatic paraeter of bow tabiiation at the intant a an arrow aunche a tring c t, N/ -000* *, 00* /, %, ,45, ,53, ,6, ,73,6 49 0,9-65 SSci eai for contribution: [email protected]

11 nternationa Journa of Sport Science and Engineering, (008), pp Fig. 6: Kineatica paraeter of the yte: ongitudina acceeration of the arrow v. tie (a); ongitudina peed of the arrow (b); bow rier ange utipied by tabiier ength (c); pure bend dipaceent of the free end of the tabiier (d). On the other hand, dynaic bend of the bea tabiier decreae a the tiffne increae. For exape, in the cae of a cyindrica tube tabiier (с t =000 N/), the bend i,6 tie grater than in the cae of four rod packet of the equa a (с t =000 N/). t the ae condition, tabiier a increae caue decreae of bow turn otion beow the reonance one. Fig. 7: Kineatica paraeter of interna baitic: an ange of attack of the arrow v. tie (a); an attitude ange (b); an anguar peed of the arrow in the vertica pane (c); an ange of the peed vector of the arrow ongitudina oveent reativey horion (d). SSci eai for ubcription: [email protected]

12 4 5. Concuion hor Zanevkyy: Modeing and Coputer Siuation of Bow Stabiiation in the Vertica Pan For rea reationhip of bow and tabiier a-inertia paraeter, we can get the bet accuracy of the ain natura frequency of the yte (near, %) uing a hypothetica function a a cobination of a inear function and the function of tatic bend of a cantiever bea oaded by a force at the free end. String and arrow coon otion (interna baitic) i accopanied with intenive ociation, which are caued by detruction of the tatic baance of force at the point of tring reeae. There are even fu cyce of ociation during the otion. Bea tabiier tiffne increae caue a ignificant decreae of bow turn otion and decreae of dynaic tabiier bend. t the ae condition, tabiier a increae caue decreae of bow turn otion in beow the reonance one. The ode decribe the proce of bow tabiiation in the vertica pane during the bow and the arrow ove together. Bow rier anguar otion cockwie i party copenating by tabiier bend countercockwie. onotone character of thee oveent tetifie a beow reonance regie of the proce. The attept to the probe of port bow tabiiation in the vertica pane propoed in the paper i addreed to practica need of appied engineering echanic and the archery port. The ode and ethod have been adapted for reaiation in an engineering ethod uing we-known atheatica CD yte. Nuerica reut of coputer iuation are preented in tabuar and graphica for, which ake it eay for porten and coache to ue. 6. Reference [] S. Eion. Controing Bow Behaviour with Stabiier [] Werner Beiter Zeigt. Highpeed fi, 99. [3] B. Stuart. Standard Bow Manufacturing Manua (ed. J. Eaton). nternationa rchery Federation (FT). Switerand: auanne [4].. Bechan,.D. Myhki, Y.G. Panovko. ppied atheatic. Ruian: Naukova Duka, Kiev [5] G.S. Piarenko,.P. Yakoviev, V.V. Matvieiev. Strength of ateria. Ruian: Naukova Duka, Kiev, 975. [6] W.Jr. Weaver, S.P. Tiohenko, D.H. Young. Vibration Probe in Engineering. John Wiey & Son, nc., 990. [7]. Zanevkyy. ode of tring-ib tiffne in the atera pane of the port bow. The engineering of port 4. Oxford: Backwe Science. 00: [8]. Zanevkyy. rcher-bow-arrow behaviour in the vertica pane. cta of Bioengineerig and Bioechanic. 006, : [9]. Zanevkyy. Bow tuning in the vertica pane. Sport Engineering. 006,: [0]. Zanevkyy. Dynaic of the panar ower pair echani with cantiever eatic bea. krainian:mechanica Engineering. 006,6: []. Zanevkyy. atera defection of archery arrow. Sport Engineering. 00,: ppendixe (oitted) 7.. MathCD progra on the ode of archery bow paraeter in the drawn ituation 7.. Matheatica coputer progra on bow tabiiation in the vertica pane SSci eai for contribution: [email protected]