MATERIALS AND SCIENC SPORTEN I S

Edited by: EH. (Sam) Froes and S.J. Haake

Materials and Science in Sports

String Materials Relatively Arrod an ww DynamicBo s Ihor Zanevksyy

Pgs. 83-93

TIMS 184 Thorn Hill Road Warrendale, PA 15086-7514 (724) 776-9000 MATERIALS RELATIVELY BOW AND DYNAMICS

Ihor Zanevskyy

Lviv State Institut Physicaf eo l Culture Kostyushko street 11 Lviv 79000, Ukraine

Abstract

A mechanical and mathematical model of the sport system has been created using non-linear ben bow'f do s limbs elastin a , c string, arro stabilizersd wan symmetria , w cbo arrangemen r resistancs flatnessai it e n i n teffecth d hysteresi A f .an eo t n modei s l were neglected mathematicaA . modee l parth f o tl includes Euler-Lagrange equation degreeo tw n si s of freedo energn a d myan equation. It has been shown that during computation of the sport bow performances it is necessary to take into consideration motions causebowstrine th y db g expansion. With other equal conditione sth increas bowstrina f eo g stiffnes decreasd an s mas it f seo resulte improvementh n i s e th f o t efficienc decreasd yan dynamie th f eo c loadin stringe th n go . Starting from this point i t coul e recommendedb e primarilus o dt y bowstring material e followinth n i s g consequence: , Dacron, Lavsan. It seems to be perspective to use high-modules carbon threads in bowstring's fabrication.

Material Sciencd san Sportn ei s Edited by F.H. (Sam) Froes IMS (The Minerals, Metals & Materials Society), 2001

84 Introduction

An archer-bow syste mdetermines f mechanicai o t lo a y db l characteristics. Efficienc vers i y y important among them. This performanc s introducewa e archero dt P.E.Klopstey yb g [1]e H , has determined it as the ratio of the kinetic energy of the arrow when it leaves the string to the energy imported to the bow by the archer as he brings it to full draw. A precision of a shot in archery in a great extent is depended by the arrow speed. The archers prefer high speed cast othee bowth s rsa things being equal highee Th . r speed result lowee sth r arroe timth f weo flight longee Th . rarroe timth f weo flight doe t furthesno shoe th r t precision. In first increase th , flighf eo t time increase probabilite sth influencf yo wina f eo d shogusa n to t precision n secondI . , lower spee d moran d e curved trajectory decreas e arroth e w stability orientation during the flight. This fact causes a variation of shots in shooting series. e Ootheth n r hand archers prefer light bows e aiminTh . n archeri g y takes place with considerable forces in archer's hands and high limb. It is clear, that a hard physical loading is e aimin t gooth r no dfo g accuracy. Archery efficiency accounts thos opposito etw e demands: e efficienc th weightw e bo hig us s n thicriterio w I a hy.n lo sspee ca wayd f e o nan dw , optimizatio archer-boe nth w system. The first scientific research in this field was done by C.N.Hickman [2]. B.G.Schurter has proved tha highese tth t theoretical leve archerf o l y efficienc beins yi arroe ratis gth a f woo mass to the arrow mass adding one third of the string mass [3]. For a typical arrow and string mass the efficiency is near 91%. But the real efficiency as result of measurements is essential lower. W.C.Marlow has taken into consideration a string elasticity [4]. He has supposed that arrow exit takes place limbw whe bo e strin sth n d stilan g l have substantial kinetic energyd an , therefore this energy is unable for kinetic energy of the arrow. Moreover, he has established thae potentiath t l energy remainin e strin th bowlimbd n gan gi s syste alsn mca o reduce eth amoun energf o t y availabl conclusios arrowe hi th t r eBu fo . n tha arroe tth w leave strine sth r gfo some time afte straight-line th r e string positio questionableo s s ni conclusios hi s a , n aboue th t kinetic energy in the bow limbs at the arrow exit instant. The aim of our research is to create mechanical and mathematical model of archery efficiency, investigato t e influenceth f bowstrino e g stiffnes efficiencyn o s preparo t d e practicaan ,eth l waves to increase efficiency in the sport archery.

Bow and Arrow System Modeling

strinmomene e th th s handlt o gw it a t f d bo o t spotse o an A th tw n o n t :archei w rbo acta o st release. According the modern-days manner of sport target shooting an archer does not grasp the bow handle. He or she only sets the handle. Hence, an archer acts to the bow with a force laterae th n i l t directionplan w ac doed t bo ean no e s onlth . n Thereforyi ignorn ca e e eth w e lateral handlee actiomodelr th ou n archen n o i A . r hold strine sth g with another hand. When he or she releases it the lateral force caused of the fingers acts to the string at the initial instant of the motion. Hence, we can model this momentary force with a mechanical linear momentum. We can neglect in the modern sport bow and arrow model a friction, an air resistance, and other energy dissipation totaA . r resistancl ai energ n a f limbsy yo eb arron stringa ,a d w an ,hav e been estimated about 1.4% of the complete system energy [4]. In the most part of wooden bows an energy loss caused by hysteresis lies somewhere between 5 and 20% [1]. For the contemporary modern sport bow higa f hso quality design this energ s lesyi s tha. Thesn3% e losses are rather smaller then the whole loss of energy transferring from limbs to the bow. We shall count up all these loss using a coefficient of efficiency below. An archer-bow system model in full draw aiming phase consists a rigid riser, bend limbs, elastic string d rigian ,d arrow e archer'Th . s arms have effece th t e riseforced th an rt a s bowstring. Potential energy has imported from the archer and accumulated in bend limbs and

85 taut string. After the string release the archer affects to the bow only at the riser. The arrow moves commo strine nth g increasin speede gth . acceleratioe Th projectionarroe m th su f wcauses uppee ne i o th th lowed f y an ro sdb r string forces on the arrow direction. The arrow leaves the string, as it becomes a straight line. A part of potential energy from limbs transfer kinetio t s whole th c t energearroe no oneth t f wyo bu . rese tTh par thif o t s energy transfer potentiao st kinetid an l c energ stringe th f yo , e limbsth d an , riser. Tsai-Chen Soong has done the analytical investigation of an optimally designed archery, which duarchee eth ric o maximart e hth l possible arrow give s speehi r nd fo physica l possibilities [5]. This research was carried out in a static positing neglecting the influence of dynamical forces that decrease reliabilite sth thif yo s work results. A complete descriptio e mathematicath f arrod no an ww modelbo l s beeha s n presentey db B.W.Koo s summarizeiha [6]e importane H . th l dal t quantitie modee th n si l which determine the mechanical action of the bow: length of the limbs, length of a grip (handle), the shape of the unstrung limbs cross-sectioshape e th th , f eo limbe l positional th t f sna o s alon limbse gth e th , elastic propertie limbse th shape f ears se o masth ,d th ean f , horn se o mas th f sso (stabilizers)a , draw length, mass of the string, elastic properties of the string, and mass of the arrow. Archery efficiency depend energe th f so y value transferre arroe th wo dt frolimbsw s mi bo t I . reasonably to estimate this part of energy at the moment of an arrow exit the string. As we assume linea forcw r bo eorde a durin f o r drawings git assumn ca e proportionaew a , l ordef o r the energy dissipation has not accumulated by the arrow until its exit. Let us consider the force acte arroe dth acceleratind wan . Thankgit wore th s k this force e arroth s w store s kinetiit s c assumn energyca ordee e eth W . thif o r s force inversely proportiona arroe th wo t l longitudinal displacement like a directly proportional draw bow force. The value of this force can be assumed from a value of kinetic energy of the arrow at the instant of its exit the string that is equa e energth l ylimb w storebo sefficiencyw n i dmultiplie bo e th y . b dKlopste s ha ] [1 g determine efficience dth ratia s kinetif oa yo c energarroe th f wyo whe t leaveni strine sth o gt the energy imported to the bow by the archer as he brings it to the full draw. The rest part of energy remains in the bow as kinetic and potential energy of a string, limbs, and other singles. bowe Th f primitivo s e design hav t efficiencego y less than 50%. Half-round cross-sectional area long bows teste Hickmay b d efficiencie ha ] [2 n s about 60% e rectangulaTh . r cross- section bows teste y Klopstegb d ] fel[1 l efficienc% int70 o y value category. Efficiency measurements were occasionally reporte populan di r literatur fald 60-85n i ean l % efficiency range [4]. The efficiency of the modern sport bows is not smaller then 75%. 11% of the energy limbw bo t ful storesa e l th dran di wbouns i kinetin i p du c energ limbe th f t arroyso a w exit% 9 , remains in kinetic energy of the string. Less than 2% energy loses with the air resistance and less wit - tha hysteresise % hnth 3 . Experimental data results relatively the correlation between string properties and an arrow speed has been undertaken by C.Tuun and B.W.Kooi [7]. They pointed out that there are two counteracting hane effects velocite on dth n O . highes yi r whe strine nth lighters gothee i th n r O . hand thicker the string, the stiffer it is, and the higher the velocity, i.e. the archery efficiency. Their results for a Fast Flight string on the Greenhorn bow type 68" 30# are in the Table.

Table Velocit Functioa s yNumbea e th f no Strandf ro s

Numbe Strandf ro s Mass g , Velocitys m/ ,

16 6.45 51.88 14 5.82 51.43 12 5.28 51.62 10 4.64 51.76

86 Mechanica Mathematicad an l l Mode Bowstrine th f o l g

We conside bowstrine th r absolutn a s ga e flexible string tha tensioneds i t . Therefore thers ei onl longitudinaya l tensile crosforcs it n esi section equationo S . motiof s o elemen n a e f no th f to strin (Fig.l)e gar :

////) = =+//(|v ) s ,(T, v V (Tco g} n V+ 'si (1) longitudinae th whers i N e l tensil angle eth forces ei betwee v ; strine nth g axiverticald san ; s distributei jU d mass alonverticae ar e string g7 th horizontad 7 an l d ; £an l displacemenf o t string axis points s graviti g ; y constant consequentlypartiae ar ) (• l derivatived an ) ( ; s with

respect to the longitudinal coordinate connectef d with the longitudinal string axis 8 and to the time consequently; the signs (+) are related the upper and lower branches correspondingly.

Figure 1: Intermediate position of a bow and an arrow during their common motion [8] (a). Scheme model of a bowstring (b).

Assuming a linear order of tension of the string, i.e. Young's modulus of its material is constant, we get the equation:

(2)

whers distributei , A e o equationtw d t stiffnese ge stringth se f froW o .s m geometrical correlation:

(3)

87 Equations (1) have been transformed with projections to the normal and tangent axis of the string consequently are:

N' + ju^£ + gjcosv - 77sinv\- 0; Nv'- //[(^ + gjsin v + r/cosv\ = 0. (4)

We can assume v = if because relative transverse displacement / is considerably smaller than the string length. Substituting the last expression and (2) in (4) we get wave equations of the bowstring motion:

(5)

fT" From the first equation (5), we get a speed of longitudinal waves in the string: I— ; and from

the second one - a speed of transverse waves: I— . According [4] for a soft Dacron bowstring

these speeds are 1700 and 220 m/s consequently. So the time of propagation of the diametrical wave equa lengte halth a l f bowstrine fho th par f o t s 10-5gi 0 times smaller tha time nth f eo common motioe bowstrin th e arrow f th r no bowstring d Fo . gan s designed with more rigid material, e.g. Kevlar, this ratio is about 100. The period of transverse vibration of the string is commensurable wit time h th commof eo n motio bowstrine th arrow e f no th d .gan bowstrine th s a r Sofa g branch durin motioe fora gth s mnha nea e straigha r w ) t «l linV ( e can introduce two assumptions. The first of them is that longitudinal waves do not cause variation of the tensile force along the bowstring branch. The second one: we can neglect the longitudinal deformation when kinetic energy of the bowstring has been determined. We put gravity forces of the bowstring at the ends as two concentrated forces equal a half of the weight of the branch.

Lumped mass of a string

Instead distributed mass of the bowstring we place particles at the nock point and at the tips. There are two conditions of this operation: preservation of common mass and common kinetic energyconsides u t Le uppee . th r r branch. Assumin straighe gth tstrine forth f mgo brance hw geexpression a t kinetif no c energ uppee th f yro branch examplr fo , e (Fig.2,a):

(6)

where Su is the length of the upper branch; gs and 7)3 are projections of the velocity of a string point to the coordinate axis. t expressionsThenge e w , projectionf o velocite th f so y using speed: T pointf d so an sA

Substitutin lase gth t expression thed an n ) integratin(6 n si gete w :t gi

88 msu K = (7)

uppe e mass i th wher f rsu o s branchem .

f]T

a

Figur : Scheme2 e mode strinf o l g mass lumpinn gi arbitrar initiae th y n situatioi l situatiod an ) n(a f no On the other hand, we can get the expression of kinetic energy using the particles:

(8)

particles e mase th ar f s o T m . d wheran A em Equaling the right parts of the expressions (7) and (8), and taking into consideration the second

condition (mA+mT = msu ) we get:

+f}A+ mA =

mT = (9)

similae Th r forms hav t expressionego masr sfo s particle lowee th f sro branch. initiae Atth l position whe archen na r release stringvelocitiee e sth th l al , zeroe sar :

89 Hence the formulas (9) become to indefinite relations 0/0. Therefore we use the method of possible motions to determine virtual masses m A and mT . Thanks the bow is immovable at this instant we can consider the handle as motionless basis for limbs and the string. Let us consider two symmetrical positions of the string relatively the initial position (Fig.2,b). Virtual velocitie strine th f gso are:

where Ax and Ay are horizontal and vertical distances between the two symmetrical positions consequently same th s ei interva t timee A ; th f .o l Substituting expressio gete w n:) (10(9 n )i

2 2 _msu Ax -t-Ay III A —— 0 Ax Ay+ 2 ^ -Ax2 -Ay2 m Ax 0 + Ay9 TM _ su T r nl nr — 3 Ax^+Ay^-Ax'-Ay2

The similar forms have got expressions for mass particles of the lower branch.

Dynamic Arroe th f swo Launch Process

bracee Ath t d position tensil e (Fig.3th , e T) ,forc constans ei t alon stringe gth :

(12) where c is string stiffness; 25^ is the length of a braced string; 2S0 is the length of a free string. A conditio equilibriuf no braceda f mo bowlim: bis

kfa =Nllsin0l9 (13)

where k is bowlimb stiffness lumped to the grip; / is a length of the bowlimb; ^ is a bowlimb angle. A geometrical equation of between dimensions of the bow is:

Sl =6 + /cos^l5 (14)

where 2b is a bowgrip length. In the situation when an arrow leaves a string (Fig.3 ,'2f), a string force increases fro noce mth k lim e poininstantaneous thath p it bti o s t i t s poin rotationf o t :

90 where strina A s i ^ g noc e forcth kt e a strinpoints i s gm ; massinitias i 2 V ;l arrospeee th f wdo when it leaves the string; 2^S2 is a string length in this situation; % is a string coordinate for integration.

O

Ni iNn//

T, T2

After integration of the last expression, we get:

(15)

Assuming here ^ = S0, we get the string force at the limb tip:

91 NL=N*+tf- (16)

Lengthenin strine th : f gis o

Substituting in the last expression (15), we get after integration:

l+l2^' (1?)

An equatio dynamif no c equilibriu lime : th bf mis o

(18)

where / is a moment of inertia of the bowlimb relatively the hinged grip point. Geometrical equatio thin i s situatio similas ni (14)o rt :

Geometrical equatio arbitrarn i y situatio: nis

b + lcos =S cosu,

strind an wher gw anglebo e e (j),var s (see Fig.3). After differentiation of the last equation we get:

. u n si u S - u s co u S - u n si u S 2 - u s co S = $Jn cosi s $+ 2

We have got according the previous u = 0, ^ = 0 when the arrow leaves the string. Assuming

here S = 0, V2 = U2S2 we get:

$m(/)2=V2. (20)

An equatio energetif no c balanc situation ei n whe arroe nth w leave strine sth : gis

w = m 2 + k +2 s 2 K( « + K^h (ti - ti) 4( 2 -sj-(sl -s0) } (21)

wherpotentias i fula n li eW dra w lw bo energ e situationth arrof s yi o a wm ; mass. Numerical results for modern sport bows in FITA (International Archery Association) version are presented in Figure 4. The archer-bow-arrow system parameters for this example are: the energy imported to the bow by the archer as he brings it to full draw is 42.9 J, bow limb

length is 48 cm, limb angle is 30°, limb mass moment of inertia is 5.28 gm, limb constant is 121 Nm, string mass is 7 g, arrow mass is 25 g. 2 92 F(N) V2(m/s 750 55

90

70d_50 80

FL 650 45 70

60 600 40

8 10 12 14 o o 16 C(kN/m) Figur Relationshi: e4 p between string elastic constan strind an gC t tension lime forcth bn arrod ei poin an wL tF knock poin spee, tFA d arrow exit efficiencVad an , yE

Conclusions bees Iha t n shown that during computatio spore performancew th f tbo n o necessars i t si tako yt e into consideration motions causebowstrine th y db g expansion othee th t r A .equa l conditions increasin e bowstrinth f go g stiffnes decreasind an s s masit f sgo result improvinn i s e th f go archery efficienc decreasind yan dynamie th f go c loadin . Startinit n go g from this poin t couli t d be recommende e primarilus o t de bowstrin th y g material e followinth n i s g consequence: Kevlar, Dacron, Lavsan. It seems to be perspective to use high-modules carbon threads for the bowstring fabrication. The model and methods have been approved for a few Ukrainian top archers.

References

1. P.E.Klopsteg ,d , an "Physicw "Bo f Americao s n Journa f Physics o l) (1943)(4 1 1 . , 175-192. 2. C.N.Hicman, F.Nagler d P.Klopstegan , , Archery e TechnicaTh : l Side (National Association, 1947). 3. B.G.Shuster, "Ballistic e Moderth f o s n Working d Arrow,Recurvean w " Bo dAmerica n Journal of Physics. 37 (4) (1969), 364-373. 4. W.C.Marlow, "Bow and Arrows Dynamics," American Journal of Physics, 49 (1981), 320- 333. 5. T-C.Soong, "An Optimally Designed Archery," (Xerox Corp., Rochester, NY, 1986). 6. B.W.Kooi, "Archer d Mathematicaan y l Modeling. Journae Societth f f o Archero ly - Antiquaries. (1991), 21-29. 7. C.Tuu B.W.Kooid nan , "The Measuremen Arrof o t w Velocitie Studentse th n si ' Laboratory" (www.stiident.utwente.nl/-sagi/artikei/speed/aiTovv.html). 8. P.Baier et al., Instructor's manual (Colorado Springs, Co: National Archery Association of USA, 1982), 127.

93