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 STRING MATERIALS RELATIVELY BOW AND ARROW 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 bow and arrow 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 archery 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: Kevlar, 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.