THREE DIMENSIONAL GOLF BALL FLIGHT
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1 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies THREE IMENSIONA GOF BA FIGHT Ansts Ino Julin Joro Todor Kbleshko Uniersity of Trnsport Sofi BUGARIA ii00@b.bg Uniersity of Chemicl Technology nd Metllurgy Sofi BUGARIA july@uctm.edu Abstrct: The three dimensionl (3) flight of golf bll t tking into ccount the Mgnus effect is studied in the pper. For this purpose it is composed system of six nonliner differentil equtions. To determine the 3 orienttion of the bll the rottions round ll three xes re gien by the so-clled Crdn ngles insted of clssicl Euler ones. The high nonliner system differentil equtions re soled numericlly by specil progrm creted in the Mtb-Simulink enironment. It is founded the lws of motion elocities nd ccelertions on ll six coordintes s well s the projections of trjectory on the three coordinte plnes. The presented nlyticl bse nd numericl results in the pper incresing nd expnding the knowledge in the theory of generl motion of sphericl solid nd leds to new more extensie reserch in this complicted re. Keywords: three dimensionl flight golf bll drg nd lift forces Mgnus effect Mtb. Introduction In current time lot of publictions bout studies of golf bll flight in ir enironment re known. A lrge prt of the ppers cn be generlized s reserch on the erodynmics (s whole) of the bll flight but the other studies cn be ssigned to some specific fields. The first of them is relted to the biomechnics of the stroke with club from the side of the golfer [ 3 etc.]. The publictions from the second field del with the impct between the club hed nd the bll nd the influence of this interction on the bll flight fter tht [4 5 etc.]. The other group of uthors put the ccent on the inestigtion of the mechnicl properties nd dmping chrcteristics of the blls mterils [6 7 8 etc.]. Furthermore in some of the studies [9 0 etc.] the focus is on the loction density size nd shpe of the bll dimples s well s their influence on the drg nd lift coefficients in dependence of the bll elocity nd Reynolds number. By the other hnd feture of most studies is tht the flight of the golf bll in the ir enironment is considered s mteril point motion [ etc.]. A reson for this gies them the fct tht the golf bll is reltiely smll in size. In the boe mentioned ppers the golf bll flight is described by three differentil equtions bout the three xes of the Crtesin coordinte system. Furthermore some of the solutions [9 0] refer to golf bll motion only in one plne (the erticl one). Such pproches re not so precise since the rel motion of the bll is three dimensionl nd becuse of which it is necessry to consider dynmic model with six degrees of freedom. One solution for such 3 model is presented in [6] t studying the tennis bll flight. Here it is importnt to mention tht with similr mthemticl models which re linerized with respect to the ngulr rottions re studied the 3 ibrtions of the indiidul moble prts of the mchine structures or een entire ggregtes connected by elstic elements [7 nd mny others]. In reltion with the boe mentioned the im of the current work is to study the 3 golf bll flight in ir enironment s the moing bll is considered s motion of the mteril body in ir enironment with erodynmic resistnces. For this purpose the chrcteristics of the golf bll re identified the mthemticl 54
2 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies model with six degree of freedom is composed the numericl solution nd results re presented nd discussed. The flight is studied without tking into ccount number of dditionl fctors ffecting the golf stroke s: type nd size of the golf-club bll design indiidul sporting nd professionl skills of the golfer etc. The prmeters of the presented solutions re selected ccording to ctul fetures of the golf gme.. Golf bll chrcteristics In order to strt study of the golf bll flight it is necessry to know the geometricl nd mechnicl chrcteristics of the bll. The rcing golf bll hs mss no more thn [kg] nd hs dimeter not less thn [m]. The golf bll must lso he the bsic properties of sphericlly symmetricl body [8]. It mens tht the bll itself must be sphericl nd must he symmetricl rrngement of dimples on its surfce. There re officil documents with dditionl restrictions such s rdius nd depth of dimples mximum lunch speed from test pprtus nd mximum totl distnce when lunched from the test equipment. Most golf blls tody he bout dimples though there he been blls with more thn Usully the golf blls re designed to be s symmetricl s possible. But symmetricl design helped the bll to chnge its trjectory during the flight. 000 dimples (Fig.). Figure : Some golf bll constructions The presence of the dimples on the golf bll surfce mkes lrger ir coer which moes simultneously with bll. By this wy t interction of the golf bll nd ir during the flight the contct re fluid-fluid rises nd the friction reduces. This leds to significnt reduction of the erodynmic resistnce (or so clled drg) nd respectiely to incresing of the length height nd time of the golf bll flight. In the lst yers different kind of dimpled golf blls ppered s they differs minly by the size shpe nd density of the dimples. Here must be emphsizing tht ccording to [0] these dimpled blls induce higher lues of the drg coefficient but C lso bigger lift-to drg rtio. The golf blls cn be differently designed (Fig.). The most used ones re two lyers blls: lrge inner core of dense mteril (rubber or similr) nd thin hrd coer (cutproof Surlyn) s second outer lyer. These blls combine durbility with mximum distnce s well s lower spinning. tely the golf blls construction is multilyered (three four lyers). Three lyered blls typiclly he Surlyn or softer polyurethne coer nd mntle lyer of rubber windings round rubber core. Four lyered blls dditionlly he n inner coer mde of blend of ionomers nd n outer coer mde of Elstomer or Blt. Hing more lyers helps to esily implement the bll spin nd Mgnus effect. Or shortly with multilyer blls it is possible to chiee smller distnce of the flight but the golfer hs greter opportunity to chnge the flight trjectory. As dditionl informtion here must be note tht in the golf gme re used nine different kinds of the golf strokes (push push slice push hook; slice stright hook; pull pull slice pull hook) which led to different kind of the bll flights trjectory. In the current pper will be considered the clssicl cse of two lyered golf body s for the study the following ssumptions re introduced: the golf bll is solid sphere with n erge rdius R 4. 0 [m] mss m. lues mss moment of inerti J [kg.m ] [kg] nd corresponding to these 55
3 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies By the kinemtic nd dynmic point of iew bsed on the boe ssumptions the flight of the golf bll cn be considered s generl motion of solid sphere (which is ccepted s perfectly rigid body) in unlterble ir enironment without wind. The ir resistnce is tken into ccount by externl to the golf bll forces nd moments. 3. Mthemticl model 3.. Elements of the golf bll Kinemtics The golf bll hs six degrees of freedom. Its lows of motion bout fixed coordinte system re presented by six functions of the time (Fig. ): O x y z x x( t ) (t ) y y( t ) (t ) z z( t ) (t ) () () From the lws of motion () which define the position of the mss bll center fter excluding the time its trjectory is obtined. Further in the text this trjectory of the grity center of the bll is clled shortly trjectory. x z O z (t) x y (t) t (t) x Figure : Golf bll lows of motion It is ccepted tht the three ngulr functions cn be ssigned with Crdn ngles in their trditionl form: rottion of ngle (t ) bout xis rottion of ngle (t ) bout xis y y nd finlly rottion of ngle (t ) bout xis z. z C z (t) x(t) (t) y y y The bll ngle elocity is defined by its lgebric projections on xes of the trnsltion moble coordinte system C which re usully clled kinemtic equtions: ( t ). sin (3.). cos.cos. sin (3.b). cos.cos. sin. (3.c) 3.. Elements of the golf bll ynmics uring the golf bll flight in the ir enironment the following forces ct on the moble object (Fig. 3). Weight force G: G m.g Aerodynmic resistnce (drg) force F : (4) This force ct in the direction opposite to the elocity nd it is determined by the squre of this elocity nmely: F.C..A.. k.. (5) F.C..A. k.. (6) Here is the elocity of the center of grity of the bll - dimensionless drg C coefficient - ir density A - re of the middle cross-section of the bll. On the other hnd the introduced coefficient k is function of the drg coefficient ir density nd the re. The ir density depends on the enironment temperture nd it is ccepted by the erge lue 05 [kg/m 3 ] t t 0 [ o C]. The middle cross-section re of the bll for the considered cse is 4 A. R [m ]. The erodynmic resistnce coefficient C or so clled drg coefficient depends on the Reynolds number Re. It is well known tht Re chrcterizes the turbulent conditions nd A 56
4 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies on the other hnd it is function of the bll liner elocity. Here must be noted tht detiled study of the drg coefficient for C dimpled nd grooed golf blls is presented in the pper [0] where this coefficient is studied in dependence of the bll motion chrcteristics (elocity Reynolds number nd reolutions per minute n ). Under the Re M F M 0 90 C problem solution the lift coefficient C cn be ssumed s lue pproximtely twice less in comprison with the drg coefficient. The coefficient depends on A nd the k C mgnitude of the ector. It is essentil to note tht tht the effect of the Mgnus force (often clled Mgnus effect) cn be positie or negtie in dependence of the directions of the liner elocity ector nd ngulr elocity ector. Furthermore it is possible lso the cse without Mgnus effect when the ngulr elocity is zero. The boe mentioned three cses relted to the Mgnus effect re presented schemticlly on Fig. 4. F G Figure 3: Forces cting of the golf bll chosen here circumflow conditions n [rpm] 0 60 [m/s] Re 5. 0 the coefficient hs 5 C reltiely constnt lues nd it is possible to ccept the erged lue of it nmely C 030. Mgnus ( lift ) force F M This force tkes into ccount the pressure difference tht is creted by the ir strem flowing round the bll in the two opposite res normlly rrnged bout the plne formed by the ectors of liner elocity nd ngulr elocity. The Mgnus force (or so clled lift force) is determined by: where F k M. C. C.. A... A. : C is the lift coefficient. (7) (8) is the ir density nd A is the middle cross-section re. (Fig. 3). As first pproch of the current Figure 4: Mgnus effect cses Aerodynmic resistnce moment M This moment (clled lso drg moment) cts in n opposite direction of the ngulr elocity nd it is defined s: where M k C. C. C.. A... A. : (9) (0) is the so clled drg moment coefficient nd in the current pper it is C 3. C. Moreoer the ccepted tht introduced coefficient k depends on A nd the mgnitude of the ector ifferentil equtions C The systems of differentil equtions tht describe the 3 golf bll motion considering the bll s solid body with six degrees of freedom hs form: 57
5 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies Figure 5: Simulink Block Scheme for soling the systems of differentil equtions () nd () m. x k. x. x y z k.. z. y ; m. y k. y. x y z k.. x. z ; m. z k. z. x y z k. y. x m g.. k () J.. ; () J.. ; k J... k Both systems of differentil equtions () nd () re strong nonliner nd they re connected to ech other. On the other hnd their solution must be crrying out together with the kinemtic equtions (3). 4. Numericl solution For the numericl integrtion of the systems of differentil equtions () nd () is creted progrm in the Mtb Simulink enironment. The block scheme of the progrm is shown on Fig. 5. As initil prmeters the progrm regulrly uses the initil conditions of the golf bll (initil position nd initil ngle; the ltter sometimes is clled in the specilized literture n eletion ngle). Moreoer the progrm works lso t preliminry defined kinemtic chrcteristics s initil liner elocity nd initil ngulr elocity. The progrm llows obtining the following kinemtic chrcteristics of the golf bll motion: the lows of motion bout the six 58
6 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies coordintes the elocity lows (t ) (t ) x(t ) (t ) x(t ) (t ) y (t ) (t ) z(t ) x (t ) (t ) y (t ) (t ) z(t ) the ccelertion lows the ngle elocity lows (t ) (t ) (t ) (t ) y(t ) z(t ) (t ) (t ) (t ) (t ) the ngle ccelertion lows (t ) (t ) (t ) (t ) s well s the three projections of the trjectory on the coordinte plnes O x z O y z nd O x y nmely z ( x ) и. z( y ) y (x ) 5. Results nd discussions In the pper re presented some groups of results bout the trjectory of the golf bll during the flight. First of the results refer to different lues of the initil liner elocity : 0 [m/s] 40 [m/s] 60 [m/s] t keeping the constnt lues of the ngulr elocity 094 [rd/s] nd initil ngle 3 [ o ]. On Fig. 6 nd 7 re shown the trjectories of the golf bll in the coordinte plnes nd respectiely. As it would be expected the lrgest (in height nd in length) trjectory refers to the cse of mximl liner elocity. The reducing of the elocity leds to decresing of the trjectory. These tendencies re equl for both of the considered plnes of motion (Fig. 6 nd Fig. 7) s the height is equl of course but the length of the trjectory on x xis is definitely lrge compred to this on xis. 0 y O x z 0 O y z Figure 6: Trjectory of the golf bll in Oxz plne t different initil liner elocities o Figure 7: Trjectory of the golf bll in Oyz plne t different initil liner elocities o The second groups of results re gien on Fig. 8 nd Fig. 9 nd they demonstrte the effect of ngulr elocity on the model of the bll trjectory t constnt initil liner elocity 50 [m/s] nd constnt initil ngle 3 [ o ]. It will be considered four ngulr elocities : 0 [rd/s]; 047 [rd/s]; 0944 [rd/s] nd 3 46 [rd/s]. These four lues of corresponds to the following lues of : 0 [rpm] 00 [rpm] 00 [rpm] 300 [rpm]. On Fig. 8 re presented four trjectories in the coordinte plne O x z t lues of : 0 n y [rd/s] 9 [rd/s] 8 [rd/s] nd 7 [rd/s]. Here represent the lgebric projections of y the boe mentioned four different ngulr elocities. On the other hnd on Fig. 9 the sme trjectories re shown but in the plne O y z t the sme ngulr elocities. But here the lgebric projections x of the considered four ngulr elocities re respectiely s follows: 0 [rd/s] -45 [rd/s] -9 [rd/s] nd -35 [rd/s]. The negtie signs here re relted to the orienttion of the ngulr elocity ector bout the coordinte xis x. The results relted to the Mgnus effect on the bll trjectory re presented on Fig. 0. In the cse of no-spin golf stroke (blck solid line) we he not the effect of Mgnus force. At under-spin stroke (red dotted line; top line) the lift force cts upwrds nd increses the mximum height significntly. The opposite 59
7 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies tendencies bout the effects of different kind of elocities on the bll trjectories re in greement with the results published in [9 4 0]. It is obsered tht here some of the trjectories profiles looks more symmetric which is bsed on the reltiely lower lues of the top-spin ngulr elocities. Figure 8: Trjectory of the golf bll in Oxz plne t different initil ngulr elocities Figure 9: Trjectory of the golf bll in Oyz plne t different initil ngulr elocities Figure 0: Mgnus effect on the bll trjectory cse is top-spin stroke (green dotted line; the lowermost line) which results in downwrd Mgnus force nd respectiely to isible decresing of the mximum height. As generl remrk to ll of the gien results cn be noted tht the boe mentioned 6. Conclusion In the pper is presented study of 3 golf bll flight s prticulr ttention is pid to the Mgnus effect. To conduct the reserch the system of six nonliner differentil equtions is composed s it is importnt to mention tht the golf bll is considered s mteril body not s mteril point. The numericl solution is crried out by new deeloped progrm in the Mtb-Simulink enironment. The progrm llows to receie ll possible kinemtic chrcteristics of the golf bll flight i.e. ll lows of motion ll trjectories ll liner nd ngulr elocities s well s ll liner nd ngulr ccelertions. Moreoer by the progrm it is possible to receie lso the generlized coordintes of the three Crdn ngles nd their first nd second derities respectiely. The effects of the initil liner elocity s well s of the initil ngulr elocity re inestigted. From the results cn be concluded tht: the incresing of the initil liner elocity reflects to rising of the height nd length of the bll trjectory; the incresing of the initil ngulr elocity (but specilly for the cse of top-spin rottion) leds to decresing of the height nd length of the bll flight. The nlysis of the results bout Mgnus force effect cn be generlized s follows: t under-spin rottion the trjectory isible incresing (in height nd in length) compred to the no-spin rottion; it is the cse of positie Mgnus effect. Opposite results re receied t the top-spin stroke where the trjectory isible reduces (negtie Mgnus effect). The significnce of the presented study is expressed in the composing nd using of the more complicted but more relistic model of the golf bll flight. The presented differentil 60
8 TEHNOMUS - New Technologies nd Products in Mchine Mnufcturing Technologies equtions nd results re linked to the 3 motion of the solid body in ir enironment not 3 motion of the projectile (mteril point) s the ltter is not so ccurte from physicl stndpoint. References [] McKenzie S.J. Springings E.J. A threedimensionl forwrd dynmics model of the golf swing Sport Engineering [] heng N. Brrentine S. W. Fleisig G. S. Andrews J. R. Kinemtic nlysis of swing in pro nd mteur golfers Int. J. Sports Med. 9 (6) [3] Hely A. Morn K. ickinson J. Hurley C. Smeton A. O Connor N. Kelly P. Hhr M. Chocklingm N. Anlysis of the 5 iron golf swing when hitting for mximum distnce Journl of Sports Sciences [4] uks T.. Computtionl modeling of the Golf Stroke Ph Glsgow Theses Serice 999. [5] u. A Study of Impct between Golf Bll nd Fce of Golf Club Hed BSc MEng Project Report [6] Mruok K. Skgmi S. md K. Nkgw N. Sekiguti. ynmic Impct Chrcteristics of Golf Bll Mterils Mterils nd Science in Sports TMS 00 pp [7] Mse T. Experimentl Benchmrking Golf Bll Mechnicl Properties Interntionl Congress nd Exposition on Experimentl nd Applied Mechnics 004. [8] Wu. Sogbe. Arimitsu. etermintion of Mteril Properties of Golf Bll nd Optimiztion of Golf Clubhed SEM Annul Conference on Experimentl nd Applied Mechnics Vol [9] Alm F. Chowdhury H. Mori H. Bry R.. Subic A. A Comprtie Study of Golf Bll Aerodynmics 7 th Austrlin Fluid Mechnics Conference Aucklnd New elnd 5-9 ecember 00. [0] Kim J. Choi H. Aerodynmics of Golf Bll with Grooes Institution of Mechnicl Engineers Journl of Sports Engineering nd Technology 04 Vol [] Ti C.H. eong J.C. in C.. Effects of Golf Bll imple Configurtion on Aerodynmics Trjectory nd Acoustics Journl of Flow Visuliztion nd Imge Processing 4 () [] Meht R.. Pllis J.M. Sports bll erodynmics: effects of elocity spin nd surfce roughness Mterils nd Science in Sports pp TMS [3] Penner A.R. The physics of golf: The optimum loft of drier Americn Journl of Physics 69 (5) [4] Brber R. Golf Bll Flight ynmics Cornell Uniersity A&EP 434 Finl Project Professor oelce 007. [5] Bek S. Kim M. Flight Trjectory of Golf Bll for Relistic Gmes Interntionl Journl of Innotion Mngement nd Technology Vol. 4 No [6] Ino A.I. Inestigtion of threedimensionl tennis bll flight Journl Mechnics of Mchines yer II book (in Bulgrin) [7] Nikolo S. Nede V. Bifurction nlysis nd dynmic behiour of n inerted pendulum with bounded control J. of Theoreticl nd Applied Mechnics ol. 46 No [8] The Rules of Golf & Golf Equipment Appendix III The Bll Ailble from: Golf/Appendices/Appendix-III-the-bll/ [9] Robinson G. Robinson I. The motion of n rbitrrily rotting sphericl projectile nd its ppliction to bll gmes Physic Script Vol. 88 N p. 6
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