Analysis of Effects of Rebounds and Aerodynamics for Trajectory of Table Tennis Ball

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1 Al f Effc f Ru Ac f Tjc f Tl T Bll Juk Nu Mchcl Scc Egg, Gu Schl f Egg, Ng Uv, Fu-ch, Chku-ku, Ng, J Ak Nkh Mchcl Scc Egg, Gu Schl f Egg, Ng Uv, Fu-ch, Chku-ku, Ng, J Yhku Hkw Mchcl Scc Egg, Gu Schl f Egg, Ng Uv, Fu-ch, Chku-ku, Ng, J RIKEN-TRI Cll C, RIKEN, 71-13, Agh, Shh, M-ku, Ng, J Ac I h, w fl l h ffc f h u c f jc f l ll. W fl l h ffc f h c wh c f vlu, wh h hlf f h l c 9 v. Fuh, h g lf cffc f ug h h l vlc vl ug h ll flg. Wh h f cffc, h lg f h l ck cl vf h c vul. S cclu fll hw. I T Bll jc, Ru Ph, c, Tl I. INTRODUCTION A hu c l f f f l wl wh h,.., h f v. Wh h f, h c uful cf f whch c f k,.g., cchg, hwg hg ll,. Thf, v uful f uc v u v hv lgh f jcv k wh v f. W l l l wh hu cl l f uc v c lg l u k f hu. F lc, c h u hw Fg. 1, wh h flg ll. I h u, h g f Fg. 1. A h flg ll. Fg.. Bll jc f h c f h ck. h c c h fllwg uk: 1) T c h f h flg ll wh v. ) T c h ll jc. 3) T h jc f h ck ch h f h hg chv ll jc. Th u 1) h g cg lgh h, h ll vlc h l vlc f h ll. Th lgh f l f h k f h c. Th u ) h c f h ll/l vlc f h ll f h k f h. Th u 3) h f h jc f h f h ck f h ll fllw jc. I h uk ) 3), h ll u f h l h ck u. Fuh, h flg ll ffc c fc. I hf c l h u h c. Su h h h ll l vlc c u h. Thf, w cc h l f ffc whch hul c h c f h ll

2 = u R =. =.5 Fg. 3. = 1.5 = 3. = 1. = 3.5 = 4. Ru f h ll f h ck u. jc. I Fg., h lu ccl h c f h. Th ck cvl h ll vlc ju f h u h h ch h c. I l cf h h vlc l h l h c f h ck g h h h c f h. I, h l vlc f u l chg u h fc. I c h ll jc f h u f h l, c c h fc f h l1. Sc, l u c h u h w h ll h ck u. Fg. 3 hw l f h u f h ll f h ck u. Th g ccl h h ll. I cf h h l vlc u h l h g chg h v c f h u. Th c cg l h fc h cfc h h c f h u. I chv ll jc f h u f h ck, c c h h1. Fll, w hul c h ffc f h c fc. Fg. 4 hw h c fc f flg ll,.., lf g. Th lf l h g f h ll. Th g u h ll l h ll vlc. Th jc f h ll chg l h w ffc c h wgh f h ll ll h vlc hgh,.g., 3. I chv ll jc f h u, c c h h. I h, w l h ffc f h u c f jc f l ll. W fl l h ffc f h c wh c f vlu, wh h hlf f h l c 9 v. Fuh, h g lf cffc f ug h h l vlc vl ug h ll flg. Wh h f cffc, h lg f h l ck cl vf h c vul. S cclu fll hw. h fc f wh h - l h l. Sc h h fc w h ll h l, h vlc v chg v. Th l w l h ll gv (5) (6)1. I v c h f h cc ug h c,.., h lg llg cc. Th c ug h g vlc gv v T := v v T + ω = v ω v + ω, (1) wh := T R 3 h cc f h ll f c R + h ll u. F h lg, w k h fllwg u: Au 1: Dug h c f h u, h f h cc w h ll l cc. Th h ffc h ll ug h c. Au : Th ffc w h ll gul u f f h u qul h ul h u. Thf, h ul f h gv P, wh P R 3 h ul h ll c. Au 3: Th fllwg l uc lh h c hl: v = v () Au 4: Th ul h c P := P P T R 3 gv P = λ v T v T, λ µ P, (3) wh µ h cl cffc f fc w h ll l. Au 5: Th cc vlc v v ju f f h u h c. Th, h fllwg l hl: v T = νv T, ν. (4) If ν, λ = µ P. F Au, h fllwg u l v1. v = A v v + B v ω (5) ω = A ω v + B ω ω. (6) wh 1 α α A v := 1 1 α, B v := α Aflw Lf Dg II. REBOUND MODELS OF TABLE AND RACKET RUBBER A. Ru Ml f Tl Fg. 5 hw h u f h ll f h l, wh (v,ω ) (v, ω ) h ll l vlc f h ll ju f f h u. Σ B Fg. 4. ω Bll Lf Dg f c.

3 ω v ω V c (15f) ) V c (9f) Aucl ll cul Bll 1) Fg. 5. Σ B v Bll vlc ju f f h u A ω := 3α 3α, B ω := 1 3α 1 3α 1 α = { µ(1 + ) v v T (ν > ) 5 (ν ) ν = 1 5 µ(1 + ) v v T. ν > h c f h lg cc ν h c f h llg cc. B. Ru Ml f Rck Ru Fg. 6 hw h u f h ll f h ck u, wh h g f h vl h SucII-A wh c h ck f Σ R ch h ck h - l h ufc. I h ffc f h lc lll h ufc, F h l, w k h fllwg u: Au 6: Th u h c cu ffc h c. Au 7: Th ul h c P R 3 l h g vlc v T P = k v T. (7) F Au 1-3, 6 7, h u l f h ck v h f (5) (6). Th cffc c fllw: v ω Σ B Vul fl f v c (15f) )Vul fl f v c (15f) Σ B B Tl P f vl Fg. 7. Tl Bll Flg c 1)Vul fl f v c (9f) Dful Aucl ll cul El flg c. A v := 1 K v 1 K v, B v := K v A ω := K ω, B ω := 1 K ω 1 K ω 1 K v := k, k ω := k I wh h cffc f u f h u. III. BALL MOTION WITH AERODYNAMICS Th ll h f-fll qu f h c. Hwv h l wl, h flg ll ffc h c. Fg. 7 hw h l c h flg c. Th Σ B h g f h l. Bll h u f h uc ll cul c f v. Th lg fquc f h gh c 9f. Th c cll hgh c. Th l ll/l vlc c h hgh c. Th lg fquc f h gh c 15f. Th c cll l c. Th f vl c h c. Th lu qu h vul fl f h Σ R ω v Σ B () Ru w h ll h u R cu Σ Β O cu Fg. 6. Rck C. Fg. 8. C f vlu

4 uc ll cul ll.1 c) ) ) 1.9 v 1.8 c g1.7 l f 1.6 c D 1.5 Th ffc f h c fc Σ 115c 69c 3c l V (9f) 1.4 Clcul flg c whu c fc Fg. 1. E cf l vlc Fg. 9. Th flg c w h u vlu h clcul vlu whu c fc. hgh c h qu hw h f h l c. Th flg c u h l. F c, h flg c h c f h f ll clcul ug h u l vlc. W u ch 4 c f ll ck ll. Th l ll vlc f h /ck - 6./-6.5/ h l l vlc f h /ck u -34/34 u h f Σ B. Fg. 8 hw h l v 9. Th f Th fluc f h c fc vlu h 9 v c l v h g chg h f h. Fg. 9 hw h l w h flg c f h ll wh whu c. Th hl/vcl h clcul/u flg c. Th l h h clcul vlu qul h u vlu. Th ccl c h c f h ck. Th ffc w h l ccl/c hw h fluc f h c fc. Th lu/ w hw h fluc h c f /ck. Th ffc f h fluc h c f ck ll h h c f. Hwv u. whch cl h hlf f h f h v. Thf, h h, c c h c fc. IV. IDENTIFICATION OF AERODYNAMICS A h l clug Ac, (8) uc 3. = g 1 C DρA ṗ ṗ C M πρ 3 ω ṗ (8) wh :, g: ccl f gv, ρ:, A: jc, : u, C D : g cffc, C M : lf cffc, : f h ll, ω: l vlc. Th vlu gv = kg, g=,, 9.8 T /, ρ=1.184 kg/ 3 (5 C), A= = I f C D C M, w u h ω c vlu. Fg. 1 hw h l cf h u. Blu f ), ) c) h vul fl f h hgh c. Th f ) lc 3c w f h c l h f ) c) w f 69c 115c, cvl. Bll h u f h uc ll cul lll. F Fg. 11 hugh Fg. 13 hw h l vlc. Th hl h cl kg f h uc ll cul h vcl h l vlc c h v. Th cl ccl h vg f l vlc ju f h wh h ll h h uc ll cul. Th lck l h v f h l vlc. Th lu l h l f h cl ccl. Th k c hw h vg f l vlc h c f ) - c). Th k l h v f h l vlc h ch c ±3σ. All h k c c h w h lck l h c f ) ). Th, h l vlc c u ug h ll flg. Th cffc f C D C M f g h fllwg c fuc: V (C; ) := 1 () (; C) (9) wh () R 3 h u ll jc (=1,,N) (; C) R 3, C:=C D C M R + lvg (8) ucll wh h l vlu f ṗ whch gv () ṗ () ( ) (), = Th l wh f ch f P (N=5). Th f C D C M 4±.74.69±.9. Th ul vf h u h fc. A l hw Fg. 14 Fg. 15. I Fg. 14 Fg. 15, h lu, g l h u ll jc h ucll ul wh whu h c. I cf h h l l cc wh h lu l. V. VERIFICATION OF BALL TRAJECTORY I l u h l f h ck, w hv c h vl f h ll whch h h ck. Fg. 16 hw h l f h vfc f

5 h ck u l, h ll h h uc ll cul h ll/l vlc ju f f h u f h ck u h hgh c. Th c f vl clcul ug h c (8) wh h f C D C M h ll/l vlc ju f h u f h u l. W vf h lg f h u f h ck h c w h clcul u f vl wh h c h 9 v. Fg. 17, Fg. 18 Fg. 19 hw h jc h f vl. Th lu l l ll/l vlc gv h hgh c. Th l l ll/l vlc gv h ck l. I h c f h, h vg.9 h v.3. I h c f h ck, h vg.4 h v.4. Bcu cf h h f vl wh ll f h clu, h l f h ck c u f h c f h ll. VI. CONCLUSION I h, w l h ffc f h u c f jc f l ll. W fl l h ffc f h c wh c f vlu, wh h hlf f h l c 9 v. Fuh, h g lf cffc w f ug h h l vlc vl ug h ll flg. Wh h f cffc, h lg f h l ck cl w vf h c vul. S cclu fll hw. A fuu wk, c c h jc f h ck f h h ll f vl. REFERENCES 1 A. Nkh, Y. Ogw, Y. Kh Y. Hkw, Mlg f Ru Ph f Rg Bll wh Fc Elc Effc, Pcg f ACC1, Bl, Ml, USA,1 (T ) / ω = cl kg Avg l vl (cul) L (cul) S v (cul) Avg l vlc (vul fl )) S v (vul fl ) Fg. 1. Rl vlc u h ll cul h f ) / ω = v (vul fl c) ) cl kg Avg l vlc (cul) L (cul) v (cul) Avg l vlc (vul fl c)) Fg. 13. Rl vlc u h ll cul h f c) A. Nkh, Y. Tu, Y. Kh Y. Hkw, A Rl- T Mug Mh f Tll/Rl Vlc f Flg Bll, Pc f IFAC S. Mch. S., Cg, Mchu, USA 1( T cc) 3 Nk OKADA, Th YABUUCHI, Tku FUNATOMI, Kh KAKUSHO Mchhk MINOH, Eg Uv M f Bll Acqug I Phcl Ml f Ig Squc, IEICE, D vl.j91-d N.1,.95-96, 8 (I J) 5 45 = / ω cl kg Avg l vlc (cul) L (cul) S v (cul) Avg l vlc (vul fl )) S v (vul fl )) Fg. 11. Rl vlc u h ll cul h f )

6 Tjc u l c Tjc whu c Tjc ul Tjc (h hgh c) P f vl (h hgh c) Tjc (h ck l) P f vl (h ck l) Fg. 17. Tjc vl h c f Fg. 14. Ml fc Tjc h c f Tjc u l c Tjc whu c Tjc ul Fg Tjc (h hgh c) P f vl (h hgh c) Tjc (h ck l) P f vl (h ck l) Tjc vl h c f Fg. 15. Ml fc Tjc h c f 1 Rck Aucll ll cul Σ B Bll V c (9f) Tl Tjc (h hgh c) P f vl (h hgh c) Tjc (h ck l) P f vl (h ck l) Fg. 16. El vf h ck l Fg. 19. Tjc vl h c f

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