Negative Magnus Force Exerted on a Back-spinning Spherical Body Measurement by Flight Experiments

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347,,..,.6 1 5 <Re<2.4 1 5 <SP <.7, C D Re, SP( )., Re SP,.,.,,. Negative Magnus Force Exerted on a Back-spinning Spherical Body Measurement by Flight Experiments Keita TAKAMI, Takeshi MIYAZAKI, University of Electro-Communications Ryutaro HIMENO, RIKEN (Received 14 April, 29; in revised form 1 July, 29) Using a high-speed video camera, we measured the trajectory and the rotation of a sphere and a hard baseball thrown by a pitching machine. We determined the drag- and lift- coefficients ( ) by analyzing the video images. The measurements were performed in the range of.6 1 5 <Re<2.4 1 5, and <SP <.7 (SP : dimensionless spin rate). The dependence of C D and on the parameters Re and SP is investigated in detail. Negative Magnus force is exerted on a back-spinning sphere in a relatively wide Re-SP parameter range, whereas only usual Magnus force works on a back-spinning hard baseball. The influence of seam patterns on the aerodynamical properties is discussed. (KEY WORDS): Negative Magnus Effect, Sphere, Back Spin, Hard-Baseball, Drag coefficient, Lift coefficient 1, 1, 2). 182 8585 1 5 1 E-mail: takami@miyazaki.mce.uec.ac.jp E-mail: miyazaki@mce.uec.ac.jp 351 198 2 1 E-mail: himeno@riken.jp,,.,.,.,,., Taneda 3), 4),

348 5),.,,,.. Watts Bahill 6) 7), 4).,. C D (C D =2F D /ρu 2 AF D :, ρ:, A:, U: ). C D, Re (Re c 3. 1 5 ) 8). Re,..,. Achenbach 9) Wieselseberger 1),, Re c., Luthander Rydberg 11), Re c.,, Adair 12)., Re Briggs 13), Re., Watts Ferrer 14) Alaways Hubbard 15). Alaways Hubbard, Re =1.4 1 5,., Watts Ferrer, Alaways Hubbard Re (SP: SP =2πrf/V, f:, r: V : )., 4), 2 4 ( 2 2 4 4 2 4 1),. 16), Re, SP, 2 4., C D (C L =2F L /ρu 2 A F L : ) 2 Re,, 4) 16),. 2, 3 C D,, 4, 5. 6, 7. 1 2 4 ( ) 2 2.1 (JUGGS ), (Nobbytech 963A2)4 3.5m. (Vison Research Phantom. 1 191. ),

349.,,. 2 1 x, y, z. 2 2.2 1 ([ ] ) (. JPN. ) 3 3 ( ):, ( ): 1 [ ( XYZAX GC6 D-34S ) ] JPN 24 1 [m] 7.2 1 2 [.43 1 3 ] 7.16 1 2 [m] 7.3 1 4 [.5 1 4 ] / 1. 1 2 [.92 1 3 ] [kg].145 [1.8 1 3 ].1349 3 3.1 (C D ) F D = ρac D u 2 /2(A A = πr 2 ), D = ρac D m (m: ), x, dx dt = u (2) du dt = 1 2 Du2 (3) ( u >> w ) u, w, x, z (2) (3), (1) x = 2 D log(1 + u D t) (4) 2 u x = u x = 2, 3, 4 t 1 t 4 2, 3, 4 (x 1 x 4 ) 2 u C D Re u. u, 1, 1., 2 u u s,.7. 4, 16) 3 4. C D 2, 2. 2 C D (Re =1.3 1 5, SP =.12, [ ] ) 3 4 25 21 C D.52 [.3].53 [.19] 3.2 ( ) F LZ = ρa u 2 /2, L = ρa m (5)

35, D, L, dx dz = u, dt dt = w du dt = 1 2 Du u 2 + w 2 1 2 Lw u 2 + w 2 (6) dw dt = g 1 2 Dw u 2 + w 2 + 1 2 Lu u 2 + w 2.,. t i z z i., t i z i., R(w,L)= (z i z i ) 2 (i =1, 2, 3, ) (7) w,l. t i, (t 1 t 3 ) (t 4 ) 1. 3.3 SP :f[rps]., 2 u, w, SP =2πrf/ u 2 + w 2 (8) SP. 4 4.1 C D C D, JPN (2,4 ) (Re =1.25 1 5, SP =.12) 2, ( 4, 3)., JPN 6, 3. JPN, C D,. ( 5 ),. JPN C D, JPN,., C D 2 4. 4.2, JPN (2, 4 ) (Re =1.25 1 5, SP =.12) 2 Number of Data 2 15 1 5 4seam 2seam Sphere.2.3.4.5.6.7 CD 4 C D (Re =1.25 1 5, SP =.12) 3 C D (Re =1.25 1 5, SP =.12,[ ] ) 2seam 4seam Sphere 25 26 18 C D.349 [.22].328 [.21].52 [.14], ( 5, 4)., 1, C D. JPN,. Re SP(Re = 1.25 1 5,SP =.12),.,. ( ),., 2 4,. C D. Number of Data 2 15 1 5 4seam 2seam Sphere -.3 -.2 -.1.1.2.3.4.5 CLZ 5 (Re =1.25 1 5, SP =.12)

351 4 (Re =1.25 1 5, SP =.12, [ ] ) 2seam 4seam Sphere 25 26 18.72 [.21].174 [.19].23 [.23] 4.3 C D =.5 Re = 1.3 1 5,SP =.12, Re =1.45 1 5,SP =.23, < Re =1.8 1 5,SP =.23 3 (2 3 )., 6, 7 5, 6. Re = 1.3 1 5,SP =.12, Re =1.8 1 5,SP =.23, 5). 4.3.1 C D, C D C D. Re =1.3 1 5,SP =.12 C D =.53, Re =1.8 1 5,SP = 5 6 C D Re( 1 5 ) SP C D [ ] Nakagawa Re =1.3 SP =.12 3 Re =1.45 SP =.23 2 Re =1.8 SP =.23 22.53 [.22].478.458 [.22].394 [.17].412., Re =1.3 1 5,SP =.12. Taneda, Re =1.3 1 5,SP =.12,, = ( 13). (5.1 ), 8 Re. Re..23 C D =.394 ( 6). C D C D 5 2 15 SPHERE (Re=1.3*1**5, SP=.12) (Re=1.45*1**5, SP=.23) (Re=1.8*1**5, SP=.23) 2 15 SPHERE (Re=1.3*1**5, SP=.12) (Re=1.45*1**5, SP=.23) (Re=1.8*1**5, SP=.23) Number of Data 1 Number of Data 1 5.2.3.4.5.6.7 CD 6 C D (Re =1.3 1 5, SP =.12 Re =1.8 1 5, SP =.23) 5 -.3 -.2 -.1.1.2.3 CLZ 7 (Re =1.3 1 5, SP =.12 Re =1.8 1 5, SP =.23) 6 7 4.3.2. Re = 1.3 1 5,SP =.12 =.22, Re =1.8 1 5,SP =.23 =.179 ( 7)., Re =1.8 1 5,SP =.23., Re( 1 5 ) SP Re =1.3 SP =.12 3 Re =1.45 SP =.23 2 Re =1.8 SP =.23 22 [ ] Nakagawa.22 [.23] -.182 -.61 [.18] -.179 [.18] -.177

352 5 JPN 8 12 14 17. 2,. 1., C D =.5 =., 13 Taneda 3). 5.1 Re Re SP =.12,.23,.35 ( 8 1). SP =.12( 8), C D.6 1 5 Re 1.7 1 5 C D =.5, 1.7 1 5 Re, Re. Re =2.1 1 5 C D =.38.,.6 1 5 Re 1.4 1 5, =.1 =. 1.4 1 5 Re 1.7 1 5, =, 1.7 1 5 Re <. Re =2.1 1 5 =.2. SP =.23( 9). C D.6 1 5 Re 1.4 1 5 C D =.5, Re, Re =1.4 1 5 Re. Re =2. 1 5 C D =.38.,.6 1 5 Re 1.4 1 5 =.2 = 1.4 1 5 Re <, Re =1.8 1 5 =.2. SP =.35( 1), Re,. C D.6 1 5 Re 1.2 1 5 C D =.6, Re =1.2 1 5 C D =.5. 1.2 1 5 Re, Re =1.8 1 5 C D =.38..6 1 5 Re 1.2 1 5 =.17 Re, Re =1.2 1 5. 1.2 1 5 Re <, Re =1.8 1 5 =.2. Re C D, SP. C D Re.6.4.2 -.2.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Re(1 5 ) SPHERE C D SPHERE 8 -Re (Sphere:SP =.12).6.4.2 -.2.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Re(1 5 ) SPHERE C D SPHERE 9 -Re (Sphere:SP =.23).6.4.2 -.2.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Re(1 5 ) SPHERE C D SPHERE 1 -Re (Sphere:SP =.35) Briggs,. 7 ( T =2, r =.35m ) SP Re( 1 5 ) f[rpm] U[ft/sec].12 1.7 117 117.23 1.4 1843 96.35 1.2 24 83, Re, C D. Briggs 13), Adair 12) C D Re ( 12rpm 18rpm 1 v 125 ft/sec)., Briggs (rpm, ft/sec) 7. SP =.12 SP =.23 Briggs., Re. Re, C D C D =.5 Re. Re,,. C D <

353. 5.2 SP Re =1.1 1 5, 1.6 1 5, SP ( 11, 12). Re =1.1 1 5 ( 11), SP.4, C D =.5, SP..4 SP.5 C D SP, SP =.5, C D =.3., SP.4 =.1, SP =.4..4 SP.5, SP =.45 =.7., SP =.5, SP. SP =.65 =.2. Re =1.6 1 5 ( 12),. SP.23, C D =.5, SP C D..45 SP, SP, C D =.3., SP.23 =. SP =.23,.23 SP.45. SP =.25 =.2. SP.25 SP, SP =.45, SP =.58 =.28. SP C D, Re. C D SP, SP. SP, C D 2. SP,.,,. SP, C D,.,,,. Re SP,. 5.3, Re SP.,, >, <., (4.2 ), <.3 =..6.4.2 -.2 SPHERE C D SPHERE.1.2.3.4.5.6.7 11 -SP (Sphere:Re =1.1 1 5 ).6.4.2 -.2 SPHERE C D SPHERE.1.2.3.4.5.6.7 12 -SP (Sphere:Re =1.6 1 5 ) 5.3.1 Taneda ( 8 12 ), >, <, = 3, Re SP 13., Taneda 3). 13, <, Taneda., Re, SP Taneda. Taneda. Taneda 7) Taneda.5, Re. Re Re +, Re Re, SP

354. Re + =(1+SP)Re (9) Re =(1 SP)Re (1) (9), (1) Re + Re 13. Re, 1 Re + =1.4 1 5, 1.7 1 5, Re =.5 1 5, 13 (2 ) (1 )., Re 1.4 1 5 <Re + < 1.7 1 5., Re (Re ). SP.7.6.5.4.3.2.1.6.8 1 1.2 1.4 1.6 1.8 2 2.2 Re(1 5 ) Taneda line Sphere < > ~ 13 5.4 Re Re SP =.12,.23 ( 14, 15). SP =.12( 14), C D 2 4,.6 1 5 Re 1.6 1 5 C D =.5. 1.6 1 5 Re Re, C D =.38. Re 1.4 1 5,2 4 C D.,2.6 1 5 Re 1.8 1 5 = Re. 1.8 1 5 Re Re, =.2. 4,.6 1 5 Re 1.6 1 5 Re =., 1.6 1 5 Re =.2. Re 1.8 1 5 2 4., Re =1.25 1 5,2 : =.72,4 : =.174. SP =.23( 15), Re SP =.12. 2 C D.6 1 5 Re 1.2 1 5, C D =.5 Re 1.2 1 5 Re Re, C D =.38. 4 C D Re Re 2 4 C D., 2,.6 1 5 Re 1.7 1 5, =. 1.7 1 5 Re Re =.25. 4,.6 1 5 Re 1.5 1 5, = =.25 Re, Re. 2 4. SP =.12, SP =.23, Re, C D C D =.5 Re. (5.1 ),., =,., 6, 1).,, Re,. Re,,.,,,.,, Re. Re Re.6 1 5, SP =.23 <,.,., C D Re Re (SP =.12 Re =1.6 1 5, SP =.23 Re =1.4 1 5 )., C D, (5.1 ).,. C D, Re, SP. SP =.12 1. 1 5 <Re< 1.6 1 5, 2 4. 4 (4.2 ). SP =.23,2 4 C D,., SP Re. Re SP

355, Re SP..6.4.2 -.2.6.4.2 -.2 4seam C D 2seam C D.6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Re(1 5 ) 14 -Re (JPN:SP =.12).6.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Re(1 5 ) 4seam C D 2seam C D 15 -Re (JPN:SP =.23) 5.5 SP, SP 16, 17. Re =.65 1 5 ( 16),2 4, SP.3 SP C D =.5 C D..3 SP SP C D =.4. SP.3, 2 4 C D.,2 4 SP.45 = SP..2 SP, 2 4,. Re =1.1 1 5 ( 17), 2 C D, SP.2 SP C D =.45..2 SP.4 SP, C D =.4.,.4 SP C D SP. 4 C D SP 2 4 C D., 2 4 SP.4 SP =. SP.4, SP =.7 =.4., SP.4 2 4. Re =.65 1 5 Re =1.1 1 5, SP SP C D. (5.2 ), SP,.,, <., Re SP,. SP,., C D SP. (5.2 ), SP C D. Re =1.1 1 5,.4 SP C D.,. C D, SP, Re. Re =.65 1 5,.2 <SP <.5,2 4, 4., Re =1.1 1 5, 2 4 C D,,.15 <SP <.3.., Re SP., Re SP..6.4.2 -.2 4seam C D 2seam C D.1.2.3.4.5.6.7 SP 16 C D,-SP (JPN:Re =.65 1 5 ) 6,., SP.23., 12 16km/h, 1.6 1 5 <

356.6.4.2 -.2.1.2.3.4.5.6.7 SP 4seam C D 2seam C D 17 C D,-SP (JPN:Re =1.1 1 5 ) Re < 2.1 1 5 ( 2 )., 145km/h,, 18m 1.7m., 4,.45m. 1.25m., 2 4., SP =.23, 1.6 1 5 <Re<2.1 1 5, 2 4., 2 4, (SP). 7,,., Re, C D,.,, ( ).., Re-SP. 1) G. Magnus : Poggendorfs annual on Physics and Chemistry,88 (1853). 2) L. Rayleigh : On the irregular flight of a tennis ball, Messenger of Mathematics,7 (1877) 14-16. Reprinted in Scientific Papers(Cambrige,1899),1 344-346. 3) S.Taneda: Negative Magnus Effect, Reports of Research Institute for Applied Mechanics(1957) 123-128 4),,, :,,25 (26) 257-264. 5),,,, :, 25 6) R.G.Watts A.T.Bahill, (1993) 7) :,,49 (1979) 51-53. 8) E.Achenbach : Experiments on the flow past spheres at very high Reynolds numbers,j.fluid Mech,54 (1972) 565-575. 9) E.Achenbach : The effects of surface roughness and tunnel blockage on the flow past spheres, J.Fluid Mech,65 (1974) 113-125. 1) C.Wieselsberger : Weitere Feststellungen über die Gesetze des Flüssigkeits-und Luftwiderstandes, Phys.Z. 23 (1922) 219-224 11) S. Luthander and A. Rydberg : Experimentelle untersuchungen uber den luftwiderstand bei einer um eine mit der windrichtung parallelen achse rotieren kugel, Zeitschrift für Physikalische Chemie 36 (1935) 552-558. 12) R.K.Adair : THE PHYSICS OF BASEBALL, 3rd ed. HarperCollins, New York (22). 13) L.J.Briggs : Effect of Spin and Speed on the Lateral Deflection (Curve) of a Baseball; and the Magnus Effect for Smooth Spheres, Am.J.Phys. 27, 589-596 (1959). 14) R.G.Watts and R.Ferrer : The Lateral Force on a Spinning Sphere, Amer.J.Physics,55,No.1(1987) 4-44. 15) L.W.Alaways and M.Hubbard : Experimental determination of baseball spin and lift, Journal of Sports Sciences, 19 (21) 349-358. 16),, :,, 27 (28) 43-49.

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