4m ago

2 Views

2 Downloads

366.76 KB

6 Pages

Transcription

Available online at www.sciencedirect.comProcedia Engineering 34 (2012) 182 – 1879th Conference of the International Sports Engineering Association (ISEA)Spin of a batted baseballAlan M. Nathana,*, Jonas Cantakosa, Russ Kesmana, Biju Mathewb,Wes LukashbbaUniversity of Illinois, 1100 W. Green St, Urbana IL 61801, USARawlings Sporting Goods, Inc510 Maryville University Drive, Ste. 510, St. Louis, Missouri 63141, USAAccepted 29 February 2012AbstractExperiments are conducted to investigate the spin of a baseball undergoing an oblique collision with a bat. Abaseball was fired horizontally at speeds up to 120 mph onto a 3"-diameter cylinder of wood that was rigidly attachedto a wall. In one experiment, a two-wheel pitching machine was used in which the backspin or topspin of the incidentball could be adjusted. In another experiment, an air cannon was used to project the ball with no spin. In bothexperiments, markers on the ball were tracked with high-speed video to determine the velocity and spin vectors,before and after the scattering. Our primary results are as follows: (1) For a given angle of incidence, the scatteredspin is nearly independent of the incident spin; (2) The spin of the scattered baseball is considerably larger thanexpected for a model whereby the ball rolls before leaving the surface. Implications for the spin of batted baseballswill be explored. 2012 Published by Elsevier Ltd.Keywords: Baseball; spin1. IntroductionOne of the major unsolved problems in baseball physics is our understanding of the spin of a battedball. It is more than just a physics problem; it also has great practical value as knowing it would greatlyenhance our ability to predict the flight of a baseball given the initial velocity vector. Low-speed ball-batcollisions have shown that the often-made sliding-to-rolling assumption is not valid [1]. Instead, the ballgrips the surface of the bat, stretching the ball in the transverse direction, resulting in a significantlyhigher spin than can be obtained by rolling. The object of the present experiment is to extend theprevious studies to much higher speed.* Corresponding author. Tel.: 1-217-333-0965; fax: 1-217-333-1215.E-mail address: [email protected]1877-7058 2012 Published by Elsevier Ltd.doi:10.1016/j.proeng.2012.04.032

Alan M. Nathan et al. / Procedia Engineering 34 (2012) 182 – 1872. Experiment and Data ReductionTwo different experiments were performed, both with the geometry shown in Fig. 1. A baseball isprojected horizontally onto a three-inch diameter wood cylinder that is rigidly bolted to a rigid surface sothat no recoil is possible. The incident velocity vector v1 is offset from the centerline of the cylinder byan amount E so that it makes an angle T with the normal to the surface of the cylinder. The ball scattersthrough an angle D and has final velocity v2. The incoming and outgoing spins are Z1 and Z2,respectively, with the positive direction along the outward normal to Fig. 1, corresponding to incidenttopspin and outgoing backspin. In the first experiment, the ball was projected with a two-wheel pitchingmachine for which the incident speed and spin were independently adjusted to be in the range 85-110mph and 1000-3000 rpm, respectively. In the second experiment, the ball was projected without spin at afixed speed of 120 mph from a pressurized air cannon. In both experiments, the scattering was viewed bya video camera operating at 2000 frames/sec. Care was taken to assure the camera axis was normal to thescattering plane. A calibration grid in the field of view of the camera was used for calibration.Fig. 1. Geometry for the scattering experiments, with the arrows indicating the positive direction for the various quantitiesFor the air cannon experiment, each baseball had four reflective markers that could be tracked with thehigh-speed video and the pixel coordinates determined for approximately 20 frames (0.01 sec). Thecamera calibration allowed the pixels to be converted to coordinates. One marker was placed near therotation axis; the others were 0.9-1.3 inch from the axis. Using the formalism described in [2], a leastsquares fitting procedure was used to fit simultaneously the locations of all markers in all frames todetermine v1, v2, Z1, Z2, D, and the orientation of the spin axis. It was confirmed that the spin axiscoincided with the normal to the scattering plane to better than 3 degrees. Neither T nor E are directlymeasured. A similar but slightly different procedure was used for the other experiment. A total of 52impacts were analyzed.3. Ball-Bat Collision ModelThe goal of the analysis is to interpret the experimental results in the context of the collision modeldescribed in detail by Cross [3]. First we denote x and y as directions transverse and normal to thesurface of the bat at the contact point, respectively (see Fig. 1). The model is described by threeparameters. One of these is the coefficient of restitution (COR) for the normal velocity component,183

184Alan M. Nathan et al. / Procedia Engineering 34 (2012) 182 – 187ey -vy2/vy1, which is approximately 0.5 for a baseball on a rigid wood surface. Another is the tangentialCOR ex, defined as the negative ratio of final to initial tangential surface velocities of the ball:ex vx 2 rZ2vx1 rZ1(1)where r is the ball radius. The third is the perpendicular distance from the center of the ball to thenormal line of force, denoted by D:I Z2 mvx 2 r I Z1 mvx1 r mD(1 ey )v y1 ,(2)where m and I 0.4mr2 are the mass and moment of inertia of the ball. The left-hand-side of Eq. 2 isthe difference between the final and initial angular momentum of the ball about the contact point, so thatD 0 implies angular momentum conservation in the collision. Eqs. 1 and 2 can be combined to obtainexpressions for the final spin and transverse velocity:§ 0.4 e ·§ 1 e · vD § 1 ey · vy1xxx1 Z2 Z1 1.4 ¹ 1.4 ¹ r r 1.4 ¹ rD § 1 ey ·§ 1 0.4ex ·§ 1 ex ·vx2 vx1 0.4 rZ1 vy1 r 1.4 ¹ 1.4 ¹ 1.4 ¹(3)(4)The primary parameter determining the final spin of the ball is ex, so it is useful at this stage to discussits physical significance [3]. For ease of exposition, we consider the simplified situation with no initialspin and with D 0, in which case Z2 (5/7)(1 ex)vx1/r, which clearly shows the dependence of the finalspin on ex. When the ball makes contact with the bat, the latter exerts a normal force N on the ball in the ydirection. If the ball has nonzero tangential velocity vx1-rZ1, it will slide along the surface of the bat sothat a frictional force F PN, where P is the coefficient of sliding friction, acts in the opposite direction toretard the tangential velocity. We consider three cases. First is the situation where F brings the sliding toa halt prior to the ball leaving the surface, in which case ex 0 and the ball rolls along the surface as itleaves the bat. Second is the situation where F is insufficient to bring the sliding to a halt before the ballleaves the bat, a condition referred to as “gross slip”. In this case the final and initial tangential velocitieshave the same sign so that ex 0 and the spin is reduced relative to the rolling case. The gross slip androlling cases are the only possibilities for a rigid baseball and were the only ones considered in [4,5].However, for a baseball with tangential compliance, a third case is possible in which the ball grips thesurface while still sliding, as elastic energy is stored in the tangential stretching of the ball. To analyzesuch a situation in detail requires a dynamic model [6,7]. Depending on the details the resultingtangential COR can be positive, so that the final spin is enhanced relative to the rolling case, a conditionreferred to as “overspin”. The low-speed experiment of [1] found a modest overspin, ex 0.16.Eqs. 3-4 together with the definition of ey give a complete description of the scattering process. Forgiven initial conditions, there are three unkn ), as developed in [1]. Inthe calculation, a baseball is projected horizontally at 85 mph onto a bat that is swung horizontally at 70mph. The impact parameter E (see Fig. 1) is varied and the post-impact velocity and spin are calculatedusing our formalism and the fitted parameters. Fig. 3 shows Z2 vs. the launch angle D for Z1 r2000 rpm

186Alan M. Nathan et al. / Procedia Engineering 34 (2012) 182 – 187and for ex 0.3 (solid) and 0 (dashed), respectively. For ex 0.3, the two solid curves are essentiallyindistinguishable for impacts for which there is no gross slip, showing that the batted ball spin is notstrongly dependent on the pitched ball spin for a given launch angle D. This conclusion is very differentfrom that found with ex 0 [4,5], as seen from the dashed curves. Moreover, for pitched balls thrown withbackspin (Z1 0), the spin of a batted baseball with positive launch angle is considerably larger withex 0.3 than with ex 0.Fig. 2. Results of our analysis, with red, blue, and black points corresponding to incident backspin, topspin, and no spin,respectively. (a) Scattered spin, normalized to 100 mph initial speed, vs. T. The curve is the expected result for ex 0.3 and D 0;(b) D/r vs. T; (c) Final vs. initial tangential surface velocity, normalized to vy1. Dashed curve is expected result for ex 0.3; dottedcurve is result for gross slip with P 0.15; (d) Ratio of transverse to normal impulses vs. the normalized initial transverse surfacevelocity saturates at the value P#0.15. The dotted curve in Fig. 2(c) is the expected result for gross slip with P 0.155. Summary and ConclusionsNew experiments were conducted to investigate the spin of a baseball undergoing an oblique collisionwith a clamped cylinder. We summarize our findings as follows:x For incident angles less than about 40o, the ball grips the surface and rebounds with considerableoverspin, characterized by a tangential COR ex 0.30r0.02.x For incident angles less than about 30o, the angular momentum of the ball about the initial contactpoint is conserved in the collision

Alan M. Nathan et al. / Procedia Engineering 34 (2012) 182 – 187x For incident angles exceeding 40o, gross slip occurs and the corresponding coefficient of friction is0.15.x The spin of a batted baseball is less dependent on the spin of the pitched baseball less than previouslythought based on the rolling scenario.Fig. 3. Calculations of batted ball spin as a function of vertical launch angle under typical game conditions. The solid and dashedcurves are for ex 0.3 and 0, respectively; blue and red curves are for incident topspin and backspin, respectivelyAcknowledgementsThe authors thank Ben Thoren, Stephen Wilhelm, and Jane Nathan for their help with the datacollection and reduction.References[1] Cross R., Nathan A. Scattering of a baseball by a bat. Am. J. Phys. 2006;74:896-904.[2] Alaways L., Hubbard M. Experimental determination of baseball spin and lift. J. Sports Sci . 2001;19:349-358. Seeparticularly Eq. 13.[3] Cross R. Grip-slip behavior of a bouncing ball. Am. J. Phys. 2002;70:1093-1102.[4] Sawicki G, Hubbard M, Stronge W. How to hit home runs: Optimum baseball swing parameters for maximum rangetrajectories. Am. J. Phys. 2003;71:1152-62.[5] Watts R, Baroni S. Baseball-bat collisions and the resulting trajectories of spinning balls. Am. J. Phys. 2006;57:40-45.[6] Maw N, Barber J, Fawcett J. The role of elastic tangential compliance in oblique collisions. J. Lubr. Technol. 1981;103:74-80.[7] Stronge W. Impact Mechanics. Cambridge: Cambridge University Press; 2000.187

Experiments are conducted to investigate the spin of a baseball undergoing an oblique collision with a bat. A baseball was fired horizontally at speeds up to 120 mph onto a 3"-diameter cylinder of wood that was rigidly attached to a wall. In one experiment, a two-wheel pitching machine