A cardioid-parametric model for the Magnus effect in baseballs
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A cardioid-parametric model for the Magnus effect in baseballs 1 · O. D´ıaz-Hernandez 2 · Filiberto Hueyotl-Zahuantitla3 · ´ ´ Mario A. Aguirre-Lopez 4 · F.-Javier Almaguer1 · Gerardo J. Escalera Santos2 Javier Morales-Castillo
Received: 17 November 2018 / Accepted: 27 March 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The Magnus effect is responsible for deflecting the trajectory of a spinning baseball. The deflection at the end of the trajectory can be estimated by simulating some similar trajectories or by clustering real paths; however, previous to this study, there are no reports for a detailed connection between the initial throw conditions and the resulting deflection by using. The only approximation about this is the PITCHf/x algorithm, which uses the kinematics equations. In this work, deflections from simulated spinning throws with random linear and angular velocities and spin axis parallel to the horizontal plane are analyzed in their polar representation. A cardioid function is proposed to express the vertical deflection as response of the angular velocity. This is based on both theoretical arguments from the ball movement equations and from the numerical solution of such equations. We found that the vertical deflection fits a cardioid model as function of the Magnus coefficient and the spin angle, for a set of trajectories with initial linear velocities symmetrically distributed around the direction of motion. A variation of the model can be applied to estimate the radial deflection whereas an extended model should be explored for trajectories with velocities asymmetrically distributed. The model is suitable for many applications: from video games to pitching machines. In addition, the model approaches to the results obatined with the kinematic equations, which serves as validation of the PITCHf/x algorithm. Keywords Magnus effect · Baseball · Aerodynamic forces · Cardioid model · Directional data Mathematics Subject Classification (2010) 62F10 · 62P35 · 65L05 · 68U20 · 70K65
Communicated by: Pavel Solin Javier Morales-Castillo
[email protected]
Extended author information available on the last page of the article.
M.A. Aguirre-L´opez et al.
1 Introduction Spinning baseballs are the most common type of pitches in baseball games. Pitchers tend to throw them because their trajectories deflect smoothly from a ballistic motion, and then they are more predictable and easier to control than the erratic knuckleballs, or non-spinning balls [1]. The force that produces the deflection of the trajectory is the Magnus force (FM ), which is originated by a pressure unbalance at opposite sides of the ball, and depends on the velocity V and the angular velocity ω so that ω×V 1 FM = ρACM V (1) 2 ω where ρ is the air density, A is the cross-sectional area of the ball, and CM ≡ CM (V , ω) is the Magnus coefficient [2–4]. Nevertheless, for angular speed values in the range (100, 310) rad/s—which are the typical minimum and maximum limits in professional pitc
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