You are to define the requirements for a Baseball Pitching machine. A user study indicated that current owners of baseball pitching machines are unhappy with the ratio of balls-to-strikes thrown and that they would be satisfied with a 95% strike rate. Your goal is to determine the nominal velocity, velocity range, and back-spin range that can achieve this strike rate. Your boss suggests (i. e. professor requires) you use a Monte-Carlo simulation technique [1] to assist your requirement development. One question she has is: Is possible to develop a baseball pitching machine given the intrinsic variability in the ball mass, air density, drag-coefficient, etc. The below table provides known parameters and their ranges to assist in the analysis. Your goal is to determine how much variability the pitching machine can have form pitch to pitch while still satisfying your customer's goal. Parameter Range Source
Pitching Machine Location X: 0.0 +0.2
Y: 0.0 +0.2 [ft] NA
Z: 3.0 +0.2
Strike Zone Location X: 60.5 +0.0
Y: 0.0 ±0.0 [ft] [2], [3]
Z: 2.83 +0.0
Strike Zone Dimensions Width: 17 +0.0 [in] [3]
Height: 34 ±0.0 [4]
Baseball Mass 145.5 ± 1.4[grams]
Baseball Diameter (D) 74 ± 0.4 mm [4]
Drag Coefficient (Cd) 0.50 +0.005 [5]
Lift Coefficient (Ct) 0.15 0.005 [6]
Air Density (p) 1.029 0.04[kg/m3] [7]
Nominal Pitch Speed 80/[mph] [8]
Nominal Back Spin Rate (w) 40 rev/sec [8]
Air Drag & Lift Models A thrown spinning ball, with linear velocity ū and angular velocity w, has aerodynamic forces exerted on it. These can be split up into drag and lift components, approximated by the following expressions [5,6,9]
Fd = -СdрpiD2/8 CD (underroot v. v) barv
Ft = 2/3 Ct pi2D3 pw(v * y)
Although their is a drag torque acting on the ball as well, we'll neglect it for this analysis.