Okubo H, Hubbard M
Department of Mechanical Systems Engineering, National Defense Academy, Yokosuka, Kanagawa, Japan.
J Sports Sci. 2006 Dec;24(12):1303-14. doi: 10.1080/02640410500520401.
A completely general three-dimensional dynamic model is presented for the motion of basketball shots that may contact the rim, the backboard, the bridge between the rim and board, and possibly the board and the bridge simultaneously. Non-linear ordinary differential equations with six degrees of freedom describe the ball angular velocity and ball centre position. The model includes radial ball compliance and damping and contains five sub-models: purely gravitational flight, and ball-rim, ball-bridge, ball-board, and ball-bridge-board contact. Each contact sub-model has both slipping and non-slipping motions. Switching between the sub-models depends on the reaction force at, and velocity of, the contact point. Although the model can be used to study shots from any point on the court, we here use it to study the sets of free throw release angle, velocity, angular velocity, and lateral deviation angle that result in success (capture), as well as underhand free throws and those using an under-inflated ball. Free throw shots with larger backspin, lower inflation pressures, and underhand release conditions are shown to result in larger capture percentages.
本文提出了一个完全通用的三维动力学模型，用于描述篮球投篮运动，该运动可能会与篮筐、篮板、篮筐与篮板之间的连接结构接触，甚至可能同时与篮板和连接结构接触。具有六个自由度的非线性常微分方程描述了球的角速度和球心位置。该模型包括球的径向柔顺性和阻尼，并包含五个子模型：纯重力飞行、球与篮筐接触、球与连接结构接触、球与篮板接触以及球与篮板 - 连接结构同时接触。每个接触子模型都有滑动和非滑动运动。子模型之间的切换取决于接触点的反作用力和速度。虽然该模型可用于研究球场上任何点的投篮，但我们在此用它来研究导致投篮成功（球被捕获）的罚球出手角度、速度、角速度和横向偏差角的组合，以及低手罚球和使用充气不足的球进行的罚球。结果表明，具有较大后旋、较低充气压力和低手出手条件的罚球投篮具有更高的捕获百分比。
