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2 Hiroki Okubo and Mont Hubbard / Procedia Engineering 3 () Both these models included radial stiffness and damping ball forces. However, neither has been used to investigate the rebounding motion of the basketball. First basketball trajectories are simulated for free-throw rebounds. In typical free-throws, the basketball moves in, or with only small deviation from, a plane normal to the backboard. We show typical rebounding ball motions in three dimensions. We use one example lateral deviation to investigate qualitatively the nature and cause of the variable rebound paths. The results of a more complete analysis including the probabilities of errors in release velocity, release angle and lateral deviation, can be used to instruct the best position for rebounders. The dynamic model has been described in previous work [7] [8]. It is able to follow accurately arbitrary sequences of contacts of the ball with the rim, backboard, and bridge between the rim and backboard, interspersed with periods of flight.. Numerical Simulation.. Simulation parameters The rules state that foul shots shall be attempted behind the free throw line located about 4. m horizontally from the hoop centre. The top of the rim is 3.5 m above the floor. Release conditions are the four input parameters: release velocity; lateral deviation angle between the vertical plane containing the initial ball path and the board vertical normal plane; release angle between the velocity and horizontal plane; and backspin angular velocity. We choose release velocity from 7 to 8. m/s in increments of. m/s and release angle from 45 to 65 deg in increments of.5 deg and Hz backspin angular velocity. We assume the release point is 4 m behind and m below the hoop centre... First ball-contact boundaries for rebounds Release angle ( ) Bridge Board Release velocity (ms ) Release angle ( ) Bridge Board Release velocity (ms ) Fig.. Shooter release conditions which result in rebound when launched from a point 4 m behind and m below the hoop centre with Hz backspin angular velocity, with no lateral deviation angle and a small lateral deviation angle of deg from the board normal plane. Rebound regions for first contact with front rim, back rim, bridge and backboard are shown in red, blue, yellow and green, respectively. Capture or air ball regions are shown in white.

3 96 Hiroki Okubo and Mont Hubbard / Procedia Engineering 3 () (c) (d) Fig.. Ball centre path in two dimensions for free throws from 4 m away and m below hoop centre, with no lateral deviation angle at the release point, a release angle of 5.5 deg, Hz backspin angular velocity, and release velocity of 7.5 m/s 7.35 m/s (c) 7.58 m/s and (d) 7.67 m/s. We use the dynamic model to study rebounds of free throws, beginning with first ball contact boundaries. Figures and show the first contact conditions of the ball in the release angle velocity parameter space for free throws with no lateral deviation and for a small lateral deviation angle of deg at the release point, respectively. The initial angular velocity is Hz backspin angular velocity. Figure shows typical ball paths with no lateral deviation for free-throw rebounds in the original ball trajectory plane. The coordinate origin is at the hoop centre. The shots are released 4 m away from and m below the hoop centre, with initial angular velocity of Hz backspin (the spin axis is perpendicular to the initial shot plane and parallel to the board surface). In Figure the ball first contacts the front rim and rebounds directly. In Figure the shot bounces sequentially off the rim and board. Figure (c) shows the rebound after the ball bounces off the back rim. In Figure (d) the shot has contact first with the back rim and then with the board.

4 Hiroki Okubo and Mont Hubbard / Procedia Engineering 3 () (c) (d) Fig. 3. Ball centre paths in three dimensions for free throws from 4 m away and m below the hoop centre, with degree lateral deviation at the release point, release angle of 5.5 deg, Hz backspin angular velocity, and release velocity of 7.5 m/s 7.35 m/s (c) 7.58 m/s and (d) 7.67 m/s. Ball centre paths in three dimensions are shown for free throws with a small lateral deviation of deg at the release point in Figure 3. The ball moves in three-dimensional space and has Hz backspin angular velocity. The initial backspin axis is perpendicular to the initial ball trajectory plane. The motion of the ball is more complicated than that with no lateral deviation at the release point. In Figure 3 the ball bounces to the right side of the hoop after contacting the upper front rim. It may bounce far away from the hoop. In Figure 3 the ball first contacts the inside of the front rim, bounces off the backboard and the rim, and rebounds. The direction for the rebound is approximately normal to the board. The ball usually drops near the hoop when it has several contacts, loses kinetic energy, and its speed decreases. In Figure 3(c) the direction of the rebounding ball is opposite to the first ball-contact side when the ball first contacts the inside of the back rim. In this case, the ball bounces far away from the hoop. In Figure 3(d) the shot rebounds after bouncing off the back rim and the board. The rebounding direction is approximately normal to the board.

5 98 Hiroki Okubo and Mont Hubbard / Procedia Engineering 3 () Rebound analysis We calculate the position and distance of the ball from the hoop centre for rebounds when the basketball first bounces off the front or back rim. Rebound position is defined at the moment when the height of ball centre equals the hoop height. Figure 4 shows the distance as a function of the tilt angle at first rim contact that characterizes the deviation of the contact point in or out of the rim. The shots were made from 4 m away and m below hoop centre, move in the board normal plane, and have Hz backspin angular velocity and release angle of 5.5 deg. A positive value of tilt angle means the ball is inside the rim. Because players use direct shots for free throws, we investigate the relationship between the first rim contact position and the rebounding distance from the hoop centre. When the tilt angle is approximately -45 deg for front rim contact and 45 deg for back rim contact, the ball rebounds far away from the hoop. The ball falls near the hoop if the ball first contacts near the top of the rim. The players should locate near the hoop when good shooters try free throws because it is unusual that the ball hits outside the hoop. Distance (m) Tilt angle ( ) Normal position (m) Parallel position (m) Normal position (m) Parallel position (m) (c) Normal position (m) Parallel position (m) (d) Fig. 4. Distance from the hoop centre for rebounds as a function of the tilt angle at first ball-rim contact when the height of the ball equals the hoop height. The shot is normal to the backboard, and has release angle of 5.5 deg, and Hz backspin angular velocity. Rebound position for free throws with lateral deviation angles of., (c).6, and (d). deg at release. Shots for rebound after bouncing off the front and back rim have release velocity from 7 to 8. m/s in increments of. m/s and release angle from 45 to 65 deg in increments of.5 deg. Rebound regions for front rim contact are shown in red and for back rim in blue.

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