1. GOLF DRIVER OPTIMIZATION COLE SNIDER, ANTHONY BOYD, FRANK RIVERA, MAX DREAGER

2. PROBLEM STATEMENT To determine the optimum club head loft and swing elevation angle to achieve the maximum drive distance. The swing elevation angle (ε) will be optimized for varying loft angles (λ), and vice versa. It was assumed that the club head and golf ball impact at the “sweet spot” of the club head Other factors considered were the mass of the golf ball and the shape of the club head.

3. SUBSYSTEMS Impact Model Inputs velocity, swing elevation angle, and loft angle Objective: Determine the velocities of the golf ball directly after impact Constraints: Elevation angle and loft angle Variables: Velocities of golf ball, club head, swing speed, mass of the club head, mass of the golf ball, shape of the golf ball and club head Aerodynamic Model Receives resultant velocities from the impact model Objective: Determine the distance the golf ball travels in the air Constraints: Elevation angle, loft angle and coefficient of drag and lift Variables: Cross-sectional area of the golf ball, acceleration due to gravity, and the density of air

4. IMPACT MODEL Z X λ ε z’ u 1-12 =velocities of the club head and golf ball θ=ε+λ Resultant velocities of the golf ball must be translated to the X,Y,Z reference frame from x’,y’,z’ r c and r b are the position vectors from their respective center of masses Club Head Golf Ball rcrc rbrb

5. IMPACT MODEL

6. IMPACT EQUATIONS

7. AERODYNAMIC MODEL Translational/rotational velocities from the impact model were used to find drag and lift forces The gravitational, aerodynamic, and inertia forces on the golf ball were set equal to zero ode45 was used simultaneously on these equations to calculate translational velocities at each time step The height of the golf ball and the distance it had travelled were updated iteratively until the ball hit the ground

8. AERODYNAMIC EQUATIONS

9. CONSTANT SWING ELEVATION – VARYING LOFT ε (degrees) Optimum λ (degrees) Maximum Distance (m) -1011.5713131.08 -54.3575134.47 -44.4011135.41 -34.4368137.47 -24.4644140.24 4.4824143.33 04.4946147.09 119.2683246.05 218.605246.44 318.02246.96 417.6815248.48 59.4692200.83 104.4166187.59 Obtained optimum values by using a meta model of the system Optimum combination of loft angle and swing elevation angle occurred between 1 and 4 degrees of swing elevation

10. CONSTANT LOFT – VARYING SWING ELEVATION λ (degrees) Optimum ε (degrees) Maximum Distance (m) 57.5236182.3 77.4428194.68 87.4111201.31 97.3827207.7 107.3428213.93 117.3060220.20 127.2875226.2 137.2593232.24 157.2053243.43 Obtained optimum values using a meta model of the system These values linearly increased over time

11. AVERAGE DISTANCE FOR CONSTANT LOFT Swing elevation varied from 0° to 10° for a constant loft angle Loft was varied from 8° to 18° for each range of swing elevation Average distance was calculated at each loft angle Optimum lambda value is approximately 16°

12. SUMMARY Objective – Optimize the swing elevation and loft angle to achieve the longest drive distance Variables – Swing elevation angle and loft of club head (constrained at 0° ≤ x ≤ 20°) The subsystems are connected by the velocities of the golf ball directly after impact Swing elevation angle has a greater impact on the distance the ball will travel Maximum distance was approximately 248 meters Loft Angle=17° Swing Elevation Angle= 4°