1. FEA of a Golf Driver and Golf Ball • Solid Mechanics - ES 240 • Adrian Podpirka • ABAQUS Project

2. Outline of Work • Introduction to Golf • Goal of Research • Theory • Modeling • Results • Discussion • Analysis • Conclusion • Citations

3. Golf • Invented in Scotland around 1450s. • Requires hitting a small ball roughly 200-500 yards into a small hole. • Different clubs are used depending on distance and arc required. • During the first shot, the golfer tries to hit it down course as far as possible

4. Goals of Project • To determine stress distributions in a golf ball and in a driver. • To determine natural frequency of the golf ball and driver. • Attempt to determine percentage of sweet spot and effect of driving distance. • Learn ABAQUS

5. FEA & Golf • Only recently has FEA been used to design clubs. • Programs are being made specifically to cater to the golf industry. • Used to analyze swings, slicing, tendency to hook, etc.

6. Theories • Golf Ball • Internal Stresses in the golf ball will arise due to sudden impact and different properties of the two materials • Ball will deform as drastically as seen in the picture to the right. • Frequency Measurement • A closer natural frequency between the ball and the club will lead to an increase in distance. • Stress Propagation • Sweet spot occurs symmetrically from propagating waves. • Allows for wave dispersion before coming in contact with the club face edge.

7. Young’s Modulus E (GNm^-2) Poisson’s Ration Density (kg m^-3) Note Butadien Rubber .0392 .45 1150 Inner Core of Golf Ball Iononer Resin .294 .40 950 Outer Core of Golf Ball Ti-6Al-4V 118 .34 4507 Standard Driver Head Material Carbon Fiber 17.2 .31 1545 Standard Shaft Material Materials • Golf Ball • Driver Head • Driver Shaft T. Iwatsubo et al B. Wang et al

8. Geometry of Equipment • Golf Ball • Golf Club - Wood Driver 40 cm 44 cm Shaft length - 1.05 m Height - 40 mm Width - 90 mm Depth - 65 mm

9. Golf Ball • loaded linearly ramping to 15000 N. • Golf Ball • Sweep meshed with 1600 elements • Modeled a 44 cm diameter area and partitioned off middle section. • Traction load placed in between 7 & 9 • Boundary Condition placed directly opposite • Results • Internal stresses develop as a result of mismatch of materials on the order of 40 kN. • Golf ball is seen to deform. This is analogous to the picture shown before.

10. Golf Ball Results

11. Natural Frequency • The closer the frequency between club and ball, the better energy transfer and therefore, farther distance. • We will test the difference between hollow and solid clubs • Golf Ball • Meshed with 124 elements • Circular edge boundary conditions • Driver • Meshed with roughly 169 & 171 elements • Pinned at top

12. The hollow bodied club face has a lower frequency then the solid body, closer matching that of the balls.

13. 2D Stress Distribution • Assume traction loading on face of of driver. • Large deformation occurs in shaft of carbon fiber. • Stress waves still occurs in driver face but much less then with coupled shaft.

14. Stress

15. 3D Stress Note: The full 3D club could not be meshed because of element assignment errors in ABAQUS. The Natural frequency of the club could not be found.

17. Analysis • Full Analysis of all data and values will be given in the paper. • The golf balls deformed as theory and practice indicated. • By tuning golf balls to different clubs, better distances can be obtained. This would require changing either the parameters on the ball or club. • Since ABAQUS was not able to mesh the merged structure, I had to forgo on the natural frequency aspect of the 3D driver.

18. Recommendations • Many of the articles could not be located since Harvard did not have a subscription to them. • Many parameters • Using different material parameters in order to optimize values. • Dynamically loading and setting contact parameters

19. Citations • Wang et al. “Modal Properties of Golf Club Wood Driver in Different Boundary Conditions” • Hocknell et al. “Hollow Club Hear Modal Characteristics: Determination and Impact Applications” • Hocknell et al. “Experimental Analysis of Impacts with Large Elastic Deformation: I. Linear Motion” • Iwatsubo et al. “Numerical Analysis of Golf Club Head and Ball” • Penner, A. “The Physics of Golf: The Convex Face of a Driver” • Newman et al. “The Dynamic Flexing of a Golf Club Shaft During a Typical Swing” • Arakawa et al. “Dynamic Contact Behavior of a Golf Ball during an Oblique Impact” • H. Kolsky. Stress Waves in Solids. Dover Publications Inc. • Axe et al. “The vibrational mode structure of a golf ball”