ANALYSIS OF NONCONFORMING COEFFICIENTS OF RESTITUTION IN GOLF DRIVERS USING A FINITE ELEMENT APPROACH Engineering Project by: Brian Hill Main Advisor: Ernesto Gutierrez-Miravete Co-Advisor: Ken Brown First Progress Report

Recall: What is Coefficient of Restitution? COR is a number between 0 and 1 that represents a percentage of energy transferred during an impact event. Usually expressed a ratio to pre and post impact velocities. COR = Vout/Vin. COR = 0 is a perfectly inelastic collision. All energy is absorbed at impact. COR = 1 is a perfectly elastic collision. All energy is retained and transferred at impact.

Recall: COR and Golf Theoretical limit of COR in golf drivers is about 0.910 – 0.930. Difference in performance for a rigid clubface (.768 COR) vs. one at the COR limit of 0.830 is about 10-15 yards less at a swing speed of 110 mph. Difference of a clubface at a COR of 0.830 vs. 0.860 is about 5.6 yards less at a swing speed of 100 mph. For slower swing speeds the difference is less. Largely only effects professionals or players with higher swing speeds.

Goals Investigate the effect clubface thickness has on COR. Investigate the effect swing speed at impact has on COR. Use USGA COR test method as a basis to create a suitable ANSYS FE model. Plot comparative results of COR vs. face thickness at different swing speeds.

ANSYS Modeling Properties 2-d model of representative clubface and 3 piece golf ball with axisymmetry Ball modeled as concentric semicircles and plate modeled as rectangles. LS-DYNA explicit 2-d solid 162 elements Quads or tris, 2d only Plastic kinematic properties for titanium plate Impact could be close to yield stress at small face thicknesses Mooney-Rivlin hyper-elastic properties for ball Core, Mantle, and Cover of 3 piece ball modeled using material properties from previous works. SI units used

Titanium Plate Geometry

LS-DYNA Analysis Options Nodal component created for the ball Initial velocity placed in –y direction on nodal component Surface defined for cover of ball, and top of plate. Contact definition: “Surface to Surface” contact in lieu of “automatic” Contact time defined as .005 seconds

Axisymmetric Model Cross Section Areas Defined Mesh Created Materials Defined

Expanded Axisymmetric Model

Current Status Meshing accomplished via trial and error with line sizing tool. ANSYS-ED has element number restrictions Element and material properties applied correctly Issues with defining contact within LS-DYNA Solution achieved, but post-processing details errors. Ball seems to pass right through plate, no reactionary forces or stresses. Incorrectly defined contact surfaces are most likely the cause.

References: K. Tanaka et al., Construction of the finite-element models of golf balls and simulations of their collisions, Proc. of the Institution of Mechanical Engineers, Part L: J of Materials: Design and Applications 220 (1) (2006), pp. 13–22. Petersen, Willem et al., Shape optimization of golf clubface using finite element impact models, International Sports Engineering Organization, December 2009 K. Tanaka et al., Experimental and finite element analyses of a golf ball colliding with a simplified club during a two-dimensional swing, Procedia Engineering, Volume 2, Issue 2, The Engineering of Sport 8 - Engineering Emotion, June 2010, Pages 3249-3254 www.fanklygolf.com, September 21, 2010. <http://www.franklygolf.com/driver-cor-springlike-effect.aspx> GOLF CHANNEL Newsroom. “USGA, R&A Rule on 'Spring-Like' Effect.” thegolfchannel. August 6, 2002. September 21, 2010 <http://www.thegolfchannel.com/tour-insider/usga-ra-rule-spring-like-effect-6874/> “What is C.O.R.? What is CT?” wishongolf. September 21, 2010 <http://www.wishongolf.com/faq_tech_answer.php?techKey=1> “Conflicts surrounding the Callaway ERC driver.” golftoday. September 21, 2010 <http://www.golftoday.co.uk/news/yeartodate/news01/callaway.html> “Coefficient of restitution.” wikipedia. September 21, 2010 <http://en.wikipedia.org/wiki/Coefficient_of_restitution> Kelley, Brent. “Clubhead Speed.” about. September 26, 2010 <http://golf.about.com/od/golfterms/g/clubheadspeed.htm>