Transactions on Engineering Sciences vol 7, 1995 WIT Press, ISSN

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Edmund McDowell
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1 The role of the engineering properties of materials on the performance of golf clubs R.D. Thompson/ A.A. Adam^ T.D. Lucas" "James Watt Engineering Laboratories, University of Glasgow, ^Unifiber Europe, Glasgow, Scotland, UK Abstract Golf equipment has evolved during the history of the game and modern manufacturers must squeeze more performance from tried and tested products. The problem is to isolate the factors which truly affect performance from those which are cosmetic or of psychological benefit. From finite element analyses of the impact of the club and ball it is possible to determine properties of the clubhead which influence spin and hence distance. It was found that both the elastic modulus and friction coefficient of the clubface affect spin but, over the range of materials commonly used, the difference was insignificant. Geometric design factors, such as size and shape and their influence on weight distribution, are more important than material. 1 Introduction Golf equipment has evolved by natural selection during the history of the game and novel design features which significantly improve performance and are a commercial success, are difficult to achieve. Design theory was well advanced by the end of the nineteenth century but its practical application was constrained by the materials and manufacturing methods available. Professionals of one hundred years ago were, for example, aware of the theoretical advantages of peripheral weighting in irons and woods but it was only the development of lost wax casting and ductile cast irons that allowed mass production of cavity backed irons and metal woods. Such dramatic design innovations are however rare and modern manufacturers must usually try to squeeze more performance from tried and tested products. But in a sport where the skill of the participant is paramount, many claimed improvements are cosmetic or of psychological benefit. Nevertheless, even small effects must be considered when they make the difference between winning

370 Contact Mechanics II and losing. The aim of the current work is to use finite element analyses to isolate design features which have an effect at the margins of performance.

2 370 Contact Mechanics II and losing. The aim of the current work is to use finite element analyses to isolate design features which have an effect at the margins of performance. 2 Golf equipment as an engineering system For design purposes, the golf club and ball can be regarded as an equipment "system". Physical experiments with the complete system are relatively easy but detailed computational models involving all known influences are not practical even with modern supercomputers. It is however possible to obtain useful results from models of subsystems or simplifications of the complete system. Whittaker at al* and Grundy, Cairns and HkxxP have thus assessed the potential for game-improvement clubs by modelling the system as an assemblage of rigid links. This work culminated in refinement of the John Letters' "Calculus", a club which matched theoretical expectations in tests. In associated stress analyses, Thomson et al\ accepting Cochran and Stobbs* finding that the shaft may influence the orientation and velocity of the clubhead at the start of the impact but plays no direct role after initial contact, modelled the club/ball system as a rigid clubface impacting a deformable elastic "ball". Computer models may promise better equipment but before this promise can be fulfilled, they must explain what makes existing designs successful. They must show which features contribute most to distance and controllability. The explanation must also be given in scientific terms and here lies a problem, for golf has evolved its own quasi-technical language. Golfers may, for example, argue about the merits of "hard" or "soft" clubfaces but will generally attest that they play differently. But since permanent deformation of the club does not occur, hardness in an engineering sense cannot affect performance. One must identify what is meant by such golfing terms and see whether these have any validity as factors in club design. 3 Computational modelling Performance factors in golf include the initial velocity and trajectory of the ball and the rate and direction of its spin. This paper focusses on backspin co, which generates lift and affects the distance of a drive. The classic experiments of Cochran and Stobbs* imply that, contrary to intuition, the friction coefficient \JL between the ball and the clubface has no effect on co. The problem is then to identify engineering properties of the clubhead which do significantly affect CD and must be included in design calculations. Other than i, the list of material properties which might affect to is small. The most likely candidate is elastic modulus, E. In this study, computer analyses have a powerful advantage over practical tests in that they admit

3 Contact Mechanics II 371 experiments on hypothetical materials which do not occur naturally; they admit, for example, a change in E for the clubhead without any change in the density p, as might be necessary in practice. To study the effect of design changes on the performance of a clubhead, it is necessary to have a computer model of the ball whose impact response is realistic. It is then appropriate to analyse a finite element model in which both the clubhead and ball are represented by a mesh of 8-noded bricks as shown in Figure 1. The ball is of two-piece construction with a thin cover surrounding a hyperelastic core. The latter is needed to properly account for the large elastic strains which develop in the core during an impact which reduces its diameter to about half its original value. Hyperelastic properties are however difficult to obtain and so data for vulcanised rubber, whose behaviour is similar to that of the core material, was used \ In contrast, the cover undergoes large displacements and rotations but sustains only small strains. Geometrically nonlinear analysis is still necessary but a linear material model is acceptable and E=0.75 GPa was chosen, to simulate Du- Pont Surlyn, a modern thermoplastic cover material. Figure 1. The finite element model of the ball and clubhead. The clubhead mesh, with its 30 loft, is simple but models the essential mechanical features of a forged 5-iron blade. To isolate the desired effect, p was held constant at 7800 kg/m^ (p for steel) while E was varied from 210

372 Contact Mechanics II GPa (steel) through 80 GPa (Al), 3 GPa (nylon), to an extremely low 0.2 GPa (LDPE). The analysis was done with Abaqus/Explicit 5.2 on an IBM RS6000 work-station.

4 372 Contact Mechanics II GPa (steel) through 80 GPa (Al), 3 GPa (nylon), to an extremely low 0.2 GPa (LDPE). The analysis was done with Abaqus/Explicit 5.2 on an IBM RS6000 work-station. The results were displayed using Abaqus/Post. 4 Results Plots of the deformed shape during impact show that an initial horizontal clubhead velocity of 50 m/s causes significant flattening of the ball (Figure 2), angular acceleration as the ball climbs the face and separation after 0.8 ms. These features are in accord with the experiments of Cochran and Stobbs\ who measured the impact duration as 0.5 ms. Later, the ball recovers its original shape as the stress state becomes more isotropic, probably as a result of multiple reflections of the stress waves, since there is no damping to cause energy dissipation. Figure 2, incidentally, is for the compliant LDPE material and shows some deformation of the clubface. Figure 2. Deformed shape of the system 0.4 ms after initial contact. Backspin is best calculated from snapshots of the motion. Figure 3 shows a snapshot for E = 200 GPa. From plots of the position 2ms after impact it was apparent that a stiffer clubhead produces greater angular displacement. This suggests that co increases with E. However this 2ms includes an acceleration time and a more accurate value of co is found by comparing angular displacements at, say, 1ms (after the ball has separated from the club) and at

5 Contact Mechanics II 373 2ms. The position of the ball at 1ms is shown on Figure 3, from which the angular displacement may be measured. It was found that the stiff club gave CD = 156 revs/s while the compliant club gave co = 145 rev/s. Figure 3. Angular position of the ball after impact with a stiff clubhead. 5 Clubface friction Given the 1000-fold difference in E, the difference in co was smaller than expected. This forces a re-examination of the role of club face friction in spite of the experimental evidence. Again computer models allow hypothetical materials, all with E = 200 GPa and p = 7800 kg/rn^ and differing only in i, to be tested. The result of an initial analysis with p = 0 is shown in Figure 4. This shows the initiation of backspin, which continued to develop during contact. It results, not from shear but from a resultant interfacial normal force which has a moment arm about the centre of mass and so induces spin, even in the absence of friction! Different values of ji were then applied and co plotted as in Figure 5. This indicates that co is an increasing function of p. However the curve has a pronounced knee at about \i = 0.1 and is almost constant between 0.3 and the upper value of 0.9 applied to the model.

374 Contact Mechanics II Figure 4. Deformed shape of the ji=0 system 0.4 ms after initial contact. 160 140 _ 120 j/» 100 1 80 (/) O 60 CO 40 20 0 0.3 0.4 05 06 FRICTION COEFFICIENT Figure 5.

6 374 Contact Mechanics II Figure 4. Deformed shape of the ji=0 system 0.4 ms after initial contact _ 120 j/» (/) O 60 CO FRICTION COEFFICIENT Figure 5. Variation in backspin o with friction coefficient i. Figure 6 is the velocity/time response of a point in the interior of the ball and one on the club-face. This shows the ball reaching the clubhead speed after 0.4ms but continuing to accelerate during elastic recovery. Fluctuations in the ball velocity after separation are due to local changes in the centre of mass caused by (undamped) stress waves travelling in the ball.

7 Contact Mechanics II 375 ~~i Time (*10**-3) Figure 6. Velocity-time response for the ball and the club From plan views of the impact it was found that i = 0 induces a hooked shot while J.L = 0.6 induces a slice. However the effect is small and is not pursued here. Side views of the impact allow the departure angle a of the ball to be determined. With ji = 0, a =30, equal to the loft of the club. This agrees with the theoretical prediction of Cochran and Stobbs *, who did however predict minimal backspin. With i = 0.6, the translational speed of the ball is as before but a = Discussion and conclusions The behaviour of the ball after impact is, where applicable, in accord with the experiments of Cochran and Stobbs\ This provides confidence in the ability of the computer model to predict the performance of novel clubhead designs. The large deformation of the ball and the consequent prediction of backspin with a frictionless interface demonstrate the need for nonlinear large-deformation analysis if the response of the system is to be accurately predicted; rigid-body and point-mass models are inadequate.

8 376 Contact Mechanics II The spin of the ball depends on the elastic modulus of the clubface but only slightly. Good estimates of spin can then be obtained from analy-ses with a rigid clubface. Since material stiffness has a small influence on spin, it may be discounted from performance calculations, allowing product designers to concentrate on the more significant effect of geometry, with its consequences for weight distribution and the shape of control surfaces. He (or she) is free to select from a wider range of materials for novel designs and to emphasise, for example, cost, manufacturability or appearance. Spin is an increasing function of friction coefficient but the curve is flat over a range which includes the materials commonly used for clubheads. This explains the counter-intuitive experimental results of Cochrane and Stobbs\ Furthermore, the curve flattens at increasing values of friction coefficient and materials with high friction coefficients will not impart more spin. This has implications for the rules of tournament golf. It also reinforces the need for the club designer to concentrate on geometry rather than material. Since a high-friction interface induces increased rotational KE in the ball but the same translational KE, it is to be expected that the velocity of the highfriction clubhead would be less after impact. This is confirmed by the computed results. That the angle of departure of the ball is dependent on friction coefficient is important: this implies that a coating of low friction material may be used to modify the angle of departure from the same clubhead casting. The use of coatings to modify clubhead performance has been implicit for many years but has not been fully exploited in design. References 1. Whittaker, A.W., Thomson, R.D., McKeown, D. and McCafferty, J. The application of CAE techniques in advanced clubhead design, Proc 1st World Scientific Congress on Golf, St Andrews, Grundy, R.E., Cairns, R.A. and Hood A.W. The effect of clubhead weight distribution on ball-club impact dynamics, Proc 1st World Scientific Congress on Golf, St Andrews, Thomson, R.D., Whittaker, A.W. and Wong, K. Computer-aided analysis of golf club/ball interaction during impact, Proc UK Abaqus Users' Conf, UKAEA, Risley, Cochran, A. and Stobbs, J. The Search for the Perfect Swing, Heinemann, London, Abaqus/Explicit Version 5.2 Example Problems Manual, HKS Inc, Providence, RI, USA, 1992.

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