Development and Comparison of 3D Dynamic Models of Golf Clubhead-Ball Impacts

The scientific understanding and modeling of impacts has led to major advancements in golf equipment performance. The latest generation of drivers allow golfers to hit the ball farther compared to previous models. Simultaneously, new clubs are more forgiving, resulting in poor shots to land closer to the target. Impact models are an important tool as they allow for computer simulations and optimizations to inform design decisions. Several driver impact models have been published for this purpose. Impulse-momentum (IM) and other analytical contact models offer fast simulation times. While finite element (FE) models are used to design clubs, they were not considered here as they require orders of magnitude more compute time. Many existing impact models have no or limited experimental validation to support their conclusions. From a literature review, several impact models were considered to develop golf-specific impact models. A significant portion of this research was identifying model parameters and quantifying the accuracy of these models. This was achieved using experimental data consisting of driver and 7-iron shots by elite players. This allowed for the accuracy of these models to be compared against each other.
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\nThree IM-based impact models were used to predict the initial ball conditions for driver and 7-iron shots. The first model was referred to as the standard IM model as it followed the general methodology in the literature. The second IM-based model adds a small amount of mass to the clubhead and moves the ball centre of gravity slightly downwards to improve accuracy. These adjustments are intended to compensate for neglecting the shaft and ball deformation. The third IM-based model replaces the pure rolling assumption to allow a prescribed amount of slip at the contact point. This was motivated from experiments in the literature. The standard and adjusted IM models have been used to predict driver impacts. However, no published IM-based models exist to predict iron shots.
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\nThree continuous analytical models were also used to model golf impacts. The first model was a volumetric normal force contact model with a two-layer ball and velocity based friction model. This model captures the tangential force reversal observed in experimental studies and FE models. From a literature review, two other non-FE models were considered as they also predict this tangential force reversal. These models were extended to be suitable for modeling golf impacts. A damping parameter was added to the normal force equations to make the predicted collisions inelastic. These models were originally developed for 2D collisions against a rigid plate. Here they were extended to a 3D multibody dynamics model to predict the collision between a golf club and ball.
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\nFor modeling driver impacts, the adjusted IM-based model was the most accurate overall. Ball speed was the most accurate launch condition with a mean absolute error (MAE) of 1%. The MAE for vertical launch angle and backspin was less than 10%. However, the MAE for sidespin and horizontal launch angle (azimuth) was between 30%-50%. The relative error of these two launch conditions was considerably greater for all impact models considered. For modeling iron shots, the two-layer ball with a volumetric normal force model was the most accurate of the models considered. The MAE for ball speed was 2%. The remaining relative errors were similar to the driver model. 
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\nFrom separate experimental testing, it was shown that amateur golfers hit the ground before the ball on nearly one-third of iron shots hit off the ground. Soil mechanics models were applied to the clubhead to extend the iron impact model to predict this type of mishit. This new model with ground reaction forces was used to predict the behaviour of iron strikes for off-centre hits. This model shows that irons follow the same trends as a driver for off-centre shots. Including the ground model results in a greater distance loss and the shot being significantly offline compared to the no ground model. While no rigorous experimental validation was performed for the ground model, the decrease in ball speed is consistent with experimental observations in the literature. This new iron impact model allows for design optimizations, previously performed for drivers, to be completed with iron type clubs.

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