2.3.1. Definition ball kinematics
Adding topspin to the ball is an essential factor in achieving putting accuracy. The object is to minimize initial skidding of the ball on the green after putter contact, deceleration force is high, and the ball will often skid. During skidding, the ball loses linear momentum while gaining angular momentum as its rolling motion increases. The skidding ends with pure rolling motion when no further skidding occurs. Past studies (Lindsay, 2000) impact-face surface properties, impact-face loft, and putter-head mass distribution.
Figure 2.5. The primary forces on a golf ball rolls on putting green.
If a golf ball is struck by a putter to produce no initial spin, it will skid for a distance (xt) until the frictional torque increases its angular velocity, finally achieving the pure rolling-without-skidding condition.
ω= Vcm (2-2) R
Currently, to formulate golf ball models are formulated to predict what will happen to a golf ball when subjected to a given set of impact conditions. It appears that no single model of golf ball impact currently can be incorporated, either for typical impacts, or oblique, which can predict the dynamic performance of different types of golf ball construction under every set of impact conditions. Modelling visco-elastic, multi-component golf balls involve complex calculations and numerous assumptions before a model can be incorporated.
The run of a golf ball consists of the bounce phase and the subsequent rolling after landing.
In the case of a drive, golfers will typically want a long run, while for shots on the green, limiting the bounce and skid of the golf ball are ideal. Several researchers (Daish, 1972; Penner, 2002) have presented models of golf balls bouncing on turf. In the case of Daish (1972), the general behaviour of a bouncing ball on a rigid surface is modelled, with the golf ball used as an example. Daish considers two specific cases. First, is the case where the ball slides over the surface throughout the impact, which, for the typical impact of a golf ball on the turf, will result in the golf ball retaining some of its backspin as it bounces from the surface.
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In the second case, the frictional force between the ball and the surface is significant enough to check the backspin and to have the ball bouncing out of the impact with topspin.
Daish determined that the minimum value of the coefficient of kinetic friction, µmin, between the ball and the surface, required to check the backspin, is given by where vix and viy are the impact velocity components, r is the radius of the ball, ωi is the backspin of the impacting ball, and e is the coefficient of restitution between the ball and the surface. Daish used this model to compute the runs for several different golf shots.
In general, however, this value would depend on the impact speed. Penner (2002) measured the dependence of the coefficient of restitution for standard impact on the impact speed of the golf ball and found that although the value of e was approximately 0.5 at low impact speeds, less than one ms−1, for higher impact speeds the value of e decreased, approaching a value of 0.12 as impact speeds approached 20 ms−1.
Putting is the most common shot in the game of golf. Several researchers have considered the motion of a golf ball as well as the interaction between the golf ball and the hole. The golf ball's initial motion is dependent on how the golfer strikes the ball. Daish (1972) found, from high-speed films, that most golfers putt the ball ‘on the up.’ For these cases, the golf ball is projected into the air and will make a series of bounces before it begins rolling along the green.
For putts that are not projected upwards, the golf ball will initially skid along the turf. Both Daish, and Cochran and Stobbs (1968), state that the golf ball will be in a state of pure rolling after the ball has travelled approximately 20% of the total length of the putt.
Lorensen and Yamrom (1992), Alessandrini (1995), and Penner (2002) have presented models of the motion of a rolling golf ball over a green. The primary difference between the models is how the frictional force acting on the golf ball is handled. Lorensen and Yamrom use two different constant coefficients of friction, one to model an initial sliding phase, and the other to model the rolling phase. This model was used as the basis for visualizing trajectories, using computer graphics and golf balls travelling over piecewise-planar models of real greens.
Alessandrini treats a putt as a two-point boundary value problem and determines the initial conditions required that would allow the trajectory to terminate at the hole with zero velocity.
In this model, the frictional force is kept constant over the total length of the putt. Penner (2002) treats the golf ball as being in a state of rolling throughout the putt. In this model, the retarding force acting on the golf ball is taken to be that which constrains the ball to roll. Using this model, Penner (2002) determined that golf balls' trajectories roll on the flat, uphill, downhill, and sideways-sloped greens.
Penner (2002) determined the launch conditions required for a successful putt by determining which golf ball trajectories along the green would lead to allowable impact speeds and parameters as determined by Holmes’ ball capture model. As with Mahoney, it was found that the probability of making a downhill putt can be significantly higher than the probability of making an equally distant uphill putt. For this downhill putt, an off-line trajectory will tend to converge back towards the target line, while the reverse occurs for uphill putts. This somewhat surprising result must be tempered by the consequences of a missed putt, as a missed downhill putt will, in general, stop much further from the hole than an equivalent uphill putt.
For most golfers, this would result in a preference for uphill putts. Penner (2002) also determined the range of allowable launch speeds and angles for putts across the slope of a green.
It was found that taking a slower uphill path, as opposed to a faster, more direct path, would increase the probability of success.
Werner and Greig (2000) did a detailed analysis of several aspects of putting. They looked at hit patterns on a putter and the stop patterns around the hole for golfers of various handicaps.
They used these results, along with a model of the putt, to determine what is the ideal distance beyond the hole that a golfer should be aiming for. For example, for a ten-foot putt on a flat green, they found that a golfer should strike the golf ball so it would stop a distance of approximately 20–40 cm, depending on the golfer's handicap and the green speed, past the position of the hole.
Lemons, Stanczack, Beasley and Cocharn (1999) looked at the effects of ball construction on both putting distance and break amounts. They found that the cover hardness played the most crucial role in the distance the ball rolls. For example, using a putting mechanical arm, they found that for the same putter head speed, a hard-covered two-piece ball rolled approximately 3–5% farther than either a soft covered three-piece or soft covered two-piece ball. It was also found that the ball construction played no part in the break of the putt.
Hubbard and Alaways (1999) considered several aspects of the interaction between a golf ball and a green. They measured surface viscoelastic properties of the turf by experimentally dropping balls onto a green and measuring their resulting accelerations and positions. Peak accelerations of about 50g were measured, and several bounces were observed from small drop heights (2 cm). They also measured the variation of the coefficient of rolling friction over the length of a putt. They found that the rolling friction increased by about 10% throughout a 4.3 m putt. This would indicate that the coefficient of rolling friction increases with decreasing velocity.
2.3.2. Essential ball kinematics to performance
Lindsay (2013) presented a theoretical and experimental study showing the selection of centre-of-gravity (CG) and moment of inertia (MOI) of the putter head around the heel-toe axis could minimize and even eliminate the initial skidding phase of a putt.
In the past, putter design has mainly focused on the inertia properties of the golf equipment, putting kinematics whereas putter kinematics like face angle, putter path and horizontal impact spot has been key to successful putting skills (Karlsen et al., 2008a; Pelz, 2000). Putter designs have been focussed on to maximize the performance on mis-hits like horizontal off-centre impact (Lambeth et al., 2018) and skid distance, roll efficiency (Hurrion and Hurrion, 2002), as well as polymer inserts on the forward spin (Brouillette & Valade, 2008). Putter design has been interested in examining how inertia properties of the clubhead in the horizontal plane, as well
as facial treatments, on off-centre heel-toe impacts. Interests in improving the performance of the putter in the vertical plane, reducing skidding distance. Pure rolling earlier in the putt trajectory, conceptually producing more stable putts via the spin stabilization effect (Karnopp, 2004).