Science of putting

Oblique Modelling

Various investigations, including those of Mindlin and Deresiewicz (1954) and Maw, Barber, and Fawcett (1976; 1981) have been built on Hertz's work in the modelling typical impacts by adding various tangential parameters to attempt to model the oblique impact of elastic spheres. While these models provide the scientific community with a good starting point into the modelling of an oblique impact, the materials used to construct golf balls behave visco-elastically, meaning that the material behaviour will vary at different strains and strain rates.

The golf ball can deform by up to 25 % (Ujihashi, 1994), depending on the impact speed.

As such, assumptions such as 'the coefficient of restitution is unity' (Maw et al., 1976) make the models difficult to modify for use in a golfing context where approximately 20% energy is lost in the ball following a high-speed (40 ms^-1) standard impact (Lemons, 2002). The presence of a tangential component would only enhance the reduction in post-impact speed, through energy used to generate backspin (Lieberman & Johnson, 1994).

Jones (2002) used the Hertzian theory to model the typical impact of a golf ball. However, the model could only replicate deformation at low inbound speeds (4 ms^-1) and failed to take into account the energy loss in the ball material at increased speed. Chou et al. (1994) determined that during oblique impact in the golf ball, energy loss was explained by hysteresis

as well as friction by calculation from force/time histories of real, oblique impacts at 45°. These indicated that the maximum force of the impact was well below the yield stress of the materials used to construct golf balls. This energy loss during oblique impact has been defined subsequently as having two different components, motion within the ball relative to the centre of gravity and elastic strain energy displacement within the material. Maw et al. (1976, 1981) and attempted to model the oblique impact of an elastic golf ball theoretically. The analogy chosen was that shown in Figure 2.1. they were modelling the ball as a series of concentric circles where slip occurred between each layer throughout the impact. FT is the direction of the tangential force, FN, the direction of the reasonable force.

(a) (b) (c)

Figure 2.1. Theoretical golf ball model. (a) The annul positions immediately before oblique impact, (b) The annuli positions immediately on oblique impact, (c).

Source: Maw et al. (1981)

Chou et al. (1994) used experimental data to verify FE models simulating oblique impact.

The impact speeds of Chou et al. (1994) were comparable to that following contact with a metal-wood (45 ms^-1), using a professional golfer to swing clubs from metal-wood (driver) through to a nine iron and pitching equipment (loft angle around 55°) at swing speeds of around 30 ms

-1. The fit from both papers, between experimental and FE backspin/loft angle data, is reproduced in Figure 2.2.

FN

FT

Figure 2.2. Backspin rate versus loft angle/club for FE and empirical oblique impact data.

Chou et al. (1994) fitted parameters for coefficients of friction and viscosity to consider friction in more detail. Chou et al. (1994) yielded an agreement with backspin magnitudes up to a loft angle of 30° (backspin magnitudes of 1500-8500 rpm). This was also encountered by Johnson (1998) in their attempts to model oblique impact using numerical methods and experimentally determined friction coefficients.

It appears that the measurement of sliding and rolling friction coefficients between putter-ball interaction, has to this point not been comparable with the tangential force magnitude, and the relationship is generated at loft angles of above 40°. The accuracy of the FE results compared to the real force/time data (determined by firing the modelled ball at an instrumented, rigid oblique (45°) target), decreased with an increase in the impact speed, particularly in the unloading of the golf ball from the clubface. This indicates that polymer viscoelasticity may not have been fully taken into account when the material parameters were selected for the model.

This is a problem with the 'black-box' nature of FE modelling as it is hard to take into account changes in material behaviour with strain rate, particularly if the role of the various ball components (core, cover, and mantle) during deformation changes at different speeds (Cochran

& Farrally, 2002).

Coefficient of Friction

The friction properties between the turf and ball affect the ball slide (skid) and roll and how ball speed changes. Drane, Duffy, Fournier, Sherwood and Breed (2014) have indicated

the static coefficient of between artificial turf conditions by incline with horizontal (θ) and Equation to calculate Coefficient numbers as below (2-1) and friction of coefficient of sliding, dynamic and incline angle for rolling (Drane et al., 2014) in Table 2.1.

tan (𝜃) = 𝜇 (2-1)

Table 2.1.

Summary of Friction Coefficient Data

Source: Drane, Duffy, Fournier, Sherwood and Breed (2014)

The method to measure the friction and coefficient between the ball and the green is called The Stimpmeter. The Stimpmeter is thirty-six inches long with a groove about notch with 30 inches from the tapered end (USGA). The ball sits on the notch at the starting position, lying flat on the ground; when the user lifts the other end of the Stimpmeter to an angle of about 22 degrees, gravity influence the ball from the notch (Yun, 2013). The average Stimp of top professional tours like PGA/ LPGA ranges from 10~12ft, as illustrated in Figure 2.3.

Figure 2.3. Speed ramp for green speed STIMPMETER from USGA.

Source: Yun (2013)

Test Conditions μ Incline Angle(θ)

Static Coefficient of Sliding

Friction 0.3 16.8

Dynamic Coefficient of

Sliding Friction 0.26 14.5

Incline Angle to Initiate

Rolling - 5.2