In Search For The Perfect Swing is mentioned the results of experimental work done on putting. The duration of impact was measured as being between 0.0005 and 0.0007 sec. Perhaps a bit of an surprise but this is very similar to the high speed impact duration. This implies that the putter head also behaves essentially as a free body. I did some calculations using for the putter head a simple uniform square bar as shown in Fig1.

During impact the ball exerts a force, -F, on the bar. Fig2 shows that this is equivalent to a force, -F, acting through the center, and a couple, having a moment, -F * d.

The velocities, V2, U2, and the angular velocity of the bar, at separation (end of impact), have been derived and are given by relationships (1), (2) and (3). The average force Fav and average moment Mav, during impact, are given by relationships (4) and (5).

Let's assume some numeric values, e=0.6, m1=0 .3 kg, m2=0.045 kg, a=0.1 m, b=0.02 m, Δt=0.0006 sec, V1=4 m/s

Let's assume an off center impact with d=.0254 m ( 1 inch). Let's use the graphs above to determine the various parameters.

From the graphs one can see that the bar separation speed V2 is equal to 81 % of the incoming bar speed V1. Furthermore the ball separation speed U2 is 27% above this incoming bar speed V1. The angular separation speed is 22.3 rad/sec.

The average force Fav exerted by the ball on the bar, during the impact time of 0.0006 sec, is 380 N (85 lbs), which the peak force likely about 2 times this average value, hence about 760 N (170 lbs).  

The average couple Mav exerted on the bar by the ball is 9.6 Nm (7.1 lbft). As for the peak force, the peak couple likely reaches 2 times this value and hence about 19 Nm (14 lbft).

For fun let's use our model to see if we can get an idea of how much deviation would result from this off center impact. One can derive the average angular acceleration to be:

The angular deviation Δφ due to the off center impact is hence:

For the values chosen this gives Δφ = 0.43 deg. If we take the putting distance to be 7 m (21 ft) this then results in a deviation of 5.2 cm (2 inch) due to the off center impact.

The simple model used above to represent an off center impact should really be developed together with experimental work. Nevertheless it is fun to look at such a simple theoretical model and see the surprising large forces/torques being developed during impact. It is equally surprising to see that off center impacts don't have that much effect on the resulting velocity of the golf ball. Also the deviation of the ball off the target line is not very great.

mandrin