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## Results and discussion

Based on the measured torques, the simplified shoulder and wrist torques were used to drive our dynamic model. The specific measured and simplified torques of the shoulder and wrist are indicated in Figure 3-5. The wrist actions used by both subjects are obviously the ones using negative torque (Figure 3-5(b) and Figure 3-5(d)). The comparisons of the speed of marker c on the shaft, arm and club rotational angles between the actual swings and computer simulation are given in Figure 3-6. Note that the swing motions from the proposed dynamic model are consistent with those from the actual swings (Figure 3-6). We should also be aware that the simplified torques applied in the simulation, are not expected to agree exactly with those from the measured ones but approximate to them.

The maximum horizontal club head speed at impact using three patterns of wrist actions are indicated in figure 3-7. It can be seen that the positive wrist torque results in the maximum club head speed at impact, 40.40 m/s for subject A and 32.18 m/s for subject B, giving the corresponding increases by 13.7 % and 12.0 % when compared with those using negative torque. The neutral wrist torque also provides an improvement in club head speed, and the increases are 11.0 % and 8.2 % for subject A and B, respectively.

The optimum ball position at impact is shown in Figure 3-8. We can see that the optimum ball position is different for the various types of wrist actions. However, for both subjects the positive torque gives the largest optimum ball position, followed by the neutral torque, followed by the negative torque.

An energy based analysis method, the same one as shown, with the ball position inconstant but being optimally determined (Maximum criterion), is used to search for how the wrist action affects the club head speed. Figure 3-9 shows the total work exerted by a golfer using three kinds of wrist actions. The application of the positive wrist torque leads to the maximum total work, and the negative gives the minimum. It should be noticed that the club head speed at impact is determined not only by the total work produced by a golfer but also by how the resulting energy delivers from the arm to club head. Figure 3-10 shows the efficiency index of swing motion η . It is shown that the positive wrist torque yields the maximum value of η , 75.4 % for subject A and 77.5 % for subject B. This phenomenon may be explained by that the positive wrist torque causes the high club angular speed, which in turn results in the large centrifugal force of golf club to retard the arm, so the arm angular speed is decreased, and as a result, the efficiency index η is increased. This point is consistent with the opinion in the work of Budney & Bellow, in which a positive wrist torque instead of a retarding torque was used to enhance the club head speed on condition of a reduced arm angular speed.

It can be seen that the ‘absorbed’ or ‘released’ bending strain energy of golf shaft at impact is exhibited to be different for various patterns of wrist actions (Figure 3-11). The negative wrist torque indicates the maximum ‘absorbed’ strain energy, 1.06 J for subject A .The neutral wrist torque, however, gives the maximum ‘released’ strain energy, -0.40 J for subject A and -0.45 J for subject B, which can be transformed into the kinetic energy of clubhead and thus the speed is increased. The positive wrist torque also offers the ‘released’ strain energy, but the values, -0.31 J for subject A and -0.35 J for subject B, are smaller than those using neutral torque.

The main purpose of this study was to examine whether different wrist actions in consideration of ball position offer a benefit to horizontal club head speed at impact by a new golf swing model. Considering the bending flexibility and centrifugal stiffening of golf shaft, a two-dimensional dynamic model, derived from a combined Euler-Lagrange formulation and assumed mode technique, was used to emulate the downward phase of golf swing. Although the torques of shoulder and wrist employed in this model are not exactly the same as those by the measurements, they are relatively close to the actual ones with necessary simplifications. Moreover, it is the good agreement of swing motions, including the speed of marker c on the shaft, arm and club rotational angles, between the actual swings and computer simulation for both subjects that leads to considerable confidence in the verity of the proposed dynamic model.

The simulation results show that the positive wrist torque, activated at the optimum ‘timing’ (α=210 degrees), provides a significant gain in club head speed when compared with those using negative wrist torque (13.7% for subject A and 12.0% for subject B). But a relatively small improvement in speed is achieved as compared to those using neutral wrist torque (2.5% for subject A and 3.5% for subject B). This finding agrees with the results from Jorgensen (0.7 %) and Sprigings & Neal (9 %), although the bending flexibility of golf shaft was not included in their models. Note that the percentage gain in club head speed is, however, different with those from their simulation. The most likely reason is that various magnitude of wrist torque was used: the maximum torque is only 2.7 Nm in Jorgensen, but 18.5 Nm in Sprigings & Neal and 8.0 Nm in the present study.

It is certainly important to know how the wrist action changes the club head speed. Some researchers, including Jorgensen and Sprigings & Mackenzie, used a rigid-segment model of golf swing to search for the answer, but the bending flexibility of golf shaft was not considered in their models; Using a model considering bending vibration of golf shaft, Suzuki & Inooka thought that the properly ‘timed’ wrist action could effectively utilize the shaft elasticity to improve the club head speed, yet the explicit explanations were not given. Through analyzing the energy transference from the input joints of shoulder and wrist to club head, the simulation results, in which bending vibration of golf shaft is considered, show that two factors facilitate the club head speed at impact:

1. the work produced by a golfer;
2. the energy transference efficiency from the arm to club head. For the positive wrist torque, both factors are higher than those using negative or neutral wrist torques, so the increase in club head speed at impact is achieved.

It is also found that the most effective transformation of bending strain energy of golf shaft into kinetic energy of club head takes place when the neutral wrist torque is used, in which the ‘absorbed’ strain energy at the initial of the downswing is almost completely released at impact and thus the correspondingly increase in kinetic energy of club head is achieved. This means that the neutral wrist torque can make the best of golf shaft elasticity to improve the club head speed. But the amount of ‘released’ bending strain energy at impact is a minor percentage of the total work produced by a golfer (0.18 % for subject A and 0.32 % for subject B). Thus, it can be concluded that the proper wrist technique (neutral wrist torque at the latter stage of the downswing) is capable of utilizing the elasticity of golf shaft by releasing the ‘absorbed’ bending strain energy to improve the club head speed, but the role in speed increase is relatively small as compared to the total work exerted by a golfer.

So far the simulation has been mainly concerned with a golf club. Three other clubs with different flexural rigidity (EI) were examined. EI was respectively set to be 20, 50 and 70 N.m2 as compared to the original 33.78 N.m2. The results show very much the same as for the original club: the maximum difference of optimum ball position among
the four clubs is only 28 mm; and the maximum distinction in club head speed is merely 0.54 m/s. This means that the shaft flexibility appears not to be dynamically significant in the golf downswing, which is consistent with the conclusion of Milne & Davis.

It should be noticed that the simulation results are based on the swing styles of two amateur players in our experiment. The strong backswing is exhibited in both subjects’ swings just like that shown in the work of Cochran & Stobbs. The new model of the golf downswing can be used as a simulation tool to emulate the swing motions of different golfers. The simulation results clearly show that, for golfers, it would be preferable to employ the positive wrist torque at the latter stage of the downswing, rather than apply the neutral wrist torque, though it can utilize the shaft bending elasticity effectively.

## Summary

This chapter uses a new two-dimensional model of golf downswing to examine whether the combination of ball position and wrist action (various patterns of torque applications) can increase the horizontal club head speed at impact. The bending flexibility and centrifugal stiffening of golf shaft are taken into account in this model, which has been verified by the actual golf swings using a three-dimensional motion analysis system and strain gauge measurements. Three different types of wrist actions (negative, neutral, and positive torque at the wrist) are studied by the maximum criterion (maximum club head speed at impact); and the corresponding optimum ball positions are
determined. The results show that the positive wrist torque can give an increased club head speed as compared with the negative and neutral torques. It is also found that the utilization of golf shaft elasticity by a properly ‘timed’ wrist torque plays a minor role in the improvement of the club head speed. On the basis of the energy transference from the input joints of shoulder and wrist to club head, we discuss the way the wrist action influences club head speed.

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