Study On The Wrist Action In A Double-Pendulum Swing Model (P2)
Energy analysis method
As far as the authors are aware, the way the wrist action alters the club head speed at impact has not been given thoroughly, although some researchers did involve this point in their work. Jorgensen attributed the role of the passive wrist action to the change of a mysterious term ‘timing’, yet the explicit explanation was not presented. Sprigings & Mackenzie used a three-segment model comprising torso, arm and golf club to identify the mechanical sources of power that are responsible for the increase in club head speed. The ball position, however, remained constant in their simulation model, and thus the influence of ball position upon the energy transference between the arm and golf club at impact was neglected. In the current study, we show how the wrist action affects the club head speed from an energy based analysis, when the ball position is not constant but determined by two criteria.
The work and power generated by a golfer are given by
where E is the total woke produced by a golfer; P1 and P2 are the power generated from the shoulder and wrist joints, respectively.
The efficiency index of swing motion, η , is introduced, which is expressed as
Where K4 is the horizontal component of the kinetic energy associated with the club head.
Results and discussion
Figure 2-2 shows the maximal horizontal club head speed at impact using three patterns of wrist actions (PW, AW and PAW), plotted against the wrist torque by the maximum criterion. The figure clearly shows that the larger wrist torque gives a better improvement in the horizontal club head speed at impact. We also note that the increase in speed can be gained for all kinds of wrist actions, as compared to NW. It is fairly clear that PAW results in a higher club head speed at impact when compared to PW and AW with the same wrist torque. For example, when wrist torque is 15 Nm, PAW gives the highest horizontal club head speed at impact, 50.3454 m/s, which is 6.9 % greater than that of NW; while PW and AW result in the increases by 1.4 % and 5.6 %, respectively.
The optimum golf ball position at impact by the maximum criterion is shown in Figure 2-3. The ball position is defined for a right-handed golfer with the horizontal displacement BC, as shown in Figure 2-4. As we can see from Figure 2-3, the optimum ball position is determined by the various types of wrist actions and wrist torques. Figures 2-2 and 2-3 show that for PW, the increase in the horizontal club head speed is achieved when the ball is far away from the center of the golfer’s stance (large ball position). This result is consistent with the conclusion of Pickering & Vickers. It is known that the large ball position needs a relatively large arm release angle (the so-called ‘late hit’) to achieve maximal club head speed when the passive wrist action is used (Pickering & Vickers). Figure 2-5 shows that a large arm release angle requires such a high wrist torque that even exceeds the limit of 30 Nm (Neal et al,). It thus appears to be an impracticable situation for golfers using passive wrist action to obtain the improvement of horizontal club head speed when the ball position is too large, since humans can never provide such a huge wrist torque to achieve the desired arm release angle. For PAW, Figures 2-2 and 2-3 show that the large ball position also gives the increased horizontal club head speed. However, it should be noted that the optimum ball position is obviously smaller than that for passive wrist action. Figures 2-2 and 2-3 also show that the relatively small ball position is able to provide an improvement of horizontal club head speed for AW.
For AW and PAW, the arm rotational angle is used to describe when the optimum timing for the activation of positive wrist torque occurs (Figure 2-6). As we can see, the optimum timing for the activation occurs when the arm link approximately reaches the angle of 210 degree. This result agrees well with that from Jorgensen and Sprigings & Neal. We also observe that the optimum activation changes slightly with the wrist torque.
As is reported by Cochran & Stobbs , the timing for the activation of positive wrist torque was able to influence the club head speed at impact. We examined this point in our simulation by advancing and delaying the optimum activation of positive wrist torque (Figure 2-7). For AW, if the activation of positive wrist torque is advanced as the arm angle arrived at o180 , the horizontal club head speed at impact is reduced by 0.8 %; when the activation is delayed until the arm angle reaches o 230 , the reduction is 0.3 %. For PAW, the decreases are 0.8 % and 0.2 %, respectively.
The simulation results also show that the horizontal club head speeds at impact by maximum and impact criteria are almost the same for NW and PW. The maximal speed difference between them is merely 0.0002 m/s for NW and 0.0008 m/s for PW. The optimum ball positions are also almost equal between them. The maximal difference of ball position is 4 mm and 8 mm for NW and PW, respectively. On the basis of these results, it can be concluded that for the golfers whose wrist actions belonging to NW or PW, the simple way to determine the optimum ball position is to put the ball at the position where the shaft is vertical at impact when viewed ‘face-on’. This theoretical finding is consistent with the actual shaft position at impact that was observed from the numerous swing photographs of professional golfers such as Hogan, Lietzke, Nicklaus, Norman, Nelson, Peete, Price, Snead, Woods (McLean).
For AW and PAW, the maximal difference in speed between the two criteria is 0.0797 m/s and 0.0785 m/s, respectively; and the maximal difference of ball position is 78mm and 73 mm, respectively. The optimum ball position at impact by the two criteria is shown in Figure 2-8. We note that the ball position by the maximum criterion is larger than that by the impact criterion, and the position difference becomes larger with the increase of the wrist torque. However, due to the small difference in the horizontal club head speed at impact by the two criteria (the maximal difference is only 0.0797 m/s), the impact criterion can be regarded as a reliable reference to obtain the optimum ball position for AW and PAW.
A set of comparisons in work and kinetic energy are undertaken among NW, PW, AW and PAW, and here, we use 15 Nm wrist torque. It should be noted that other wrist torques show different comparison results, but the overall way the wrist action affects the club head speed is not altered in nature. Table 2-2 shows the work and kinetic energy of arm and club head at impact by the maximum criterion. It can be seen that all the total work for PW, AW and PAW are increased as compared to that for NW. For PW, the higher total work obviously results from the increase of work produced by shoulder joint, because the negative wrist torque maintains the wrist-cock angle constant and thus offers zero work. For AW and PAW, the sum of work exerted by the shoulder and wrist joints gives the increased total work, although the work by shoulder joint is smaller than that for NW. The work by gravitational force is almost the same, even though various patterns of wrist actions are employed. Table 2-2 also shows that the ratio of the club head kinetic energy to total work is enhanced for PW, AW and PAW. This means that the efficiency of the swing is improved, especially for AW and PAW where the positive wrist torque is used.
Through analyzing the energy transference from the input joints of shoulder and wrist to club head, we find that two factors determine the club head speed at impact: (1) the work produced by the golfer; and (2) the efficiency index of swing motion η .It is evident that the larger the two factors are, the faster the club head speed at impact is. For the wrist action using the positive torque (AW and PAW), both factors are enhanced as compared with those for NW, and thus the improvement of club head speed at impact can be achieved.
Table 2-3 shows the comparison of the work and kinetic energy between the two criteria. It can be observed that a very small distinction is found, which means that for the two criteria, the energy flowing into the swing system and the energy distributing at impact are almost the same. Thus the club head speeds at impact are highly close for the two criteria.
It should be noted that the values of maximal horizontal club head speed and optimum ball position are obviously affected by different values of model parameters and initial conditions. To examine the influence of small changes in model parameter values on the above results, the equations of motion of golf swing are re-written in another pattern.
The maximal horizontal club head speed and optimum ball position are investigated again by the two criteria, with increasing only one parameter such as SMA by ten percent and maintaining the others at their original values.
The numerical results demonstrate that both the maximal horizontal club head speed and optimum ball position change slightly with the ten percent increases in parameter values. For example, when 15 Nm wrist torque is used, the maximal changes in club head speed and ball position are no more than 1.2913 m/s (2.6 % as compared to the original) and 78 mm, respectively. It is of great interest that the impact criterion can still determine the optimum ball position, even though the relative model parameter values are increased by ten percent (Table 2-4). It is fairly clear that the differences in club head speed and ball position between the two criteria are very small for all the seven increased parameters. For NW, the maximal difference in speed is merely 0.0002 m/s; and the maximal distinction in ball position is only 4mm. For PW, the maximal differences in speed and ball position are 0.0004 m/s and 6mm, respectively. For AW and PAW, the relatively large differences are exhibited, but can not influence the effectiveness of impact criterion because the maximal distinction in club head speed (0.0279 m/s) is no more than 0.057%, as compared to that by the maximum criterion.
The same procedures are repeated, when the initial angles of the arm and club are changed by ten percent, respectively (θ1(0)= o 99 or θ2(0)= o 99− ). Similar results are observed: the maximal changes in club head speed and ball position are no more than 1.1746 m/s (2.5 % as compared to the original) and 35 mm, respectively; the maximal distinction in club head speed between the two criteria is only 0.0282 m/s.
We should also note that the simulation results, including maximum club head speed and optimum ball position, depend on what is fed into the downswing model. So far the simulation has been mainly concerned with constant torque pattern. Different torque patterns are also used in the calculation. For example, the shoulder torque is applied as a ramp function with rise time 110 ms and maximum magnitude 110 Nm, and the positive wrist toque is increased linearly in time from 0 with a constant slope of 93.7 Nm/s (the maximum wrist torque is 15 Nm). The results show very much the same as for the constant torque pattern: the impact criterion still works well for various types of wrist actions to determine the optimum ball position (the maximal distinction in club head speed is only 0.0277 m/s between the two criteria); PAW gives the highest club head speed as compared to NW, PW and AW; the positive wrist torque enhances both factors in determining the club head speed.
The purpose of this chapter is to examine whether the combination of ball position and wrist action (different types of torque applications) can increase the horizontal club head speed at impact. A 2-dimentional double-pendulum model of golf downswing is used to determine what extent wrist action increases the club head speed in a driver, and affects the optimum ball position. Three different patterns of wrist actions (negative, positive, and negative-positive torque at the wrist) are investigated; and two optimization methods (maximum and impact criteria) used to assess their effectiveness – maximum horizontal club head speed and club head speed as the shaft becomes vertical when viewed ‘face-on’. The simulation results indicate that the horizontal club head speed at impact can be increased by these patterns of wrist actions, and the optimum ball position can be determined by the impact criterion. Based on the analysis of the energy flow from the input joints of shoulder and wrist to the arm and club head, we discuss the way the wrist action affects the club head speed. The sensitivity of the results to small changes in model parameter values and initial conditions is investigated. The results are also examined under different torque patterns.