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## Optimization method

Based on the above part (3.1 Mathematical model), the bending displacement of club head, yc , is given by

Where φ1(a3) and φ2(a3) are the first and second mode shape functions of bending vibration for the end of golf shaft, respectively.

The horizontal component of club head velocity, v(h) , is written as

Substituting Eq.(3-33) into Eq.(3-34), and then differentiating the result with respect to time, it is found that h v will reach a maximal value when

The fourth-order Runge-Kutta method at intervals of 5 1001 − ×. s was used to solve Eq. (3-32), and the left-side expression of Eq. (3-35) was evaluated at each time-step. The optimum time t(o), at which the horizontal club head velocity arrives at a maximal value, was achieved when the left-side expression of Eq. (3-35) is most close to zero. Then the corresponding values of y(c),,,, βα βα and y(c) & at t(o) could be calculated. The optimum ball position, p(h) , is also calculated at t(o) :

## Wrist action simulation

Both Jorgensen and Sprigings & Neal have suggested that the optimal ‘timing’ of the activation of positive wrist torque occurred when the left arm was about 30 degrees below the horizontal line through the shoulder joint ( o = α=210 degrees in this chapter). This conclusion is also consistent with the result obtained from Chapter 2. Therefore, the neutral and positive wrist torques, activated from the optimal ‘timing’ mentioned above, are used to re-examine whether the club head speed could be improved by means of the optimization method (maximum horizontal club head speed at impact). In the present study, the positive wrist torque is not constant but increased linearly with time from 0 to 8 Nm, as muscles could not be activated to their full torque magnitude instantaneously.

## Experiment

Two amateur golfers were analyzed in the experiment. Both subjects were right-handed and labelled as A and B, with handicaps of 10 and 28, respectively. The golf club used for the experiment was a wooden-clubhead driver with titanium alloy shaft (351 g in mass and 110 cm in length). The swing motions of the subjects’ arms and golf club were recorded by a 3-D motion analysis system (MotionAnalysis EVaRT 4.6) with 4 digital cameras (Hawk camera) at a rate of 200 frames per second and shutter speed of 1000 us (Figure 3-2).  Five reflective markers were placed on the subjects and golf club. The specific position distribution of the reflective markers is shown in Figure 3-3. One marker was placed on the center point between the shoulders as the shoulder jointo; one marker was put on a place on the grip near the grip end of the club as the wrist joints; the third marker c was situated at a place near the end of the golf shaft; the last two markers a and b, with the negligible-mass shaft attaching on the golf shaft, were used to measure the orientation of the club during the downswing. Three groups of foil strain gauges (Kyowa, KFRP-2-120-C19L3M2R) were boned to the shaft and each group included 2 single-axial type strain gauges. Two groups of strain gauges were bonded to the shaft near the grip to measure two bending moments, which are parallel and normal to the clubface, respectively. The tension of the shaft was measured by the other group of strain gauges situated at the middle of the shaft. The forces and moments were obtained from the appropriate calibration, applying static loadings for the club as fixing the grip in a cantilever manner. The strain gauge data was recorded from at address to the follow-through at the sampling rate of 500 Hz, and the whole course lasted 6 seconds. The strain gauge data was first transmitted by the strain gauge amplifiers (Kyowa, HSC-20BS) and then fed to a personal computer by an A/D convertor (CONTEC, AD12-16(PCI)). The configuration of the strain data collection system is shown in Figure 3-4.

As both subjects rotated the club at the latter stage of the downswing to square the clubface at impact, the bending moments, parallel and normal to the clubface, were transformed into two moments: one is in the swing plane and the other in the normal swing plane by the orientation of golf club. This is because that only the in-swing-plane bending moment is considered in our simulation model. It is clear that the wrist torque could be achieved from the measured bending moments near the grip; and the shoulder torque could be gained from the measured shaft tension and hand force along the tangential direction to the arm. Parameter values of arm, including mass, center of mass and moment of inertia, were calculated using the formula given by AE et al. [20]; parameter values for the club were obtained by the actual measurement and experiment  modal analysis.

Both subjects were asked to swing 5 times after enough practice until they felt comfortable with the test situation. One swing of each subject A and B was just used to give results since other swings of both subjects could not alter their overall swing styles in nature. As the downswing was our focus, the initial time of the downswing, estimated by the camera system, was chosen as 325 ms and 435 ms before impact for subject A and B, respectively. The initial angles and angular velocities of the arm and golf club were estimated by the motion analysis system; the initial bending displacement of golf shaft was gained from the strain gauge measurement.

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