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## Mathematical model

The mathematical model of golf downswing is considered as a 2-dimensional double pendulum, which is the same one as that proposed in Chapter 2. For simplicity, we consider the arm link and golf shaft as continuous distribution of mass and the club head as a tip mass. It is also assumed that m2 = ms+mh and mh = 2/3 m2 . Here, ms and mh are the masses of the golf shaft and club head, respectively. Jorgensen  considered that the gravity effect upon the motion of a vigorously swung club could be negligible and this effect could be approximately regarded as a small increase in torques applied at the arm and club links, so the effect of gravity is ignored in our calculation. The dynamics of the golf downswing can be described by the following two differential equations:

and

We assume that there are two groups, named A and B, play golf. Their input torques on the arm links are equivalent (G1 = 100 Nm) and the initial conditions for the differential equations are shown as below:

and

## Results and discussion

According to the above results, the following findings at impact are observed for golfers A1 and A3 when the same ratio of club length to arm length k1 is applied ( t(imp) denotes impact time).

The equations of the horizontal club head speed vh at impact for A1 and A3 are expressed as

and

We also assume that there are a group B comprising three golfers named B1, B2 and B3 with the same arm length 0.60 m but different masses 5 Kg, 6 Kg and 7Kg respectively, and they play the same length 1.11 m and equivalent k2 golf clubs. The reasoning is carried out just as that for group A on condition of k1 = 1.85 and k2(B1) = k2(B3). The results are shown below:

Similarly, Substituting Eq. (4-13)-(4-16) into the expression of the horizontal club head speed at impact, which is the same one as Eq. (4-10), we obtain

Although only two of the three golfers in group A and B are studied, it is no doubt that the results are also verifiable to any two of them.

Numerical simulation is carried out for group A and B in order to explicitly indicate the influence of the interactions on the downswing. The simulation results indicate that different mass or length ratios of clubs to arms lead to various impact time (Figure 4-1 and Figure 4-2). For group A and B, the larger the mass or length ratio of clubs to arms, the larger the impact time is. This means that the impact time will become large when the mass or length of golf club is increased. The golf club position at impact is defined in Figure 4-3. As we can see from Figure 4-4 and Figure 4-5, the same golf club position is exhibited at impact when the mass or length ratios of clubs to arms are equal. This point can be used to explain the phenomenon that different-arm golfers are able to obtain the almost same golf club position at impact even if they hold different golf clubs during the golf-playing. Figure 4-6 shows that the horizontal club head speeds at impact for the three members in group A are equal with the same length ratio of clubs to arms, even if their arm lengths are different. The horizontal club head speed at impact is shown to be enhanced for every golfer with the increase of the length ratio. This means that the lengthening of the golf club effectively increases the horizontal club head speed at impact. Figure 4-7 indicates that the same mass ratio of clubs to arms results in different club head speed at impact. So it is clear that the arm and club masses are two important factors to determine the final club head speed at impact. This point can be confirmed by Figure 4-8 and Figure 4-9. Figure 4-1 Impact time when different length ratios of clubs to arms are applied. Figure 4-2 Impact time when different mass ratios of clubs to arms are applied