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## Mathematical model Figure 2-1 Double pendulum model of golf downswing. c1, c2 are the centers of the mass of arm and club, respectively. The X axis is parallel to the target line on the ground

As shown in Figure 2-1, the mathematical model of golf downswing is deemed to a 2-dimensional double pendulum. This model consists of a rigid arm link and a rigid club link. It is assumed that the swing takes place in a plane tilted φ to the ground. The assumption of the planar movement of the downswing is well supported in the work of Cochran & Stobbs and Jorgensen.

The downswing is separated into two phases: phase 1 and phase 2. In phase 1, the two rigid bodies rotate as one body with a constant wrist-cock angle. In phase 2, the various types of wrist actions are employed.

The following notation is applied:

• G1, G2: Torque on arm and club, respectively
• m1, m2: Mass of arm and club, respectively
• a1, a2: Length of arm and club, respectively
• l1, l2: Length from shoulder joint to c1 and from wrist joint to c2, respectively
• I: Moment of inertia of arm about shoulder joint
• J: Moment of inertia of club about c2
• θ 1, θ 2: Angle of arm and club, respectively
• φ: Inclination of plane of downswing
• g: Acceleration of gravity

The parameter values of the arm and club applied in the calculation are those given by Lampsa. All the parameter data is shown in Table 2-1 and the club data is considered to be appropriate for a driver. The equations of motion of the model are derived from the Lagrangian method. The detailed equations in two phases are given as follows.

## Optimization method

Two optimization methods are used in our simulation: one is the maximum criterion, and the other is the impact criterion. The maximum criterion is used to achieve the maximum horizontal club head speed at impact; and the impact criterion is utilized to obtain the optimal golf club position at impact.

### Maximum criterion

First, the criterion of maximal horizontal club head speed at impact (maximum criterion) is used to investigate the effects of different kinds of wrist actions on the golf downswing. The following equation indicates the maximum criterion.

After obtaining the swing motions of the arm and club at impact, the maximal horizontal club head speed Vm is achieved

### Impact criterion

It has been reported by McLean that the shaft positions of many professional golfers are always maintained vertical at impact when viewed ‘face-on’, the impact criterion (vertical club shaft at impact) is thus applied to examine the role of the wrist action.

The impact criterion is given by.

## Wrist action simulation

Three patterns of wrist actions including passive, active and passive-active are studied by the maximum and impact criteria. Six positive constant wrist torques are employed (5 Nm, 10 Nm, 15 Nm, 20 Nm, 25Nm and 30 Nm). Here, the maximal wrist torque (30 Nm) is given by Neal et al,, who measured this upper value of wrist torque from a low handicap amateur using inverse dynamics.

For the passive wrist action (PW), the arm release angle r 1 θ (given by Eq.2-10), that denotes when the wrist joint can be turned open, is delayed in every degree from the ‘natural release point’ until the point at which the regulated negative wrist torque is reached (the absolute value of the regulated negative wrist torque is the same as that of the positive wrist torque but with an opposite sign. For example, the regulated negative wrist torque is -30 Nm, if the positive wrist torque is 30 Nm).

Where θ is the integer part of n 1 θ; n 1 θ, in this chapter, is the arm rotational angle when the ‘natural release point’ is reached; p 1 θ is the arm rotational angle when the regulated negative wrist torque is reached.

For the active wrist action (AW), the onset and termination of the positive wrist torque are determined when the arm rotational angle satisfies Eq.2-11 and Eq.2-12, respectively.

where θ is the integer part of n 1 θ ; o 1 θ and t 1 θ are the arm rotational angles when the positive wrist torque is activated and deactivated , respectively;  θo1(timp) is the arm rotational angle when impact occurs;

For the passive-active wrist action (PAW), the negative wrist torque is applied, and then followed by the positive wrist torque. The application of the negative wrist torque is the same as that in the PW, in which the negative torque keeps the wrist-cock angle constant until the desired arm release angle r 1 θ is reached. The onset and termination of the positive wrist torque, following the passive wrist action, are given by Eq.2-13 and 2-14, respectively

The natural release wrist action (NW), where no wrist torque is employed after the ‘natural release point’, is also studied for the purpose of comparison with the three wrist actions as mentioned above.

We assume that the simulation process commences when the golfer had just completed his backswing and was about to begin his downswing. Lampsa thought that a pause usually occurred at this moment, indicating that the angular velocities of arm and club are zero. Thus we chose the following initial conditions for the downswing.

It has been noted that there are many types of torque functions of shoulder joint applied in the previous research. Jorgensen and Pickering & Vickers considered that the shoulder input torque was constant during the downswing; Milne & Davis used a ramp as the torque function and Suzuki & Inooka set the torque function as a trapezoid. In the present study, the input torque of shoulder joint 1 G is assumed to be constant during the downswing as that in Jorgense and Pickering & Vickers. The value of 1 G is chosen as 110 Nm which make the swing like that of a professional golfer (the horizontal club head speed at impact can reach 47.0741 m/s using NW).

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