Parametric acceleration II

Having looked previously at the basic mechanism of parametric acceleration, the next logic step is to try to get some idea of its impact on clubhead speed. We will use, like Dr. Miuara, a double pendulum model. Whereas Dr. Miura limited himself to vertical motion we will analyze it for a variety of motions for the center pivot.
  
The analysis is summarized below by a series of figures showing respectively the motion of the double pendulum and the clubhead speed. Starting with no motion for the center - to be used for comparison when various motions are imposed on the center. The numeric clubhead speeds given in the b Figures correspond to the clubhead speed when passing through the vertical dashed red line.

In Figs 1a and 1b there is no movement of the center and they constitute a reference for the remaining figures.

In Figs 2a and 2b we have the ideal situation of the radius shortening. This is readily done mathematically but not quite so for a golfer. But it is interesting to have the ideal case for comparison with more realistic cases. Increase in clubhead speed of  6.2 %.

In Figs 3a and 3b we have a more realistic case where the motion of the center is as shown in Fig 3c. The lead shoulder could possibly be used to produce a somewhat similar motion pattern for the center. Mathematically it is equivalent to the ideal case of shortening the radius. Indeed  the clubhead speeds for case 2 and 3 are the same.
  

In Figs 4a and 4b we have eased up a bit and simply imposed a vertical motion for the center.  Again the resulting clubhead speed is the same as for the ideal case.

In Figs 5a and 5b we have the results for a diagonal motion. Less gain in clubhead speed, still 4.1 %. It shows that parametric acceleration is not very critical with respect to the motion of the center.

To complete the analysis I have for fun analyzed a horizontal motion for the center, Figs 6a and 6b. As expected there is no parametric acceleration effect and even a slight decrease in  clubhead speed.  The clubhead's motion indicates a slight increase in its instantaneous radius.

Conclusions

A planar math model does not account for all the complexities of a real golfer, but for the present case it is more than adequate to give some reasonable ballpark figures of what can be expected from parametric acceleration. The gain in clubhead speed, from the analysis above, is fairly modest. The motion of the center is not very critical.

In real life the swing takes place in 3D on an inclined plane. The analysis above is really about what happens in this plane. Therefore the 'in' plays equally a role in addition to the 'up' motion. Hence swinging left through impact can also be considered to induce parametric acceleration.

mandrin