Prof. Jorgensen has been the first to analyze in 'The Physics of Golf' the flexing of the shaft using his two rod model. He used a very simple approach by replacing the flexible shaft by a stiff shaft with all the flexing put in a spring positioned just below the grip. A more challenging task however is to try to use the more appropriate application of continuous systems to model the shaft as a truly flexing slender beam. I tried earlier this year, didn't quite succeed, but recently my efforts lead to a seemingly useful model.

The flexibility of golf shafts has been an interesting challenge,i.e.,to formulate its behavior in mathematical form.There is a considerable activity on the subject of flexible beam-like structures due to its presence in many cases such as for instance helicopter rotor blades, airplane wings, flexible robot arms, long span bridges, high rise buildings, etc. However most of it seem to confine itself to small deflections/vibrations and it took me quite a while to get a satisfactory mathematical formulation for very large deflections such as occur in a whippy tempomaster.

Some think that by bending the shaft at the onset of the downswing we are "loading" the club with energy and that this is released at impact.. This illusion is effectively supported by the apparent behavior of the shaft, bending back and then seemingly 'kicking' forward.  I don't believe this to be true. The two main objections are (1) mismatch between natural frequency of the golf shaft relative to the time for a typical downswing, (2) mechanical impedance between hands and grip introduces so much damping that the "springiness" of the shaft as an energy storage element is to be ignored.

Even if the flexing of a shaft can't be used to store energy to be released at impact with a conventional golfclub, it remains interesting to look at a hypothetical Whippy TempoMaster type golf club and see if there is clubhead speed to be gained by virtue of appropriate flexibility of the shaft. To start, I have used a simple model as illustrated in Fig1. Two in-line segments and a mass M3 at the tip. The first experiment simply consists of letting the ensemble drop from the horizontal position. The second segment is either rigid or quite flexible.

The question I like to elucidate with this experiment is does the flexing of the shaft lead to greater velocities whilst expending the same amount of effort? The assembly is pinned at P0 and freed at t=0. Only gravity is exerting a torque. The flexural rigidity is chosen to give a natural frequency for the flexible shaft such that the half-period is equal to the time for the shaft to rotate to the vertical position. The result are shown in Figs2 and 3, comparing a rigid shaft with a flexible shaft.

Figs 2a,b,c show results for a rigid second segment, Figs3a,b,c show results for a very flexible second segment.
Figs2b,3b - red curve gives velocity for mass M3, brown curve represents velocity of P1.
Figs2c,3c - dashed curve is total kinetic energy (equals work expended), red->mass M3, blue->inner segment, green->flexible segment.

Comparing the velocity and kinetic energy show clearly some interesting results.
-1- The dashed curves for the total kinetic energy are identical for the two cases, hence equal work expended.
-2- The maximum velocity at the bottom is about 29% greater for the flexible beam.
-3-  Figs 3,b,c show very clearly the energy transfer taking place form the inner segment to the outer segment.

Conclusion - The spring type feature of the flexible shaft allows, in this particular experiment, to obtain substantial larger velocities for the same effort employed.

mandrin