I had meant to analyze the bending of the shaft under the influence of centrifugal force but almost forgotten about if it had not been for some recent posts which reminded me of my posted intention to analyze the relation between shaft bending and centrifugal force. I like to first dispel of the myth of the rebounding of the shaft or the kick furnished by the shaft increasing the club head velocity at impact. This idea, commonly accepted, is based on concept that the shaft behaves like a spring which is loaded during the first part of the down swing and unloads, bends, vibrates or kicks back towards the bottom part of the down swing.
This false belief is based on misunderstanding the concept of mechanical impedance. Put the handle of a driver securely into a vice and deflect the head, it vibrates vigorously. Put the handle into sand or mud and try to vibrate the shaft, out of luck, it will not vibrate. It is the interface between handle and vice or handle and sand/mud, scientifically expressed by the concept of mechanical impedance, determining if oscillations occur. Intuitively it is probably clear now that it is related to rigidity and amount of mass of the 'media' holding the handle. The hands holding on to the club are much closer to mud/sand than to a vice.
Bending of the shaft can occur from both ends of the club - due to a torque transmitted via the hands or due to a torque transmitted via the club head by centrifugal force. Loading of the shaft is most effectively done utilizing the recoil at the transition at the top. Soon however it is the rapidly increasing centrifugal force, creating a torque via the offset of the center-of-mass of the club head, which starts to dominate and reversing the direction of the bend of the shaft. It is the high speed photographic evidence of this reversing of bending that let people believe that the shaft actually behaved like a spring. As explained here it is actually two different phenomena at work, but no spring, just that 'non-existing' centrifugal force doing quietly its job so as to not disturb JK and his ideas.
To analyze bending due to centrifugal force I needed to develop a more accurate model of the golf club than a simple uniform rod which is mathematically very convenient for many things but not appropriate for studying bending due to centrifugal force. Fig1 shows the model developed and used, consisting of grip, shaft and club head. Fig 2 and Fig3 show more details and with Fig4 give an idea how a bending torque is developed by the centrifugal force, acting on the line going through the wrist cock axis and the center of mass of the clubhead, with the latter having a offset relative to the shaft.
The 'swing' used for the derivation of the torque is shown in Fig5. The angle was held till 60 deg before impact at which angle a constant torque of 20 Nm for the wrist was added.
The centrifugal torque derived for this swing, acting at the end of the shaft, is given below. It is directly proportional to the mass M3 of the clubhead and almost so to the radius of gyration L3. The club head mass of M3=200 gram is assumed to be concentrated into a point having a 1 inch offset (L3) from the shaft.
![[Graphics:Images/Centrifugal_Bending_Shaft_gr_4.gif]](Centrifugal_Bending_Shaft_gr_4.gif)
The maximum bending moment (torque) occurs at impact and is about 8 lbft. This seems quite small, however is large enough to deflect a driver shaft a considerable amount. Take a driver and clamp the last 5 inches of the handle against a table surface. Load the club with a dumbbell of 2 lbs close to the head and notice the deflection. It is considerable and this represents only 6 lbft.
The component of the centrifugal force acting in line with the shaft, F, and the x and y components, Fx and Fy, are shown in Fig7. The peak force of 96 lbs when multiplied with the offset of 1 inch results in a 8 lbft for the torque at impact. F is almost equal to the centrifugal force acting through the center of mass of the club head. Force and torque were derived with totally different approaches. It is then nice to see that they mutually reinforce their validity with this simple check.
![[Graphics:Images/Centrifugal_Bending_Shaft_gr_7.gif]](Centrifugal_Bending_Shaft_gr_7.gif)
The rendering of the centrifugal force can be made more attractive plotting it as a force vector field at the end of the shaft as shown in Fig8. Once the down swing underway the centrifugal force gets quickly almost in line with the club shaft. At impact the length of the vector represents for this particular swing a force of 96 lbs.
It is interesting to show more in detail the behaviour of the centrifugal force at the beginning of the down swing. (The scaling is different from Fig 8.) Notice in particular that the vector and the club shaft are, at one moment, perfectly aligned. This corresponds to the exact moment when the torque acting on the wrist changes direction.
I hope through the use of appropriate graphics to have given a fair idea of the centrifugal force and the centrifugal torque as they occur dynamically in a golf swing acting on the end of the shaft where it joins the club head. So whenever TV commentators, articles or books refer to the vibrational kick of the club shaft we chuckle quietly since we know better.
mandrin
