I'm a newbie here, but the question of stiffness of a shaft does fall into my training as a mechanical engineer. The bending stiffness of a shaft is basically the product E*I where 'E' is the modulus of elasticity of the material and 'I' is the area moment of the shaft cross section, a property of the geometry of the shaft. For steel shafts it is almost exactly this simple because steel is linearly elastic, isotropic and homogeneous, at least within the range we use it for in golf shafts. It is more complex for composite shafts because the material is neither isotropic nor homogeneous. In either case, however the stiffness is the same in both directions (unless you have a really weird material whose modulus in compression is different than in tension, which can be the case for composites near failure). The area moment is taken about a line about which the bending moment is applied at a cross section which, for a shaft that is not moving, is the 'neutral axis' and is determined by the cross section of the shaft, and the area moment is the same for that portion of the shaft above this neutral axis and below it (this is hard to describe without a picture).
The stiffness of a shaft can vary azimuthally around the axis of the shaft (ergo the 'cpm' differences) but within a bending plane (to which the neutral bending axis is normal) the stiffness is the same in both directions. Where the concept of 'strong side' and 'weak side' came from is not clear. A shaft can have a 'weak' plane (in which the axis of the shaft lies) and a 'strong' plane, but not a weak or strong 'side'. And actually, the 'stiffness' is only defined at a single point on the shaft. For a uniform shaft it can be the same all along the shaft, but for a tapered shaft, or one where the cross section or material changes it varies along the length of the shaft. A shaft's resistance to bending is determined by the aggregate (integral) of the stiffness of the 'elements' along its length.
I hope this helps.
Alan Brooks
Livermore, CA
At 01:52 PM 12/26/02 -0500, you wrote:
Hey John,
Glad to see you had enough time to take away from your golf trips to be home for the holidays. You are not the Grinch I thought you to be.
A dummies question about your example below that says the 500 grams works on both sides of the shaft equally. What if, on one side of the shaft there is a thick side/spot/area, which would stiffen that side. And to make it obvious, let's assume a thin spot/area on the opposite side. This would give us a "spine" effect. Does physics say it still takes the same force, on both sides to move it a cm.? Just trying to complete 9th grade physics. Thanks.
Al
At 12:56 PM 12/26/2002, you wrote:
Hi Folks, Christmas is over the kids are getting ready to head back to KC and I thought I'd try to respond to some of the comments I found when I logged on today.Dave, thanks for your support! Bernie, I guess 15 seconds is a bit of a stretch! That's what it took when I ran the test but I'd guess it's more typically in the 15 to 30 second region. I like to use my Club Scout IV for this purpose because it reads to a tenth of a cpm ( and is quite repeatable to a tenth, sorry for the blatant product shill). I simply rotate the shaft until I get a minimum frequency reading. I'm not all that concerned about FLO because it must always occur at the min frequency plane. FLO is just something that happens when you only excite one of the shaft's two natural frequencies without exciting the other. This only happens in the S1/S2 and N1/N2 planes. It's not the FLO that effects club performance it's the shaft being bent in something other than it's weakest plane. When a shaft is bent a torque is generated which tries to rotate it into the weak plane. I assume the torque is a function of the differential frequency and the amount of the bending force. I'm guessing at this because I couldn't find it in a text book. Maybe Dave can help me. I've tried to get across that a shaft always has 180 degree symmetry when it comes to stiffness. If I bend a shaft an inch and it requires say 500 grams of force, if I now rotate the shaft 180 degree it will also take exactly 500 grams of force to bend it an inch. Consequently when aligning a shaft to a club head I put the N1/N2 plane in the target plane. It doesn't make a bit of difference if N1 or N2 is pointed at the target . It's a plane in the shaft not a hard side or soft side. Regarding bent shafts, I ran a quick test yesterday before the herd showed up at our house. I took a TT S300 and put it in a spinefinder. It was a homemade one I put together out of a two inch piece of water pipe. The bearings were salvaged years ago from the inner roll gimbal of a Titan Ballistic missile inertial guidance system that I worked on in my youth. If the pentagon pays $500 for a hammer my spinefinder is easily the most expensive one on the planet! ( I did stumble into something kinda neat. I usually clamp the SF in my big bench vise since the body it steel. Instead I slid it into my CS analyzer clamp. It held it very nicely.) I bent the shaft downward and with a little help it found it's preferred orientation very solidly. I put a grease pencil mark on the top of the shaft. I then placed the shaft in my spin indexer and rotated it to find the direction of it's natural bend. As I rotated it the shaft I founds it's lowest point relative to the bench top and the grease pencil mark was exactly on top, just where I thought it would be. There have been comments about these steel shaft rolling very smoothly on a table top even though when rotated in the indexer the tip obviously traces out a nice circle. I'm wondering if the weight of the shaft lying on the table doesn't tend to straighten out the shaft. Typically a shaft might have a tenth of and inch runout yet when I hang a 200 gram test weight on it, it will typically droop about .6 to .7 inches. Since a shaft weights about 100 to 125 grams maybe that's enough to appear to straighten it out when it's lying on the table. Don't know. BTW I ran a bunch of tests where I attempted to prove the text books wrong by producing a shaft that didn't have 180 symmetry. I failed. I used a 1/2" aluminum rod for a shaft. I think I reported my results on Spinetalk. I wrote up a paper and supposedly it will be in the next issue of the PCS Journal. One last comment. Dave mentioned my "bowtie" plots. I must confess these have always confused me. If I plot stiffness in grams per inch vs. angle and use a polar plot to display the results I get a near perfect circle for extremely uniform shafts. As the differential frequency gets larger the plot turns into an oval or ellipse. I don't understand the "Bowtie". It's a ellipse with a constriction in the middle. In theory I don't think that should happen. It think it should always be a pure ellipse. I never went back and did a detailed study of the data to see why I got the distorted ellipse. I guess I should generate a polar plot of the weird asymmetrical 1/2" aluminum shaft I mentioned above. Cheers and a happy, healthy and prosperous New Year, John K
