# Frequency

NOTE 1: Why is frequency a measure of shaft stiffness? This seems kind of intuitive but the technical details are interesting. Starting with a simple example; think of a spring attached to the ceiling with a weight attached to the end of the spring. If the weight is pulled down and released it will bob up and down. Its motion is fully described by a second order differential equation. The solution of which indicates the weight will oscillate up and down at a precise frequency equal to the square root of K/m where K is the stiffness of the spring or spring constant and m is the mass of the weight. (Lets just call it weight) Spring constant is a term like pounds per inch of stretch. It is obvious therefore that frequency in this case is a measure of the stiffness of the spring, K. In other words the frequency will increase as K increases. Frequency on the other hand is inversely proportional to the m mass. As the mass increases the frequency will decrease. These results are pretty easy to visualize.

Now instead of a coil spring let's think of a leaf spring held at one end with a weight on the other end such as a cantilever beam configuration. A diving board is a good example. This is a variation of the above spring/mass problem. The differential equation is messy but the solution is somewhat similar. The frequency of oscillation of the beam if the tip is snapped is a function of the stiffness of the spring, the mass and in addition, the length of the beam. Obviously a long beam will oscillate more slowly than a short one. The exact equation is:

E is the modulus of elasticity or strength of the material (is the beam made of wood or spring steel?). I is a term based on its cross sectional shape. An "H" beam is stiffer than a piece of flat stock even though they may be made of the same material. In the denominator, m is the mass of the leaf spring or beam and M is the mass on the end of the beam. The overall length is L. Looking at the equation for a while it makes some sense. If the beam is stiffer (EI is high) it will oscillate faster whereas if the masses and or length are increased it will oscillate more slowly, since those terms are in the denominator. These are fairly obviously deductions and are very similar to the simple spring mass system described earlier. As in the coil spring example the frequency of vibration of the beam is therefore directly related to the stiffness of the beam.

It must be obvious by now that the cantilever beam just described is exactly what a golf shaft is when clamped in a frequency analyzer. If mass and length are constant, frequency is a direct measure of the shaft's "EI" which is its stiffness and can most easily be defined or related to as cycles per minute.

With such a precise means of defining a shaft's characteristics it's a shame the manufacturers continue to label their shafts X, S, R etc. or more recently "firm" or "strong". Is firm or strong more stiff than stiff? To me one of the most important functions of a frequency analyzer is to figure what these terms mean when you get shafts from the various manufacturers. I've gotten R shafts which were like fence posts and more often S shafts that were like wet noodles. Fortunately some manufacturers will give the exact frequency when requested.