## Vibrational Frequency

If you bend, compress, or stretch most solids, they will try to "spring" back to their original shape when you remove the pressure that is bending them. While not all solids react this way, most of those that comprise a completed golf club certainly do. Let's look at what happens when you "pluck" such a solid (deform it, then let it go suddenly).

The object vibrates back and forth. This is due to a combination of:
• The force on the object due to the spring effect, which wants to accelerate the object, and...
• The mass of the object, which resists acceleration.
Let's look at it in more detail. We'll use an example that most clubmakers see all the time: a golf club held in a vise at the grip and plucked at the head. Then we'll go back and look at other examples.

A more interesting question than how long it takes for the vibrations to die out is how fast the vibrations are, and why. The standard way of describing this is how many complete "cycles" of vibration occur in a standard time interval; e.g.- cycles per minute or cycles per second. A complete cycle is the movement from a peak to the opposite peak and back. The measurement of cycles per minute or cycles per second is known as the "frequency" of the vibration. It is reasonable to talk about a "frequency" because the duration of a cycle is constant for a given club, regardless of how big the amplitude of the oscillation.

If we could hold a stopwatch and count the number of cycles of our vibrating golf club, we'd see somewhere between 200 and 350 cycles per minute. What determines this frequency?

• A stiffer shaft material will give a higher frequency, because its elastic force gives more acceleration of the mass.
• A stiffer shaft geometry will give a higher frequency, for the same reason. (A thicker shaft or a thicker shaft wall is a stiffer geometry. A longer shaft is a more flexible geometry.)
• A heavier clubhead will give a lower frequency, because the increased mass takes longer to accelerate to a given speed.
• A heavier shaft is a mixed bag; the increased mass slows things down a little --  but not as much as the head's mass, because its mass is closer to the grip. And a heavier shaft is usually associated with a stiffer geometry which will speed things up.
So the vibrational frequency of a golf club is a combined measure of the shaft stiffness (due both to material and geometry), club length, and weight. The weight involved is mostly head weight, but there is a small component of shaft weight there, too. We'll have more to say about this in the chapter on Shaft Flex, because frequency is a rather precise way of expressing a measurement of flex. (In fact today, unlike the years when these notes were first published, a frequency meter is in most clubmakers' shops as the prime instrument to measure shaft stiffness.)

Before we leave the topic of vibrational frequency, I'd like to pick on a couple of additional examples, and illustrate the relationship between frequency and response times. Both of these issues are non-trivial in learning things about golf clubs and the golf swing. First the examples:

• Put two similar drivers in a vise (same length and flex), pluck them both, and compare how long it takes for the vibrations to die out. If one has a steel shaft and the other graphite, the steel-shafted driver will probably vibrate longer and truer. This is because there is more internal damping in a graphite shaft. (That also contributes to the different "feel" of graphite, with less of the vibration of impact reaching your hands.)
• Tap the face of a metalwood. You'll hear a "pinging" sound, usually clearly enough that you could match it to a musical note. I've done this with some of my clubs, and determined their vibrational frequency this way. (Many scientific/engineering handbooks give the frequencies of all the notes of the piano keyboard.)
• Tap the face of a wooden wood. Bet you don't hear a note, just a "click". This is because wood has much more internal friction (damping) than steel, so the vibrations die out before they have a chance to get organized at one frequency.
• For this reason, you can tell how much foam (or structural) damping there is in a metalwood by noting the duration and clarity of the note you get by striking it. The longer/clearer the note, the less damping in the clubhead.
There's one more important issue to be covered on the topic of vibrational frequency: the relationship between frequency and response time. Since frequency is measured in cycles per second, it is the inverse of time (which is measured in seconds). Another way of putting it is that the inverse of frequency (1/f) is some sort of "natural response time" of the object.

This brings us to the notion of a "time constant" of an object, a concept common in electrical engineering and useful in other technical disciplines as well. The time constant of an "underdamped vibrating body" (the plucked club or ringing metalwood) is 1/(2*pi*f); we can usefully approximate it as 1/(6f). Consider the time constants of some of the vibrations in our examples:

Example
Frequency
Time Constant
Plucked driver
250cpm
40msec
Plucked 5-iron
320cpm
31msec