Design Notes - Physical Principles p3

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.

We begin by pulling the clubhead up above the position it would have with a straight shaft. (The blue centerline in the diagram is the position the club would have if the shaft were straight.) Because the shaft is bent upward, it is exerting a downward force on the clubhead -- to try to return it to the straight position.
The position in the diagram shows the situation at the instant we release the clubhead. The clubhead is accelerating because there is a force on it, but it has not yet started moving -- so its velocity is still zero. Velocity will come; the acceleration will create it over a little bit of time.

As time (seconds? milliseconds? microseconds? we'll see soon) passes, the head picks up some speed in the direction of a straight shaft. That speed is caused by acceleration.
In the diagram, the head has moved about halfway to the "straight" position. It has some velocity. But it has lost acceleration. Why? Because acceleration comes from force. The shaft isn't bent upward as much as before, so it isn't putting as much downward force on the clubhead. Thus the acceleration is less. But there is still some acceleration so the velocity is still increasing, just not as fast as before.

The clubhead will still be moving when it reaches the point of a straight shaft. In fact, it is moving faster than ever. That's because the acceleration has been there the whole way down; it may have been decreasing, but it was there. And acceleration in the same direction as velocity will increase the velocity.
At this point, the situation changes considerably. The shaft is straight, so it exerts no force on the clubhead. But the clubhead is a moving mass, so -- as Newton told us -- it will continue to move in the direction it was going.

Here the clubhead has moved past straight, and is bending the shaft downward. This means there is an upward force on the clubhead, which produces an upward acceleration.
So here we have a downward velocity and an upward acceleration. Remember what this does? It reduces the downward velocity. That's called "deceleration".

As the shaft bends more, the deceleration increases. This deceleration -- the reduction in downward velocity -- continues and increases as the shaft bends more away from straight. Eventually the shaft stops moving altogether.
At this point, it is in exactly the opposite position to where it started. It is just as far below the straight-shaft position as it was above it when we started. If that is true, then it should return to the starting position the same way it got here from the starting position.

This continues again and again. The shaft will oscillate back and forth for tens or even hundreds of cycles.
The internal frictions in the shaft dissipate a little energy with every oscillation, causing the amplitude of the swings to gradually die out. How much internal friction exists depends on the material. Graphite composite has more friction than steel. We call this being more "lossy", because the friction results in a loss of kinetic energy. How long the oscillations continue depends on how lossy the shaft is.

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
Ringing head	500cps	0.33 msec

Why should anyone care about time constant? Because in any interaction between objects (like a hand swinging a club, or the clubhead hitting the ball) will depend on how the duration of the interaction compares with the time constants of the objects.

If the interaction lasts more than about 3-6 time constants, then the object's elasticity has time to respond. The impinging object will "feel" the whole object respond.
If the interaction lasts less than about 1/3-1/6 of a time constant, the impinging object will only "feel" the immediate fragment of the object that it touches. The rest is too "loosely coupled" to the point of impact.

To make the concept concrete, let's consider a driver hitting a golf ball. It is well known that the ball stays in contact with the clubface for only about a half a millisecond. We can draw the conclusions:

The time constant of the "plucked driver assembly" is about 40 milliseconds. That's almost 100 times longer than the contact between clubhead and ball. So anything that happens at the grip of the club during contact has no more chance of effecting the ball than if the club were being swung on a string.
The time constant of the ringing clubhead itself is about 1/3 millisec. That's shorter than the time of contact, but a similar magnitude. So all of the clubhead "gets into the act", but the face probably has more effect on the ball than the back of the clubhead. If it were made a little stiffer (say, 0.1 millisec, or a natural frequency of 1300 cps), the whole clubhead could be considered a single rigid body. In practice, most analyses -- including ours -- consider the clubhead to be rigid; but most aren't perfectly rigid. However, they're much more rigid than the ball, so the approximation gives only small errors.

Last modified Oct 6, 2006