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.

 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.



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
straightshaft 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 nontrivial 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 steelshafted 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 5iron

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 36 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 1/31/6 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
