Analysis of stick movement

In this article the movement of the drumstick and the various ways one can use this movement to produce more efficient playing, are discussed.
Don't let the formulas and graphics fear you!

The length of the arc (I call it the stroke path) between a en a' (or b en b') is given by:

• gamma is the length from the fulcrum point to the tip
• alpha is the angle of the stick from its striking point (80 degrees in full stroke)
• pi is our well know friend 3.14...

With a stick length of 41 cm (16") the fulcrum point is at 28cm (11") and therefore the length of the stroke is about 39 cm (15.3"). The loudness of a stroke is calculated from the force with which the stick hits the head of the drum. This force (P) itself is calculated from the speed of the stick (V) times the mass (the "weight" M) of the stick:

P = V x M

Because we cannot change the stick easily while we are playing (at least not in the midst of a series of fast strokes!), we only have to deal with the stroke speed (V) when comparing different stroke heights and loudness levels.
Is this stroke speed a constant speed?
No, most of the time the stroke speed is increasing during its way downward along the stroke path. If you use shorter sticks, then you'll have shorter stroke paths and hence less distance to build up stroke speed (and sound level).
The increase can be linear (a fixed amount is added each unit of time) or non-linear or logarithmic (it is multiplied by a constant factor each unit of time). You cannot easily control the type of acceleration. It's mainly done by finger and wrist control but some of it falls beyond our possibilities.
Here you see a graphic representation of the two types of acceleration:

The numbers along the axes are just for illustrative purposes, they weren't measured! The X-axis denotes time, the Y-axis speed.
The red line denotes a linear acceleration and the blue line is the logarithmic type. One can easily see that the blue line reaches about twice the speed of the red one in the same time. This is of course dependent of the used formulas.

The longer the stroke path, the more you can accelerate the stick and hence produce a louder sound. Using a long stroke path (playing full strokes) when you have to play soft, is normally a waste of energy and can only be done if you have to play slowly too, because you need to have enough time between strokes with the same hand to travel the longer stroke path.
If you have to play soft, you should use a shorter stroke path.

Do the following experiment:

0. Play slowly with one hand a series of strokes, not faster than, say, 60 beats per minute (bpm) and start all the strokes from the same height (keep your other hand at that position and let the stick travel between that hand and the drum).
1. Try to vary the speed of the stick and listen to the sound, both the loudness and the quality are important. If you take longer periods of rest by keeping the stick in the up position and moving fast down at the last moment, you'll notice that this produces a louder sound, but it's almost always a better sound too!
2. Try to move the stick up and down between the two positions with the same speed and listen again: softer but with a bad quality of sound.(The stick seems to "stick" to the head).

Here we learn an important point:

The sound-quality improves if the stroke speed is not a constant speed but increasing with time. Playing "snap wise" with wrist and/or finger stroke control to get a fast stroke speed, produces a good sound.

Now start increasing the tempo gradually to 160 bpm while keeping the same stroke length. you'll notice the increase in loudness but you'll also notice that you no longer have much control over the way you are moving the stick. The time between the different strokes is to short to allow for much variation and the stick seems to be all the time in motion, abruptly changing its direction at the endpoints (a en a').

Some remarks

If you have enough time between the strokes, you can control almost all aspects of the sound in a number of ways (slow stroke speed and greater distance or fast speed along a shorter stroke path).
If the time between successive strokes is short, you have only one way to control all aspects: change the stroke length.
You cannot control the stroke speed any more because that is now dictated by the tempo!
More exact: if the time between two successive strokes is shorter than the time to let the stick travel along its stroke path, you cannot longer control the stroke speed in all ways.

We can learn from this that playing fast soft strokes is difficult to achieve because the time between the strokes forces the sticks to have greater stroke speed. We can only diminish this stroke speed by playing closer to the drum, creating a shorter stroke path. The stick has now a lesser distance to build up its speed, hence the sound will be softer.
But you cannot diminish this distance unlimited, you have to have some distance to produce any sound at all!
One other way to play softer is to use shorter sticks. If you don't change anything else, then because of the shorter stroke path, you'll end up with less volume! But now the tone color might have changed! (See: Stick Anatomy)
If you use two pairs of sticks of the same length, but one pair lighter than the other, then you have to increase the stroke speed with the lighter pair to get the same loudness as the other pair. But because the heavier pair also produces much more lower frequencies, you will also hear color differences between the two pairs. There is nothing to be done about that!

Playing fast and loud seems to be much easier, but here we also can expect some problems to solve. If you have reached maximum volume by using the maximum stroke path, you can only increase the volume by trying to find better ways to increase the stroke speed. Using combinations of finger/wrist control to reach maximum acceleration.
If you need still more volume you can increase the stroke path by using your arm together with your wrist, but now you will suffer from the fact that you cannot move your whole arm at the same speed as you could move your wrist! Hence, the tempo will drop!

Here we see how matters are complicated when we take the movement of the arm into account. The line f is the semi circle along which the tip of the stick would travel if the wrist didn't move!
Semi circle g is the semi circle that describes the path of the wrist alone. But because the wrist itself is used to let the stick travel along its own, independent path h, the final road the tip of the stick takes to reach the drum d, is hard to describe precisely.
This is one of the things where the movement of the stick changes from linear accelerated to non-linear accelerated.

Conclusion

1. because the loudness level of the sound of the stroke on a drum is directly related to the speed that the stick has at the moment of impact
2. because the various and interacting ways the movements of arms, wrist and fingers have on this speed
3. because of various natural borders that cannot be crossed

it is important that you learn to control the stroke speed in great detail so that at all times you get the result that you want with the least amount of energy. Every movement your arms make that do not directly have effect on the sound, or do not produce any sound at all, is a waste of energy.
If you want to play loud, then try to achieve that by moving your sticks as fast as possible downward, not by lifting your arms up to the ceiling! This has another great benefit: you get much more control when you have to play really soft!