Drumstick Rebound: Physics Behind Stick Stroke & Rotation Point

[Rudolfs]

Hello,

I'm trying to figure out what effect does the rotation point where you choose to hold a drumstick have on the rebound of the stroke. Drummers usually find this point by feel, or by fiding out at which rotation point the stick produces the most rebounds. I'm curious to know what are the physics behind this.

So far I've tried to look at this problem as a straight rod with a lenghth l certain mass m and volume falling freely in a rotational movement to the rebound surface from an angle α≤15°. The distance from the mass center of the stick to the rotation point is d. I chose a small angle, so that I could assume the torque on the stick, given by m*g*d*cosα remains constant throughout the motion, thus yielding a constant angular acceleration ε, just to make life easier. By changing the rotation point, it is clear that the moment of inertia I of the stick also changes, being lowest at the center and highest at the very end of the stick.

At first I thought that the stroke with the highest kinetic energy at the collision point (E=0.5*I*ω², where ω - angular acceleration) would yield the best rebound, but I did some calculations and found that the kinetic energy of the stroke is the greatest when the stick rotates about the very end point, and I know from practice that that's not the best place to hold the stick.

So then thought - what else has a role in the rebound? I tried to think of the rebound as just having a dampening effect of the total energy of the stick (that it takes away, for example, 10% of the total energy), thus after the rebound the angular velocity is reduced and the same acceleration ε slows down the movement of the stick. Then I tried to calculate the greatest angle traveled after the rebound, but the results made no sense - it was the same angle for all rotation points.

So, in the end, my question is - what forces play a role in the rebound of the stick? What am I missing? Should I think of the rebound surface as a spring, perhaps?

I hope I posted this in the right place.
Thank you!

 

[gsal]

Well, I am not drummer, but I can see a couple of things going on here.

First, you don't mention anything about the drummer's hand! ..the way I see it, the drumstick point of rotation is possibly between the thumb and pointing fingers of the drummer; then, the remaining fingers continuously apply a force and allow some spring back too...but I am thinking the back fingers are the ones that produce rotation more than gravity alone.

By the way, I presume that when you are hitting a drum like that, the point of rotation is not stationary and, instead, it also moves a bit closer and closer to the drum in an attempt to continue to introduce some energy into the system.

Like, I said, I feel like in your description of the problem, you left drummer out of the picture.

 

[Rudolfs]

Well, I am not drummer, but I can see a couple of things going on here.

First, you don't mention anything about the drummer's hand! ..the way I see it, the drumstick point of rotation is possibly between the thumb and pointing fingers of the drummer; then, the remaining fingers continuously apply a force and allow some spring back too...but I am thinking the back fingers are the ones that produce rotation more than gravity alone.

By the way, I presume that when you are hitting a drum like that, the point of rotation is not stationary and, instead, it also moves a bit closer and closer to the drum in an attempt to continue to introduce some energy into the system.

Like, I said, I feel like in your description of the problem, you left drummer out of the picture.

Thanks for the reply!
I left the drummers hand out of my calculations on purpose, because it has no effect on the rebound. Well, ok, it can have an effect, but the point is that there is a place on the stick which offers the best rebound in all techniques. You can check for good rebound by just letting the stick rotate freely about the rotation point until it hits the surface, no external force added. When you rotate the stick about its end, there is next to no rebound, but when the rotation point is in the right spot between the center of mass and the end of the stick, ir produces many bounces. Then, when you add the forces from your fingers whilst holding the stick in this spot, it feels much easier to play and bounce the stick.
Yes, the point of rotation does move slightly, but in this case, to not make things too complicated, I think you could assume it doesn't (and, for example, in the so called 'french grip' it doesn't move much at all).

 

[256bits]

Have you looked at the centre of percussion ( centre of oscillation ).
http://en.wikipedia.org/wiki/Center_of_percussion

Except the pivot point and COP for the drumstick are switched from the figure in the Wiki.
One would hold the drumstick at or near the COP and strike at the pivot point.

Imagine your finger is at the point labeled COP as seen in the picture.
You let the drumstick fall and lower your finger at the same rate as gravity is pulling down the drumstick.
The pivot point hits the drum.
The strike would cause a translation upwards, with no rotation, of the drumstick, and the drummer then has to deal with only one mode of motion (translation), rather than 2 modes (translation + rotation) on each rebound.
The drummer has to only lift his finger up at the same rate as the drumstick is rebounding in pure translation.

Try it with your drumstick to get to see some of the the pure translation on the first bounce.

Of course that is ideal if one wants pure translation only ( or rather non ideal for subsequent bounces ), since with each bounce, the next bounce is less, so one has to compensate by adding a force upwards to the stick with your finger.
If you add a force at the COP, then the stick will rotate, as only a force at the COG ( centre of gravity ) causes a pure translation.

So it seems to me it is best to hold at the COP on the downstroke, and at the COG on the upstroke.

Or maybe fabricate the drumstick so that the COP and COG are a bit closer together, and hold it somewhere other than those two locations, taking into account the size of fingers, hand, rotation and translation of the hand and arm, position of thumb, and the forcing of energy into the drumstick as the hand moves up and down for the major next blow.

Interesting problem.

 

[bahamagreen]

This is the reverse of the situation of where to hit a baseball along the length of the bat.

In that situation, the sweet spot is where the strike on the bat displaces it (bends it) to set up the simplest of wave motion of the bat - the strike point being the single node with the portion of the bat before and beyond the strike point taking taking part in a single curve... this is the elastic deformation that provides the most force back to the ball.

For the drummer, the same applies in reverse - he wants to hold the stick at the point where hitting the drumhead causes the simplest deformation of the stick, flexing forward and behind the node where he holds it, each flex being half of an "s" curve.

With the bat, the flexing on either side of the node is toward the same direction, with the drumstick the flexing of the stick on either side of the node is in opposite directions, but the same preservation of maximum power applies because the single node provides the simplest curve, free of other movements that would damp the force applied.

 

[Curiousgeorge77]

Well, I am not drummer, but I can see a couple of things going on here.

First, you don't mention anything about the drummer's hand! ..the way I see it, the drumstick point of rotation is possibly between the thumb and pointing fingers of the drummer; then, the remaining fingers continuously apply a force and allow some spring back too...but I am thinking the back fingers are the ones that produce rotation more than gravity alone.

By the way, I presume that when you are hitting a drum like that, the point of rotation is not stationary and, instead, it also moves a bit closer and closer to the drum in an attempt to continue to introduce some energy into the system.

Like, I said, I feel like in your description of the problem, you left drummer out of the picture.

The "sweet spot" effect is true for a stable fulcrum. You can observe it without a hand at all.

As far as including the drummer, excellent point. Drummers do apply downward pressure at the fulcrum to add energy to the entire system as well as rotate the stick at the fulcrum by applying upward pressure with the fingertips or downward pressure forward of the fulcrum with the thumb or the metacarpal/phalanges (depending on grip - whatever happens to be on top of the stick and forward of the fulcrum)

Most pro drummers don't use a constant fulcrum, btw, but constantly shift between fulcrums, use a double fulcrum, etc. Whip mechanics apply to actual drumming as well for the most effective techniques.I wonder if this is why the Center of Percussion doesn't seem to apply? (baseball bat studies show the fulcrum is actually behind the bat itself, not on it - so the CoP ignores the batter, as you say)