Drum Head Vibrations

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knight427 Offline
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Drum Head Vibrations

Posted: Thu Sep 23, 2010 12:51 am



Every drummer knows that hitting a drum directly in the center gives a much different sound than hitting a drum off-center. This difference in sound quality is most extreme on large drums such as timpani. While a technical understanding of wave mechanics is not necessary to play a drum properly, it is helpful to understand the basics of how your drum is producing the sounds you hear.

Ideal Case Study: circular membrane modal behavior

A drum head is a circular membrane. While most drums have both a top and bottom head, it is really only necessary to consider the top head to achieve a sufficient level of understanding for our purposes. Below you will see a high-speed video of a circular membrane undergoing sympathetic vibration with the subwoofer shown in the same shot. While the video is slowed down to show the motion, the audio is dubbed over at normal sped so you can hear the actual tones being produced by the speaker. The first one is approximately 82 Hz, which means in reality, both the speaker and membrane are completing an up-down (or in-out) motion 82 times per second.

In the first shot, you are seeing the membrane vibrate at its fundamental mode. This is the slowest vibration the membrane can sustain, which means it is the lowest pitch produced (by the head) when you strike it. This pitch is primarily determined by the diameter of the membrane and the tension. This mode can only be started if the speaker is producing 82 Hz. It does not matter if the speaker also produces 41 Hz (an octave below) or some other random frequency, only the 82 Hz tone within whatever sound is produced by the speaker will be driving this mode. This may sound obvious at this point, but it will be important later on to understand that a mode can only be excited by a driving force which contains the modal frequency.



Now note the mode shapes in the video above. The fundamental of any system with fixed endpoints will cause the entire system to move in unison. This will make more sense when you reach 0:23 in the video when the pitch changes to 158 Hz. In this mode, you can see that the membrane is essentially split in half and those 2 halves move in opposition to one another. At about 0:48 you will see 217 Hz which cuts the membrane into quarters. At 1:06 the pitch goes up to 227 Hz and the motion changes dramatically. This mode is actually closely related to the fundamental. It is circular mode just like the fundamental. This is another detail that will be important later, we will explore this a bit more than you may think is necessary at this time.

Nodes
A node is place of no motion. The fundamental mode of a membrane has a node along the entire perimeter (rim). In fact, every single mode of the membrane will have this circular node since the rim prevents the membrane from moving here. This is the only node of the fundamental mode, and its shape is a circle, so it considered a circular mode.
Image <fundamental mode (circular)

The mode viewed at 0:23 has another node running down the middle of the membrane (no motion along this node line), this mode has one circular node (rim) and one linear node (so it is NOT a circular mode).
Image <(1,1) mode

At 0:48 you see another non-circular mode with two nodal lines cutting the head into quarters (again, no motion at the nodes).
Image <(2,1) mode

At 1:06 we see another circular mode because both nodes are circles (one at the rim, the other half way into the circle).
Image <(0,2) mode (circular)

The really important idea here is to see that circular modes will always have motion in the center of the membrane, while non-circular modes always have one or more nodal lines running through the center point.
Animations courtesy of Dr. Dan Russell, Kettering University

Striking a Circular Membrane
Now that we have a basic understanding of modes and nodes, we can move along to a more detailed investigation of circular membrane modes. Specifically, what happens when we strike the circular membrane at different locations.

First I will point out the obvious. Striking a drum head with a stick is very different from exciting modes of vibration with a pure tone produced by a speaker. The first difference is that when you strike the head, you almost create an impulse. An impulse has a very specific definition in acoustics, but basically it is a very short pulse with large amplitude that contains all frequencies. As mentioned above, in order to excite a particular mode, it must be driven by a signal that contains the specific modal frequency. We are in luck here then because an impulse contains all frequencies, so all modes are capable of being excited by a single down stroke. We don’t have to tune our drum stick or time our pulses.
Second, I will remind you that all modes have nodes (locations of no motion). If you strike a node, you cannot excite the mode which contains that node. Remember that all non-circular modes have one or more nodal lines running through the center of the drum.
Putting this information together, we can start to understand what really happens when we strike the middle of the drum head. Your impulse is capable of exciting all possible modes supported by the head. But if you hit it dead-center, none of the non-circular modes will be excited, thus producing far fewer frequencies. But if you move slightly off-center, no important nodal lines run through that location and you can excite most of the modes and thus produce more frequencies.
Try playing around with this application. When you click on the link, the app should open in a new window and look like this:
Image
The boxes on the lower half indicate modes. The fundamental mode is on by default and you see a familiar mode shape on the membrane. You can click the boxes to turn modes on and off. You should be able to find the previous modes listed and many more (only turning one box a time will make them easier to see).

Now adjust a few settings. Click the sound box and you hear a simulation of the membrane sound (keep in mind this is a simple model and it will not really sound much like a real drum). Change the drop-down menus to:
Mouse = Poke membrane
Display 2D+3D

Now you can tap the membrane in different location, hear a simulation of the sound, and see which modes are excited. The picture below shows a tap near the center. Note that only a few of the modes are excited. If you already played around with the modes, you might also notice all of the modes in the first column are circular modes.
Image

Now tap off-center and note how many more modes are excited. Now you have many circular and non-circular modes going, giving you a more complex sound (more frequencies).
Image

Playing a Real Drum
Finally we can check to see that all these simulations relate to a real drum with 2 heads and a shell (which complicates things considerably). Watch this high-speed video of an actual drum head being played. If you watch it straight through, it is difficult to discern much relevant information. However, if you repeatedly watch the first strike at 0:01, you can see the stick strikes the head nearly in the middle of a quadrant and primarily excites the (2,1) mode (see .gif animations above). At 0:12, in the lower frame the stick strikes pretty close to the center, and you mostly see the fundamental. Finally, the last strike at 1:02 hits just off-center and you can see both circular and non-circular modes being excited.


Conclusion
When a drum head is played in the center, only a small portion of the modes are excited. Specifically, only the circular modes will be excited because all of the non-circular modes have nodes in the center. Therefore only a small portion of the frequencies are produced, and the drum sounds “dead”. By striking the drum head off center, most of the modes can be excited and we can hear the full timbre of the drum.
Last edited by knight427 on Mon Sep 12, 2011 11:35 am, edited 4 times in total.


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Re: Drum Head Vibrations

Posted: Thu Sep 23, 2010 3:33 am



Wow, I thought I went into scientific detail when explaining things.

This is fantastic stuff sir.
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optsyn Offline
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Re: Drum Head Vibrations

Posted: Thu Sep 23, 2010 7:18 am



Excellent article.


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Re: Drum Head Vibrations

Posted: Thu Sep 23, 2010 10:17 pm



I'm glad you guys liked my post. I've had several of these websites bookmarked for quite some time and was finally inspired to cobble them together into an article after finding snarescience this summer. I find that I more fully understand something after I teach it. Unfortunately I never had an interested audience with which to discuss drum head vibrations until I got here.

I'm now looking into stick-drum interaction. I know a lot less technical detail on this front. I've found some interesting stuff though. I'll post the links if you are interested. Perhaps sometime in the future I'll have found enough separate sources (and cool videos/applications) to justify another summary article similar to the one above.

How to Pitch-Match Drumsticks - 10,000 Times a Day
The Anatomy of a Drumstick
Movements and grip influence the contact force and sound in drumming
Analysis of Drumbeats – Interaction between Drummer, Drumstick and Instrument


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Re: Drum Head Vibrations

Posted: Thu Sep 23, 2010 10:19 pm



i read this and shat


thank you
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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 2:40 am



Wow, welcome to snarescience and amazing posts! Bricks have been shat, mind has been blown, etc. Really great, descriptive, and informative.

One question. Does it affect the bottom head in a similar manner? (if top head struck at a node, affects similar nodes as the top head on the bottom) I tried to observe from the video, but I'm not sure.
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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 7:48 am



Yeah, it does.... any circular membrane acts in a similar manner. But it's being perturbed differently. Whereas the top head is being struck by a stick at a point... the bottom head is moving as a result of the air column of air being forced downward in the shell.


knight427 Offline
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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 9:13 am



That's a tough question to answer. I'm not completely sure, but I can make some guesses.

The bottom head will have the same mode shapes as the top (all circular membranes will). Of course the difference is that those shapes will be exited by different frequencies since it will likely be tuned differently. First we need to consider how the bottom head is excited. The top head moves the air in the drum, which in turn moves the bottom head. The question I have is how much of the original impulse makes it to the bottom head. I think most of the impulse (which has all frequencies in it) will pass through the top head, pass through the drum and then then hit the bottom head. The top head is only going to resonate at certain modal frequencies, but the initial pulse should go through. Now when the impulse hits the bottom head, I think it will excite all modes. The reason I think this is because the impulse hitting the bottom head is a pressure wave that should hit the entire area of the bottom head (while on the top, the impulse was mechanically applied to a very specific location on the head).

However, after that first impulse, the top and bottom head will communicate with another. So if you hit the center of the top head, the limited resonance of the top head should have some effect on limiting the resonance of the bottom head. I think it would be very hard to see such complex behavior on a high-speed video though.

Finally, I just wanted to point out again, that I am pretty much just guessing here.


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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 11:51 am



Good stuff man! Welcome to Snarescience!
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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 1:34 pm



Thanks for sharing this with us!

This is definately snareSCIENCE. :D

Welcome to the board!
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Re: Drum Head Vibrations

Posted: Fri Sep 24, 2010 2:30 pm



Dude, this is so awesome. Thanks for gracing us with your knowledge haha


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Re: Drum Head Vibrations

Posted: Mon Oct 11, 2010 9:12 am



I finally got around to reading this thread. I love it! Now this is what snarescience is all about. :)

I really enjoyed the "Movements and grip influence the contact force and sound in drumming" article. I am really surprised that the stick / head contact duration is longer in a rebound stroke than it is in a controlled down stroke. This is really counter intuitive to me.
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Re: Drum Head Vibrations

Posted: Mon Oct 11, 2010 11:31 am



Gary Gvistad gave a lecture on this at Ohio State.

It's very interesting, especially when applied to keyboard instruments and cymbals.
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Re: Drum Head Vibrations

Posted: Mon Oct 11, 2010 4:01 pm



.....that's it. I'm outta here......

actually, we all deserve honorary Masters in Percussion Science just for reading that article...
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knight427 Offline
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Re: Drum Head Vibrations

Posted: Mon Oct 11, 2010 9:38 pm



teh_Guardian wrote: It's very interesting, especially when applied to keyboard instruments and cymbals.
I was planning on covering that at some point.


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Re: Drum Head Vibrations

Posted: Tue Oct 12, 2010 12:03 am



dude thats fricken mind blowing!!! does the tension that the heads are tuned affect the vibration at all like to say if a marching snare head tuned at a high tension react the same? does it have the same excursion height but it just fades faster because of the high resistance the tension applies??
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knight427 Offline
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Re: Drum Head Vibrations

Posted: Tue Oct 12, 2010 9:19 am



tenor_7x7x7x wrote:dude thats fricken mind blowing!!! does the tension that the heads are tuned affect the vibration at all like to say if a marching snare head tuned at a high tension react the same? does it have the same excursion height but it just fades faster because of the high resistance the tension applies??
Let me start by saying that I don't for sure b/c I haven't seen any high speed videos of a marching snare. Certainly the mode shapes are identical, they are just happening at a higher frequency. Intuition strongly suggest the displacement will be much lower though. Just take a string and hold it taught enough to get a sustained vibration when plucked. Now pull harder on the string and the displacement will be reduced. The higher tension provides a stronger restoring force, requiring greater applied force to achieve similar displacement.


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Re: Drum Head Vibrations

Posted: Tue Oct 12, 2010 1:09 pm



knight427 wrote:
tenor_7x7x7x wrote:dude thats fricken mind blowing!!! does the tension that the heads are tuned affect the vibration at all like to say if a marching snare head tuned at a high tension react the same? does it have the same excursion height but it just fades faster because of the high resistance the tension applies??
Let me start by saying that I don't for sure b/c I haven't seen any high speed videos of a marching snare. Certainly the mode shapes are identical, they are just happening at a higher frequency. Intuition strongly suggest the displacement will be much lower though. Just take a string and hold it taught enough to get a sustained vibration when plucked. Now pull harder on the string and the displacement will be reduced. The higher tension provides a stronger restoring force, requiring greater applied force to achieve similar displacement.
I wonder how kevlar weaves affects resonance patterns. I'm assuming that the really nice looking mode shapes in your animated gifs are simulated with an isotropic material (isotropic = same material properties when measured in any direction.) A mylar head would be isotropic, whereas a kevlar head would not. For example, if you had a kevlar weave where the threads lay 90 degrees to one another (I think this is how most are manufactured), you would not measure the same material properties directly along one of the thread directions vs 45% to both thread directions. The effect on mode shapes is probably small, but I bet it is measurable.
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Re: Drum Head Vibrations

Posted: Tue Oct 12, 2010 3:21 pm



snarescience wrote:I wonder how kevlar weaves affects resonance patterns. I'm assuming that the really nice looking mode shapes in your animated gifs are simulated with an isotropic material (isotropic = same material properties when measured in any direction.) A mylar head would be isotropic, whereas a kevlar head would not. For example, if you had a kevlar weave where the threads lay 90 degrees to one another (I think this is how most are manufactured), you would not measure the same material properties directly along one of the thread directions vs 45% to both thread directions. The effect on mode shapes is probably small, but I bet it is measurable.
Good point. If I only knew someone with a high-speed video camera.


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Re: Drum Head Vibrations

Posted: Sat Nov 06, 2010 1:42 pm



Here is an amazing visual demonstration of nodes in a vibrating surface. The salt settles to the into the stationary nodes of the plate. At each harmonic the node pattern is different, just as in the drum head harmonics.

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Re: Drum Head Vibrations

Posted: Sat Nov 06, 2010 8:29 pm



Those are called Chlandi patterns. He was doing this in Europe a few centuries ago, attracting many famous people to his lectures including Napoleon. He used a violin bow instead of a speaker.

Anyway, these modes are certainly the same idea as a membrane, but there is an important difference. The restoring force in a membrane is the tension, in plates the restoring force is the stiffness. Also, because a membrane is under tension, it must have nodes at the perimeter. Plates have stiffness regardless of their boundary condition, so they will undergo resonance weather the ends are clamped or free (though the boundary condition will effect the mode shapes).

Finally, I just wanted to point out that just as the membrane is a simplified model of a drum head, these Chlandi plates are simplified versions of violin top/bottom plates, piano sound boards, etc.

Oh, one more thing actually. Earlier you asked about Kevlar fibers and how much it might impact mode shapes. I was brushing up on Chlandi plates and browsing violin acoustics. Apparently spruce is a preferred wood in violins. Because wood has grains, it is much stiffer along the grain than against (called anisotropy). Spruce has a very high ratio of anisotropy which apparently makes it sound better. The best synthetic material to approach spruce has a core of cardboard overlaid with graphite* fibers set in epoxy. So yeah, the fibers in Kevlar are probably important not just to increase the amount of tension it can endure, but they will also have an impact on the modes.

*my text is dated, I think carbon fiber is now being used in synthetic violins...much like we are seeing from Pearl in their Carbonply


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