What is the Equation for Determining the Resonant Frequency of a Drum Shell?

[SIMON1]

Hi there I am currently doing a major thesis on the procedural generation of an acoustic drum kit, using PD (puredata). Now what I am actually doing is physically modelling the drum kit piece by piece, starting with the drum shell. I know that every aspect of the drum contributes to its complex sound and is not dominated by a single factor, for example the the batter head & resonant head exciting resonant frequencies, the depth & diameter of the drum determining the pitch and amplitude of the drum, also the lugs, sticks and bass hole (on a bass drum) all adding to the very complex sound that is perceived as one singular sound.
They are not the problem in this equation but finding the resonant frequency the shell is to start the workings of the kit, now damping occurs by the density of the wood of course and all the other factors but the question i propose is; what equation can i do to determine a cylinders resonant frequency if their is given dimensions for that specific drum? Then to work out the rest of the equation once the other factors have been added?

I know only basics of physics but I am certainly a quick learner. I just haven't found anything online that could point me in the right direction.

Any help on this matter would be greatly appreciated & I would be happy to forward the paper onto the site as well for critique and analysis for whom may be interested.

 

[nasu]

It will depend on what kind of modes are you looking at. Are these the flexural modes of the cylinder's walls or the modes of the air in cylinder?
The boundary conditions will matter too.

 

[Andy Resnick]

Hi there I am currently doing a major thesis on the procedural generation of an acoustic drum kit, using PD (puredata). <snip>

Good luck with that- have you done any research at all into how the major manufacturers (Roland, Yamaha, etc) model drums? Hint- it's not based on first-principles calculations of resonant frequencies.

 

[BG1]

Not a physicist but am a drummer who has done some work with acoustics. You also need to factor in humidity for the drum head and for the drum shell if it is wood. There are also metal shells which will respond much less to likely levels of humidity. Also in a snare drum the snares add their own frequencies.

How will you determine your "sound" for test purposes? The tightening of lugs and choice of sticks (some with nylon tips for instance) and the head itself all impact the sound and there is no single sound but there are desired effects, such as crisp or more dull. There a large number of variables here.

 

[SIMON1]

It will depend on what kind of modes are you looking at. Are these the flexural modes of the cylinder's walls or the modes of the air in cylinder?
The boundary conditions will matter too.

The flexural modes of the cylinder walls and the modes of the air in the cylinder preferably the equation to determine all, the process I am researching is to model the cylinder (drum shell) as one medium and then to incorporate the remaining features of the drum after. there was an equation that I came across that only took into account the dimensions of the drum shell being that of depth & diameter but I am not sure if that would be correct to use as the resonant frequency of the shell?

 

[SIMON1]

Good luck with that- have you done any research at all into how the major manufacturers (Roland, Yamaha, etc) model drums? Hint- it's not based on first-principles calculations of resonant frequencies. -

I read a small amount and gathered that they are mainly sample based, with numerous variables added in behaviour modelling to give the player a more realistic feel. For the purpose of the project is to mimic that of an acoustic drum kit using solely procedural generated audio, the approach I have took is to physically model the kit including every variable from scratch, thus being very difficult but not impossible. To some extent this is a proposed question that does that need to be successful it is the process that counts and is marked accordingly.

 

[SIMON1]

How will you determine your "sound" for test purposes? The tightening of lugs and choice of sticks (some with nylon tips for instance) and the head itself all impact the sound and there is no single sound but there are desired effects, such as crisp or more dull. There a large number of variables here.

The variables that I have chosen to model are the basics first and then to dwell on more complicated matters afterwards. The shell, head, snare wires, depth & diameter, being the most important to achieve here first. I am well aware of the variables on the overall sound but as previously stated there is a tolerance on the project to be deemed successful, the test purposes will be a listening test to drummers as well as an audience of audio students. Upon reviews of the project we have spoken about reaching a point that exceeds ability and the timescale given but ideally i would like to add all variables for tuning and selection of sticks etc.

 

[SIMON1]

It will depend on what kind of modes are you looking at. Are these the flexural modes of the cylinder's walls or the modes of the air in cylinder?
The boundary conditions will matter too.

The flexural modes of the cylinder walls and the modes of the air in the cylinder preferably the equation to determine all, the process I am researching is to model the cylinder (drum shell) as one medium and then to incorporate the remaining features of the drum after. there was an equation that I came across that only took into account the dimensions of the drum shell being that of depth & diameter but I am not sure if that would be correct to use as the resonant frequency of the shell?

 

[nasu]

It will depend on what kind of modes are you looking at. Are these the flexural modes of the cylinder's walls or the modes of the air in cylinder?
The boundary conditions will matter too.

The flexural modes of the cylinder walls and the modes of the air in the cylinder preferably the equation to determine all, the process I am researching is to model the cylinder (drum shell) as one medium and then to incorporate the remaining features of the drum after. there was an equation that I came across that only took into account the dimensions of the drum shell being that of depth & diameter but I am not sure if that would be correct to use as the resonant frequency of the shell?

The flexural modes are not simple, as a simple formula. And it will depend on model (thin wall, thick wall) and boundary condition.
You can start with this paper, maybe, to get an idea to what are you getting into:
http://pme.sagepub.com/content/167/1/62.full.pdf+html

 

[SIMON1]

The flexural modes are not simple, as a simple formula. And it will depend on model (thin wall, thick wall) and boundary condition.
You can start with this paper, maybe, to get an idea to what are you getting into:
http://pme.sagepub.com/content/167/1/62.full.pdf+html

I couldn't access that paper directly but i did find this one in comparison: http://rspa.royalsocietypublishing.org/content/royprsa/197/1049/238.full.pdf - you are right it definitely is not simple! My supervisor on the project has stated that their are certain routes which would include too much maths & physics, and to choose a different route that wouldn't lead to the project timeline being majorly based on equations from physics & maths. I am getting help with maths from stepmother but its physics I am struggling with, if we can forget about the flexural modes for a minute, is there a method that does not include total accuracy but yet could still be used to determine the resonant frequency, given we had a tolerance?

 

[nasu]

I couldn't access that paper directly but i did find this one in comparison: http://rspa.royalsocietypublishing.org/content/royprsa/197/1049/238.full.pdf - you are right it definitely is not simple! My supervisor on the project has stated that their are certain routes which would include too much maths & physics, and to choose a different route that wouldn't lead to the project timeline being majorly based on equations from physics & maths. I am getting help with maths from stepmother but its physics I am struggling with, if we can forget about the flexural modes for a minute, is there a method that does not include total accuracy but yet could still be used to determine the resonant frequency, given we had a tolerance?

You may be able to find and use the results of these complicated calculations, without actually doing the calculations.

 

[SIMON1]

http://facweb.cs.depaul.edu/sgrais/DrumMakerproposal3_drum_shells.htm

L = (c / 4f) - (0.61r)

L represents the length of the shell, c represents the speed of sound within the given medium (wood, metal, etc.), and f represents the frequency of the drumhead's fundamental. Also, this equation takes account of the fact that the reflection of standing sound waves within the shell does not take place precisely at the end, but slightly beyond. Stopped pipes, therefore, act slightly longer than they truly are, so r, representing the radius of the end of the tube, is used in "end correction," a process of making up for this anomaly of "acting" length.

The fact that the stopped pipe resonates when it is only one fourth of the wavelength of the sound which sets it to vibrating is particularly useful in the practice of building drums. A pipe open on both ends would have to be twice as long as a stopped pipe to resonate the same fundamental.

Another useful attribute of the stopped pipe is its natural tendency to strengthen the fundamental. Since a stopped pipe only resonates at odd-numbered partials, one half of all the overtones do not receive any of its benefits, so the fundamental, already somewhat stronger than the rest, is further reinforced.

Therefore, through vibrating sympathetically with a drumhead, each shell strengthens the fundamental both by its own resonance as well as its omission of one half of all overtones in that resonance.
- so this makes sense to me but yet is it accurate enough say a 22 inch in depth bass drum the resonant frequency is 150.3 Hz?

 

[SIMON1]

You may be able to find and use the results of these complicated calculations, without actually doing the calculations.

That would be great I am trying to gain access into that paper through my uni just now see how it goes hopefully I can as you get the results without doing the hard part lol