Uncovering the Forces on Drum Lugs: An Engineering Design Exploration

[omurphy436]

TL;DR Summary

Attempting to mathematically model the forces on acoustic drum hardware based on ideal fundamental frequency of the drum head

I am currently designing an alternative to normal acoustic drum lugs and I need to find the forces that are felt by the tuning rods of a drum for the engineering background to my design. I have been able to use the 2D wave equation assuming uniform tension on the membrane and found values of the radial tension in the drum head based on the ideal fundamental frequency for the drum in question. However, I am stuck translating this radial force to the vertical reaction force felt by the drum hoop at the boundary and then to 6 tuning rods, or how I would go about finding this. There are lots of resources online for finding the tension on ideal circular membranes but I cannot seem to find anything that looks at the drum hardware. I have been able to use Abaqus CAE to create a model of the lug frame I have designed but I just don't have any force values to input.

If someone could help that would be great.

 

[bob012345]

Welcome to PF! First a question. In a typical tuned drum, do the tension rods hold all of the tension or does the hoop hold most of it and the rods just add some more?

 

[Baluncore]

The membrane is stretched over the circular end of the drum.
That fixes the diameter, and so the circumference, of the membrane.

Surface tension has the dimension of force per unit length.
The total tension in the rods will apply tension over the circumferential length of the membrane.

Since the forces are opposed across the drum, they are balanced, so I would not be surprised if there was a factor of two involved somewhere.

 

[pinball1970]

Welcome to PF! First a question. In a typical tuned drum, do the tension rods hold all of the tension or does the hoop hold most of it and the rods just add some more?

Very little. The ring of the head is not very malleable. In position the head sits on the drum cylinder but there is no tension in the head.
The edge maintains the shape
Tension only begins once the hoop is in place and the tension rods are tightened.
My process is in pairs finger tight then use the drum key, again in pairs, half turn.
Objective is even tension, playable tension and a note best fit to the head and in tune with the resonant head.

 

[omurphy436]

The membrane is stretched over the circular end of the drum.
That fixes the diameter, and so the circumference, of the membrane.

Surface tension has the dimension of force per unit length.
The total tension in the rods will apply tension over the circumferential length of the membrane.

Since the forces are opposed across the drum, they are balanced, so I would not be surprised if there was a factor of two involved somewhere.

Hi, thanks for the reply,

I have a value for the surface tension at 587.15 N/m when the drum is tuned to a frequency of 123.47 Hz. To convert that to a stress tension uniform around the circumference of the membrane, is it acceptable then to just multiply it by the circumferential length of the boundary? If so, how can I relate that to the vertical tension that the hoop exerts on the edge of the head to apply that radial tension to the skin.

As the tension is assumed constant, Is it possible to take a sectional view of the head/hoop/drum contact and resolve the forces with trig? Alternatively, could I use a similar process to the one that is used to calculate the longitudinal stress in a cylindrical pressure vessel scenario with the radial stress as a pressure?

Thanks for the help

 

[omurphy436]

Welcome to PF! First a question. In a typical tuned drum, do the tension rods hold all of the tension or does the hoop hold most of it and the rods just add some more?

Hi, thanks for the reply,

@pinball1970 explained it well, the hoop itself is not fixed and will just sit on the head until the tension rods are inserted and are screwed down, applying the vertical force which pulls the hoop downwards with the drumhead, stretching the skin and applying the surface tension for tuning. Essentially, I am looking at the vertical reaction force that the stretched skin has on the hoop due to it's tension, then hopefully I can resolve that to find the distribution of the same tension on the individual rods.

Cheers

 

[Baluncore]

If so, how can I relate that to the vertical tension that the hoop exerts on the edge of the head to apply that radial tension to the skin.

The skin is dragged around the corner by the tension of the rods.
Any vibration will take it closer to equilibrium.

I think we need a link or a diagram.

 

[bob012345]

Hi, thanks for the reply,

@pinball1970 explained it well, the hoop itself is not fixed and will just sit on the head until the tension rods are inserted and are screwed down, applying the vertical force which pulls the hoop downwards with the drumhead, stretching the skin and applying the surface tension for tuning. Essentially, I am looking at the vertical reaction force that the stretched skin has on the hoop due to it's tension, then hopefully I can resolve that to find the distribution of the same tension on the individual rods.

Cheers

As it is being explained to me it would seem to be not unlike a rope turned around a pulley 90 degrees. The tension is the same but the direction is changed. Then to first order, the total tension at the hoop edge translates to a force on each rod which if all the rods are tuned to the same tension would be the total force computed around the periphery of the edge (the magnitude of course as the radial direction makes the vector force sum to zero) divided by the number of rods.

 

[omurphy436]

The skin is dragged around the corner by the tension of the rods.
Any vibration will take it closer to equilibrium.

I think we need a link or a diagram.

So I am not currently looking at the vibration in the skin, the purpose of the wave physics was to determine the surface tension at the boundary to produce X Hz f01 frequency in the centre. I have been trying to model the solution in abaqus with an experimentally determined downwards displacement of the hoop of 1mm, stretching it over the rim. I have linked an image. Is it possible to theoretically determine the reaction force on the hoop knowing that downwards displacement of the skin, the elastic modulus of the material and the angles and lengths of the skin extrusion to the hoop?

 

[omurphy436]

As it is being explained to me it would seem to be not unlike a rope turned around a pulley 90 degrees. The tension is the same but the direction is changed. Then to first order, the total tension at the hoop edge translates to a force on each rod which if all the rods are tuned to the same tension would be the total force computed around the periphery of the edge (the magnitude of course as the radial direction makes the vector force sum to zero) divided by the number of rods.

Hi, it is not dissimilar to that concept, however the angle over the rim is about 45 degrees, not 90. It is also complicated by the fact that the rope is in tension across opposite ends, and also the elastic properties of the Mylar material that is used to make these drumheads. Do you know if those assumptions will sill work with those things factored in?

 

[bob012345]

Hi, it is not dissimilar to that concept, however the angle over the rim is about 45 degrees, not 90. It is also complicated by the fact that the rope is in tension across opposite ends, and also the elastic properties of the Mylar material that is used to make these drumheads. Do you know if those assumptions will sill work with those things factored in?

According to post #4 there is little or no tension in the skin till the rods are tightened so to first order assume all the tension is from the rods. Then take your value for the tuned drumhead tension and multiply that by $2 \pi$ and divide into six parts, one for each rod. That should be the approximate vertical force on each rod. I would assume the force on the rods is essentially vertical and not worry about the 45 vs 90 degrees for now. A rope on a pulley changes direction over some radius as well. Then you can work on doing a more detailed numerical simulation.