Air Friction energy loss on a trampoline

[Sherwood Botsford]

TL;DR

Moving air out of the way is the largest energy loss on a trampoline. How do I calculate the change is energy loss with different mesh jumping surfaces?

Consider a 5 m diamter trampoline. It has an area of not quite 20 m2

If a jumper sinks 1 meter into the mat at center, the volume of the displacement cone is 6.5 m3 (1/3 base of cone times height)

Because air as to move in over the top, as well as get out from under, the air moved per jump is double this, or 13 m3. And on the return, going back up, the process is reversed. So 27 m3

Air has a density of 1.25 kg/m3. So we are moving 33 kg of air out of the way. This is an appreciable fraction of the mass of the jumper.

If the jumper has a height of 2 m he's hitting the mat with a speed of about 6m/s, slowing down to 0 over a distance of 1 meter, giving an average deceleration of 2 g. Force on the jumper however increases at very close to the cube of the depression angle of the mat. (within a few percent to angles up to 45 degrees)

Now trampoline mats come in various forms.

Early mats were in essence sail cloth, with fairly low porosity.
Current common trampoline mats have a porosity of 5-15%, mostly of holes under 0.5 mm

Mats have been made using meshs of 25 mm web with 25 mm openings (25% porosity) 10 mm web with 20 mm openings (44% porosity) 6 mm web with 12 mm openings (again 44%) and 4 mm web with 12 mm openings (56% porosity and 2 mm string with 6 mm openings same 56% again.

From personal experience I cannot tell the difference between the web mats. I've used the 6 mm 4 mm and 2 mm. But I'm only reaching jump heights of about 2 meters. The string mats have very close to the same characteristics as the 4 mm mat. Moving from a mat with fine holes to the 6 mm mat gives an almost instant 25% increase in jumping height. More critically, it gives a big decrease in energy required to maintain that height.

There are a few effects in play here:

Small holes have a lot more hole edge to hole area. This is going to mess things up. For the woven mats common in backyard trampolines, this effect dominates. "high flow" mats still have a porosity of under 20%.

At larger sizes, the edge effects get small, but the transport distances increase. E.g. on the 10 mm web air has to move sideways 5 mm to find a hole, where on the 6 mm web it only has to move 3 mm. My mental model of this says that there is a scaling factor at work here. We can consider the air affected to be some N * L where N is a small number 2-5 and L is the center to center distance between openings. N will be smaller for higher porosity meshes. So without edge effects the finer mesh will have lower drag.

For a given mesh size, as the porosity increases (e.g. 5 mm mesh on 10 mm spacing vs 5 mm on 15 mm spacing with 25% and 44% respectively) at what point do you get no benefit from an increase in porosity.

Similarly I would expect to see a sharp increase in resistance as the holes get smaller.

How do I model this.

(Practical points: larger openings are finger and toe grabbers. One of the potential approaches is a two layer composite with a fairly fine mesh (5 mm) of material like grocery store onion bags supported by a coarser mesh )

 

[jrmichler]

Try searching aerodynamic drag mesh fabric and air permeability fabric. Both of those search gave [what looked like] good hits related to air resistance vs air flow rate.

I'm not sure how to model it because the air pressure difference across the fabric is a function of the permeability, fabric velocity, and flow field around and below the entire trampoline. I suspect that you will end up with some correlation between perceived effort and mesh permeability. You might be able to do an experiment by dropping a heavy weight on different fabrics and measuring the bounce height or number of bounces.

 

[berkeman]

TL;DR Summary: Moving air out of the way is the largest energy loss on a trampoline. How do I calculate the change is energy loss with different mesh jumping surfaces?

Because air as to move in over the top, as well as get out from under, the air moved per jump is double this, or 13 m3. And on the return, going back up, the process is reversed. So 27 m3

Air has a density of 1.25 kg/m3. So we are moving 33 kg of air out of the way. This is an appreciable fraction of the mass of the jumper.

Paging @boneh3ad to see if he has thoughts on this interesting aero problem...

 

[rcgldr]

Competition trampoline mats (beds) used to be 1/4 inch canvas strips with about 3/4 inch openings. Current ones are paired strings, evolved from ozzie | Aussie | Australian mats, which are even more porous, almost like nets, probably still used in acts like Cirque Du Soleil. On the ozzie net like mats, once at about 2 meters, it felt like you were spending almost no energy to maintain height. If web is 4 mm and opening is 12 mm, then each 14 mm square has 2 mm borders, and 12 mm x 12 mm opening, is that 56% porosity?

The deflected shape is a concave cone.

 

[256bits]

TL;DR Summary: Moving air out of the way is the largest energy loss on a trampoline. How do I calculate the change is energy loss with different mesh jumping surfaces?

Consider a 5 m diamter trampoline. It has an area of not quite 20 m2

If a jumper sinks 1 meter into the mat at center, the volume of the displacement cone is 6.5 m3 (1/3 base of cone times height)

How does one know that the air movement is the largest energy loss? Stating this must mean that there has already been done studies on trampolines asserting this as fact.

There is hysterisis loss in the webbing, the springs, the frame, in addition to an aerodynamic friction loss from the jumper herself as she moves through the air.

The trampoline system itself is an underdamped system, where the oscillations decay after some time.
This is evident in that if the jumper does not input energy into the system during the cast off from the net, the jump height decreases, and she eventually comes to a stop on the net - a built in safety net ( pardon the pun ).

Whether the energy loss from the movement of the air around the net as it deforms overwhelms the other hysteris losses is not evident. A simple knapkin analysis yielded about a 5 pound force loss from the air movement for a completely closed net given the 1m deflection of the net and 6 m/s jumper velocity.
An open net would necessarily involve a decrease, and for competion with a tuned trampoline vs home jumping from sloppy over the counter model( ie less height, more effort - again safety for non-professional jumpers ) , the open netting is probably welcome, although there is still the pressure drop across the open net to contend with.

Underdamped oscillation relaxation:
https://en.wikipedia.org/wiki/Damping

 

[256bits]

Paging @boneh3ad to see if he has thoughts on this interesting aero problem...

I would think that @russ_watters with his experience in HVAC air supply systems.
In fact this is no different than the pressure drop through say perforated ceiling tiles ( ASHRAE ), filters and/or orifices.

 

[256bits]

From this,
https://www.accurateperforating.com/resources/technical-information/pressure-loss

one can get an idea of what is being looked at.
Note that this is for a perforated plate in an enclosed duct, so thus not directly applicable.
Movement around the perorated plate is not possible.
It would be as if the bottom portion of the trampoline is fully enclosed.
If one uses 6 m/s air velocity ( only at the centre of the mat and only at the moment of impact from the falling jumper ), a vertical line from 1100 gives the pressure drop for various cases of open area of plate.
1 inch water column = 0,04 psi.

 

[erobz]

I would want to try find a relationship for a piece of impermeable trampoline fabric first.

We shouldn't see much notable compression in the flow. What is the "outlet" area though? I picture the outlet flow propagating outward radially from under the trampoline. The volumetric flowrate would also be a more than challenging element to determine due to the velocity distribution over the trampoline surface (as it deforms) and the fact that everything about this is transient, etc...

Better just experiment first...

 

[Baluncore]

What is the drag coefficient of a net?
Calculate the total length of cord, ignoring knots. Model the drag coefficient of cord as cylinders.
What size, and how many knots are there? Model the knots as spheres.

 

[Dullard]

"Moving air out of the way is the largest energy loss on a trampoline"

I'm not disputing that, but also see it as a potentially incorrect assumption. I'm suspicious because:

Losses associated with 'stretching' the fibers in the net exist.
Losses associated with fiber-fiber friction exist.

Materials and Construction could matter (a lot). I don't have a feel for how large that loss is WRT to the loss due to pumping air.

Both of these losses (net friction, air pumping) would tend to vary inversely with mesh size. The 'hairiness' of the physical situation is such that modelling is unlikely to allow proper separation of the pepper and fly-dang.
A little time in a large vacuum chamber would allow (mathematically) simple isolation of the quantity in question.

 

[256bits]

I would want to try find a relationship for a piece of impermeable trampoline fabric first.

We shouldn't see much notable compression in the flow. What is the "outlet" area though? I picture the outlet flow propagating outward radially from under the trampoline. The volumetric flowrate would also be a more than challenging element to determine due to the velocity distribution over the trampoline surface (as it deforms) and the fact that everything about this is transient, etc...

Better just experiment first...

The pressure on top of the net pushes down.
On the bottom it pushes up.
No need for volumetric flow.

As the net moves down, it creates a dynamic pressure.
Integrate over the entire net area, and over the time span of movement of the net.
Result is the force differential from movement of the net.

 

[erobz]

The pressure on top of the net pushes down.
On the bottom it pushes up.
No need for volumetric flow.

As the net moves down, it creates a dynamic pressure.
Integrate over the entire net area, and over the time span of movement of the net.
Result is the force differential from movement of the net.

I look forward to seeing the calculations.

 

[rcgldr]

There is hysteresis loss in the webbing, the springs, the frame,

Hysteresis loss in a steel spring is very small. Hysteresis loss in nylon webbing with rubber coating is fairly small. Frame is essentially a rigid body. Bounces to around 30 feet are done in the Olympics, and higher still on Aussie trampolines and custom trampolines.

Multi-bouncing can be used for extreme heights:

 

[boneh3ad]

I would suspect that air "friction" is actually a very small portion of losses here. A larger source of loss from the air would simply be the work required to continually move air around above and below it. That's a purely inviscid phenomenon. This would be mitigated with a more porous surface, which would allow mass to pass through but also incur more viscous losses.