Watching a person on a trampoline they can get higher for about 4-5 bounces then they are losing as much energy as they are putting in. A 70 kg person bouncing 3 meters is roughly 2100 J
What are the energy loss mechanisms?
- Spring losses.
- Elastic losses on the matt
- Air displacement.
I think that the largest of these is air displacement. The professional mats on olympic trampolines are nets, with only about 10% of the surface blocked. They can reach heights of 16 feet (5 m) This is close to double the energy.
A 5m circular trampoline with a 1 meter displacement, would (if a perfect cone) have a volume of about 6 cubic meters, or about 8 Kg of air. Air gets moved on both sides, so there is 16 kg of air involved.
To bring him to a stop in 1 meter is going to take more than 1 g. I think it will be square root of the distance ratio or 1.7 g. 3m corresponds to a velocity of about 8 m/s
If the landing accelerated 8 kg of air to 8m sec twice (does this on the rebound too) it would be mv2 = 512 J. This is a huge oversimplication, but I can't see it as being much larger than this, and likely smaller. Only the center of the trampoline is moving that fast, and only at the moment of impact. Moving air on both sides however doubles it.
Springs generally have a low hysteresis. Sanity check: If it takes 4 jumps to get close to maximum height, then the jumper is pumping in about 500 J/jump. 100 springs at 100g = is 10 kg of metal. 1 jump per second (too fast) would be 500W or about 5W per spring. This may be detectable, but they aren't going to glow.
The mat isn't deforming much. It will have wrinkles running from the spring attachments to the center
The final component is the jumper himself. If he starts his squat for the next jump as he touches the mat he is extracting some energy from the system, and also making his deceleration non linear.
Mind you: If he's pulling 1.7 g's, there may be an issue of him being unable to pump more energy into the system. If this is the case, then designing a trampoline to have a larger displacement and smaller forces would increase the jumping height.
This can be checked: Can a more fit athlete jump significantly higher on the same unit?
There is some sort of impedance matching happening here. Someone who is too light has a hard time getting much bounce. Two effects: Small mass doesn't deflect the mat much. Small mass also means that for a given deflection, the g forces will be larger.
So where is the energy going.
One answer says, "look at the noise" But trampolines aren't that loud. Even a watt of actual sound energy is LOUD.