pws trampoline

Hello,
We are Melissa and Rhodé; we are in the sixth grade of bilingual VWO at Lyceum Oudehoven in Gorinchem. We are doing our PWS (in English) on balls bouncing on a trampoline. We are looking into resonance, the interaction of forces and energy transfer. We've made graphs of the movements of the balls, the person who jumps on the trampoline and the trampoline mat, using Coach and Excel. Furthermore we are planning to make a model in Coach.
We came across a relevant source called "People Bouncing on Trampolines: Dramatic Energy Transfer, a TableTop Demonstration, Complex Dynamics and a Zero Sum Game” , however we do have some questions about the mathematical formulas. The source is:
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0078645
Unfortunately we could not upload a file, so we have to describe it here: the formulas on the vertical movement of the masses and the energy in the spring are the ones that we do not fully understand. So everything under the heading: "A mathematical model: passive bouncing of two masses" , so only 2 times 4 formulas.
Could you explain the formulas and how they are derived?
Thank you in advance!
Kind regards,
Melissa and Rhodé

Hi Melissa and Rhodé,
That is a pretty advanced topic you picked for your PWS. I'm not sure what your specific question is about the formulas, but I can explain them in general:

The first four equations are based on Newtons law of motion: F = ma. In this case, there is only motion in the y direction. The derivative of the position y with respect to time (twice) will give the acceleration a, which is the y(dot dot) term on the left hand side of the equal sign. So the left hand side gives the ma term, which means the right hand side gives the forces. In the P0 case, there is only gravity: mg. In the other cases, there is also a spring force from the tension in the trampoline. As you know, spring force is F = Cu, with C the spring constant and u the deflection. In this case, u is given by y1 and y2 given in Figure 3. The constant C is not directly given, but the Tension T is. Ill come back to that in a second.

These are four energy equations. The formulas compute the spring energy: E = 0.5Cu^2. Again u is given by y1 and y2. C is again not given, but T is. Now how do you go from T to C? T is the tension in the trampoline, in Newton, so just a force. C is the Force per unit of length, so we need to divide by some length. In this case, the length is given by a, b, and c.
Quite a story, but I hope it is clear.
If you have any additional questions, let me know!
Best regards,
Matthijs


Hello Matthijs,
Thank you very much for your quick response!
It helped us enormously!If we'll have any other questions, we'll let you know!
Kind regards,
Melissa and Rhodé