Thank you for the compliment. Our group would not be this far without your help. I was actually trying the graph idea, but I couldn't get anywhere. It seemed this is dependent on that, and that is dependent on this. And it looped.
For this entire post, pretend that there is no inner chamber; when the air exits out of the tube, it is always exiting into a 14.7 psi atmosphere. Later, the inflation of the inner chamber will be incorporated. For now, pretend the inner chamber is the atmosphere.[/color][/size]
As I understand, this is what happens as the air enters the tube:
The Three Factors Of Pressure (j, k, m, r, q, v)
1) Let j(t) be the function for the rate at which pressure rises because of the air that flows in at a rate of r(t).
2) Let k(t) be the function for the rate at which pressure lowers because of the air that flows out at a rate of q(t).
3) Let m(t) be the function for the rate at which pressure lowers because of the increasing of the tube's volume at a rate of v(t)
The Instantanous Pressure Available (f, F)
Let f(t) be j(t) + k(t) + m(t). j(t) will be positive since pressure is rising. k(t) and m(t) will be negative, since pressure is lowering. The integration of f(t) in respect to dt would give the instantaneous pressure at t. Let F(t) be the integration of f(t) in respect to dt.
The Instantaneous Pressure Required (p)
Let p(t) be the pressure required to lift the weight. p(t) is 200 pounds divided by how much surface area is in contact with the tube. p(t) changes as the volume of the tube changes. It is hard to explain, so I'll ask Brad (the frame's main CAD designer) to export an image eventually. At peak inflation, 212 square inches will be in contact. At complete theoretical deflation ("theoretical" because in the real world, the tube is sure to fold and twist and leave gaps), 612 square inches will be in contact. As the tube fills with air, less surface area is available, and thus, more pressure is required to lift.
Acceleration Of Weight Due To Net Instantaneous Pressure
Whenever F(t) [instantaneous pressure] is greater than p(t) [instantaneous pressure required to lift], a force is available going upward. Thus, the 200 pound object experience upward acceleration related to p(t) subtracted from F(t). From this, an acceleration of the object (and thus, the height of the tube) can be found. With plenty more contemplating and bottles of aspirin, v(t) [mentioned before as the rate at which the volume of the tube increases] is concluded to be dependent on F(t) - p(t). However, m(t), one of the three parts of f(t), is dependent on v(t). This is the first loop that confuses me. Maybe the loop doesn't matter, and something can be done. But I don't know how.
First Summary
So far, I have defined all I know about the following functions:
f(t), F(t), m(t), p(t), v(t)
What is left?
j(t), k(t), r(t), q(t)
Function Detail: j(t)
j(t), to repeat, is the function for the rate at which pressure rises because of the air that flows in at a rate of r(t). The pressure is the instantaneous volume of air (at STP) inside the tube... divided by the instantaneous volume of the tube.
The integration of r(t) gives R(t), the instantaneous volume of air at (STP) that has entered the tube. The integration of v(t), the rate at which the volume of the tube changes, is V(t), the instantaneous volume of the tube. The integration of q(t), the rate at which the air leaves the tube, is Q(t), the instantaneous amount that the air has left the tube. Q(t) subtracted from R(t) gives the amount of air at STP that is contained within V(t). Therefore...
j(t) = {14.7 * [R(t) - Q(t)] / V(t)} - 14.7
The 14.7's have to do with ratios and absolute pressures. j(t) is the rate of gauge pressure.
Function Detail: k(t)
I just realized that j(t) takes k(t) into consideration. To repeat, k(t) is the function for the rate at which pressure lowers because of the air that flows out at a rate of q(t). Well, I think j(t) takes care of that. If anyone disagrees, please let me know.
Function Detail: m(t)
I've already discussed this above. It is part of the loop. It seems to be dependent on itself, in some twisted way. This is the part I need most help with.
Function Detail: r(t)
This has to do with the pump curve. For now, I'd like to keep it constant. But as pressure increases inside the tube, the harder it is for the leaf-blower to pump air inside the tube. And thus, the rate decreases.
Function Detail: q(t)
This is determined by the very first equation that FredGarvin told me. However, the delta-pressure in that equation is dependent on F(t). This is the second loop that I see.
Second Summary
I think I've outlined everything in my head. I could go into more variables such as "h" for the height of the tube, etc. But I'm already having trouble just remembering the function names.
Can someone please check through this and confirm that my logic is correct? Are there any flaws? Anything I'm missing?[/color]
As always, every comment is appreciated. Thank you.
PS - I am not proof-reading this. It is 1:15AM and I am at the point of delirium. If something really doesn't make sense, let me know, and I'll look into it tomorrow.[/size]