[Matt H.]

In 2008 I did some calculations for 1726 regarding the optimal precharged length of a pneumatic cylinder. As many teams are considering the same approach this year I figured it would be helpful to post the math here.
If you just want the answer see the bold portion of the post.

Note: Precharging a cylinder refers to locking the piston of a pneumatic cylinder extended some distance d and then allowing the piston to fill with air before removing the latch.

Warning: This process requires a little calculus!

Variables:
P: The pressure as the system kicks.
P_int: The initial pressure.
l: The position of the piston as the system kicks (ranges from d to L)
L: The length of the cylinder
d: The amount by which the piston is extended before the system is latched
A: The cross sectional area of your cylinder.
F: The force exerted by the cylinder as it expands
W: The work done on the ball (This is what we want to maximize).

The math:
We assume the kick is an Isothermal process and that the amount of air entering the cylinder during the kick is negligible.
so
P_int*A*d=constant
This implies that as the cylinder expands
P=P_int*d/l (The cross sectional area cancels)

The force exerted by the cylinder is given by P*A
so
F=P_int*A*d/l
The total work done by the system on the soccer ball is given by integrating the force with respect to distance
So
W=P_int*A*d *the integral with respect to l of 1/l from d to L
W=P_int*A*d*ln(L/d)
where ln is the natural logarithm

At this point anyone with a graphing calculator can find the maximum of the function (d=1/e*L), but it is much more fun to derive.

dw/dd=P_int*A*(ln(L/d)-1)
which is zero when
ln(L/d)=1
which implies
L/d=e
so
d=L/e~=.368*L
Note we determine that this is a maxima using the first derivative sign test (or by realizing that if d=L the cylinder doesn't move so W=0 and if d=0 there is no air in the cylinder so W=0)


In short a precharged cylinder will do the most work for you when the cylinder is latched at .368 or about one third of its total length.

Remember that other design consideration may trump this optimization and that variables other than energy (such as size) may be more important to you.

If you have any questions about the math feel free to ask.