Piston Extention Rates:
While working on the design of a Vex Catapult robot, I took some data on Vex dual acting cylinder extension rates. We used a high speed camera to measure the position of the cylinder at 300 frames per second. Two data points were taken:

2)0.5 kg load pull against gravity: 28/300 seconds or .093 sec

The piston extends 2 inches.

So the average extension rates are:

No load: 54.5 in per sec

.5kg load: 21.4 in per sec

Vex tubing elastic spring coefficient:

With a force sensor, the force vs distance of a four strand tubing spring. ( 10 foot length of black tubing was folded over twice to get 4 strands).

Here is the data:

Millimeters Newtons

900 18.866

800 16.537

700 14.209

600 11.164

500 7.433

400 1.672

The curve is slightly nonlinear. The best fit linear line has a slope of 1/29.1 N/mm or about 0.2 lbs/in.
The zero force mm point on the linear curve fit is 311.4 .

An assesment of this data for a catapult is posted on my vex blog :

A Mathematical model is science.
Checking the model with experimental results is engineering.
Your updated comments on power vs energy, and concept of recharging the tubing spring with a piston sound intriguing.
I’d like to see the piston energy and power calculated more similarly to the spring. Otherwise, your 2inches x 12 lb from piston seems same as 24 in-lb from spring.

The piston 24 in-lb is only achievable under static or very low piston rates as you might guess. Once the pistons move quickly the pushing chamber must work against the exit camber which acts like a damper. Both chambers are flow restricted so the net pressure across the piston approaches zero for a good portion of the extension time. This is particularly true under the no-load case where the piston is at steady state speed most of the time.

So the spring can deliver the stored energy without much loss where as the piston cannot. The pressure differential during the extension has a burst and remains actually quite small (<5psi) for most of the time.

I completed my excel model of the actuator after some headaches caused by my choice of slug,in,psi units. It models the pressures,volumes and mass flow based upon input and output pipe tube lengths/diameters and viscosity. I adjusted the tube lengths to generate mass flow rates that would reproduce the average extension rates for both loaded and unloaded cases. I now have predicted time histories of the speeds and it appears that the instantaneous terminal speeds are about 1.5 times the average speed. This compares well with new frame data that we took off the movie today. See enclosed doc file.

The force generated by the shafted side is smaller due to the lower effective piston area. I measured the shaft to be about .157 in. This gives an area of .1 si vs the .12 si on the unshafted side. So we would expect about 10 lbs of force vs 12 lbs. My unloaded case was with the 12 lb side whereas the loaded pull was with the 10 lb side.