## What is a thrust curve?

Thrust curves are obtained experimentally for solid-propellent rocket motors by placing the motor on a test stand, igniting the propellent, and recording the thrust force as a function of time. This allows one to know how the motor will perform when placed in a rocket. The key parameter for rocket motors is the motor impulse--the integral of the thrust curve--because this is a simple measure of how much velocity a given rocket will have by the end of motor burnout.

For obvious reasons, water-rocket thrust production is different than for chemically propelled rockets. Water rockets have very high acceleration during thrust (often exceeding 200G) and very short thrust time (often less then 0.1 second). As described on my high-speed video page, there are distinct phases to the thrust curve depending on what is occupying the throat of the bottle: launch tube, water, and gas. The peak thrust generally occurs during the water-thrust phase. As described on my theory page, a different set of equations must be used to describe each phase. I have come up with a pretty sophisticated rocket model, implemented as a Java applet, that predicts exotic looking rocket-acceleration curves. The model needs to be validated with real-world data. This page describes my validation attempts and the things I'm learning along the way.

## Validating the thrust model

The best way to validate a water-rocket model is by recording data from an actual flying rocket, rather than from one on a test stand. This is because the flying rocket has substantial acceleration that impacts the water-phase thrust, as well as the stability of the interface between the water and gas. My model is not sufficiently detailed to treat the interface stability (in effect, it assumes the interface is perfectly stable and flat). However, the former effect can be assessed using my water-rocket model. As an example, one can approximate the effect of the test stand by running the simulator with the rocket mass set to an unrealistically large value--in other words, the combination of test stand and rocket is like a very heavy rocket. Shown below are two simulated thrust curves for a 2-liter rocket with no launch tube. The only thing that is changed between the two curves is the mass of the rocket.

Note: strictly speaking I fudged on the above thrust curves. In each case I took the rocket acceleration curve (output from my simulator) and multiplied by the empty rocket mass. Both rockets had 500 g of water onboard at time zero, and so if I had done the calculation strictly correctly the 100-g rocket curve would actually be above the 2500-g rocket curve at short times. The "effective impulse" calculated from the above curves is more convenient because it is based entirely on the empty rocket mass. In any case, regardless of the accounting method, the empty mass of the rocket does make a difference on the impulse. Obtaining a thrust curve for a water rocket mounted to a test stand is like having an infinitely heavy rocket and the resulting data would not give a fully accurate picture of what happens during flight.

## How do I collect the flight data?

Small electronic data-logging rocketry altimeters are readily available. Most will not do for my purposes for the following reasons:

- They evaluate the altitude of the rocket using a barometric (pressure) sensor. The thrust phase of water-rocket flight is generally finished before the rocket has reached several meters above ground. Even the most sensitive pressure sensors cannot achieve reliable sub-meter accuracy.
- To get from altitude data to thrust data requires a taking a second derivative of the altitude data--this introduces substantial additional noise.
- Most low-cost data-logging altimeters do not record data fast enough. In my opinion, at least 1000 samples per second is required.

Thus I was very excited to discover in December 2005 a new series of very small and capable altimeters at www.picoalt.com. The AA1 model is an accelerometer-based altimeter that records up to 1000 samples per second. I initiated a dialog with the very helpful proprietor of the site, who agreed to sell me a modified model for my purposes and let me beta test it. The modified AA1 has an accelerometer with a 120G limit (ADXL193 chip), rather than the stock 50G limit.

## January 2006: Preliminary test with the Picoalt AA1 Altimeter

Below is a picture of the rocket hastily put together for the first test, shown with the nose detached. The body is built from a 1-liter Perrier soda bottle. The padded nose is build from a section of "pool noodle" wrapped in vinyl electrical tape. There is a small rectangular cutout that forms a compartment for the altimeter and 9-volt battery. The rocket with altimeter and battery weighs 155 g prior to adding water and pressurized air.

Normally the Perrier bottle makes for a speedy rocket. I had to launch with much reduced pressure (less than 40 psig) in order to stay within the acceleration limits of even the modified AA1. The recorded data are compared to the corresponding simulation curve in the graph below. As before, please note that these are "effective-thrust" curves because I simply took the respective rocket acceleration curves and multiplied by the empty-rocket mass.

### Results

Indicated on the graph are the three boost phases. The simulation and experiment agree very well on the length of time for thrust and the total impulse. This is encouraging.

Why do the two curves differ? Besides that the simulation model simplifies the fluid physics somewhat, there are two possible causes of the discrepency. The first is that the accelerometer board was press-fit into the padded cavity and was not secured rigidly. It therefore likely bounced around a bit, leading to erroneous oscillations in the recorded rocket acceleration. The second potential problem is that the accelerometer chip has a 400 Hz low-pass filter. This means that the filter smooths out otherwise rapid acceleration changes, and that the device will produce data with a time lag behind the actual curve when there are large swings in acceleration. In future tests I can likely eliminate the bouncing-board problem; the lagging-filter problem I will have to live with. In any case, while these kinds of problems degrade the resolution of the thrust curve, they do not change the calculated total impulse.

To be continued...