# How would I calculate the extra change in velocity due to residual thrust?

I'm trying to script a hover slam in a simulated environment using generic kinematic equations to calculate the correct throttle values I should be matching. I cut the throttle of my engine when my altitude is close to 0, at this stage my vertical velocity is also close to 0. Though the problem starts when I cut my engines at said altitude/velocity (just above the surface and velocity being very close to 0). The throttle takes some time to spool down so I end up having some extra impulse which causes the rocket to go back up... and eventually crash down.

To save headache I will introduce some restraints to the problem:

1. The rocket is in deep space. Meaning that there are no forces of attraction from any body acting on it. And thus the velocity of the rocket is conserved when the engines are not active.

2. Total spool down time is a constant directly proportional to throttle value. So if the throttle was 100% then the spool down time would be t seconds. If it was 25% then the spool down time would be 0.25*t.

3. The mass flow rate is a constant directly proportional to the throttle value. So if the mass flow rate was 20kg/s for a throttle value of 100%. Then the mass flow rate for a throttle value of 5% would be 1kg/s.

4. The engine has an extremely high deep throttle capability, into the range of nanonewtons.

5. The exhaust velocity is directly proportional to its throttle value.

Import - When I'm referring to throttle, I'm actually referring the ratio of the current engine thrust to the maximum engine thrust. They are not the same since if the throttle is set to 100% then in the next micro second set to 0%, it will take much longer for the thrust of the engines to spool down to 0%.

Things to be account for :

1. The mass of the vehicle is not constant during the burn.

Here is an animation I made that may make it easier to understand the question I am trying to ask: https://i.stack.imgur.com/JVIav.jpg

So my final question would be : For a given throttle value that produces a thrust N. If I was to suddenly cut off the throttle how would I calculate the total deltaV caused the the extra impulse generated when the engine is spooling down ?

What you want is a model thas been called in various places a thruster build-up/trail-off model. Trail-off models how much time it takes for a thruster to go from from full thrust to zero thrust, and models the nature of the curve between, for example, valves commanded closed and zero thrust. Build-up is the flip side of trail-off, modeling how much time it takes for a thruster to start from zero thrust to full thrust, and models the nature of the curve between, for example, valves commanded open and full thrust.

I've looked on the internet for literature on thruster build-up/trail-off models, and didn't have much luck. Google gets confused by interpreting build-up to mean the process of building a rocket engine (or something that has nothing to do with rocket engines). It also gets confused by interpreting trail-off to mean the trails that appear to come off of a firing rocket engine ((or something that has nothing to do with rocket engines).

Even Google Scholar gets confused. The few articles I found that are relevant and do address build-up/trail-off do so in words only, with no details. The few articles that I did find are not worth citing.

Both the control subsystem in your flight software and the models used in your simulation will need to address build-up/trail-off to some extent. At a minimum, you could be modeling build-up/trail-off as a simple lag. At a maximum, there is no maximum. There exists lots and lots of scientific literature on controlling laggy systems. This is the topic of many journal articles, book chapters, and even entire books.

Developing a build-up/trail-off model is one of the many reasons engine developers use static firing tests in testing their engines. The models are very engine-specific, and least in terms of specific parameter values. A somewhat simple model is that it takes a certain amount of time for a command to even begin taking effect, followed by a linear ramp-up to full thrust in the case of build-up, and a linear ramp-down to zero thrust for trail-off. Other models are significantly more complex.

• Thats unfourtunate. So in summary no analytical solutions exist for this problem?
– Sam
Jun 18, 2022 at 20:29
• @Sam there are many analytical models for this problem, but I wouldn't say that any of the ones I've used or seen constitutes a true "solution". Some are fancier than others, but they're all approximations to empirical data. Jun 20, 2022 at 22:53