Analytic drag equation for rocket under acceleration

[AlexApps99]

Hello, this is my inaugural post on the KSP forum.

To learn some more physics and calculus, I'm trying to create an analytical equation for the path of a solid fuel model rocket fired vertically.

I first made an equation assuming no atmosphere and no orbital mechanics, which I am satisfied with.

I'm currently making an equation that accounts for drag, and assumes there is no change in atmospheric density and no wind.

The equation I am using is from Wikipedia, specifically vertical motion upward and vertical motion downward.

I am not sure how to adapt this equation to work while the thrusters are still active.

If anybody can help me with this, I would be extremely grateful.

 

The work I have done so far is here: https://www.desmos.com/calculator/ncmd38rosu

The Variables folder contains all variables, the No Drag folder contains a complete equation that I have made personally, and the drag folder contains what I have done so far to account for drag.

I am more than happy to explain my thought process/reasoning on how I have come up with what is on there so far.

 

Thanks, and I look forward to spending more time with the KSP community.

Edited October 11, 2020 by AlexApps99
Added tags

[Hotel26]

Welcome to the forum!!

Well done embarking upon the course of hands-on learning and kudos for selecting an ambitious project...

Take a look at: Runge-Kutta, Euler's method, predictor-corrector and Simpson's rule.  There's another technique related to those that is tickling my memory^[1]  While none of these may be exactly what you want, reviewing them will lead in interesting directions.  If you have access to a good (university-level) library, Chapter 4 of "Fundamentals of Astrodynamics" may be very helpful.

Good luck.

[1] ah, "leapfrog integration"

Edited October 12, 2020 by Hotel26

[K^2]

You are already making an assumption that rocket's TWR is maintained constant, so it's moving up with a constant acceleration. That's not strictly speaking true for typical ascent profile, but if you are going to make that simplification, then you can use the formula you already have for falling with drag and use it for ascent with drag. Simply substitute a = (Δ_v - g t₁) / t₁ in place of g and flip the sign so that you're ascending.

[linuxgurugamer]

This mod will maintain a constant TWR, it will control the throttle to do so:

 

[sevenperforce]

17 hours ago, AlexApps99 said:

The equation I am using is from Wikipedia, specifically vertical motion upward and vertical motion downward.

I am not sure how to adapt this equation to work while the thrusters are still active.

If anybody can help me with this, I would be extremely grateful.

If you want an analytical equation of state, you're going to need to know atmospheric density. Density is a function of altitude. In your equation, altitude is going to be equal to displacement, which is ultimately a function of velocity and time.

You're also going to need to know the coefficient of drag for your rocket. That's difficult because the drag coefficient is velocity-dependent, so you'll either need to take an approximation and treat it as a constant or you'll need to use a discontinuous function.

[Guest]

On 10/12/2020 at 1:08 PM, sevenperforce said:

If you want an analytical equation of state, you're going to need to know atmospheric density. Density is a function of altitude. In your equation, altitude is going to be equal to displacement, which is ultimately a function of velocity and time.

You're also going to need to know the coefficient of drag for your rocket. That's difficult because the drag coefficient is velocity-dependent, so you'll either need to take an approximation and treat it as a constant or you'll need to use a discontinuous function.

I think it's more than likely that there already are analytical, semiempirical methods for estimating that. OP should probably look for those in a rocket/missile design book.

But also, OP, I don't think you can get an analytical solution to this problem... you'll probably have to go numeric. I've tried doing the same you're doing before but modeling an aircraft takeoff instead, and given that thrust, drag and landing gear friction were all in function of velocity (while velocity also was a function of these forces!), really, there is no way to achieve an exact analytical solution with a finite amount of terms. At least that I know of. You might achieve an approximation if you make a few extra assumptions, but by then... will it really be useful? The numerical method for simulating this isn't at all computationally expensive and would get a higher fidelity solution than the analytical approximation either way, I don't see it being adopted.

Edited October 19, 2020 by Guest