How can I accurately model the launch trajectory of a Saturn V rocket?

[davehatton01]

hi, I'm an undergraduate from the University of Newcastle Upon Tyne, and I'm trying to develop an accurate model of a launch of a saturn V rocket for a project.

i have access to the technical specs like fuel capacities, consumption rates, etc. but i need help on some of the harder mathematics, like how to account for the variable gravity, variable mass, and how to model the curved flight path etc etc.

any and all help appreciated.

Dave

Changed the title to something more descriptive. -enigma

 

[Clausius2]

First advice: post your doubt in the Mechanical&Aerospace Forum. There, flying around, there is an inhabbitant called "Enigma". He is the "Master of the Rockets" here. Surely he will advice you.

Second advice: the unique thing I can say to you is about the rocket equation (Tsoilkovsky equation) of rocket dynamics. I don't know if it is enough accurate for you. Surely there is another Tsoilkovsky equation that has aerodynamic drag into account. You only have to write the 2nd Newton Law taking into account the mass variation in time, the gravity force and the aerodynamic drag:

-gravity force: it varies with the height, and the law of variation is given by the Gauss theorem. I think $$g\approx \frac{1}{r^2}$$ without constants.

-variable pitch angle: sorry but here I cannot help you. The curved path is established in the user's manual of the rocket for a correct entrance angle into orbit, but I don't know how to calculate such angle given some orbit.

Once again, I encourage you to talk with "Enigma". If not, please post me something you want to have clearer.

 

[enigma]

There, flying around, there is an inhabbitant called "Enigma". He is the "Master of the Rockets" here. Surely he will advice you.

Thanks for the kind words, Clausius.

variable gravity, variable mass, and how to model the curved flight path etc etc.

Hi Dave, welcome to PF!

A teammate and I did a similar project last year for a design project we were participating in. Unfortunately, we didn't get it working optimally, and had to scrap it for our final report (we just assumed that 9.5km/sec delta V would get us to orbit, waved our hands, pulled out the smoke and mirrors, and let it go with that).

The variable gravity and variable mass (we also took into account variable atmospheric density for drag and change in thrust due to the changing motor Isp due to atmospheric pressure) can be taken care of quite easily with a little calculus and Matlab's ode23 function. Basically, you'll need to set carefully define your coordinate systems such that the differential of the local gravity changes as height from the center of the Earth changes, and the differential of atmospheric pressure and density change WRT altitude. You couple those equations along with mass, thrust, position, velocity, etc.; code the ode function and let MATLAB do the numerical analysis for you. IIRC, we had about 10 terms. You'll need to run several piecewise ode's to account for when you drop of the Saturn V's first and second stages. Pass the ending conditions of the previous run into the next stage with removing the mass from the previous stage, and changing the nozzle specs when you switch from the 5 F-1's to the 5 J-2's to the single J-2.

I'd restrain the motion into a 2-d plane if it's possible. It's very tricky to do the motion in an IJK coordinate system.

Our major problem we had will probably not be an issue for you. We were trying to optimize the trajectory to minimize the delta-V requirement. This will not be an issue for you, because you already know the rocket got into orbit. If you know certain points (downrange distance vs. altitude) for the ascent, you can make a simplifying assumption that it was traveling in a single orbital plane (and as a result, following 2d motion), and then curvefit the points to obtain an 'optimal' trajectory. If you've got the optimal trajectory, you don't really have to worry about a control system, which is what stuck us up. Just feed in the optimal path to the ode solver (angle vs. altitude or time... you'll have to figure which works better) and have the thrust point in that direction for all times. We needed to code a multiple input-multiple output controller which wasn't covered in our controls system class (a graduate course covers it). By the time we got to the point that I realized my simple proportional controller wasn't doing what I wanted it to, we were too far down the semester with too many other systems to design to worry about trying to figure it out.

Hope that helped some.

[moderator hat on]
I'm going to change the title of the thread to something more descriptive.
[/moderator hat on]

 

[enigma]

What year are you, Dave?

Have you had a propulsions class and a course which teaches Matlab (or similar) yet?

Do you have partners for this project, or is it solo?

 

[Clausius2]

Thanks for the kind words, Clausius.

No thanks needed. The only thing I hope of you now is you could answer "collaterally" another question besides the dave's question. Right?


(say "yes" and I will ask you my question here).

 

[Bobster]

Hi there,I am Dave's coursemate as well.We are in Stage-1.
I am doing a BSc in Astronomy With Astrophysics [hopefully].The softwares we use now are Transmath and Maple both which are not that hardcore.

 

[enigma]

Hi there,I am Dave's coursemate as well.We are in Stage-1.
I am doing a BSc in Astronomy With Astrophysics [hopefully].The softwares we use now are Transmath and Maple both which are not that hardcore.

Hi there,

what does Stage-1 mean? 1st year?

I'm not familiar with Transmath, and Maple and I don't communicate too well. You'll want to find the ordinary differential equation solver on whichever software package you're going to use (that's what ode is in Matlab). It basically breaks down a motion into small steps, and generates the new coordinates based on the previous locations and the differentials (which is what you code into the function).

Clausius, if the question is very closely related, I suppose it can go here. Otherwise, start a new thread. Thanks
[/mod hat]

 

[Bobster]

Stage-1 is nothing but Year-1.Most students here go straight to Stage-1 after completing their A-Levels.But I decided to start from Stage-0 which is basically a kind of Foundation Year where we did basic Maths,Physics and Computing[like how to use search engines and go to a website and stuff lol !].The main reason I decided to start from there was coz I did my A-Levels in India.

 

[enigma]

How long is this project supposed to last? Is it a whole semester thing, or is it due in a few weeks?

To get any real level of detail, you'll be biting off a lot as freshmen (what 1st year is called state-side). You can certainly make a bunch of simplifying assumptions, but Dave said you're shooting for realism.

How much math, physics, and computing have both of you had?

 

[Bobster]

Well as far as Physics is concerned in India we did a bit almost everything.There are no classifications like astrophyiscs,nuclear physics etc..All students aspiring to be doctors or engineers/scientists had to do physics and it included the basics of all branches of Physics : Mechanics,Nuclear Physics,Astrophysics,Rotational Dynamics,Matter and Energy,Waves and Oscillations,Forces and Fields,Optics etc...

Maths ? Well mainly Integration and Differentiation,plus differential equations,and Vectors.Stats was there as well but we ignored it for most of the year :zzz: and only concentrated on a few questions relevant for the exams .I hate Stats anyways.

Computing is my favourite pastime,and I chose it as my main subjects for A-Levels.I did mainly C++ and i really enjoyed it. .Transmath and Maple seem quite easy but I haven't used Matlab yet.

Anyway I don't want the thread to veer out of topic ,and I am not doing that project,its Dave's.

 

[enigma]

Oh. I thought the two of you were teammates.

Dave, did my first post make sense to you, or was I talking over your head?

The project is do-able, but the math may be more than you're used to.

 

[davehatton01]

model

Enigma, thanks for your help...
i realize it seems a lot for a 1st year to bite off right now, but it's something I've been wanting to do for ages.
As for the maths, hit me with as much math as you like! i did further maths at A level, so that's covering things like 2nd and 3rd order differential equns, integrals out of all proportion etc.

have you got any specific equations as starting points? i have a copy of the ideal rocket equation but as that's an "ideal" equation, i am not sure how much that can help me... as you said, I'm going for realism to the highest possible level.

you seem to really know your stuff on this one, any help could prove invaluable.

thanks

Dave

 

[enigma]

Well, your best bet will be to ignore the ideal rocket equation. You want to go straight to Newtons Second Law, and do stepwise calculations.

Basically something sort of like this only convert the gravity into a polar coordinate system:
$$x_1=x_0+vx_0*dt$$
$$y_1=y_0+vy_0*dt$$
$$vx_1=vx_0+(Thrust_{x0}/Mass_0)*dt$$
$$vy_1=vy_0+((Thrust_{y0}/Mass_0)-g)*dt$$
$$Mass_1=Mass_0-burnrate*dt$$

etc.

You then repeat this step over and over again, constantly updating the individual values. Basically, you're doing an Euler differential equation solution. Matlab has several of these built in (ode23 and ode45), only they do a Runge-Kutta (sp?) method to make the stepsize appropriate regardless of changes in thrust and velocity.

You'll need to model that the acceleration will be changing because you're throwing off mass. You'll need to model the gravity in both X and Y coordinates based on position (changing it due to altitude would help too, but the effect is really small for Earth to LEO, and could probably be ignored). You'll need to consider the three different stages. You'll need to consider the angle which you're going to apply the thrust (this you'll need to do some sort of curvefit to researched flight profile information).

For a freshman level project here in the states, that would be more than enough for a good grade.

Depending on how much detail you really need to get a good grade on the project (you don't want to bite off more than you can chew), you may want to model the atmosphere and take drag into effect. You may also want to look at thrust variations due to atmospheric pressure (thrust is larger in a vacuum than at sea level). You can model the initial rotation of the Earth to give you a 'kickstart' in the velocity components. For accuracy there, you'll need to do some trig with the Lat/Long of Kennedy Space Center. You can research when and how much they throttled the engines during the ascent, and model that into thrust and burnrate. There are others.

When you're all done with that, you can go back and run the ideal rocket equation for multiple stage rockets and compare to see how bad the losses were. You should expect losses on the order of 1000-2000 m/s.

How much time do you have for the project? My advice would be to go with the basics first, and then go back in and put in bells and whistles.

 

[davehatton01]

Curved Launch

Hey, Help is again needed.
i have done most of the mathematics required for variable mass, but before i get my variable acceleration formulae i need to know the path the rocket took.
I tried looking online, and could not find any specific references, should i model it as a circle of set radius that goes to a straight line at 75 degrees from the horizontal? if so what radius?

thanks
Dave

 

[enigma]

This is going to be the hardest part of the project, probably.

You're going to need to scour the web looking for information, plot the points, and do a curve-fit.

I saw a few comments which could be used as data points in this transcript of the http://history.nasa.gov/ap15fj/01launch_to_earth_orbit.htm

I did a search for 'Saturn V downrange altitude' on NASA's http://www.hq.nasa.gov/office/pao/History/search.html.

You could also search NASA's technical reports server.

 

[enigma]

Yeah. That history page is the way to go.

Under 2.07 Boost on http://history.nasa.gov/ap15fj/csmlcindex.htm

I'm not sure if the nautical mile (nm) readings are altitude, range, or some combination. Convert them into cylindrical coords and play arond until you find something which looks like a launch profile.

The final orbit can be found in the communications. They read off two nm numbers. Those are apogee and perigee of the final orbit.

 

[davehatton01]

Curve Points

hey, I've been browsing that site you gave me and I've found a table full of velocities and angle of pitches... http://history.nasa.gov/ap15fj/csmlc/2-03.gif
a model involving a series of straight lines between known points would be undoubtedly the simplest, although is there another one i can use?

Dave

 

[enigma]

Plug the values into Excel (or similar) and have it give you a curvefit to the points.

You'll probably want to go up to the 2 degree data point at 7:30, and then do straight lines. It looks like the last 4 minutes they were pointing it downward to straighten out their orbit.

 

[davehatton01]

Downrange

Hey, is there any "sensible" way of calculating downrange other than some hefty trig between all the given points on that pitch-time table?
thanks
Dave

 

[CharlesP]

I copied the first link, and then got nowhere after changing the second two searches to "downrange altitude". My intent here is to just get the answer from someone who has already measured it. I want to know what the orbital insertion curve looks like for a typical vehicle. All I want is any vehicle getting into orbit and x, y, z, vs t every second or so, for the first half way around the planet. I would even be happy with the 2 Dimensional simplification. I think this would make a fine picture when plotted.
I first pursued this problem when I worked for Boeing on Space Station at MSFC to no avail. I did not have the right contacts.