Mathematics of rocketry?

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skybuster

skybuster

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Hi everyone!

It wasn't until a while ago until I began to think about the math that is involved in rocketry, but I'm lost on what all is used. I imagine that there's equations for altitude, CP, CG, drift. I was wondering if anyone might know what the equations are and what other type(s) of math and physics is involved with rockets.

Thanks
 
You can find many equations for model rocketry here:

https://www.rocketmime.com/rockets/rckt_eqn.html

You can find some equations in "Handbook of Model Rocketry" by G. Harry Stine, especially if you go to the appendices. You could also survey old NAR and Estes Technical reports and perhaps search Apogee newsletters.

A good book for the professional rocketry world is "Rocket Propulsion Elements" by George Sutton.
 
The HBOMR has the best assembly of calculations used in stability determination.

It's far easier to get one of the design programs (RockSim, OpenRocket, SpaceCAD) to check out your designs for stability and flight performance but reading the chapters in the HBOMR as well as the Estes and Apogee tech reports mentioned by others gives you a basic understanding of how the programs arrive at their answers.

As far as the actual math, for the most part it tops out at roughly senior-HS level algebra. It certainly helps to go higher, into calculus, etc etc., but if you have a good grasp on HS algebra you can work out most basic stability equations.

The Handbook is real good at giving an explanation of how the basic flight processes work: thrust (impulse) vs mass for acceleration, CP vs. CG for stability, so on and so forth.
 
+1 on the HBOMR for the basics. Anyway, most of the topics you would be interested in would be covered in the 1st semester of college physics. To really get a handle on how the math in the flight simulations works, you really need to learn calculus.
 
One of the reasons that I join many others in promoting rocketry as a fun and educational hobby for young kids is that I learned a lot about math and physics long before I took classes that covered those topics because of my interest in rockets. I was having fun and .... oh my ... learning stuff at the same time!

One resource that I relied upon was the collection of Technical Reports from Estes. They mailed me a large package of them after I used rockets in a Science Fair project. They are available online now:

https://www.google.com/search?q=technical+reports+site:estesrockets.com

A problem with them, though, is that some use US customary units (feet, pounds) instead of SI (meters, kilograms, Newtons). SI units make the math much simpler to use and understand.

-- Roger
 
skybuster said:
Hi everyone!

It wasn't until a while ago until I began to think about the math that is involved in rocketry, but I'm lost on what all is used. I imagine that there's equations for altitude, CP, CG, drift. I was wondering if anyone might know what the equations are and what other type(s) of math and physics is involved with rockets.

Thanks
Click to expand...

What level math are we talking about?
 
jadebox said:
One of the reasons that I join many others in promoting rocketry as a fun and educational hobby for young kids is that I learned a lot about math and physics long before I took classes that covered those topics because of my interest in rockets. I was having fun and .... oh my ... learning stuff at the same time!

One resource that I relied upon was the collection of Technical Reports from Estes. They mailed me a large package of them after I used rockets in a Science Fair project. They are available online now:

https://www.google.com/search?q=technical+reports+site:estesrockets.com

A problem with them, though, is that some use US customary units (feet, pounds) instead of SI (meters, kilograms, Newtons). SI units make the math much simpler to use and understand.

-- Roger
Click to expand...

That's one reason I am kinda sad to see optical altitude tracking slipping out of use -- altitude calculation is a great hands-on way to teach kids basic trigonometry. I sailed through geometry and basic trig in HS because of stuff I had learned as a model rocketeer when I was 10-11 years old.
 
JohnCoker said:
I'd say basic calculus would cover most of our needs.
Click to expand...

OK then. I'm going to be unpopular, when I say there are substantial objective errors in the treatment of this topic as presented in the Handbook chapter, "How High Will it Go?" These errors are not just quantitative; they lead Stine to make false qualitative statements as well.

Not everything reduces to an equation. There are approximate equations that work in single-stage LPR, but most computations depend on numerical solutions: computer simulations. The approximate equations are known as the Fehskens-Malewicki equations. Tom Beach has an Applet that uses these. His web site lists the underlying equations and assumptions.
https://www.la.unm.edu/~beach/altitudeHelp1.html

(Aside: The Estes TR-10 technical report is now available at Amazon for $185.00)

A more advanced beginner’s reference can be found here:
https://www.amazon.com/dp/0262632780/?tag=skimlinks_replacement-20

The first chapter, on stability, is challenging. (But it’s an excellent treatment of an inherently difficult subject.) The later sections, on simulation, are simpler, and the first section is not a prerequisite for them. There are more advanced methods of numerical integration than those used, but this book lays the foundations for vertical, two-dimensional, and 6-dof simulation. They also derive the approximate equations.

Caution: Take the DATCOM method of drag coefficient estimation with a large grain of salt.

First year calc and physics are prerequisites for the simulations stuff. Diff Equ for the first chapter.
 
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And it is always a good idea to have some understanding of the basic calculations before you blindly use someone else's simulation software

You can find the basic Barrowman calculations for center-of-pressure in several places online (for free):
https://my.execpc.com/~culp/rockets/Barrowman.html

If you have any problems I will be happy to answer qstns, just PM me and I will step you through

And I add my 2cents to recommending Stine's Handbook of Model Rocketry (any edition, although the latest is the best), this is a great all-around source for how to perform many of these calcs.
 
Larry Curcio said:
(Aside: The Estes TR-10 technical report is now available at Amazon for $185.00)
Click to expand...

If that is the going price, can I assume correctly that it is no longer available from Estes (at a more reasonable price)?

And if it is out of print, has the copyright expired?

And if someone scanned it and posted it, how much trouble would they be in?

($185 is pretty outrageous, IMHO)
 
WillMarchant said:
A soft copy is here https://plans.rocketshoppe.com/pubs/Estes/estTR-10/TR-10.pdf and the index page is https://plans.rocketshoppe.com/publications.htm
Click to expand...

I've got an original TR-10 copy. It looks like Will's reference for an electronic copy has all the same stuff.

IMG_1785.jpg

The derivation uses calculus to solve a second order differential equation. An analytical solution is possible if there is no thrust or if the thrust is assumed to be constant (use an average thrust). If the the thrust is assumed to vary with time, an analytical solution is not possible. For a constant thrust (or thrust being zero) this non-linear differential equation is classified as a Riccati equation. It looks the solution has been around for some time. Although Malewicki's report came out in 1967, I have found the solution in the book "An Introduction to Non-linear Differential and Integral Equations" by Harold T. Davis, (p. 60) that was published by Dover in 1962.
 
powderburner said:
And it is always a good idea to have some understanding of the basic calculations before you blindly use someone else's simulation software

You can find the basic Barrowman calculations for center-of-pressure in several places online (for free):
https://my.execpc.com/~culp/rockets/Barrowman.html

If you have any problems I will be happy to answer qstns, just PM me and I will step you through

And I add my 2cents to recommending Stine's Handbook of Model Rocketry (any edition, although the latest is the best), this is a great all-around source for how to perform many of these calcs.
Click to expand...

I should have been more specific in my criticism. Sorry.

To be objective, Stine makes four primary mistakes:

First, he uses an average mass approximation, which works over very small intervals, but normally not over the entire operating interval of a motor. He is saved, in his LPR computations, by the very small mass fraction of propellant. That is, his methods were poor (and the methods are the object here), but they made no significant difference in his results in this case. (A simple remark that this is appropriate because of the low mass fraction would be an adequate correction, here!)

Secondly, he forgot to include the effect of gravity in the boost phase. This makes a significant error in his calculations, and it’s also a flaw in his method. (This is easily corrected.)

Thirdly, he compares the results of his computations with those of his RASP simulation program – but the thrust curve in that simulation program has a different impulse from the one he was using in his computations. He blames the discrepancy on the thrust curve shape. In fact, the thrust curve shape makes extremely little difference if you do things right. This is a qualitatively wrong inference. (Simply changing the working impulses would correct this problem.)

Finally, his simulation program isn’t very good (because of his coarse intervals for small motors, and his use of velocity at least interval to compute drag) so it makes a poor reference. (Using a shorter interval for small motors would help here.)

The second and third problems tend to cancel each other to a certain extent – but two errors don’t make a correct result.

This stuff is easy to correct. Given that the book (which is very good!) is such an important document, it really should be corrected. That's all I'm sayin'...

Best Regards (Not comin' atcha, and not thinkin' you were comin' at me) C.
 
Larry Curcio said:
I should have been more specific in my criticism. Sorry.


Finally, his simulation program isn’t very good (because of his coarse intervals for small motors, and his use of velocity at least interval to compute drag) so it makes a poor reference. (Using a shorter interval for small motors would help here.)
Click to expand...

Sorry. Now I'm wrong... Stine's computations don't involve drag. (He was using Cd=0 in his simulations.) Nevertheless, his simulation program gave poor results. IIRC. I had to use another to get close correspondence.
 
I suppose we may be coming up on the Golden Anniversary Edition of the HBOMR (the First Edition came out in 1965) so maybe the calculation/simulation sections of the appendices will be polished up using SOTA methodologies.

My understanding was that G. Harry's RASP simulation program was written in the late 80s/early 90s when sim programs were in their infancy so it's not surprising it has some significant gaps. I do remember when you worked it through for basic Alpha-style sport models you got results which were wildly optimistic compared with real-world results (mainly due to failure to incorporate drag effects).

I also remember doing pencil-and-paper altitude predictions in junior high school in the early 1970s, filling up entire notebook pages with arithmetic and coming up with results which sounded utterly awesome -- single-stage rockets reaching 2,000+ feet on B motors, 3-stage C-C-B rockets breaking Mach 1, etc etc. -- and then realizing, "Maybe I will have to deduct 10% or something for air drag. That sounds about right."

Ha, ha. Little did I realize.

:lol: :eyeroll: :facepalm: :surprised:
 
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aerostadt said:
I lot of information was reported on this thread. Did Skybuster find what he wanted?
Click to expand...

Wow, a lot of inforamtion indeed, could almost write a textbook! Thank you to everyone who has posted. Fairly suprised that most the equations are simple algrbra, or what you would encounter in a first year calculus course
 
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If you have any interest in computing (as many nowadays who have an interest in maths or physics do!), there's also some fun programming to be done too with all these equations. It is not too difficult to code up a program to solve these equations numerically (which, as has been said, is really the only way to solve them), and create your own basic version of Rocksim / OpenRocket. Granted you will probably never use it if you have access to a more serious program but it will help you understand on a much deeper level what is going on inside those software (and indeed what factors go into a rocket's performance!)

If you've never programmed before this would probably be a great starter project!
 
mattk91 said:
If you've never programmed before this would probably be a great starter project!
Click to expand...
In Model Rocketry Magazine, circa 1970, there were two or three articles on numerical altitude simulations in FORTRAN. As a freshman with my first access to a computer I laboriously punched those programs into punch cards, ran them, modified them, expanded them, and spent many nights in the computer lab neglecting other studies. So yeah, I learned programming via model rocketry.
 

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