It's not rocket surgery...
 

Well I guess it is really. OK, here's the setup, you're designing a multi-stage rocket and want to use as little fuel as possible by mass.

These are your initial arbitrary parameters:[LIST][*]The rocket has 4 stages.[*]Each stage is composed of 85% propellant by mass.[*]Exhaust velocity for each stage is 3,000 m/s.[*]Target delta-v is 10,000 m/s[*]Payload is 1,000 kg.[/LIST]
Two questions, what is the minimum mass of the rocket stack on the launch pad, and what delta-v does each stage impart?

The rocket equation will be quite useful for this question:
[url]http://en.wikipedia.org/wiki/Rocket_equation[/url]

Hint: [SPOILER]The payload mass doesn't matter when calculating delta-v for each stage.[/SPOILER]

Are the stages supposed to be identical (in mass)?

[QUOTE=axn;220849]Are the stages supposed to be identical (in mass)?[/QUOTE]
I suppose that the mass of each stage shall be optimized so that the total mass comes out minimal.

We can get rid of physical units by dividing all physical masses by the mass of the payload. Let's call those dimensionless stage masses m1...m4. The dimensionless payload mass is 1 then.
The overall Delta-v divided by the exhaust-v shall be 10/3.
This must equal the sums of the ln((total_mass_at_begin_of_stage_i)/(total_mass_at_end_of_stage_i)).

Using a Lagrange multiplier lambda for this constraint, you have to minimize
[code]m1+m2+m3+m4+lambda*(
ln((m1+m2+m3+m4+1)/(0.15m1+m2+m3+m4+1))
+ln((m2+m3+m4+1)/(0.15m2+m3+m4+1))
+ln((m3+m4+1)/(0.15m3+m4+1))
+ln((m4+1)/(0.15m4+1))
-10/3)[/code]by varying m1, m2, m3, m4, and lambda.

Is this to be g-force limited? Will it carry people or other sensative cargo vs. a solid conic craft?

[QUOTE=axn;220849]Are the stages supposed to be identical (in mass)?[/QUOTE]No.

[QUOTE=ccorn;220851]I suppose that the mass of each stage shall be optimized so that the total mass comes out minimal.[/quote]Correct.

[QUOTE=ccorn;220851]Using a Lagrange multiplier lambda for this constraint, you have to minimize
[code]m1+m2+m3+m4+lambda*(
ln((m1+m2+m3+m4+1)/(0.15m1+m2+m3+m4+1))
+ln((m2+m3+m4+1)/(0.15m2+m3+m4+1))
+ln((m3+m4+1)/(0.15m3+m4+1))
+ln((m4+1)/(0.15m4+1))
-10/3)[/code]by varying m1, m2, m3, m4, and lambda.[/QUOTE]Hah, I have no idea what a "Lagrange multiplier lambda" is, but I ended up with something similar to what you have in this code block.

[QUOTE=Uncwilly;220856]Is this to be g-force limited? Will it carry people or other sensative cargo vs. a solid conic craft?[/QUOTE]None of this matters, the rocket equation does not take into account thrust or acceleration. The delta-v is the same for a small weedy rocket motor or a big beefy rocket motor (provided the have the same mass ratio and exhaust velocity). This WOULD matter when determining a launch profile, but that's an entire other can of worms.

[QUOTE=lavalamp;220863]Hah, I have no idea what a "Lagrange multiplier lambda" is, but I ended up with something similar to what you have in this code block.[/QUOTE]
Cf. [url]http://en.wikipedia.org/wiki/Lagrange_multiplier[/url]. Used for (local) optimization with several variables that are not independent but must fulfill some equation (the [I]constraint[/I]).

Indeed, currently reading the "Level set" page, since the "Very simple example" turned out not to be simple enough.

I am going to assert, without proof, that the optimal design has each stage providing the same delta-v.

[QUOTE=axn;220866]I am going to assert, without proof, that the optimal design has each stage providing the same delta-v.[/QUOTE]
Without proof, but not without thought? Please share. How do you figure?

[QUOTE=axn;220866]I am going to assert, without proof, that the optimal design has each stage providing the same delta-v.[/QUOTE]I also reached this conclusion, I don't know why it is the case, but it very much seems to be true.

ccorn, try out the same example, but with only two stages, it's much easier to graph and see the values for the minimum mass. Not that plotting 5 dimensional graphs isn't possible, it's just tricky. Or impossible, one of the two anyway.

Now for phase 2 of this puzzle, the generalisation. Suppose instead of the mass ratios and exhaust velocities being equal ... they are not! Ne! This is a much more realistic scenario for a rocket launch.

Can you find a method for detemining the optimum masses/delta-vs of each stage in this case? This part of the puzzle kept me tied up for quite a bit longer than the first part, but then you lot are all smarter than me so maybe it won't be so hard after all. I hope it won't be [url=http://www.mersenneforum.org/showthread.php?t=11281]instantly solved[/url] in any case, always a little disappointing when that happens.

OK, so it could be that everyone is experiencing friday evening/night AWAY from the computer (heathens), or a general apathy for the puzzle, but since there have been no posts in over 24 hours, I'll post the method I used. I've also uploaded some data related to what's in the black below.

[SPOILER]I wrote some code to determine the optimal stage masses for 2, 3 and 4 stage rockets where all stages had the same exhaust velocity, n-1 stages had the same mass ratio, and the final stage had a varying mass ratio. I've attached the data I gleaned from these tests so you don't have to run them (unless you really want to). I recommend at least playing around with some code though, I pretty much discovered the relationship entirely by accident, but then I compared it to all the data and it fit exactly.

There are 4 files in the attached ZIP, an Excel file, and three CSV files for anyone who cannot open the Excel file. There are two columns per data set, the first is the mass ratio for the final stage, the second is the delta-v for that stage divided by 10000, so it is basically the fraction of the total delta-v supplied by that stage. Since the other stages have the same mass ratio and exhaust velocity as each other, they will have equal delta-vs that can easily be calculated. The mass ratio for the other stages is included in the column header.[/SPOILER]