Maximizing payload ratio

[Nich]

In college a professor showed us a way to maximize dv for a payload using ratios but I can for the life of me find it on the internet or derive it my self.  

Given a Payload of P then the dv for a 3 stage rocket would be

dv0 = 9.8*ISP0 * log(m0/m1)

dv1 = 9.8*ISP1*log(m1.1/m2)

dv2 = 9.8*ISP2*log(m2.1/m3)

Unlike in KSP mass of a real decoupler would be dependent on the mass of the payload (or next stage) and Final TWR of the previous stage

thus

m1 = m1.1 + decoupler mass

m2 = m2.1 + decoupler mass

Thus

dv0 = 9.8*ISP0 * log((m0+ decoupler1+ decoupler2)/(m1.1 + decoupler1+ decoupler2))

dv1 = 9.8*ISP1*log((m1.1+ decoupler2)/(m2.1 + decoupler2))

dv2 = 9.8*ISP2*log(m2.1/m3)

Given the fact that all tanks in KSP have the same fuel/mass ratio of 9:1 we know

ft/et = 9/1 or ft = 9et

m3 = payload + engine mass + empty tanks or p + em2 +et2

m2.1 = payload + engine mass + full tanks or p + em2 + ft2

m2.1/m3 = (p+em2+9et2)/(p+em2+et2)

m1.1/m2.1 = (p+em2+9et2+em1+9et1)/(p+em2+9et2+em1+et1)

m0/m1.1 = (p+em2+9et2+em1+9et1+em0+9et0)/(p+em2+9et2+em1+9et1+em0+et0)

Thus

Total dv = 9.8*ISP0 * log((p+em2+9et2+em1+9et1+em0+9et0+ decoupler1+ decoupler2)/((p+em2+9et2+em1+9et1+em0+et0+ decoupler2 + decoupler1))

               + 9.8*ISP1*log((p+em2+9et2+em1+9et1+ decoupler2)/((p+em2+9et2+em1+et1) + decoupler2)) 

               + 9.8*ISP2*log((p+em2+9et2)/(p+em2+et2))

ISP0, 1, 2 are known

p is known

em0, 1, 2 are known

decoupler 1, and 2 are known

The only variables are e0, e1, and e2 so if we differentiate these and find the position that slope = 0 will have a max or min.  The problem is it has been 10 years since I took calc 3 and I have forgotten how to differentiate a 3 variable problem.  Also this will give us the absolute maximum dv possible how would we constrain the problem to give us the min weight with a specified min dv?  Would you simply add et0+et1+et2 = some arbitrary number?

Edited March 15, 2016 by Nich

[Nich]

you would probably want to add in constraints for min TWR

engine thrust 0/ (p+em2+9et2+em1+9et1+em0+9et0 + decoupler1 + decoupler2 )*9.81 = 1.7

engine thrust 1/ (p+em2+9et2+em1+9et1 + decoupler2 )*9.81= .8

engine thrust 2/ (p+em2+9et2) 9.81= .3

Edited March 15, 2016 by Nich

[FancyMouse]

Differentiation is not a problem - just for each variable, take its partial derivative (view the current variable as variable and others as constants). Then solve the systems of equations of 3 partial derivatives equaling zero - the unfortunate part is most likely it's not easily solvable. If that is the case, you're probably better off by just numerically solving it using some sort of software.

[surge]

I suck at maths, but this looks like something I found at:

http://disciplinas.stoa.usp.br/pluginfile.php/66104/mod_resource/content/1/OrbitalMechanicsForEngineeringStudents-AerospaceEngineering.pdf

Note: Scientific information is *supposed* to be public domain, but this looks like an overpriced student textbook from the 1980s, so it might be horribly illegal to look at, or even post about. I found it via google, during a search for some kOS scripting I was doing, so go talk to google or that brazillian uni if you have problems with it...

If people lose their sh.it, and it gets taken down, they relied on something called "Lagrange multiplier" (put it in google, maybe? Oh, the irony!) to find the maximum the stage +all other stages above it could support and used a bunch of other maths I don't understand to find the convergence point. Dunno if that would help.

Like I said, I suck at maths, sorry. If I didn't, I'd be playing Orbiter instead... well, maybe. Kerbal shenanigans are hilarious, and the physics are close enough that it's worth a bit of money.

Edited April 9, 2017 by surge

[Nich]

Ha ha I used to own that book until it was stolen