Staging Optimally: wanna help me with math?

[Reusables]

1 hour ago, FancyMouse said:

For your theorem it is easy if you know topology - continuous function maps compact set to compact set. And compact set on real line is bounded and closed, so max and min exists.

EVT is trivial as Well, given that one know relevant concepts pretty well.

I think understanding these concepts is hard.

1 hour ago, FancyMouse said:

OTOH this has nothing to do with the original question - "shouldn't have local non-global maxima" is literally equivalent to "global maxima that is not on a boundary, is a local maximum", and it's trivial just based on the property that it's a "global" maximum.

I posted the theorem without thinking. 

Btw, local non-global maxima sounds like 'a local maximum which is not global maximum'.

[Cunjo Carl]

@FancyMouse Er, I believe @Abastro was actually referring to my n-variable unbounded case not having a bunch of local maxima is hard to prove. Edit: Missed Abastro's reply, n/m!

Fortunately, it seems to have only one max with some beautiful tapering near it in all directions. Solver central!

Hey! Speaking of which, the new spreadsheet optimizer is up. Link

I'm planning on beautifying and triple checking it Sunday (perhaps Sat) before posting it.... in discussion, I guess? I was thinking to link back here, and also link to prominently your solver when it's ready @IncongruousGoat if you'd like?

Edited May 20, 2017 by Cunjo Carl

[IncongruousGoat]

17 hours ago, Cunjo Carl said:

I was thinking to link back here, and also link to prominently your solver when it's ready @IncongruousGoat if you'd like?

Sure, sounds good to me. In news of the solver, things are going well, although there are still one or two things to work out. In particular, the math seems to go squirrelly where Nervs are concerned. I should have it all working and with a proper release tomorrow.

[GoSlash27]

This whole "goodness ratio" concept has me thinking. Perhaps I can use it to solve for more generalized cases in a different way. Thanks for taking the time to chase this approach!

Best,
-Slashy

Edited May 22, 2017 by GoSlash27

[Reusables]

I tried to find the optimum for the general case, but I got kinda lost with what kind of restriction I should put on.

Should I restrict engines by mass? Also should I optimize mass ratio for dv, or dv for mass ratio? What kind of conditions users would need?

This is hard..

[GoSlash27]

Abastro,

 The way I see it, users will need to use DV as an input and mass ratio as an output. I think most cases will be modeled at 0.5g minimum acceleration, although this value would need to be adjustable.

Best,
-Slashy

 

[GoSlash27]

I found some time to play with the math today. After several false starts and head- scratching moments, I came up with this...

Wet-to-dry ratio= Rwd = e^(DV/9.81Isp)

Fraction of rocket that is neither fuel nor tankage= (Rft+1-Rwd)/RftRwd where

Rft = the ratio of the mass of the fuel in a tank to the mass of the tank itself. 8 for LF&O tanks.

Fraction of the rocket that is engine = 9.81RtwMeng/T where

Rtw is thrust to weight ratio
Meng is engine mass in tonnes
T is engine thrust in kiloNewtons.

It stands to reason that the payload fraction is left when you subtract the engine fraction from the fraction that is neither fuel nor tanks, so

Fp= [(Rft+1-Rwd)/RftRwd]-[9.81RtwMeng/T]

 

I then plugged this into a spreadsheet and multiplied by DV to give a "goodness" figure of merit, and it popped out a good looking curve with a local maxima and a payload fraction that hits zero at the DV limit of the engine.

Finding the DV where the "goodness" is equal to that at 1/2 the DV yields the staging point, while the maximum occurs at the ideal DV for that engine and t/w. For my plot of a Terrier at 0.5g it hit a maxima at 2,400 m/sec DV and the staging point was around 3,200 m/sec.

HTHs,
-Slashy

PS I left my paperwork and spreadsheet at work, so I'm writing all this from memory. I may have something incorrect here. I'll verify it tomorrow.

 

 

Edited May 24, 2017 by GoSlash27

[GoSlash27]

I ran a few comparison charts using the above method. I think they're pretty interesting, so I'll share them here.
Wall of pics

Spoiler

 

Here's the Terrier vs. the NERV when used as an interplanetary stage. The Terrier is actually the lighter choice for trips under 1,500 m/sec, while the NERV is lighter above that. The tic marks represent the optimal operating range for a single stage. The NERV should be staged for trips above 5,000 m/sec, while the Terrier should be staged for trips above 3,200 m/sec.

The higher t/w demands of an upper launch stage severely hamper the utility of the LV-N. Here the Terrier is lighter for stages that are 2,600 m/sec or less. Since the typical DV budget for an upper stage to orbit is only 1,600 m/sec, the LV-N is not well- suited for this job.

The LV-N vs. Dawn as an interplanetary stage. Even the 0.5g t/w of a typical interplanetary stage is really too much for the Dawn, and this doesn't count the additional mass required to generate power. The Dawn starts to perform well at 0.35g or below.

Poodle vs. Aerospike. The moral of this story is to not worry about small differences.

A specialized vacuum motor vs. non- specialized. The vacuum motor is superior, but not by all that much.

 

 

I made a list of the staging points for all motors, but left them at work because I'm addled. Pretty much all of the LF&O engines were in the 2800-3400 m/sec range.

Best,
-Slashy

[GoSlash27]

Answers for generalized cases using the above method:

5km/ sec is the staging point for the LV-N at .5g minimum acceleration. This is where a 2 stage rocket will weigh the same as a single stage.

3.4km/sec Poodle, Rhino
3.2km/sec Terrier, Aerospike, Skipper
3.0km/sec Ant, Swivel, Twin-Boar, Vector
2.8km/sec Spark, Reliant.

Best,
-Slashy

 

 

[Cunjo Carl]

@GoSlash27

Ooh, it's cool to see your approach rederived from the base principals! I'm plugging through it at the moment. I like that you're using the term "goodness" for your figure of merit, that term has always made me happy somehow. One thing I'll suggest is to use is a goodness factor in terms of deltaV times 1/ln(1/payloadFraction) rather than times normal linear payloadFraction. The place it becomes important is when saying your results for a stage should also apply to that stage in a rocket.

I hand-waved right across the distinction in my analysis for the original question, but for a few days before posting I struggled with the concept, having gotten some very promising looking answers that didn't stand up to reality because I also used a linear payload fraction in the goodness factor. So what's different between the two options? Why would we want to use one vs the other? I think it's easiest to describe through case-and-point.

... Let's say we have a stage with a given goodness factor (from its given TWR, engine, payload fraction, and deltaV/stage) . If we build a rocket using two stages of this configuration, that rocket should have the same goodness factor as the individual stages. This is what lets us say that the goodness of a stage will make goodness for our rocket, which is the end goal. Otherwise we can't compare the two. So let's look at our options for goodness and see how they fit this description.

 And just for visual simplicity, let's rewrite the logarithmic goodness factor:    deltaV*1/ln(1/pyldFrac)  = -deltaV/ln(pyldFrac)                    

characteristic                     1 stage     2 stage rocket
  deltaV                                   X                  2X
  pyldFrac                               Y                   Y²
  deltaV*pyldFrac                  X*Y               2X*Y²                               ... The goodness changes when using the same stage twice, that can't be right.
  -deltaV/ln(pyldFrac)         -X/ln(Y)         -2X/ln(Y²)  =  -X/ln(Y)           Success! The goodness stays the same.

 

So the realworld interpretation of this logarythmic goodness factor is that it's a number that when multiplied by ln(RocketWetMass/FinalPayloadMass) will give you the rocket's deltaV... I've been calling it DeltaV per e-scale. Given the logarythmic nature of the rocket equation, it kinda feels intuitive that the masses should be tucked in a log... well, intuitive after a bit of hand waving perhaps .

I hope that makes sense and doesn't muddy things. I'm happy to talk further, and I'm looking forward to seeing what you come up with next!

Also, I like that you're comparing LV-N, Dawn and LV-909. It's the classic feud! (Well, the classic feud after they nerfed the 48-7S spark at any rate )

 

 

On 5/22/2017 at 6:38 PM, Abastro said:

I tried to find the optimum for the general case, but I got kinda lost with what kind of restriction I should put on.

Should I restrict engines by mass? Also should I optimize mass ratio for dv, or dv for mass ratio? What kind of conditions users would need?

This is hard..

 

I've been stuck at the same point for about 2 weeks! I have a couple heuristic restrictions I've been tinkering with, and I'll pass them along if they come to anything.

Edited May 30, 2017 by Cunjo Carl

[Cunjo Carl]

Somehow this posted before it was ready. Avert thine eyes!

Edit:

The topic's fairly different, so doublepost! I did some thinking and worked together an equation exclusively for sizing drop tanks. The idea is we have a stage that starts at TWRmin, burns for a while, drops a tank and then finishes its burns. In this way, the drymass of the tank doesn't hold you back, and you can coax a tiny bit more tasty deltaV out of whatever mass you've got at your disposal. The question is, how large should this droptank be relative to the stage it fuels. If it's too small, it's just not worthwhile, but if it's too big then it drops just before the next stage and there's not enough time to enjoy it being gone! The answer should depend on the mass of the engines, which I happen to handle as the min TWR requirement of the craft divided by the TWR of the engine (like from KSP wiki).

The easiest way to handle it is to consider mainstage and droptank as two separate stages: first the drop tank, then the main stage. The drop tank stage gets to include its engine into its 'payload' mass fraction, because you get to keep on using it. Then the follow-up mainstage needs to size its engines to be large enough to start the droptank stage with enough TWRmin. It works out to:

p = m_{mainStage /} m_full = payload fraction of droptank 'stage'
q = m_{payload /} m_mainStage =  payload fraction of main 'stage'
 

            ΔV_craft                                        ln(¹/₉ + ⁸/_{9 *} p)   +   ln(¹/₉ + ⁸/₉*(1/p)*TWR_min/TWR_eng + ⁸/_{9 *} q)            
                                         =     g_{0 *} I_{sp *}                                                                                                                                                                                                           
      ln(m_full/m_payload)                                                                     ln(p*q)

(or its equivalent in e-scale, which I've been using!)

So I discovered on paper what I'd been finding in practice, that you only really get like 3-5% boost to your deltaV when using drop tanks like this at 1g-ish TWR, and that droptanks are best sized to be larger (heavier) than the mainstage when TWR_min/TWR_eng is greater than about .11, and smaller when less. The optimum is quite broad, so there's no real reason to get picky about sizing them precisely. I'm happy to send derivations or spreadsheets if folk would find them useful. Have fun!

Edited May 29, 2017 by Cunjo Carl

[Wcmille]

On ‎5‎/‎15‎/‎2017 at 4:13 AM, Cunjo Carl said:

Maybe I finally got it? It might all be correct after all, and I was just looking for a decay-to-zero that was way farther out there than I expected. I just realized that if we're plotting efficiency relative to a logarithmic mass ratio, the value actually shouldn't decline very quickly. We should only be able to see the decline on an exponential scale, so I replotted in terms of one, and got something I'd really like to believe!
 

  Hide contents

 

define x = ln(m_full /m_payload) so that x is the log scale of each stage in our rocket 

   (m_full /m_payload) = e^x     We can rearrange for a more convenient form and plug it in to our equation from part 4.

 

            ΔV_craft                         - g_{0 *} I_{sp *}  ln(¹/₉ + ⁸/₉*TWR_min/TWR_eng + ⁸/_{9 *} e^-x )   
                                         =                                                                                                                                                                   
      ln(m_craft/m_pod)                                                    x

Y axis: DeltaV for each e-fold scale of the craft in m/s  -> ΔV_craft / ln(m_craft/m_pod)    ...  X axis: Relative size of each stage to the one it pushes on an e-fold scale -> ln(m_full /m_payload) or, newly coined, x.

From KER, the asymptotic limit on deltaV in KSP starts showing up at a stage-to-stage mass ratio of 7x (2 e-fold), it becomes very apparent by a stage-to-stage mass ratio of 20x (3 e-fold), and it really hits home at a stage-to-stage mass ratio of 50x (4 e-fold) ... These values seem to match.

 

 

It seems to be working out, but I'd still love a second set of eyes on all this. If the urge to take on some math strikes anyone, some corroboration (or rebuttal) would be hugely appreciated.. Otherwise I'll probably mark it as solved in a few days.

What is meant by "e-fold"?

[Cunjo Carl]

On 12/31/2017 at 5:44 PM, Wcmille said:

What is meant by "e-fold"?

  e-folds (I should have been using the plural) is kinda a funny one. It's the imaginary units you would get out of a natural log, and only really gets used when ReallyImportantThing= ln(A/B) and you're talking about A and B. If you happen to be familiar with the concept of orders-of-magnitude, e-folds are exactly the same thing just using base e rather than base 10. If you happen not to be, I think it's best to explain it through example.

Saying A is X-fold bigger than B means that:
             A is X times B

so 440 is 2-fold bigger than 220          , and
      880 is 2 2-folds bigger than 220!    Why would we do that???

Well, 2-folds happens to be very important in music, where it's called an 'octave'. Returning to our example: 440Hz (middle A) is an octave higher than 220Hz (low A), and 880Hz (high A) is 2 octaves higher than 220Hz (low A). The concept of 2-fold lets us connect it together! While the numbers are much messier, we can use the concept of e-fold to connect wet mass and dry mass to the deltaV of our rocket. From the rocket equation deltaV = g₀*Isp*ln(m_wet/m_dry) so...

if m_wet is e-fold bigger than m_dry           our craft will have a deltaV of      g₀*Isp
if m_wet is 2 e-folds bigger than m_dry      our craft will have a deltaV of 2 * g₀*Isp

Telling us how many e-folds bigger m_wet is than m_dry is exactly what ln(m_wet/m_dry) does, so that's why we use it! This is why

On 12/31/2017 at 5:44 PM, Wcmille said:

DeltaV for each e-fold scale of the craft in m/s  ->   ΔV_craft / ln(m_craft/m_pod)

 

I hope that helps! Let me know if you have follow up questions.