It's not rocket surgery... - Page 2

science_man_88 · 2010-07-10, 01:51

according to the equation the final mass must be about 3.57% of the original mass assuming no payload this has the full stage as about 20.22% of the original mass. assuming this is even close the answer to the question about equal mass per stage is trivially no even to a dummy like me.

never mind I actually read more lol.

first stage = 88-89%of total mass ?

lavalamp · 2010-07-10, 02:55

I don't know what the question is that that is the answer to. But in relation to this part of your post:

Quote:

Originally Posted by science_man_88

equal mass per stage

No, the stages are not equal in mass.

science_man_88 · 2010-07-10, 11:10

Quote:

Originally Posted by lavalamp

I don't know what the question is that that is the answer to. But in relation to this part of your post:No, the stages are not equal in mass.

learn to read i clearly state if what i believe is accurate even to a dummy like me it's trivially no.

and i also state what I believe to be the percentage of the weight to be the first stage.
oh sorry that's the mass of the fuel in the first stage by my guess i realize that now lol so 88-89% total mass is in the fuel of the first stage violating your limit. but if you look at it 10000/3000=ln(100/100-x) which if i did the math correctly means e^(10/3) = (100/100-x)
which makes the denominator between 100-88 and 100-89 which violates your rule.

lavalamp · 2010-07-10, 16:10

OK, some clarification. In the first example I posted, with each stage having a fuel mass 85% and dry mass 15%, that means each stage taken individually.

The minimum mass of the rocket is given when each stage imparts a quarter of the desired delta-v to the remaining rocket stack, 2500 m/s, and these are the masses of each stage:

Code:

stage 1:
dry mass        = 7939.2 kg
propellant mass = 44988.8 kg
total mass      = 52928.0 kg

stage 2:
dry mass        = 2658.2 kg
propellant mass = 15063.2 kg
total mass      = 17721.4 kg

stage 3:
dry mass        = 890.0 kg
propellant mass = 5043.5 kg
total mass      = 5933.5 kg

stage 4:
dry mass        = 298.0 kg
propellant mass = 1688.7 kg
total mass      = 1986.7 kg

complete rocket stack:
payload         = 1000 kg
dry mass        = 12785.4 kg
propellant mass = 66784.1 kg
total mass      = 79569.6 kg

So on launch when the first stage fires, m0 = 79569.6 kg, m1 = 34580.8 kg:

Code:

delta-v = v_e * ln( m0 / m1 )
delta-v = 3000 * ln( 79569.6 / 34580.8 )
delta-v = 3000 * ln( 2.3009763 )
delta-v = 2500 m/s

In the first stage, the propellant accounts for 44988.8 / 52928.0 * 100 = 85% of the mass. For the rocket as a whole, the first stage propellant accounts for 44988.8 / 79569.6 * 100 = 56.54% of the mass.

For the rocket as a whole, the propellant accounts for 66784.1 / 79569.6 * 100 = 83.93% of the mass.

The first stage accounts for 52928.0 / 79569.6 * 100 = 66.52% of the mass of the rocket, ths number isn't really relevant, but you mentioned it in your posts as being 88 to 89% of the mass.

science_man_88 · 2010-07-10, 16:45

Quote:

Originally Posted by lavalamp

OK, some clarification. In the first example I posted, with each stage having a fuel mass 85% and dry mass 15%, that means each stage taken individually.

The minimum mass of the rocket is given when each stage imparts a quarter of the desired delta-v to the remaining rocket stack, 2500 m/s, and these are the masses of each stage:

Code:

stage 1:
dry mass        = 7939.2 kg
propellant mass = 44988.8 kg
total mass      = 52928.0 kg

stage 2:
dry mass        = 2658.2 kg
propellant mass = 15063.2 kg
total mass      = 17721.4 kg

stage 3:
dry mass        = 890.0 kg
propellant mass = 5043.5 kg
total mass      = 5933.5 kg

stage 4:
dry mass        = 298.0 kg
propellant mass = 1688.7 kg
total mass      = 1986.7 kg

complete rocket stack:
payload         = 1000 kg
dry mass        = 12785.4 kg
propellant mass = 66784.1 kg
total mass      = 79569.6 kg

So on launch when the first stage fires, m0 = 79569.6 kg, m1 = 34580.8 kg:

Code:

delta-v = v_e * ln( m0 / m1 )
delta-v = 3000 * ln( 79569.6 / 34580.8 )
delta-v = 3000 * ln( 2.3009763 )
delta-v = 2500 m/s

In the first stage, the propellant accounts for 44988.8 / 52928.0 * 100 = 85% of the mass. For the rocket as a whole, the first stage propellant accounts for 44988.8 / 79569.6 * 100 = 56.54% of the mass.

For the rocket as a whole, the propellant accounts for 66784.1 / 79569.6 * 100 = 83.93% of the mass.

The first stage accounts for 52928.0 / 79569.6 * 100 = 66.52% of the mass of the rocket, ths number isn't really relevant, but you mentioned it in your posts as being 88 to 89% of the mass.

I am an idiot when it comes to this but I finally figured that out the part i was missing was to work backwards once I did that i figured out the ranges for the mass of each so should I even try the second ?

I don't see how you got the masses I know about the percentages a bit care to explain it more ?

lavalamp · 2010-07-10, 17:53

Well now you know you know, so go for it. Here's a big hint.

Look at the delta-v for each stage individually, ignoring payload and other stages. Compare this to the delta-v the stage provides in the actual rocket. The data I uploaded previously will help here.

science_man_88 · 2010-07-10, 18:07

Quote:

Originally Posted by lavalamp

Well now you know you know, so go for it. Here's a big hint.

Look at the delta-v for each stage individually, ignoring payload and other stages. Compare this to the delta-v the stage provides in the actual rocket. The data I uploaded previously will help here.

no I don't and second you realize both m0 and m1 can be scaled down by 16 each at least by my math that would mean it's not the minimum(oh wait you'd need scaled payload(62.5 kg isn't worth it)).

lavalamp · 2010-07-10, 18:22

When finding the masses, start at the top of the rocket, in other words, figure it out from the last stage first. Using the first example again, you know the payload is 1000 kg, you know the stage should impart 2500 m/s, and you know if has a fuel fraction of 85% and an exhaust velocity of 3000 m/s, so use the rocket equation:

Code:

delta-v = v_e * ln(m0 / m1)
2500 = 3000 * ln((m+1000) / (0.15m+1000))
5/6 = ln((m+1000) / (0.15m+1000))
e^(5/6) = (m+1000) / (0.15m+1000)
(0.15m+1000)e^(5/6) = (m+1000)
0.15m*e^(5/6) + 1000e^(5/6) = m + 1000
1000e^(5/6) - 1000 = m - 0.15m*e^(5/6)
2300.976 - 1000 = m - 0.345m
1300.976 = 0.655m
m = 1300.976 / 0.655
m = 1986.7 kg

That gives you the mass of the final stage including fuel. Then you repeat the process to find the mass of the second stage. It looks a bit long winded since I included every step of the process instead of condensing it a little, but it's easy enough to type up in Excel or Pari/gp and have it calculate stage masses automatically when given a delta-v and fuel mass fraction.

The important part is finding the correct values for m0 and m1. m0 is the full mass of the stage PLUS the mass of any stages higher up and the mass of the payload. m1 is the empty mass of the stage plus all that other stuff too. That is why you need to start at the top and work down, because you need to find the masses of the higher stages first and include them in the calculations for the lower stages.

science_man_88 · 2010-07-10, 18:29

Quote:

Originally Posted by lavalamp

When finding the masses, start at the top of the rocket, in other words, figure it out from the last stage first. Using the first example again, you know the payload is 1000 kg, you know the stage should impart 2500 m/s, and you know if has a fuel fraction of 85% and an exhaust velocity of 3000 m/s, so use the rocket equation:

Code:

delta-v = v_e * ln(m0 / m1)
2500 = 3000 * ln((m+1000) / (0.15m+1000))
5/6 = ln((m+1000) / (0.15m+1000))
e^(5/6) = (m+1000) / (0.15m+1000)
(0.15m+1000)e^(5/6) = (m+1000)
0.15m*e^(5/6) + 1000e^(5/6) = m + 1000
1000e^(5/6) - 1000 = m - 0.15m*e^(5/6)
2300.976 - 1000 = m - 0.345m
1300.976 = 0.655m
m = 1300.976 / 0.655
m = 1986.7 kg

That gives you the mass of the final stage including fuel. Then you repeat the process to find the mass of the second stage. It looks a bit long winded since I included every step of the process instead of condensing it a little, but it's easy enough to type up in Excel or Pari/gp and have it calculate stage masses automatically when given a delta-v and fuel mass fraction.

The important part is finding the correct values for m0 and m1. m0 is the full mass of the stage PLUS the mass of any stages higher up and the mass of the payload. m1 is the empty mass of the stage plus all that other stuff too. That is why you need to start at the top and work down, because you need to find the masses of the higher stages first and include them in the calculations for the lower stages.

believe it or not I finally understand lol.

ccorn · 2010-07-11, 05:57

Quote:

Originally Posted by lavalamp

I also reached this conclusion, I don't know why it is the case, but it very much seems to be true.

Rewrite the original problem a bit, then it becomes clear.

Define Mi as the dimensionless total mass at the beginning of stage i:

M1 = m1+m2+m3+m4+1
M2 = m2+m3+m4+1
M3 = m3+m4+1
M4 = m4+1
M5 = 1

Thus, for optimization purposes, you are looking for a stationary point of

M1+lambda*(
ln(M1/(0.15M1+0.85M2))
+ln(M2/(0.15M2+0.85M3))
+ln(M3/(0.15M3+0.85M4))
+ln(M4/(0.15M4+0.85))
-10/3)

with respect to M1...M4, and lambda.

Rewrite further by defining the mass ratios Xi = M(i+1)/Mi and the additional constraint M1 X1 X2 X3 X4 = 1. Abbreviate the core expression for the per-stage delta-v as f(Xi) = -ln(0.15+0.85Xi). Now you are looking for zero partial derivatives of

M1 + lambda * (f(X1)+f(X2)+f(X3)+f(X4)-10/3) + mu * (M1 X1 X2 X3 X4 - 1)

with respect to M1, X1...X4, lambda and mu.
We have two constraints for M1, X1...X4, and therefore there are two corresponding Lagrange multipliers lambda and mu. Note that this to-be-optimized expression is symmetric in X1...X4.

Looking for a zero of the partial derivative with respect to M1 gives
1 + mu X1 X2 X3 X4 = 0,
hence mu = -1 / (X1 X2 X3 X4).

Looking for a zero of the partial derivative with respect to Xi yields
lambda * f'(Xi) + mu M1 X1 X2 X3 X4 / Xi = 0, i.e.
-lambda * 0.85/(0.15+0.85Xi) - M1/Xi = 0, thus
0.85Xi/(0.15+0.85Xi) = -M1/lambda
(assuming nonzero lambda because otherwise its constraint would not be effective)

We could go on further, but you already see that all Xi must have the same value. Consequently, all f(Xi) and thus all delta-v-per-stage must have the same value. This generalizes to any number of stages, as long as exhaust-v and propellant-mass-percentage are the same for each stage.

science_man_88 · 2011-01-11, 15:37

I was going to reopen with a speed of light test but now I can't see how the highlighted steps work out.

Quote:

Originally Posted by lavalamp

Code:

delta-v = v_e * ln(m0 / m1)
2500 = 3000 * ln((m+1000) / (0.15m+1000))
5/6 = ln((m+1000) / (0.15m+1000))
e^(5/6) = (m+1000) / (0.15m+1000)
(0.15m+1000)e^(5/6) = (m+1000)
0.15m*e^(5/6) + 1000e^(5/6) = m + 1000
1000e^(5/6) - 1000 = m - 0.15m*e^(5/6)
2300.976 - 1000 = m - 0.345m
1300.976 = 0.655m
m = 1300.976 / 0.655
m = 1986.7 kg

never mind I figured it out lol

2010-07-10, 01:51	#12
science_man_88 "Forget I exist" Jul 2009 Dumbassville 2⁶×131 Posts	according to the equation the final mass must be about 3.57% of the original mass assuming no payload this has the full stage as about 20.22% of the original mass. assuming this is even close the answer to the question about equal mass per stage is trivially no even to a dummy like me. never mind I actually read more lol. first stage = 88-89%of total mass ? Last fiddled with by science_man_88 on 2010-07-10 at 02:14

2010-07-10, 16:10	#15
lavalamp Oct 2007 Manchester, UK 2²·3·113 Posts	OK, some clarification. In the first example I posted, with each stage having a fuel mass 85% and dry mass 15%, that means each stage taken individually. The minimum mass of the rocket is given when each stage imparts a quarter of the desired delta-v to the remaining rocket stack, 2500 m/s, and these are the masses of each stage: Code: stage 1: dry mass = 7939.2 kg propellant mass = 44988.8 kg total mass = 52928.0 kg stage 2: dry mass = 2658.2 kg propellant mass = 15063.2 kg total mass = 17721.4 kg stage 3: dry mass = 890.0 kg propellant mass = 5043.5 kg total mass = 5933.5 kg stage 4: dry mass = 298.0 kg propellant mass = 1688.7 kg total mass = 1986.7 kg complete rocket stack: payload = 1000 kg dry mass = 12785.4 kg propellant mass = 66784.1 kg total mass = 79569.6 kg So on launch when the first stage fires, m0 = 79569.6 kg, m1 = 34580.8 kg: Code: delta-v = v_e * ln( m0 / m1 ) delta-v = 3000 * ln( 79569.6 / 34580.8 ) delta-v = 3000 * ln( 2.3009763 ) delta-v = 2500 m/s In the first stage, the propellant accounts for 44988.8 / 52928.0 * 100 = 85% of the mass. For the rocket as a whole, the first stage propellant accounts for 44988.8 / 79569.6 * 100 = 56.54% of the mass. For the rocket as a whole, the propellant accounts for 66784.1 / 79569.6 * 100 = 83.93% of the mass. The first stage accounts for 52928.0 / 79569.6 * 100 = 66.52% of the mass of the rocket, ths number isn't really relevant, but you mentioned it in your posts as being 88 to 89% of the mass.

2010-07-10, 18:22	#19
lavalamp Oct 2007 Manchester, UK 2²·3·113 Posts	When finding the masses, start at the top of the rocket, in other words, figure it out from the last stage first. Using the first example again, you know the payload is 1000 kg, you know the stage should impart 2500 m/s, and you know if has a fuel fraction of 85% and an exhaust velocity of 3000 m/s, so use the rocket equation: Code: delta-v = v_e * ln(m0 / m1) 2500 = 3000 * ln((m+1000) / (0.15m+1000)) 5/6 = ln((m+1000) / (0.15m+1000)) e^(5/6) = (m+1000) / (0.15m+1000) (0.15m+1000)e^(5/6) = (m+1000) 0.15me^(5/6) + 1000e^(5/6) = m + 1000 1000e^(5/6) - 1000 = m - 0.15me^(5/6) 2300.976 - 1000 = m - 0.345m 1300.976 = 0.655m m = 1300.976 / 0.655 m = 1986.7 kg That gives you the mass of the final stage including fuel. Then you repeat the process to find the mass of the second stage. It looks a bit long winded since I included every step of the process instead of condensing it a little, but it's easy enough to type up in Excel or Pari/gp and have it calculate stage masses automatically when given a delta-v and fuel mass fraction. The important part is finding the correct values for m0 and m1. m0 is the full mass of the stage PLUS the mass of any stages higher up and the mass of the payload. m1 is the empty mass of the stage plus all that other stuff too. That is why you need to start at the top and work down, because you need to find the masses of the higher stages first and include them in the calculations for the lower stages.

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2010-07-10, 17:53	#17
lavalamp Oct 2007 Manchester, UK 2²·3·113 Posts	Well now you know you know, so go for it. Here's a big hint. Look at the delta-v for each stage individually, ignoring payload and other stages. Compare this to the delta-v the stage provides in the actual rocket. The data I uploaded previously will help here.