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Old 08-24-2013, 06:29 PM   #21
Anthony
 
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Okay, some simplified math:

To stay up with a rocket, you need a vertical force equal to gravity (actually, gravity * (1-(V/Vorbit))^2, but at launch that's gravity). The rest of your force is applied as a horizontal effect. However, these are perpendicular forces, and we're dealing with vector addition, so the total thrust is the hypotenuse of a right triangle, and your horizontal acceleration is sqrt( A^2 - G^2 ), which means a 2g ship has 1.73g horizontal acceleration (86.6% efficiency), a 3g ship has 2.83g horizontal (94.3% efficiency). Including the reduction in lift requirements for higher velocity, extra delta-V requirements for orbit are about 18% for 150% of local gravity, 8% for 200% of local gravity, and 4% for 300% of local gravity. In addition to this, you need to deal with the atmosphere, which is usually handled by going straight up and is more complicated to solve; usual paths just get out of atmosphere as fast as practical.

With wings, on the other hand, you have to stay in atmosphere, but you can use the wings to generate lift, instead of using your rockets. Doing this produces drag, with two parts, one which gets larger as velocity increases, one which gets smaller. This means you have an ideal velocity at which drag is minimized; this velocity is inversely proportional to the square root of atmospheric density, with considerable modifications for mach number. An airplane's lift/drag ratio is the ratio of its weight to its drag. Subsonic aircraft can achieve L/D ratios of better than 20, supersonic aircraft usually are less than 10, and hypersonic airframes are often less than 5 (e.g. 4.5 for the space shuttle at subsonic speeds, 1 for the space shuttle at hypersonic speeds). While drag is much smaller than gravity in most cases, it's also horizontal rather than vertical and thus subtracts directly from acceleration; for a hypersonic airframe with a best L/D ratio of 5, a 1G ship needs 25% extra delta-V, a 0.5G ship needs 67% extra delta-V, and a 0.2G ship can't reach orbit at all.
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Last edited by Anthony; 08-25-2013 at 12:50 AM. Reason: remove dangling sentence
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Old 08-24-2013, 07:06 PM   #22
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by Anthony View Post
Okay, some simplified math:

-snip-
Thank you, good sir!
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Old 08-30-2013, 08:27 PM   #23
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

If you write out a simple formula that I can use with any kind of aerospace designs, I'll put it in VEHICLE DESIGN spaceship rules and stick it into SPACESHIPS next time I have a chance to modify the rules.

For Vehicle Design, the aircraft's weight, drag area (which would be the same as coefficient of drag x wing area in a real vehicle) and its lift area (= wing plus lifting body area) are known. If you suggest how L/D could be derived from that and plugged into an equation, feel free.

If it's too complex to derive numbers from gamable data, then I'll have to stick with the simplifications.

The "0 for wings" is indeed a simplification; I got it from the original GURPS Space books, and just ported over. I don't remember Anthony or anyone else suggesting an alternative simplification during the SPACESHIPS playtest, but there were a lot of posts so it may have slipped trough the cracks.
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Old 08-30-2013, 09:06 PM   #24
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by David L Pulver View Post
For Vehicle Design, the aircraft's weight, drag area (which would be the same as coefficient of drag x wing area in a real vehicle) and its lift area (= wing plus lifting body area) are known. If you suggest how L/D could be derived from that and plugged into an equation, feel free.
This is far beyond my knowledge level. :/ Looks like Anthony may know though. *crosses fingers*

Quote:
Originally Posted by David L Pulver View Post
If it's too complex to derive numbers from gamable data, then I'll have to stick with the simplifications.
I think a small table showing how (low) vehicle thrust adds to required delta-V for orbital launch would be a strong addition to Spaceships. This would be similar to how the rocket equation is handled with a table under the Fuel Tank entry.

For aerospace designs you might need two axises (thrust/local gravity and L/D) but if you simplify by assuming near optimal L/D then you could fold the two tables together (with a note indicating that spacecraft below a certain thrust need wings).

Quote:
Originally Posted by David L Pulver View Post
The "0 for wings" is indeed a simplification; I got it from the original GURPS Space books, and just ported over. I don't remember Anthony or anyone else suggesting an alternative simplification during the SPACESHIPS playtest, but there were a lot of posts so it may have slipped trough the cracks.
I don't blame you (or anyone else) for leaving complex gravity drag and air resistance formulas out of Spaceships. Even Atomic Rockets barely touches on it.
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Old 08-30-2013, 10:49 PM   #25
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by David L Pulver View Post
For Vehicle Design, the aircraft's weight, drag area (which would be the same as coefficient of drag x wing area in a real vehicle) and its lift area (= wing plus lifting body area) are known. If you suggest how L/D could be derived from that and plugged into an equation, feel free.
Well, it's fairly straightforward to do from the equations for lift-induced drag and parasitic drag; some elementary calculus will demonstrate that total drag is minimized when both values are equal. If you know (Di / Dp) for some fixed velocity, you can also multiply that velocity by (Di/Dp)^1/4 to determine ideal velocity (this math only works where Cd is constant, so it won't work for transsonic or supersonic craft).

In practice, it's easiest to just use lookup or an average value (0.2 or more for anything actually capable of atmospheric hypersonic flight, as little as 0.02 for some gliders), though it's not too hard to accumulate constants to make those drag functions look easier (it turns out that you don't even need to know wing area -- aspect ratio is defined as wingspan^2 / wing area, so wing area * aspect ratio = wingspan^2).
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Last edited by Anthony; 08-30-2013 at 10:53 PM.
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Old 08-31-2013, 12:04 AM   #26
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

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Originally Posted by MatthewVilter View Post
I think a small table showing how (low) vehicle thrust adds to required delta-V for orbital launch would be a strong addition to Spaceships. This would be similar to how the rocket equation is handled with a table under the Fuel Tank entry.

For aerospace designs you might need two axises (thrust/local gravity and L/D) but if you simplify by assuming near optimal L/D then you could fold the two tables together (with a note indicating that spacecraft below a certain thrust need wings).
No idea what you're saying. Had trouble parsing Anthony's earlier formula; if the values in a formula aren't defined with it, I get lost.. (A^2? But what was A?) Anyway, based on the numbers suggested it looked like a suggestion was:

"If acceleration is less than about 4x local G (g on the table below), multiply delta-V required to reach orbit as follows:

Accel No Wings
0.2 x g ... no lift off!
0.5 x g ... no lift off!
1.0 x g ... ???
1.5 x g ... x1.18 delta V
2.0 x g ... x1.08 delta V
3.0 x g ... x1.04 delta V
4.0 x g ... x1 delta V

Accel. ... With Wings
0.2 x g ... no lift off
0.5 x g ... x1.67 delta V
1.0 x g ... x1.25 delta V
1.5 x g ...???
2.0 x g ... ???
3.0 x g ... ???
4.0 x g ... x1

(it falls between use the higher/lower/nearest/goldfish>

example: Local gravity is 1 G. Acceleration is 1.5G. Delta-V required to lift off is therefore multiplied by x1.18 if you have
no wings or by an amount I'm not yet sure of if you have wings.

Though as I couldn't parse the formula, I'm not sure how that was derived.
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Last edited by David L Pulver; 08-31-2013 at 12:13 AM.
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Old 08-31-2013, 12:24 AM   #27
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by Anthony View Post
Well, it's fairly straightforward to do from the equations for lift-induced drag and parasitic drag; some elementary calculus will demonstrate that total drag is minimized when both values are equal. If you know (Di / Dp) for some fixed velocity, you can also multiply that velocity by (Di/Dp)^1/4 to determine ideal velocity (this math only works where Cd is constant, so it won't work for transsonic or supersonic craft).

In practice, it's easiest to just use lookup or an average value (0.2 or more for anything actually capable of atmospheric hypersonic flight, as little as 0.02 for some gliders), though it's not too hard to accumulate constants to make those drag functions look easier (it turns out that you don't even need to know wing area -- aspect ratio is defined as wingspan^2 / wing area, so wing area * aspect ratio = wingspan^2).
Quick question: anyway to get a short cut to finding a L/d ratio if I already known an aircraft's stall speed? Calculating induced drag, parasitic drag, wing efficiency and so probably too much work...
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Old 08-31-2013, 12:40 AM   #28
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by David L Pulver View Post
"If acceleration is less than about 4x local G (g on the table below), multiply delta-V required to reach orbit as follows:

Accel No Wings
0.2 x g ... no lift off!
0.5 x g ... no lift off!
1.0 x g ... ???
1.5 x g ... x1.18 delta V
2.0 x g ... x1.08 delta V
3.0 x g ... x1.04 delta V
4.0 x g ... x1 delta V

Accel. ... With Wings
0.2 x g ... no lift off
0.5 x g ... x1.67 delta V
1.0 x g ... x1.25 delta V
1.5 x g ...???
2.0 x g ... ???
3.0 x g ... ???
4.0 x g ... x1

(it falls between use the higher/lower/nearest/goldfish>

example: Local gravity is 1 G. Acceleration is 1.5G. Delta-V required to lift off is therefore multiplied by x1.18 if you have
no wings or by an amount I'm not yet sure of if you have wings.
Yes, that is exactly what I mean!
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Old 08-31-2013, 12:43 AM   #29
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by David L Pulver View Post
Quick question: anyway to get a short cut to finding a L/d ratio if I already known an aircraft's stall speed? Calculating induced drag, parasitic drag, wing efficiency and so probably too much work...
How do you even know the stall speed? I once asked how to calculate stall speed for Spaceships, and IIRC the answer was fuzzy. (VE2e was another matter.)
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Old 08-31-2013, 01:06 AM   #30
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Default Re: EuroSpace E950 Minerva Executive Transatmospheric Vehicle

Quote:
Originally Posted by David L Pulver View Post
Quick question: anyway to get a short cut to finding a L/d ratio if I already known an aircraft's stall speed?
I think you can do it based on information about wing shape and drag coefficient. Lemme do some algebra:
Di = L^2/(0.5*rho*V^2*S*pi*e*AR)
Di = L^2/(0.5*rho*V^2*wingspan^2*pi*e)
Dp = 0.5*rho*V^2*Cd*A.

Setting Dp = Di and solving for V:
V^4 = L^2/(0.25*rho^2*wingspan^2*pi*e*Cd*A)
V^2 = L/(0.5*rho*wingspan*sqrt(pi*e*Cd*A))
Dp = 0.5 * rho * L * Cd * A/ (0.5 * rho * wingspan * sqrt( pi*e*Cd*A ) )
Canceling out duplicate terms, we get:
Dp = L * sqrt(Cd*A/(pi*e)) / Wingspan.
Di = Dp, so drag = 2 * L * sqrt(Cd*A/(pi*e)) / Wingspan. Rearranging:
L/D = 0.5 * Wingspan * sqrt( (pi*e) / (Cd*A) )

Since Cd*A for any given design tends to scale as Ws^2 * Form Factor, that works out to L/D = 0.5 * sqrt( (pi*e) / Form Factor ), meaning you can just list standard values based on airframe type (with adjustments for mach number).

Note that it would be fairly simple to produce a table of expected values for spacecraft; it would take some research on supersonic drag coefficients, but then you just come up with a table, and note that air density can be treated as adjusting SM: a SM +6 craft at 1/10 atmospheric density flies just like a SM +12 craft at normal atmospheric density.
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Last edited by Anthony; 08-31-2013 at 01:35 AM.
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