Books for aspiring rocket scientists?
Ian Stirling
of a multi-stage very small launch vehicle. (Payload under 5Kg)
Can anyone suggest other books than
Suttons "Rocket propulsion elements", or "Modern engineering design
of liquid-propellant rocket engines"
I'm interested in:
Construction of 10-1000+N bipropellant peroxide/fuel regeneratively
cooled engines, as well as possibly radiation cooled, for the smaller sizes.
Aerodynamics of a fairly simple (cylinder+nosecone, and little more)
as it goes through the atmosphere, drag, torques at various AOA.
Something that also covered parafoils would be a bonus, though unlikely.
High temperature engineering, refractory metals, carbon-carbon composite etc.
How much of this is covered in sutton, and "engineering design"?.
Are there better books or resources available, or ones covering these areas.
And lastly, does anyone have any older editions, that they might want to
part with, if they would be usefull, and are no longer wanted?
Thanks.
--
http://inquisitor.i.am/ | mailto:inqui...@i.am | Ian Stirling.
---------------------------+-------------------------+--------------------------
"Looks like his brainwaves crash a little short of the beach..." - Duckman.
Henry Spencer
Ian Stirling <Inqui...@I.am> wrote:
>Can anyone suggest other books than
>Suttons "Rocket propulsion elements", or "Modern engineering design
>of liquid-propellant rocket engines"
Books on these topics tend to be scarce, and often old and hard to find.
For engine design:
+ Sutton is the right place to start. The 6th edition, still in print, is
probably the best. The 5th is junk. The 4th is second best, because
while it pre-dates various recent work, and also the switch to modern
(metric) units, it also pre-dates the dumbing-down which produced the 5th
edition. The 3rd is of some interest, same reasons. The 2nd and 1st are
too old, historical curiosities only.
+ Huzel&Huang is definitely the next stop. The AIAA edition, "Modern
engineering for design of liquid-propellant rocket engines", is only
lightly revised from the original NASA SP-125... but there is just enough
new material there that it is nevertheless preferable, especially since
it's in print and the original is hard to find.
+ The third stop is various NASA SPs in the 8000 series. They are mostly
now available on-line, which is good because they're unfindable otherwise.
+ A useful supplement is Hill&Peterson, "Mechanics and thermodynamics of
propulsion", 2nd ed 1992. Better coverage of some areas of theory than
Sutton. but watch out for unit-conversion errors in some of the engine
specs given as examples. Still in print last time I checked, but that
was a while back.
For launcher design in general:
+ Wood, "Aerospace vehicle design, vol. II: spacecraft design", 1964 is,
despite the title, almost entirely a launcher book, and the only real
tutorial overview of launcher design. Long out of print and quite scarce.
(Beware, vol. I is an aircraft-design book; it was a standard text in its
time and is okay, but it has nothing to do with launchers, so know what
you're buying.)
+ Seifert&Brown eds., "Ballistic missile and space vehicle systems", 1961,
is of uneven quality -- in particular, the chapters on liquid propulsion
are nearly useless -- but some of it is excellent, e.g. a chapter on
predicting ascent losses. Long out of print, but findable.
+ Leondes&Vance eds., "Lunar missions and explorations", 1964, is not
primarily a launcher book but has a long and excellent chapter on launcher
design, and a somewhat shorter one on launch facilities. Somewhat biased
toward very large launchers but still very much worth having. Long out
of print, but findable.
+ Koelle ed., "Handbook of astronautical engineering", 1961, again is not
launcher-specific, but has much relevant material, terse summaries and
collections of tables, graphs, and reference material not found elsewhere.
Long out of print, findable but you may not like the price.
+ Isakowitz, Hopkins, & Hopkins eds., "Space launch systems", 3rd ed.,
1999. Reference book on existing launchers, but there's quite a bit of
interesting technical material, and knowing your competition is useful.
In print, from AIAA. 3rd ed. is much better than earlier editions.
There are some other books of interest for specific topics, but that's
a first memory dump on the subject...
Oh, and one un-recommendation: there's a relevant-looking new book by
Hammond from AIAA, I forget the exact title. Don't waste your money,
it's worthless junk.
>And lastly, does anyone have any older editions, that they might want to
>part with, if they would be usefull, and are no longer wanted?
You'll have a hard time finding anyone who's still interested in the field
who's willing to part with any of these... However, Internet used-book
stores like Bibliofind and Alibris are your friends. :-) You may have to
lurk in wait for a while to get things at the prices you want, though.
--
Microsoft shouldn't be broken up. | Henry Spencer he...@spsystems.net
It should be shut down. -- Phil Agre | (aka he...@zoo.toronto.edu)
Ian Stirling
>In article <XO%m5.894$z91....@news-east.usenetserver.com>,
>Ian Stirling <Inqui...@I.am> wrote:
>>Can anyone suggest other books than
>>Suttons "Rocket propulsion elements", or "Modern engineering design
>>of liquid-propellant rocket engines"
>Books on these topics tend to be scarce, and often old and hard to find.
>For engine design:
>+ Sutton is the right place to start. The 6th edition, still in print, is
>probably the best. The 5th is junk. The 4th is second best, because
>while it pre-dates various recent work, and also the switch to modern
>(metric) units, it also pre-dates the dumbing-down which produced the 5th
>edition. The 3rd is of some interest, same reasons. The 2nd and 1st are
>too old, historical curiosities only.
<snip many, many useful looking references.>
I note that the 7th edition is due in november, (http://www.amazon.com/
says so anyway).
I suppose the major expansion might be in ion engines, and similar
up and coming fields.
Plus, it's slightly cheaper than the 6th edition ($99).
Thanks for the version guide, will be very helpful, if I'm shopping around.
Though I think I'll probably spring for the 7th edition, when it's published.
When I add up all the references I've found in the "would be handy" list,
it's coming to several thousand $...
But then, I never expected this sort of textbook to be cheap.
I've also found a number of useful references to internet resources
that can be hard to find if you don't know they are out there.
I'll be sticking all of this on a web page, so others can benefit too.
Still hacking on my simulator.
Henry Spencer
Ian Stirling <Inqui...@I.am> wrote:
>>+ Sutton is the right place to start. The 6th edition, still in print, is
The 7th edition has been promised for quite some time; I wouldn't take any
current estimated release date too seriously. Sooner or later it will
appear...
>I suppose the major expansion might be in ion engines, and similar
>up and coming fields.
The 6th does have some treatment of those, which the 5th lacked. More
would certainly be better...
>When I add up all the references I've found in the "would be handy" list,
>it's coming to several thousand $...
>But then, I never expected this sort of textbook to be cheap.
Unfortunately true. These are specialized books, expensive to produce and
with quite limited audiences, even disregarding the ones that are out of
print and scarce.
Arthur Ed LeBouthillier
wrote:
>Still hacking on my simulator.
What are the specs on your simulator? I've written one
but I'm not happy with it. It's a one-dimensional (straight up)
simulator that uses a standard atmosphere model. It doesn't
account for the trans-sonic region but I could easily make
a modification that would.
How are you doing yours? Want to talk about it?
Cheers,
Art Ed LeBouthillier
Ian Stirling
>On Tue, 22 Aug 2000 12:15:15 GMT, Ian Stirling <Inqui...@I.am>
>wrote:
>>Still hacking on my simulator.
>What are the specs on your simulator? I've written one
>but I'm not happy with it. It's a one-dimensional (straight up)
>simulator that uses a standard atmosphere model. It doesn't
>account for the trans-sonic region but I could easily make
>a modification that would.
That's the equivalent of the crappy script I wrote first, command line.
>How are you doing yours? Want to talk about it?
Initially it'll be fairly basic, assuming no wind, and that the vehicle
remains pointed into the direction of flight at all times, and the
control system is perfect.
This will allow at least preliminary sizing of everything, and optimisation
of launch trajectory, with a dumb goal-seeking program.
Has to include drag model, as well as gravity and atmosphere.
This is the fairly simple one, but I'm trying to make it easily extensible,
so I can break the rocket into parts, all interacting.
For example:
The aeroshell will impart torques and drags on the rocket, and can lead to
a "fail" if the stresses are too large.
The engine uses fuel from the fuel tank, imparts thrust.
The fuel tank can slosh, again imparting torque, fuel-flow
is calculated, and the changing amount of fuel is used to work out
changes in the CG.
Also, a universe simulator, that enables testing of sensor suites and
control laws, to make sure (for example) that star-trackers can actually
sense position no-matter where they are pointed, and that gravity has
been taken into account by the acceleromter system.
It's not in a really reviewable condition at the moment, too much on paper.
--
http://inquisitor.i.am/ | mailto:inqui...@i.am | Ian Stirling.
---------------------------+-------------------------+--------------------------
"Melchett : Unhappily Blackadder, the Lord High Executioner is dead
Blackadder : Oh woe! Murdered of course.
Melchett : No, oddly enough no. They usually are but this one just got
careless one night and signed his name on the wrong dotted line.
They came for him while he slept." - Blackadder II
Arthur Ed LeBouthillier
wrote:
>>What are the specs on your simulator? I've written one
>>but I'm not happy with it. It's a one-dimensional (straight up)
>>simulator that uses a standard atmosphere model. It doesn't
>>account for the trans-sonic region but I could easily make
>>a modification that would.
>
>That's the equivalent of the crappy script I wrote first, command line.
I've been working on a 2D version lately. It seems to
"calibrate" against trajectory information I've been
able to obtain by about +/- 5%.
>Initially it'll be fairly basic, assuming no wind, and that the vehicle
>remains pointed into the direction of flight at all times, and the
>control system is perfect.
>This will allow at least preliminary sizing of everything, and optimisation
>of launch trajectory, with a dumb goal-seeking program.
>
>Has to include drag model, as well as gravity and atmosphere.
I think I finally have a good drag model (anyone who knows of
a better one, I'd be interested in hearing about).
I use the Standard Atmosphere model provided by some U.S. Gov't
agency for density and speed of sound for altitude.
Mach = Rocket_Speed/Speed_of_Sound
If Mach < 1.0 Then
Modifier = Sqr(1 - Mach * Mach)
Else
Modifier = Sqr(Mach * Mach - 1)
End If
Drag = - 0.5 * Modifier * Cd * (diam * diam / 4 * 3.14159)
* Density * Velocity*Velocity
In my 2D version, the above values are vectors and scalars. I have to
estimate the Cd for now, but I think I'm O.K. there.
The Mach modifier I got from the *D*A*R*K* site but it also checked
with graphs on a NASA site. It seems to relate well enough for bodies
with low Cd.
An earlier model I had for atmospheric density was based on fitted
points and I came up with:
[density in pounds per cubic foot, altitude in feet]
density = 0.075 / Exp(Altitude / 26000.0)
It seems to get similar results to the standard atmosphere table. I
kind of like it better because it is "infinite" and has drag at high
altitudes unlike the Standard Atmosphere which goes completely
to zero above a certain altitude. According to a table in
"Fundamentals of Astrodynamics," even the miniscule drag
at orbital heights can cause pretty quick orbital decay. That's
why I kind of like the exponential model. Maybe I'll use both
models in the future: standard atmosphere within its range and
my model at orbital altitudes.
For a gravity model, I've been using:
GravityAccel = 1.402250451e16/(radius*radius)
>This is the fairly simple one, but I'm trying to make it easily extensible,
>so I can break the rocket into parts, all interacting.
>For example:
>The aeroshell will impart torques and drags on the rocket, and can lead to
>a "fail" if the stresses are too large.
Yeah, I've been checking atmospheric drag forces against the
rocket mass; if the rocket decelerates too fast, then the rocket
is dead.
Actually, I've been using my models in a Genetic Algorithm
simulator that breeds a population of rockets in order to derive
a more optimal design from an original one. It seems to work pretty
well but the only problem is the accuracy of the simulation and the
trajectory (the more accurate the simulation, the better the results
from the genetic algorithm). The results make me believe that
genetic algorithms can play an important role in initial rocket
design. They weed out a lot of designs that have problems
that you wouldn't even think of; they're also able to get more
optimal velocities than it seems I'm able to do from
intuition and math.
>The engine uses fuel from the fuel tank, imparts thrust.
>The fuel tank can slosh, again imparting torque, fuel-flow
>is calculated, and the changing amount of fuel is used to work out
>changes in the CG.
Yeah, my simulator does all of that stuff; that's the easy stuff [
except the slosh]. I've been looking at a 3D simulator but I need to
bone up on my vector math better. [fuel in pounds, throttle 0.0 to
1.0, thrust in pounds, Isp in seconds]:
Fuel = Fuel - Throttle * Engine_Thrust/Isp
>Also, a universe simulator, that enables testing of sensor suites and
>control laws, to make sure (for example) that star-trackers can actually
>sense position no-matter where they are pointed, and that gravity has
>been taken into account by the acceleromter system.
>
>It's not in a really reviewable condition at the moment, too much on paper.
Hmmm... sounds like a lot of work. Have you seen the AGI website? They
have a free program that does some of that stuff.
Do you have any better equations than the ones above? If so, I'd like
to see them.
Cheers,
Art Ed LeBouthillier
apen...@earthlink.net
http://home.earthlink.net/~apendragn/atg
Ian Stirling
>On Wed, 23 Aug 2000 14:41:15 GMT, Ian Stirling <Inqui...@I.am>
>wrote:
>>>What are the specs on your simulator? I've written one
>>>but I'm not happy with it. It's a one-dimensional (straight up)
>>>simulator that uses a standard atmosphere model. It doesn't
>>>account for the trans-sonic region but I could easily make
>>>a modification that would.
>>
>>That's the equivalent of the crappy script I wrote first, command line.
>I've been working on a 2D version lately. It seems to
>"calibrate" against trajectory information I've been
>able to obtain by about +/- 5%.
Not bad.
<snip>
>I use the Standard Atmosphere model provided by some U.S. Gov't
>agency for density and speed of sound for altitude.
I'm using the international reference atmosphere 1986, 0-120Km.
IIRC, from ftp://ftp.nasa.gov/models atmospheric/cira/cira86ascii
I haven't bothered extrapolating a curve, linear interpolation is
more or less OK, so far.
> Mach = Rocket_Speed/Speed_of_Sound
> If Mach < 1.0 Then
> Modifier = Sqr(1 - Mach * Mach)
> Else
> Modifier = Sqr(Mach * Mach - 1)
> End If
I'd say that it might be worth starting the drag increment a little lower,
mach 0.9 or so, as that's where the dramatic increase in drag starts.
It could be argued that varying it with Cd (Cd=1, drag starts increasing
at mach 0.5, Cd=.1, 0.95).
> Drag = - 0.5 * Modifier * Cd * (diam * diam / 4 * 3.14159)
> * Density * Velocity*Velocity
>In my 2D version, the above values are vectors and scalars. I have to
>estimate the Cd for now, but I think I'm O.K. there.
>The Mach modifier I got from the *D*A*R*K* site but it also checked
>with graphs on a NASA site. It seems to relate well enough for bodies
>with low Cd.
The NACA research notes are usefull in this, especially as some of them
deal with regimes interesting to me. (the test missiles are often
more or less the same size as my hypothetical first stage)
<snip>
>It seems to get similar results to the standard atmosphere table. I
>kind of like it better because it is "infinite" and has drag at high
>altitudes unlike the Standard Atmosphere which goes completely
>to zero above a certain altitude. According to a table in
There are also lots of other useful models in the directory mentioned
earlier.
>For a gravity model, I've been using:
<snip>
>Actually, I've been using my models in a Genetic Algorithm
>simulator that breeds a population of rockets in order to derive
>a more optimal design from an original one. It seems to work pretty
>well but the only problem is the accuracy of the simulation and the
>trajectory (the more accurate the simulation, the better the results
>from the genetic algorithm). The results make me believe that
>genetic algorithms can play an important role in initial rocket
>design. They weed out a lot of designs that have problems
>that you wouldn't even think of; they're also able to get more
>optimal velocities than it seems I'm able to do from
>intuition and math.
That's what I was aiming at.
For small rockets, varying thrust as the rocket ascends looks to have
big paybacks.
<snip>
>>It's not in a really reviewable condition at the moment, too much on paper.
>Hmmm... sounds like a lot of work. Have you seen the AGI website? They
>have a free program that does some of that stuff.
No, hadn't seen it.
Currently looking.
>Do you have any better equations than the ones above? If so, I'd like
>to see them.
Not really, possibly a more accurate atmospheric model, but that's about
it.
--
http://inquisitor.i.am/ | mailto:inqui...@i.am | Ian Stirling.
---------------------------+-------------------------+--------------------------
Two parrots sitting on a perch. One asks the other, "Can you smell fish?"
Ian Stirling
>On Wed, 23 Aug 2000 14:41:15 GMT, Ian Stirling <Inqui...@I.am>
>wrote:
>>>What are the specs on your simulator? I've written one
<snip>
>I've been working on a 2D version lately. It seems to
>"calibrate" against trajectory information I've been
>able to obtain by about +/- 5%.
<snip>
>Hmmm... sounds like a lot of work. Have you seen the AGI website? They
>have a free program that does some of that stuff.
Not ideal, it seems it's more aimed at things that are already in orbit,
whereas I'd like to also do things on their way.
Also, they state that they won't send CD's to non-commercial international
people, and the one module that looked relevant (the missile one) is
US only.
I happen to have found something that at first glance looks interesting.
If only for ideas.
Look at http://www.multimania.com/launcher/Applet/angl/software.htm
--
http://inquisitor.i.am/ | mailto:inqui...@i.am | Ian Stirling.
---------------------------+-------------------------+--------------------------
Prosperity and ruin issue from the power of the tongue.
Therefore, guard yourself against thoughtless speech.
Robert Lynn
Arthur Ed LeBouthillier wrote:
>
> On Wed, 23 Aug 2000 14:41:15 GMT, Ian Stirling <Inqui...@I.am>
> wrote:
>
> >>What are the specs on your simulator? I've written one
> >>but I'm not happy with it. It's a one-dimensional (straight up)
> >>simulator that uses a standard atmosphere model. It doesn't
> >>account for the trans-sonic region but I could easily make
> >>a modification that would.
> >
> >That's the equivalent of the crappy script I wrote first, command line.
>
> I've been working on a 2D version lately. It seems to
> "calibrate" against trajectory information I've been
> able to obtain by about +/- 5%.
>
> >Initially it'll be fairly basic, assuming no wind, and that the vehicle
> >remains pointed into the direction of flight at all times, and the
> >control system is perfect.
> >This will allow at least preliminary sizing of everything, and optimisation
> >of launch trajectory, with a dumb goal-seeking program.
> >
> >Has to include drag model, as well as gravity and atmosphere.
>
> I think I finally have a good drag model (anyone who knows of
> a better one, I'd be interested in hearing about).
>
> I use the Standard Atmosphere model provided by some U.S. Gov't
> agency for density and speed of sound for altitude.
>
> Mach = Rocket_Speed/Speed_of_Sound
>
> If Mach < 1.0 Then
> Modifier = Sqr(1 - Mach * Mach)
> Else
> Modifier = Sqr(Mach * Mach - 1)
> End If
same as:
Modifier=sqr(abs(mach*mach-1))
>
> Drag = - 0.5 * Modifier * Cd * (diam * diam / 4 * 3.14159)
> * Density * Velocity*Velocity
>
> In my 2D version, the above values are vectors and scalars. I have to
> estimate the Cd for now, but I think I'm O.K. there.
>
> The Mach modifier I got from the *D*A*R*K* site but it also checked
> with graphs on a NASA site. It seems to relate well enough for bodies
> with low Cd.
Mach No. Modifier
0.1 0.994987437
0.125892541 0.992043884
0.158489319 0.987360692
0.199526231 0.979892485
0.251188643 0.967938152
0.316227766 0.948683298
0.398107171 0.917338913
0.501187234 0.865338868
0.630957344 0.775817523
0.794328235 0.607488811
1 0
1.258925412 0.764783102
1.584893192 1.229587911
1.995262315 1.726578033
2.511886432 2.304251168
3.16227766 3
3.981071706 3.853431189
5.011872336 4.91109604
6.309573445 6.229824801
7.943282347 7.880084673
10 9.949874371
This really can't be right. For example it would imply that drag drops
to zero at M=1 which is actually the high point in the drag profile.
For the V2 (from Sutton pp126) the drag coefficient (based on frontal
area) started out at about 0.15 at M=0-0.6, increased to a peak of 0.42
at M=1.2 then decreased back towards .15 beyond M=5 (though that seems a
bit strange) - could be to do with the tapered aft body. In general the
0.15-0.2 figure is about right for initial drag coefficient, but from
other reading the drag coefficient seems to drop to only about 2-3 times
this value at higher mach numbers (could be because of non-tapered aft
body) with a peak occuring at or just above mach 1.
In summary for most rockets a reasonable linear piecewise approximation
could be something like:
M 0-0.6 Cd=0.15
M 0.6-1.2 Cd increasing linearly from 0.15 to 0.42
M 1.2-1.8 Cd decreasing linearly from 0.4 to 0.36
M 1.8-5.0 decreasing linearly from 0.36 to 0.3
The final figure is probably in the range 0.3-0.4 for rockets with
non-tapering aft bodies.
At very high velocities and low pressures the Cd increases up towards 2.
I found quite a lot of useful papers on the nasa reports server, try
looking at:
http://naca.larc.nasa.gov/reports/1947/naca-tn-1350/naca-tn-1350.pdf
There are lots of other interesting reports on this type of stuff as
well.
>
> An earlier model I had for atmospheric density was based on fitted
> points and I came up with:
>
> [density in pounds per cubic foot, altitude in feet]
> density = 0.075 / Exp(Altitude / 26000.0)
>
> It seems to get similar results to the standard atmosphere table. I
> kind of like it better because it is "infinite" and has drag at high
> altitudes unlike the Standard Atmosphere which goes completely
> to zero above a certain altitude. According to a table in
> "Fundamentals of Astrodynamics," even the miniscule drag
> at orbital heights can cause pretty quick orbital decay. That's
> why I kind of like the exponential model. Maybe I'll use both
> models in the future: standard atmosphere within its range and
> my model at orbital altitudes.
It gets really strange at orbital heights and really can't be rellied
upon. I've created some piecewise polynomial curvefits with good
accuracy up to 86km. They are in the form of Excel visual basic
functions. Just cut and paste into the Visual Basic Editor that you
find under Tools => Macro to create functions that you can call from
worksheets simply by typing (for example) =Airdens(50000) into a cell
on a worksheet.
'Created by Robert G Lynn, 3rd August 2000
'Calculates air temperature in Kelvin as a function of altitude in
metres, valid for Altitude in range 0 to 86000m
'Derived from "U.S Standard Atmosphere, 1976". Maximum error from that
dataset = +/-0.06K (.025%).
Function Airtemp(Altitude As Double) As Double
If Altitude < 0 Then Altitude = 0
If Altitude < 11008.96106 Then
Airtemp = 288.1418478 - 0.006489427 * Altitude
ElseIf Altitude < 20080.50741 Then
Airtemp = 216.7
ElseIf Altitude < 32138.1882 Then
Airtemp = 196.8786083 + 0.000987096 * Altitude
ElseIf Altitude < 47363.47197 Then
Airtemp = 139.74 + 0.002765 * Altitude
ElseIf Altitude < 51384.8659 Then
Airtemp = 270.7
ElseIf Altitude < 71742.73164 Then
Airtemp = 411.8731579 - 0.002747368 * Altitude
ElseIf Altitude >= 71742.73164 Then
Airtemp = 354.9752381 - 0.001954286 * Altitude
End If
End Function
'Created by Robert G Lynn, 3rd August 2000
'Calculates air pressure in Pascals as a function of altitude in metres,
valid for Altitude in range 0 to 86000m
'Derived from "U.S Standard Atmosphere, 1976". Air pressure was first
normalised by dividing by a logarithmic
'least squares fitted curve before creating a set of piecewise least
squares fitted equations to predict the value of
'this normalised function over shorter spans of altitude. The resulting
maximum error for calculated air pressures
'compared to the dataset used is on the order of +/-0.06%
Function Airpres(Altitude As Double) As Double
If Altitude < 0 Then Altitude = 0
If Altitude < 10000 Then
logpres = 0.992515980220825 + -2.35767563391663E-05 * Altitude ^
1 + -5.0181110777017E-10 * Altitude ^ 2 + 2.70320552765292E-12 *
Altitude ^ 3 + -1.718851263848E-15 * Altitude ^ 4 + 5.79771615465955E-19
* Altitude ^ 5 + -9.32463813961912E-23 * Altitude ^ 6 +
-2.80214168849701E-30 * Altitude ^ 7 + 2.48568140644992E-30 * Altitude ^
8 + -3.33826083439437E-34 * Altitude ^ 9 + 3.0382195665408E-39 *
Altitude ^ 10 + 1.73214251036964E-42 * Altitude ^ 11 +
1.64173519522019E-46 * Altitude ^ 12 + -5.17922365831393E-50 * Altitude
^ 13 + 3.8049207659164E-54 * Altitude ^ 14 + -9.50966668981887E-59 *
Altitude ^ 15
ElseIf Altitude < 20000 Then
logpres = 48.8697781642692 + -1.95719617104864E-02 * Altitude ^
1 + 2.53906359971286E-06 * Altitude ^ 2 + 1.64761669110829E-11 *
Altitude ^ 3 + -3.456888880296E-14 * Altitude ^ 4 + 3.00613584446108E-18
* Altitude ^ 5 + -2.76666375442831E-23 * Altitude ^ 6 +
-8.27435750155133E-27 * Altitude ^ 7 + 2.42093142750114E-31 * Altitude ^
8 + 1.71442168792786E-35 * Altitude ^ 9 + -7.24266500131218E-40 *
Altitude ^ 10 + -3.33103877610279E-44 * Altitude ^ 11 +
2.51691264063566E-48 * Altitude ^ 12 + -4.09065884243178E-53 * Altitude
^ 13 + -3.76938425310477E-58 * Altitude ^ 14 + 1.19571134007853E-62 *
Altitude ^ 15
ElseIf Altitude < 30000 Then
logpres = 37.1177631100275 + -1.37822791636131E-02 * Altitude ^
1 + 2.21159238838588E-06 * Altitude ^ 2 + -1.93455699321869E-10 *
Altitude ^ 3 + 9.69029551334301E-15 * Altitude ^ 4 +
-2.40641347861316E-19 * Altitude ^ 5 + -6.88699408411082E-25 * Altitude
^ 6 + 2.43555957217186E-28 * Altitude ^ 7 + -7.76073043247129E-33 *
Altitude ^ 8 + 1.03202733448463E-37 * Altitude ^ 9 +
2.63304434399837E-42 * Altitude ^ 10 + -2.69042490029893E-46 * Altitude
^ 11 + 1.04706363339788E-50 * Altitude ^ 12 + -2.18780943676924E-55 *
Altitude ^ 13 + 2.34492917669051E-60 * Altitude ^ 14 +
-9.89703995782364E-66 * Altitude ^ 15
ElseIf Altitude < 58000 Then
logpres = 36.4676912002006 + -3.2866079799342E-03 * Altitude ^ 1
+ 8.37759503242387E-08 * Altitude ^ 2 + -2.20451607426656E-12 * Altitude
^ 3 + 2.44347317913712E-16 * Altitude ^ 4 + -8.0557284316765E-21 *
Altitude ^ 5 + -7.23608998048658E-26 * Altitude ^ 6 +
5.05064506983434E-30 * Altitude ^ 7 + 1.26853721785736E-34 * Altitude ^
8 + -7.80976791361223E-39 * Altitude ^ 9 + 9.71973108831772E-44 *
Altitude ^ 10 + 3.8650051559839E-49 * Altitude ^ 11 +
-9.1318889043665E-54 * Altitude ^ 12 + -1.80926996442671E-58 * Altitude
^ 13 + 3.71794257302445E-63 * Altitude ^ 14 + -1.77622846544708E-68 *
Altitude ^ 15
ElseIf Altitude >= 58000 Then
If Altitude > 86000 Then Altitude = 86000
logpres = 280.585681102118 * Altitude ^ 0 +
-5.04354905379207E-02 * Altitude ^ 1 + 3.25180329844521E-06 * Altitude ^
2 + -1.01769105340614E-10 * Altitude ^ 3 + 1.58340236733899E-15 *
Altitude ^ 4 + -8.3452330337524E-21 * Altitude ^ 5 +
-6.22394993095915E-26 * Altitude ^ 6 + 2.81920732678786E-31 * Altitude ^
7 + 1.15040286811985E-35 * Altitude ^ 8 + -4.02242213420458E-42 *
Altitude ^ 9 + -2.64722692487413E-45 * Altitude ^ 10 +
2.99442357543224E-50 * Altitude ^ 11 + -1.62632098067191E-55 * Altitude
^ 12 + 9.07930002593565E-61 * Altitude ^ 13 + -5.66513295471543E-66 *
Altitude ^ 14 + 1.59270650670915E-71 * Altitude ^ 15
End If
Airpres = 100566.676232844 * 0.999857111047131 ^ Altitude / logpres
End Function
'Created by Robert G Lynn, 28th July 2000
'Calculates air density in kg/m^3 as a function of altitude in metres,
valid for Altitudes in range 0 to 86000m
'Derived from "U.S Standard Atmosphere, 1976". Max error for calculated
air density compared to dataset is about 0.07%
Function Airdens(Altitude As Double) As Double
Airdens = Airpres(Altitude) / Airtemp(Altitude) / 287.1
End Function
>
> For a gravity model, I've been using:
>
> GravityAccel = 1.402250451e16/(radius*radius)
>
> >This is the fairly simple one, but I'm trying to make it easily extensible,
> >so I can break the rocket into parts, all interacting.
> >For example:
> >The aeroshell will impart torques and drags on the rocket, and can lead to
> >a "fail" if the stresses are too large.
>
> Yeah, I've been checking atmospheric drag forces against the
> rocket mass; if the rocket decelerates too fast, then the rocket
> is dead.
>
> Actually, I've been using my models in a Genetic Algorithm
> simulator that breeds a population of rockets in order to derive
> a more optimal design from an original one. It seems to work pretty
> well but the only problem is the accuracy of the simulation and the
> trajectory (the more accurate the simulation, the better the results
> from the genetic algorithm). The results make me believe that
> genetic algorithms can play an important role in initial rocket
> design. They weed out a lot of designs that have problems
> that you wouldn't even think of; they're also able to get more
> optimal velocities than it seems I'm able to do from
> intuition and math.
>
> >The engine uses fuel from the fuel tank, imparts thrust.
> >The fuel tank can slosh, again imparting torque, fuel-flow
> >is calculated, and the changing amount of fuel is used to work out
> >changes in the CG.
>
> Yeah, my simulator does all of that stuff; that's the easy stuff [
> except the slosh]. I've been looking at a 3D simulator but I need to
> bone up on my vector math better. [fuel in pounds, throttle 0.0 to
> 1.0, thrust in pounds, Isp in seconds]:
>
> Fuel = Fuel - Throttle * Engine_Thrust/Isp
>
> >Also, a universe simulator, that enables testing of sensor suites and
> >control laws, to make sure (for example) that star-trackers can actually
> >sense position no-matter where they are pointed, and that gravity has
> >been taken into account by the acceleromter system.
> >
> >It's not in a really reviewable condition at the moment, too much on paper.
>
> Hmmm... sounds like a lot of work. Have you seen the AGI website? They
> have a free program that does some of that stuff.
>
> http://www.stk.com/
>
> Do you have any better equations than the ones above? If so, I'd like
> to see them.
>
Robert Lynn
sound can be found (In Excel) by typing in the equation:
=sqrt(401.8*Airtemp(altitude in metres))
Robert Lynn
Robert Lynn
coefficient only dropped to 0.3-0.4 after the transonic peak of 0.45.
This paper below shows that for a sharp nosed cone it can drop back to
as little as 0.1 at Mach 7, while for a Roton type shape it drops back
to about 0.15 at Mach 7. However For a blunt nosed vehicle (nose radius
more than half maximum) it can be up around 0.25 or greater at Mach 7.
All this from paper at:
http://naca.larc.nasa.gov/reports/1952/naca-rm-a52b13/naca-rm-a52b13.pdf
Through trial and error I've come up with a function that seems to be a
very good fit for the V2 data in Sutton throughout the whole range:
Cd(M)=0.14*(3/(M^2+M^-10)+1)
Where Cd(M) is the drag coefficient (taken on frontal area) as a
function of Mach number M.
This function could be easily adapted to other vehicles (the exponents
of M should probably be left alone), the key data points for a vehicle
are the low speed subsonic Cd (generally in range 0.1-0.15) The max Cd
(for most cases around 0.4-0.45 at M=1.2), and the high speed Cd which
tends towards a figure that seems largely dependant on the nose shape,
with a couple of examples given above. Using those three data points it
should be relatively easy to adapt the equation for other vehicles.
Hope that helps some people out there.
Robert Lynn
Arthur Ed LeBouthillier
So, here's the equation for the Mach modifier that I'd been using
before trying the DARK equation:
Modifier = 1 + 1 * Exp(-10 * Abs(1 - Mach))
It's just a hump around the speed of sound that doubles the
drag. I may try to modify it with a line equation so that it increases
in the upper mach range.
Cheers,
Art Ed LeBouthillier