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bertibott

How to get the most out of R.A.P.I.E.R. Engines?

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Hi,

so I am once again at that point in my career where I want to build an SSTO spaceplane (optimally one that is capable of of a round-trip to Minmus).

So I have come up with a design that is powered by two R.A,P.I.E.R. engines (one shock cone air intake for each) for ascent and two Nerf nuclear engines for when it's in space. The whole aerodynamics and stability side of things is not really an issue. The design is rather stable in most situations (it gets a little wobbly at around 30km altitude). I am also very sure, that the engines (let's disregard the Nerfs for now as they should not become active until the thing is in space) are powerful enough. When (or IF) the plane reaches a speed of 450 - 500 m/s at around 10 km there is no stopping it any more. The engines start to really kick in and get the whole thing to about 30 km and 1400m/s at wich point I can switch modes and voila!.. Space!

The problem is to get over that magical threshold... I generally peak at 300m/s at 10km in level flight. Any suggestions? More air? SRBs to kick me over the threshold? Weird flight profile?

 

EDIT: A screenshot of the ship in question can be foudn here: https://www.dropbox.com/s/bb5hpwehonkz3as/Agathon.jpg?dl=0

I am using Procedutal WIngs and Procedural Fueltanks

Edited by bertibott

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The usual problem with spaceplanes that have trouble getting there is that they don't have quite enough engine on them.  Add an engine or two, problem solved.  Not saying that's necessarily the problem in your case, but that's often it.  If your ship can't get past 300 m/s in level flight at 10 km, my guess is that it's under-engined.

The other possibility is that it's not under-engined, it's over-draggy.  Turn on the aero overlay with F12 and make sure you don't have any unexpected drag sources.  KSP's aero model can be a little idiosyncratic, and sometimes there's a disconnect between how much drag a part looks like it should cause, and how much it actually does.

Admittedly I don't fly a lot of spaceplanes (I'm more of a rocket guy), but that's my impression based on the limited spaceplane flights I've done.  Perhaps a spaceplane expert (there are tons of them around here!) will chime in.  :)

In any case... all of the above are just guesses.  Would really help if you could post a screenshot of your ship-- there's a good chance that would make the issue obvious at a glance to experienced folks, who could then offer very specific suggestions for improving the design.

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Reducing weight makes you go faster. It sounds like you have more oxidizer in your plane than you need.

Reducing drag makes you go faster. Removing control surfaces and drag-based stabilizers. Using more MK1 parts and fewer MK2 and MK3 parts. Fewer nacelles. Tricks for closing open nodes. Tricks for choosing less draggy parts to accomplish the same task.

Tweaking the wing incidence can make your plane go faster.

There is a chance that if you do your acceleration at 0km to 5km, instead of 10km, that your rapiers will have enough guts to get you over Mach 1.

Rapiers in general are pretty gutless around Mach 1, and whiplashes may get your plane to a decent speed, where your rapiers can't.

You can go into a shallow dive from that 10km altitude -- sometimes that can be enough to get you over the hump. But you need something like 370 m/s, and if you are only at 300 then that will probably not be enough.

And you can always temporarily turn on your nukes at that 10km altitude -- there is a chance they will give you enough of a speed boost to get you over the Mach 1 hump.

 

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Okay so I uploaded the ship into my dropbox. Any major obvious designflaws? Other than that... from what I gathered so far I need to have more power? Add moar BOOSTERS... :)

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The weight should not be a problem. My SSTO also has 2 Rapier at 2 Nukes, but weights 10 tons more.

5PoG8ZY.jpg

 

But you have to reach 300 m/s much earlier. At 5km you should start to fly shallow. I have 700-800 m/s at 10 km. The Nukes I switch at 15km, with the Rapiers. Otherwise I will not get into orbit.

 

Edit:

Perhaps he should also angle the wings.

Edited by astroheiko
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Quote

The problem is to get over that magical threshold... I generally peak at 300m/s at 10km in level flight. Any suggestions? More air? SRBs to kick me over the threshold? Weird flight profile?

You problem its to past the transonic region  where drag peaks. You can either increase the thrust (switch to rocket mode, activate the nervs) or dive to use the gravity in your favour, also try to keep your craft pointed close to prograde to reduce drag as much as possible. As soon as you go trough the sound barrier everything become easier.

 

 

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1 hour ago, bewing said:

Tweaking the wing incidence can make your plane go faster.

 

This is always my go-to advice for planes that can't break the sound barrier. Especially Mk2 planes. Pick your acceleration range (I just burn hard at sea level for maximum rapier thrust), then tune your wings so that the fuselage is as close to perfectly level in horizontal flight at Mach 1 as possible. This is typically 1-3 degrees of incidence, depending on weight and wing size. Two rapiers can get some surprisingly large craft to orbit if you minimize drag. Assuming you can break mach 1 at low altitudes, I'd recommend running to 600m/s or so then just aiming at 10 degrees pitch and flying straight to orbit. 

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just to be clear... not a native speaker here... "incidence" means the angle at wich the wings are slanted backwards?

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48 minutes ago, bertibott said:

just to be clear... not a native speaker here... "incidence" means the angle at wich the wings are slanted backwards?

piper-9a.jpg

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This might not be directly related to your problem,  but there's no real reason to wait to light up your nukes until orbit.   By 20km altitude or so,  when the rapiers start to have trouble,  the nukes' ISP will be much greater than the rapiers in rocket mode.  Usually nukes lack the thrust to get you to orbit alone , but they can save fuel by taking some load off less efficient engines.  And they add what amounts to free TWR since they're on your plane regardless 

If you go this way,  you'll want to check your relative levels of LF and oxidizer.  Since nukes don'the use the latter,  you may want and need less of it.

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Your ship has a lot of mk2 fuselage parts, which are extremely draggy, and the wings are quite small, which means it can't fly up into thinner air where the fuselage drag might be less efficiently.      Thinner air means you need more AoA to make lift, but if you go beyond 5 degrees difference between prograde and where the nose is pointing, you get a lot of drag.

f0104-01_zpsuxu3knf0.gif

Getting through the sound  barrier is a major obstacle for  two reasons -

 

400px-Transonic_Range.png

400px-CR-7_R.A.P.I.E.R._Engine_velocity_

So,  taking an existing design, the method is

1.  climb as high as you can without being forced to adopt a draggy AoA to get lift.

2.  abruptly ease off on the nose up input so you arc over into a shallow dive and switch the nukes on until over 400 m/s

3.  gently pull out 

In pics

Spoiler

 

Note - the exact amount of pitch trim you need to generate a 5degree AoA depends on your plane's handling characteristics.

Note 2 - how high you can go without crossing the sound barrier depends on how much lift the design has

If you want to test fly the ship i pictured, download here - 

https://kerbalx.com/AeroGav/Basic-Tourist-SSTO

20161222094336_1_zpsjgecg5ir.jpg

20161222094436_1_zpsnnlbi5pe.jpg

20161222094538_1_zpsraqt85rn.jpg


 

Video version (note i'm cheating with Mechjeb!)

Spoiler


.

 

In terms of improving your design,  you got four choices

1) moar boosters (each engine weighs 2 tons though, so it hurts your delta v a lot)

2) moar wing  (you can go higher where fuselage drag is less)

3) add incidence to the wing and fly at a lower body aoa  (a bit harder to balance, an advanced construction technique)

4) forget about horrible mk2 fuselages altogether, build a mk1 with an inline cockpit.

Mark 1 will not have any overheating problems provided you use inline cockpit and try to have it two or 3 pieces back from the nose of the aircraft.         For example,   fly by wire hub,  then an NCS adapter, then engine pre-cooler intake (one per rapier), then the crewed bits.

Also if you are generous with the wing area, it'll rise quicker into thinner air where the heating is less.

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okay... this is extremely helpful! so thanks a lot everyone!

Is there a sepcial reason why the Mk2 parts are so draggy? Or is it just their shape? Since one of the tasks I had in mind for the thing is to carry payloads I feel that Mk 1 isn't really an option. Should I consider going bigger rather than smaller and build my craft around the Mk 3 parts?

Wouldn't adding incidence to the wings also cause drag? (I am pretty sure that it would in real life) So that I would only trade the drag from the fuselage against that of the wings....

I have changed the original design a little, added a Whiplash and removed some of the wings in the front to reduce drag (I think I remember reading about something called a "Mach Cone" does that get modelled?)

https://www.dropbox.com/s/8qe201jg3e2l1dm/Agathon Mk II.jpg?dl=0

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4 minutes ago, bertibott said:

Wouldn't adding incidence to the wings also cause drag? (I am pretty sure that it would in real life) So that I would only trade the drag from the fuselage against that of the wings....

 

Wings have to have some AoA to provide lift anyway.  If you have no incidence then to get that lift you have to have your nose pointing above the airstream, then you are getting drag from both wings and fuselage

 

Add incidence and you can minimize the drag from the fuselage while getting lift from the wings

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18 minutes ago, bertibott said:

Is there a special reason why the Mk2 parts are so draggy?

We argue about whether it is a bug or not. I think it is.

18 minutes ago, bertibott said:

Since one of the tasks I had in mind for the thing is to carry payloads I feel that Mk 1 isn't really an option.

You can use MK1 parts in the front, then an MK2 adapter, then your cargo bay in the back. This is a little bad for your CoM (the cargo bay should be at the CoM), but it does reduce drag.

18 minutes ago, bertibott said:

Should I consider going bigger rather than smaller and build my craft around the Mk 3 parts?

That costs a lot more, and it's harder to fly, and it doesn't look as good. You generally want to go as small as you can while still completing the mission. If all your cargo fits in an MK2 cargo bay, then no -- don't go to MK3.

18 minutes ago, bertibott said:

Wouldn't adding incidence to the wings also cause drag? (I am pretty sure that it would in real life)

Wings create a lot of lift which reduces drag. They also create more drag directly. So it is mostly an even trade. Usually adding moar wing will reduce overall drag a little bit.

 

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41 minutes ago, bertibott said:

Since one of the tasks I had in mind for the thing is to carry payloads I feel that Mk 1 isn't really an option.

 

I just gravitated to doing everything with spaceplanes, and mostly ignored "payload".    Rather than lift a probe in a spaceplane, i make the spaceplane carry instruments and be interplanetary itself.  ISRU,  decoupling jet engines when they flame out, drop tanks, whatever it takes to get the delta V.   Anything but deal with cargo bays.     

 

Quote

Wouldn't adding incidence to the wings also cause drag? (I am pretty sure that it would in real life) So that I would only trade the drag from the fuselage against that of the wings....

Vastly more drag comes from the fuselage than the wings in KSP, so adding incidence, and increasing the wing area,  both reduce overall drag by allowing you to keep a low body AoA while flying higher.

Quote

I have changed the original design a little, added a Whiplash

Adding a Whiplash can help low TWR designs get above mach 1.  For example, a 120 Ton SSTO I built, used two Rapiers, two Whiplash, and three nukes.  Something that size should not need them.

Quote

and removed some of the wings in the front to reduce drag (I think I remember reading about something called a "Mach Cone" does that get modelled?)

No, KSP does not model wave drag.   A biplane with straight wings would have the same lift and drag characteristics as mach number increases as a highly swept delta.    

Looking at the Agathon ,  you can reduce drag by removing those 4 way thruster blocks.  Use reaction wheels instead, if you need to translate up down left right etc.  then put them in the cargo bay, that way they don't create any drag till the doors are opened in space.

Also, you appear to have a set of tanks strapped to side of rear fuselage, in addition to the rapier nacelles. Could they be moved onto the rapier nacelles to reduce the number of parallel stacks, or is it a weight/balance issue (that's a legit reason).  Also I see radiators , and  solar panels outside.  Get this stuff in the cargo bay !

I might download your ship but i need to reinstall B9 procedural parts.

Have a look at my XKos interplanetary - very comparable to yours.  One rapier, two nukes is all it needs, because of generous wing area, 3 degree incidence angle on wings, and absolutely no radial parts that couldn't go in a cargo bay.

https://kerbalx.com/AeroGav/ASES-SP2-Xkos-Dual-Nukes

20161121161935_1_zpsubvnyqzj.jpg

ps.  also, change that cockpit for an inline one.   Once you get past mach 1, you'll find it overheats way too easy.

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6 minutes ago, AeroGav said:

I might download your ship but i need to reinstall B9 procedural parts.

Okay... I will check if there are any other mods involved (and remove the parts if possible... the RCS thrusters are frome another mod.. but I am not sure wwhich one exeactly... not willing to look it up)

The craft file is here: https://www.dropbox.com/s/bvx2skbvbvuuepn/KSS Agathon Mk II.craft?dl=0

 

I made some minor changes. Most importantly removing the payload, wich was some life support supplies for a space station (to reduce the amount of mods needed). The centre of mass may be a bit off now.

 

The wings are this mod:

The fuel tanks are this one:

 

I also desigend a new vessel (strongly inspired by AeroGav's) I used a lot of elevons (I locked them so they wouldn't be used for steering) to balance out the centre of lift. Is there an difference in the "Lift per Drag"-Ratio? Or in other words is it better to use fewer, but bigger parts?

On my first few tests I didn't notice any difference in performance for different incidences. It did make it into orbit but only had about 500m/s of fuel left so Minmus was out of the question (not sure if it could have reached my station which is at 600km).

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5 hours ago, bertibott said:

Have  you tried to fly this craft since removing the payload?    

There are things I could do to improve performance/reduce drag, but with two rapiers and a whiplash in a small ship there is plenty of power. 

The problem is it is uncontrollable.  The mod called CorrectCoL has a stability analysis, it shows the ship is stable in pitch at negative AoA, but becomes increasingly unstable in pitch at positive ones until near the stall angle, when it becomes stable again.

So basically, once you get over a certain speed, or pull the nose more than a couple of degrees above prograde, or get too high (forcing you to raise the nose to get enough lift)  the nose suddenly snaps up to 30 degrees uncontrollably,  then it drops again after stalling, and the cycle repeats.

I spent a bit of time tweaking stuff, but honestly it's got me beat - about the first handling issue i wasn't able to solve in a long time.  I can only guess that 

a) Your plane is mostly fuselage , ahead of CoM, and mostly wing behind it.   Perhaps they gain lift at different rates with increasing AoA, causing the pitch up?  It does have an unusually long fuselage.

b) some wierd bug with the procedural parts

Simply moving the CoL aft didn't correct this tendency, until it was so far aft the plane could no longer pitch up to take off.  As it is, you're doing over 150 m/s on the runway.  My XKOS craft can lift off at one quarter that speed.

Try flying the XKOS craft I linked to see how airplanes are supposed to respond in pitch.   This one has wing incidence and will fly ok if you just set prograde.  Or you can turn SAS off and use pitch trim (alt + S, alt + W) and it will stabilize around a particular angle of attack depending on how the trim was set.

Edited by AeroGav

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5 hours ago, bertibott said:

Okay... I will check if there are any other mods involved (and remove the parts if possible... the RCS thrusters are frome another mod.. but I am not sure wwhich one exeactly... not willing to look it up)

The craft file is here: https://www.dropbox.com/s/bvx2skbvbvuuepn/KSS Agathon Mk II.craft?dl=0

 

I made some minor changes. Most importantly removing the payload, wich was some life support supplies for a space station (to reduce the amount of mods needed). The centre of mass may be a bit off now.

 

The wings are this mod:

The fuel tanks are this one:

 

I also desigend a new vessel (strongly inspired by AeroGav's) I used a lot of elevons (I locked them so they wouldn't be used for steering) to balance out the centre of lift. Is there an difference in the "Lift per Drag"-Ratio? Or in other words is it better to use fewer, but bigger parts?

 

No, all wing parts have the same lift drag ratio.  It's just about keeping the part count down and rigidity up.

In terms of the stock aero parts,  parts with a 100% control surface rating (ie. all moving tailplanes/canards) weigh twice as much as pure wing parts that don't deflect - their "lift rating" - ie. wing area - is half what you'd expect for the mass.

Stuff like the Shuttle tail fin are about 35% control surface if i remember correctly , so have more mass than a pure wing of the same size.

Also they don't store any fuel.

Max deflection angle and the authority limiter in tweakables are important.

For example, let's say you're using a stock "advanced canard."  .  Max deflection is 10 degrees.  You've set authority to 100%.

Your space plane flies to orbit at 5 degrees AoA and to hold it there,  you're using 50% of available pitch authority.

50% of 10 degrees is 5 degrees,  add that to the 5 degrees the whole airplane has, and this canard is at 10 degrees relative to the airflow.   That's OK, but beyond that drag really starts to ramp up.   If you're using bigger angles than that in cruise flight, you'd have lower drag with a larger control surface working at a smaller angle.

Limiting authority in the tweakable basically limits max deflection angle.  So the basic all moving tail fin (max deflection 30 degrees) behaves like an advanced canard when set to 33% authority.    Of course you could just leave it at 100%, and never use more than a third of the max available nose up authority in flight, that'd be the same thing.

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17 hours ago, bertibott said:

Hi,

so I am once again at that point in my career where I want to build an SSTO spaceplane (optimally one that is capable of of a round-trip to Minmus).

So I have come up with a design that is powered by two R.A,P.I.E.R. engines (one shock cone air intake for each) for ascent and two Nerf nuclear engines for when it's in space. The whole aerodynamics and stability side of things is not really an issue. The design is rather stable in most situations (it gets a little wobbly at around 30km altitude). I am also very sure, that the engines (let's disregard the Nerfs for now as they should not become active until the thing is in space) are powerful enough. When (or IF) the plane reaches a speed of 450 - 500 m/s at around 10 km there is no stopping it any more. The engines start to really kick in and get the whole thing to about 30 km and 1400m/s at wich point I can switch modes and voila!.. Space!

The problem is to get over that magical threshold... I generally peak at 300m/s at 10km in level flight. Any suggestions? More air? SRBs to kick me over the threshold? Weird flight profile?

 

EDIT: A screenshot of the ship in question can be foudn here: https://www.dropbox.com/s/bb5hpwehonkz3as/Agathon.jpg?dl=0

I am using Procedutal WIngs and Procedural Fueltanks

Basically, you just need to crack the transonic threshold and you're away. There are a few tricks:

1) Minimise drag. Get rid of as much surface junk as possible; anything that generates a red arrow when you hit F12 in flight. Cargo bays are good, service bays are good. Just using the part translation tools to visually hide things inside fuel tanks doesn't work.

2) Minimise drag. You want your fuselage pointing prograde during your final speed run; build in as few degrees of wing incidence to allow this.

3) Minimise drag. Keep the fuselage as narrow as possible, don't carry any more intakes than you need.

Sensing a theme?

However, also:

1) Level off for your speed run, and do it low enough that the engines are getting good rich air and you can maintain level flight with minimal AoA. Crack transonic, then climb. High powered jets can get away with climbing during acceleration, but lower powered ships need to level off.

2) As always: moar boosters. You can send a brick to orbit if you strap enough RAPIERs on it. However, a 2xRAPIER/2xLV-N build is perfectly workable if you do it right.

Edited by Wanderfound

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Nobody's mentioned the tailcone. That looks to be the main source of drag in this design. It's not attached to a preceding node, and therefore not occluded. It's basically like a 1.25m plate hanging out in the breeze. I think that's why you can't break Mach 1.

*edit* also FWIW, the Nerv isn't worth the additional mass and hassle for such a low DV trip. You'd be better off to just run the RAPIER. Less parallel nodes makes for an aerodynamically cleaner design, which will allow you to use less t/w in airbreathing mode. The mass of the NERVs can be replaced by fuel, which will make for a lighter ship, even with the RAPIER's relatively poor Isp.
 My rule of thumb is "If you don't need 2km/ sec of DV, don't use nukes".

HTHs,
-Slashy

Edited by GoSlash27

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On ‎11‎/‎02‎/‎2017 at 3:04 PM, GoSlash27 said:

Nobody's mentioned the tailcone. That looks to be the main source of drag in this design. It's not attached to a preceding node, and therefore not occluded. It's basically like a 1.25m plate hanging out in the breeze. I think that's why you can't break Mach 1.

*edit* also FWIW, the Nerv isn't worth the additional mass and hassle for such a low DV trip. You'd be better off to just run the RAPIER. Less parallel nodes makes for an aerodynamically cleaner design, which will allow you to use less t/w in airbreathing mode. The mass of the NERVs can be replaced by fuel, which will make for a lighter ship, even with the RAPIER's relatively poor Isp.
 My rule of thumb is "If you don't need 2km/ sec of DV, don't use nukes".

HTHs,
-Slashy

Which version of the file did you download?   I took this one, which could be cleaner but nevertheless easily has the power to hit mach 1.  The problem is it's barely controllable , I can't get it to pitch within 10 or 15 degrees of my intended flight path.

re: NERV usage - 2k dV  - even allowing for the fact you're going to lose 800dv or so  getting from the speedrun to LKO, that's a lot - enough to go to Minmus and back.   Is that really more easily done on a RAPIER only ship?     On my 30 ton designs, burning 2 tons of LF/O in the upper atmosphere when the RAPIERs switch mode, only adds about 300 m/s velocity, though we are in a climb so I suppose there's losses.

In a RAPIER only design you're looking for a higher TWR and fuel fraction,  on a NERV ship,  supersonic lift/drag ratio is really important and you obviously can't hope to hit the TWR and fuel fraction you could with only RAPIERs.    I guess a high TWR RAPIER only design with a small payload fraction and which only goes to LKO  will better tolerate bad airframe,   but if you're looking to do something special,  you need a good airframe so you can cut back on the amount of engines and leave more mass for fuel or payload.      This is true whether it's a chemical only ship built by yourself or Rune,  or the kind of NERV-heavy spaceplanes I like to make.

 

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is it at all possible to have a craft that is pitch stable AND has the CoM and CoL in the same (not talking about "desired" or "useful") position?

I had a look at at the ASES SP2 Xkos... on of my own designs is somewhat similar but however much i twek it is either completly unstable or the CoL is way out there...

guess i'll juts go back to rockets

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29 minutes ago, bertibott said:

is it at all possible to have a craft that is pitch stable AND has the CoM and CoL in the same (not talking about "desired" or "useful") position?

 

Yes, when you  have incidence /angled wings, things get confusing.

The XKOS (it was actually someone else's spaceplane i fixed up, so not my concept) might actually show the CoL and CoM to be right on top of each other in the VAB.

That is because the main wing is angled up at 3 degrees of incidence but the front canard is at 5 degrees or more.  The front canard is working harder and this brings the CoL forward.

But , when you start pitching up,  the canard gets into diminishing returns before the main wing does, and will stall first.   So it is actually pitch stable, the more you pitch up, the more the centre of lift moves aft and the more it resists further efforts to pitch up.

Building planes with built in incidence on the wings is a dark art.     If you are new to aircraft,  I'd say don't bother with incidence  for now.

If you stick to mk1 fuselage parts the drag penalty is not so bad.   Also having lots of wing area will reduce the fuselage drag penalty because it'll help keep your aoa down and make the plane fly higher,  both of which reduce body drag.

As for heat, like i say use an inline cockpit and try to have it at least  3 modules back from the front of the aircraft.  Service bays,  engine pre-coolers, fuel tanks, ISRU converters can all go in front and keep the heat cone away from the pilot.   And generous wing area helps here too,  your plane will ride higher on thinner air where the heat is less bad.

edit - see how fast this guy went on  a mk1 cockpit becuase there is stuff in front of it.  He was trying to not get higher hence tiny wings.  I'd probably stick two pairs of big S deltas on something like this, but that's just me.

 

http://images.akamai.steamusercontent.com/ugc/80339584930689592/5E1F04EAAB58221CA6D591B1C700FD5CC19946E7/

Edited by AeroGav
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but what does one actually do against instability?  my way of thinking is have leverage. have a stabalizer as far back as possible. So after going out for a beer in frustration I came up with this obscenity:

https://www.dropbox.com/s/52mx7b9u88ji50z/Tyrol Mk I.craft?dl=0

It can fly... it can break the sound barrier... but that tail makes it more or less impossible to land... and take off isn't pretty either.

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57 minutes ago, bertibott said:

but what does one actually do against instability?

 

 

Well, my answer is to forget about tails and stabilizers and just go with canards. There are two kinds of stability, passive and dynamic.

For passive stability, you want lots of drag at the back, and very little at the front, You want the CoL vertically above the CoM. You want dihedral on the wings. And you must have the CoL a little behind the CoM horizontally.

Dynamic stability is about taking a slightly unstable plane, and having a computer (SAS system) constantly jiggle the control surfaces to keep it balanced and pointing in the right direction. This still requires more drag in the back than in the front, because there are flight regions where you don't have enough aerodynamic or gimbal authority to control orientation, so you are limited to the reaction wheels. But you can have the CoL and the CoM.at basically the same spot. The key to stability if you go this route (which seems like the way you want to go) is control authority, You need more of it. If your plane is still dynamically unstable, add more canard area. Add a second set. Add bigger rear stabilizers. And bigger reaction wheels.

The canards you were using in your Agathon don't have enough area in the moveable surface to enforce dynamic stability.

 

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