Boring maths on parachutes in 1.2

[Gaarst]

AFAIK the parachute calculator thingy is not updated since a long time so I've bothered doing this stuff. If you wonder how many chutes you need to slow down a craft to a given speed then take a look: I've done a few measurements and have derived a few formulas for parachutes mechanics.

 

First, let's start with a bit of physics before the maths. The terminal velocity of an object in air is obtained by equating drag and gravitational pull and is:

, v is the terminal velocity of the object, m its mass, g the gravitational acceleration, ρ is the density of the air, C_x is the drag coefficient and A is the effective area.

In practice, C_x and A are always grouped together since dissociating one from the other is hard, especially in KSP. From now on, g will be taken to be 9.81 m/s² and ρ to be the air density ASL on Kerbin (about 1.116 kg.m^-3), so the results will only be valid at the surface of Kerbin, but adapting them to other bodies would be simple as you would just need to change these two parameters.

 

For a single chute we can then write the terminal velocity as: , defining the constant B as  (the greater B is, the faster the body will fall).

B is a constant that is valid for a given parachute part and on the surface of Kerbin, it has a different value for each parachute in the game that can be found by reversing the equation above:

• Mk16 chute: B = 0.029
• Mk2R radial chute: B = 0.039
• Mk16XL chute: B = 0.020
• Mk25 drogue chute: B = 1.099
• Mk12R radial drogue chute: B = 1.065

 

The above is valid for a single chute. When you consider multiple chutes, the only part that changed in B is the parameter C_xA which depends on the surface of parachutes: you can consider several chutes to be a single larger chute. When trying stuff in game, you find that the stack chutes behave nicely when adding chutes: 2 chutes means C_xA is twice greater, B is twice smaller and your craft falls √2 times slower.

Radial chutes are annoying. As is turns out, when you place several radial chutes in symmetry their area does not scale linearly, ie: 2 in chutes symmetry doesn't mean B becomes 2*B, instead you find that B becomes about 2^1.5*B. The total area of several radial chutes placed in symmetry scales with their number to the power 1.5. This means that 2 radial chutes placed in symmetry are more efficient than 2 identical parachutes placed separately (same thing for more chutes).
Note that if you place several radial chutes not in symmetry (placing them separately) they add up linearly, like stack chutes.

For several chutes, the equation to find the terminal velocity of a vessel becomes: .

B is the "parachute constant" from earlier, unchanged even for several parachutes, m is the mass of the vessel, n is the number of parachutes and α is a number that depends on the type of chute used: it is 1 for stack chutes and radial chutes placed independently, and it is 1.5 for radial chutes placed in symmetry (to account for the non-linear increase).

If you have two different types of chutes, your terminal velocity is then:  (the summation is done this way because math), and so on for more types of chutes: .

By rearranging the equation, you can solve other parachute related problems:

• The maximal mass that can be slowed to under a given speed by a given number and type of parachutes is: , or if you have several types of parachutes: 
• The number of parachutes of a given type needed to slow a given mass to a given speed is: 

 

I am not physically (and mentally) capable or writing web pages or making mods so if anyone wants to take these stupid equations and make something useful out of them, feel free to.

 

EDIT:

I did a few measurements for the other bodies of the Kerbal system, to help people for interplanetary parachute landings. Use the constant B for another body with the same equations above, nothing else changes.

	Kerbin	Duna	Eve	Laythe
Mk16	0.029	0.079	0.009	0.036
Mk2-R (radial)	0.039	0.105	0.012	0.047
Mk16-XL	0.020	0.054	0.006	0.024
Mk25 (drogue)	1.099	2.967	0.340	1.331
Mk12-R (radial drogue)	1.065	2.877	0.330	1.291

All measurements were taken ASL (~370m for Duna) so your craft will fall a little faster than predicted if you're higher up.

If someone feels like getting values for the Sun or Jool go ahead.

Edited February 12, 2017 by Gaarst
Added values for other bodies

[Rath]

I can't understand but this page got me out of being caught on the forums in the middle of class ("Oh, looking at extra credit equations!  What a good student").

[paulprogart]

Neat, have always wondered how to calculate that properly rather than just slapping parachutes on until it "feels like enough".  Will have to try using this next time I get back to playing seriously...

I assume this is for standard spread angle (7)?  Would be nice to see it take that into account if it's not too difficult.  I always crank up mine to 10 to get the little bit of extra slowdown...

Some relevant threads in case they're useful:

On a related note (hopefully this isn't a hijack), does anybody know why spead angle isn't 10 by default?  It'd be more noob-friendly.  I'm far from an expert but with my limited knowledge I have trouble seeing what it could affect negatively other than maybe the safe deployment speed, and if you're flirting with that then it's probably not going to matter unless you really know what you're doing anyhow.

[GoSlash27]

Good work, Gaarst!

 I wonder what's up with symmetry altering the drag of the chutes. Sounds like a bug...

Best,
-Slashy

[Gaarst]

3 hours ago, paulprogart said:

Neat, have always wondered how to calculate that properly rather than just slapping parachutes on until it "feels like enough".  Will have to try using this next time I get back to playing seriously...

I assume this is for standard spread angle (7)?  Would be nice to see it take that into account if it's not too difficult.  I always crank up mine to 10 to get the little bit of extra slowdown...

Some relevant threads in case they're useful:

On a related note (hopefully this isn't a hijack), does anybody know why spead angle isn't 10 by default?  It'd be more noob-friendly.  I'm far from an expert but with my limited knowledge I have trouble seeing what it could affect negatively other than maybe the safe deployment speed, and if you're flirting with that then it's probably not going to matter unless you really know what you're doing anyhow.

I haven't tried messing with the spread angle (because I kinda forgot this even existed) so I don't know whether or not this affects the drag caused by the parachutes.

 

1 hour ago, GoSlash27 said:

Good work, Gaarst!

 I wonder what's up with symmetry altering the drag of the chutes. Sounds like a bug...

Best,
-Slashy

I don't think it's a bug. The results from both radial chutes (regular and drogue) seem very consistent with a 1.5th power law. I wouldn't be surprised that in real life some interactions between the parachutes could make it that the air disturbance extends further than the physical surface of the parachutes, but I'm not really sure.

[MalevolentNinja]

3 hours ago, paulprogart said:

On a related note (hopefully this isn't a hijack), does anybody know why spead angle isn't 10 by default?  It'd be more noob-friendly.  I'm far from an expert but with my limited knowledge I have trouble seeing what it could affect negatively other than maybe the safe deployment speed, and if you're flirting with that then it's probably not going to matter unless you really know what you're doing anyhow.

 

27 minutes ago, Gaarst said:

I haven't tried messing with the spread angle (because I kinda forgot this even existed) so I don't know whether or not this affects the drag caused by the parachutes.

 

This got me curious, so I did a quick test. On this vehicle (with 2 radial chutes) there was no performance difference between the default of 7 and a spread angle of 10. In both tests, once the chutes fully deployed, the vehicle descended at 5.6-5.7m/s.

 

So I'm pretty sure spread angle is just there 'cause it looks pretty.

 

Edited February 12, 2017 by MalevolentNinja

[Gaarst]

32 minutes ago, MalevolentNinja said:

This got me curious, so I did a quick test. On this vehicle (with 2 radial chutes) there was no performance difference between the default of 7 and a spread angle of 10. In both tests, once the chutes fully deployed, the vehicle descended at 5.6-5.7m/s.

(snip)

So I'm pretty sure spread angle is just there 'cause it looks pretty.

Do you use FAR or other mods changing the atmosphere? Because according to my numbers, your craft is twice heavier than it should to fall at this speed. I'd like a confirmation before re-doing everything from the start. 

[MalevolentNinja]

24 minutes ago, Gaarst said:

Do you use FAR or other mods changing the atmosphere? Because according to my numbers, your craft is twice heavier than it should to fall at this speed. I'd like a confirmation before re-doing everything from the start. 

No FAR, yes to DRE. I tested in my main game, which is the stock Kerbin system scaled up to 6.4x. So Sigma Dimensions is changing the atmosphere height, maybe it has changed other properties of the atmosphere? Ven's Revamp changes the chute textures and other properties. There's also 65 other mods...

OK, to make this easier, I recreated the test in a completely stock game  

Using the same exact vessel parts and configuration. With a default spread angle of 7 It now descends at ~11.0m/s (it fluctuates a little). With a spread angle of 10, this drops to ~10.0m/s. The ship weighs 9.8t when it's descending.

So whaddya know? My apologies for misleading in my first post. I know better than to test stock functionality in a heavily modded game.

 

 

Edited February 12, 2017 by MalevolentNinja

[GoSlash27]

1 hour ago, Gaarst said:

I don't think it's a bug. The results from both radial chutes (regular and drogue) seem very consistent with a 1.5th power law. I wouldn't be surprised that in real life some interactions between the parachutes could make it that the air disturbance extends further than the physical surface of the parachutes, but I'm not really sure.

Gaarst,

 But you did say that they add normally when placed without symmetry, right? Ergo, the difference in drag is due to symmetry rather than interference...

Best,
-Slashy

[John FX]

4 minutes ago, GoSlash27 said:

Gaarst,

 But you did say that they add normally when placed without symmetry, right? Ergo, the difference in drag is due to symmetry rather than interference...

Best,
-Slashy

Maybe KSP only calculates interference when parts are placed in symmetry. If I were a lazy programmer I would do it that way as calculating all the individually placed chutes in combination would be hard. The difference in drag could still be due to interference but only for symmetrically placed chutes because of lazy programming.

 

Edited February 12, 2017 by John FX

[Gaarst]

6 minutes ago, GoSlash27 said:

Gaarst,

 But you did say that they add normally when placed without symmetry, right? Ergo, the difference in drag is due to symmetry rather than interference...

Best,
-Slashy

I wouldn't be surprised that "some Unity excuse" forced the developers to implement interference only for symmetry with parachutes unable to interact with the ones that were placed independently. EDIT: John FX said it better than I did

I don't recall ever reading anything like this in the recent patch notes, so either this is very old and I somehow never noticed, or it's indeed a bug (or I just have a terrible memory).

Edited February 12, 2017 by Gaarst

[GoSlash27]

I just ran a test, and found the opposite result; chutes placed in symmetry on an otherwise identical testbed were slightly *more* effective than chutes placed without symmetry. The symmetrical setup had a terminal velocity of 3.15 m/sec, while the non-symmetrical setup was 3.6.

I'm running 1.22.1622, all stock.

Also, I can confirm the lack of interference when chutes aren't placed using symmetry.

Best,
-Slashy

Edited February 12, 2017 by GoSlash27

[Gaarst]

1 minute ago, GoSlash27 said:

I just ran a test, and found the opposite result; chutes placed in symmetry on an otherwise identical testbed were slightly *more* effective than chutes placed without symmetry. The symmetrical setup had a terminal velocity of 3.15 m/sec, while the non-symmetrical setup was 3.6.

I'm running 1.22.1622, all stock.

Also, I can confirm the lack of interference when chutes aren't placed using symmetry.

Best,
-Slashy

Which is exactly what I found in my measurements, I think you understood it the wrong way around  (I just realised that I never worded that finding correctly so I'm correcting this now)

[Nathair]

2 hours ago, MalevolentNinja said:

No FAR, yes to DRE.

Is there a valid 1.2.x DRE?

[MalevolentNinja]

https://github.com/Starwaster/DeadlyReentry/releases

[GoSlash27]

Gaarst,

 You should copy the OP to the tutorials section. Valuable info.

Best,
-Slashy

[Gaarst]

33 minutes ago, GoSlash27 said:

Gaarst,

 You should copy the OP to the tutorials section. Valuable info.

Best,
-Slashy

Done!

[DBT85]

I've never quite understood why some basic info isn't included in the text for the parts. Even just "Mk16 - Capable of safely landing a 1,000kg vessel on Kerbin" would help a bit.

[Zeiss Ikon]

1 hour ago, DBT85 said:

I've never quite understood why some basic info isn't included in the text for the parts. Even just "Mk16 - Capable of safely landing a 1,000kg vessel on Kerbin" would help a bit.

Well, for different components impacting the ground first, different descent rates are "safe" -- and likewise or landing in water (at reasonable velocities of, say, 3-15 m/s, water is a lot softer than ground, albeit not so much when velocity is, say, 100 m/s).  Probably better to rate this as "single parachute descent rate ASL of 5.4 m/s with 840 kg load."

[paulprogart]

Thanks for reconfirming spread angle does make a difference, @MalevolentNinja and @Gaarst. I'd discovered it the Kerbal way:  "Oh crap I'm descending too fast... hmm what's this do... woohoo that reduced my velocity by couple of tenths, maybe it'll be enough!  *splat* ... nope."

It's interesting to me that spread angle in KSP still seems somewhat mysterious and unexplored.  Obviously it's not all that important if you can do the calculations for the default spread because then you just use that many, and if it theoretically "wastes" one or two then who cares.  It's just surprising given how KSP players like to dissect everything about the physics to death and then dissect the dissections. 

I went poking around Google briefly on the subject and found this paper, which if I'm understanding correctly shows how spread angle ("rigging angle" in the paper) affects angle of descent, but I dunno if that's actually useful info here or not as I'm way out of my element.

[Sharpy]

Waait.

8 chutes in symmetry: 8^1.5 - equivalent to 22 chutes.

20x symmetry from EEX... nearly 90 chutes equivalent. This is getting insane.

[Gaarst]

5 minutes ago, paulprogart said:

Thanks for reconfirming spread angle does make a difference, @MalevolentNinja and @Gaarst. I'd discovered it the Kerbal way:  "Oh crap I'm descending too fast... hmm what's this do... woohoo that reduced my velocity by couple of tenths, maybe it'll be enough!  *splat* ... nope."

It's interesting to me that spread angle in KSP still seems somewhat mysterious and unexplored.  Obviously it's not all that important if you can do the calculations for the default spread because then you just use that many, and if it theoretically "wastes" one or two then who cares.  It's just surprising given how KSP players like to dissect everything about the physics to death and then dissect the dissections. 

I went poking around Google briefly on the subject and found this paper, which if I'm understanding correctly shows how spread angle ("rigging angle" in the paper) affects angle of descent, but I dunno if that's actually useful info here or not as I'm way out of my element.

@MalevolentNinja I'll do a few measurements on spread angles whenever I have time (so either tomorrow or next week-end) to try to put some numbers behind it.

1 minute ago, Sharpy said:

Waait.

8 chutes in symmetry: 8^1.5 - equivalent to 22 chutes.

20x symmetry from EEX... nearly 90 chutes equivalent. This is getting insane.

I don't know if larger numbers of symmetry than stock are affected by this, maybe the bonus is only coded for up to 8 parachutes, that'd be an interesting thing to look at.

[Sharpy]

43 minutes ago, Zeiss Ikon said:

(at reasonable velocities of, say, 3-15 m/s, water is a lot softer than ground, albeit not so much when velocity is, say, 100 m/s).

When you use tough parts on the bottom, impacting ground at 40-50m/s is quite survivable. Impacting water? That specific part will survive, the remainder of the craft will "splash down hard".

[DBT85]

2 hours ago, Zeiss Ikon said:

Well, for different components impacting the ground first, different descent rates are "safe" -- and likewise or landing in water (at reasonable velocities of, say, 3-15 m/s, water is a lot softer than ground, albeit not so much when velocity is, say, 100 m/s).  Probably better to rate this as "single parachute descent rate ASL of 5.4 m/s with 840 kg load."

Yeah, that would be fine. 

[Lunar Sea]

What does spread angle affect?