Formula for estimating missile performance

[saisran]

I found some really cool stuff about math and missiles wonder how can i use it to correct the missile performance in SF2? those ASM are flying at Mach 11!!!

Subject: A simple formula for estimating missile performance.

dwightlooi 3/15/2008 10:28:12 PM

 

I want to take this opportunity to introduce everyone to a very simple formula that

can be used for estimating the performance of a missile. It goes like this:-

 

Change in Velocity (Delta V) = 10 x Specific Impulse x LN (initial weight / final weight) m/s

 

This assumes that all the fuel is used to get the missile as fast as possible and

none is used to provide just enough thrust to sustain a given velocity.

In otherwords, it assumes an all-boost motor not a boost sustain motor.

 

For example, let'a take a look at the AIM-120A AMRAAM which we have some decent info on...

 

Launch weight = 335 lbs (Published stats)

Motor weight = 156 lbs (WPU-6/B HTPB rocket motor weight as per Raytheon)

Approximate specific impulse = 245 seconds (typical of HTPB solid motors)

Approximate fuel fraction of motor = 85% (typical of robust aluminum cased aerospace rocket motors)

 

OK... if 85% of the motor's mass is the fuel, we have about 132 lbs of fuel in the AMRAAM-A

-- roughly a 39.4% fuel fraction (sounds about right). So let's run the numbers...

 

Delta V = 10 x 245 x LN(335/(335-132)) = 1227 m/s

 

The formula predicts that the AMRAAM will go about 1227 m/s (~Mach 3.7) faster than it started.

If it is launched at say Mach 1.5 it'll be going Mach 5.2.

In reality the AMRAAM doesn't go that fast.

The reason is that not all the fuel is used to get it as fast as possible.

The AMRAAM's motor is a boost-sustain design.

It is probably grained to take the weapon to abut Mach 2.5~2.8 faster than it started at

(Mach 4+ in a typical Mach 1.5 release).

The rest of the fuel is shaped to burn much more slowly to keep it's velocity at

or near the achieved maximum out to a longer range before the motor burns out.

 

 

Well, for any given fuel fraction and specific impulse,

a designer can decide how fast he wants to go and how long he wants to stay at

or near the peak velocity achieved. For instance, if a missile carries 40% of its launch weight

as fuel and uses the typical a modern HTPB propellant motor, it can:-

 

(1) Spend 25% to get an approximate Mach 2.1 delta V and 15% on sustaining that speed for a relatively long while.

(2) Spend 30% to get an approximate Mach 2.7 delta V and 10% on sustaining that speed for a shorter while.

(3) Spend 40% to get an approximate Mach 3.8 delta V have no sustain burn time at all.

 

BTW, in reference to the above comment on deceleration... it doesn't really work that way.

If a missle starts at Mach 4 at burn out and decelerates 25% to Mach 3 after 10~15 seconds,

it WILL NOT decelerate to Mach 2 (another 33% from Mach 3) after 20~30 seconds.

This is impossible because aerodynamic drag (Fd = Cd x A x 0.5 x P x V^2) is a function of

the square of velocity.

As velocity decreases, drag force decreases exponentially in relation to it.

Hence, if the drag for at Mach 4 causes a 25% loss in velocity in 10~15 seconds,

there is no way a much lower drag force at Mach 3 will cause a 33% loss in velocity after

another 10~15 seconds.

What happens is that deceleration is non-linear;

you start off steep and the slope flattens out over time as velocity and hence drag drops.

It'll take a missile a heck of a lot longer to decelerate from Mach 4 to Mach 2 compared to

say Mach 2 to Mach 1 for instance.

 

 

 

Actually it also depends a heck of a lot on altitude (air density)...

Let's plug some numbers shall we?

 

Question: How much thrust is needed to sustain Mach 3.0 in an AAM like the AMRAAM?

 

Drag force (Newtons) = 0.5 x P x V^2 x Cd x A

 

P = Density of Air (kg/m^3) ; ~1.29 kg/m^3 @ sea level; ~0.232 kg/m^3 @ 12,000 m

V = Velocity (m/s) ; Mach 1 = 340 m/s @ sea level; ~295 m/s @ 12,000 m

Cd = Co-efficient of Drag ; ~ 0.6 to 0.95 for rockets depending mostly on finnage,

nose and tail profile

A = Sectional Area (m^2) ; ~ 0.025 m^2 for a 7" diameter missile.

 

For an AMRAAM like AAM going at high altitudes (40,000 ft)...

 

Drag Force @ Mach 3 = 0.5 x 0.232 x (295x3)^2 x 0.70 x 0.025 = 1590 Newtons = 357 lbs

Drag Force @ Mach 2 = 0.5 x 0.232 x (295x2)^2 x 0.70 x 0.025 = 707 Newtons = 159 lbs

Drag Force @ Mach 1 = 0.5 x 0.232 x 295^2 x 0.70 x 0.025 = 177 Newtons = 39.8 lbs

 

The same missile going Mach 3 at Sea Level...

 

Drag Force @ Mach 3 = 0.5 x 1.29 x (340x3)^2 x 0.70 x 0.025 = 11,744 Newtons = 2640 lbs

Drag Force @ Mach 2 = 0.5 x 1.29 x (340x2)^2 x 0.70 x 0.025 = 5,219 Newtons = 1173 lbs

Drag Force @ Mach 1 = 0.5 x 1.29 x 340^2 x 0.70 x 0.025 = 1,305 Newtons = 293 lbs

 

Assuming that there is no sustainer,

the deceleration experienced at Mach 3 by the 203 lbs (empty) missile is

 

Deceleration @ Mach 3 = -F / mass = -1590 / (203 x 0.454) = -17.3 m/s^2 = - Mach 0.059/sec @ 40,000 ft

Deceleration @ Mach 2 = -F / mass = -707 / (203 x 0.454) = -7.67 m/s^2 = - Mach 0.026/sec @ 40,000 ft

Deceleration @ Mach 1 = -F / mass = -177 / (203 x 0.454) = -1.92 m/s^2 = - Mach 0.0065/sec @ 40,000 ft

 

Deceleration @ Mach 3 = -F / mass = -11744 / (203 x 0.454) = -127 m/s^2 = - Mach 0.39/sec @ sea level

Deceleration @ Mach 2 = -F / mass = -5219 / (203 x 0.454) = -56.6 m/s^2 = - Mach 0.17/sec @ sea level

Deceleration @ Mach 1 = -F / mass = -1305 / (203 x 0.454) = -14.2 m/s^2 = - Mach 0.042/sec @ sea level

 

OK... enough of the math and the formulas... what does all these mean?

Well, it means that while coasting at Mach 3 an AAM is going to lose about less than 2% of

its velocity a second at high altitudes while it stands to lose about 13% of its velocity at

sea level! Huge difference isn't it?

Remember though that the rate of deceleration actually DECREASES as the

missile's velocity decreases.

It is easy to see that one can claim that a missile can burn out burn out its booster

and sustainer and be effective out to over 100 km at high altitudes or be useful only

against helos after 10km on the deck!

 

Also, we can make a pretty educated guess as to how much thrust the sustainer has to make.

An AMRAAM class missile with a 400 lbs sustain thrust will be able to stay

above Mach 3 at high altitudes and stay about Mach 1.2 at sea level.

An AMRAAM class missile carrying about 10% of its launch weight as sustainer

grained propellant will be able to keep this level of thrust lit for 20.5 seconds

in addition to whatever the boost time was using the 30% of its fuel to get a

roughly Mach 2.7 Delta V after launch.

A missile like this when fired at Mach 1.5 will reach Mach 4+ and keep

above Mach 3 for the duration of the sustainer at high altitudes.

It will also reach about Mach 2.5 and keep above about Mach 1.2 at sea level.

A motor grained for this thrust profile can have a 10 second boost at ~ 2460 lbs thrust and

a 20 second sustain burn at 400 lbs thrust -- this is a 5:1 boost sustain ratio.

This is also about right for thrust profiles of star grain vs

core burn solid propellant burn rate profiles.

 

 

 

 

Another rough rule of thumb:-

 

The time it takes for a missile to lose 25% of its velocity after burn out at supersonic speeds.

 

Never @ > 100,000 m (~300,000 ft) ; in space

~150 seconds @ 24,000 m (~80,000 ft)

~70 seconds @ 18,000 m (~ 60,000 ft)

~25 seconds @ 12,000 m (~ 40,000 ft)

~10 seconds @ 6,000 ft (~20,000 ft)

~5 seconds @ Sea Level

 

Remember, fractions over time are not additive.

In otherwords, if a missile loses about 25% of its velocity in 10 seconds,

in the 10 subsequent seconds (t =20s) the missile loses approximately another 25% of

the remaining 75% not a 100%. Total velocity loss is ~43.75% not 50%.

 

This is highly collated to the fall in air density.

Drag = 0.5 x P x V^2 x Cd x A.

Holding everything else constant Drag falls proportionally to density.

Drag also falls exponentially with Velocity which accounts for the loss in velocity

in the given time slices being about 25% instead of closer to 40%.

[orsin]

there is a special tool by HoneyFox that evaluates missile performance in download section.

 

Most of missiles need adjustments in order to align with their real world performance.

[usafphantom2]

Well all I know is the 1st & 2nd Gen Russian ones Stink!

[saisran]

there is a special tool by HoneyFox that evaluates missile performance in download section.

 

Most of missiles need adjustments in order to align with their real world performance.

 

How come i were not aware of that tool. thanks for the heads up

 

Edit: Ok what's the name of the tool?

Edited January 26, 2013 by saisran

[+SupGen]

Here: http://combatace.com...ange-simulator/

 

To be honest, I don't really see how this thing works; what is sonic speed? What does the gravity entry signify? There are separate entries for Gs of acceleration. What about the IAS/TAS entry; is that a ratio or a constant? Some pointers would be appreciated.

Edited January 26, 2013 by SupGen

[Fubar512]

The only constant that the game engine recognizes for missile acceleration is Gs. And it has no way of "varying" that acceleration over time (as would happen in RL). I once presented TK once with a set of tables gleamed from a web site, that determined acceleration over time and distance, giving the end produect as a terminal velocity. TK's argument was "Well, that may be so in a vaccum..." Huh, where did you go to school, TK? Sustained G-rate over a measured distance will still arrive at the same terminal velocity (velocity at the end of that acceleration run) whether in a vacuum, in an atmosphere, or in an ocean. Surface drag from media in which one is accelerating will lessen the rate of acceleration (Gs), but the end result will still be the same. .

[EricJ]

Which boils down to shooting at stuff and noting how it works and remembering it. Granted a great read so far but sometimes just getting in game and testing how it works is a different story or is the best way to understand performance because most of the time the hard data isn't always available.

Edited January 26, 2013 by EricJ

[saisran]

True, i just wanted to know long and howmuch boost and sustainer a missile should have so that it doesn't accelerate to mach 20. Probably will just do as Eric says and view the missiles in flight with the Debug. 3rd party weapons are mostly ok but the R-37 goes to 9+mach R-77 at 5M and R-27 goes to mach 5.5 half a sec after launch and sustains to target. His AIM-54 works quite well but still a bit over mach.only problem is the narrow FOV of the missile granting how large a radar it has.

Edited January 27, 2013 by saisran

[EricJ]

Actually my "debug" is the radar symbology but whatever works

[+SupGen]

Well, I did a little mooching around and found myself here: http://search.yahoo....avebooks.co.uk/ . Cleave Books. Conversions for just about anything. Turns out the Gravity entry is the value for 1g in Meters/Second/2. The Sonic entry I think is the value for Mach 1; it's 343 M/S, Wiki gives Mach 1 at Sea Level as 340.3. I think the IAS/TAS is a ratio. This thing is actualy kinda cool, I'm gonna take EricJ's advice and go shoot some stuff up and see how well it really works.

Edited January 27, 2013 by SupGen

[Fubar512]

Anything I code is as accurate as the game engine allows it to be. You'll not find any amateurish physics models or any such nonsense in any of my work. .

Edited January 27, 2013 by Fubar512

[EricJ]

I don't think anybody (I don't) takes your work for granted... it's just sometimes I get the more simple Soldier opinion of seeing how it works for real and base your opinions and observations on that.

 

Such as a 5.56mm round has a maximum range of 3,200m or so. Cool to know but doesn't mean anything to me. Now if you said you can kill somebody at 500m then that's more relative and more informative.

Edited January 27, 2013 by EricJ

[Fubar512]

IAS is not only effected by altitude, but by temperature as well. That's why Mach level is used. An aerodynamic body will generally behave the same at a given Mach number, irregardless of altitude. That's why the aerodynamic tables are all expressed in Mach numbers.

Edited January 27, 2013 by Fubar512

[+SupGen]

True, i just wanted to know long and howmuch boost and sustainer a missile should have so that it doesn't accelerate to mach 20. Probably will just do as Eric says and view the missiles in flight with the Debug. 3rd party weapons are mostly ok but the R-37 goes to 9+mach R-77 at 5M and R-27 goes to mach 5.5 half a sec after launch and sustains to target. His AIM-54 works quite well but still a bit over mach.only problem is the narrow FOV of the missile granting how large a radar it has.

 

Yeah, using this tool (I think correctly), the R-37 tops out at Mach 7.54. I think things would be starting to melt, there. The third-party AIM-54s hit almost Mach 6, so far it seems like I can get realistic speeds/ranges by balancing the Booster and Sustainer entries. For the AIM-54 it isn't necessary for the tool to show full maximum range as it doesn't model loft profiles. Using Booster G/Duration of 16.85/6 and Sustainer G/Duration of 4.0/50 for the AIM-54CE gave me a top speed of 1139.44 M/S, or Mach 3.44. The range given by the tool was only 75277.88 Meters-BUT I was able to fire at 99+ miles with a 60% kill ratio. That seems pretty "realistic"-or at least pretty much in the ballpark of what we've been led to expect. Try this thing out, it's one cool tool.

[saisran]

Yeah, using this tool (I think correctly), the R-37 tops out at Mach 7.54. I think things would be starting to melt, there. The third-party AIM-54s hit almost Mach 6, so far it seems like I can get realistic speeds/ranges by balancing the Booster and Sustainer entries. For the AIM-54 it isn't necessary for the tool to show full maximum range as it doesn't model loft profiles. Using Booster G/Duration of 16.85/6 and Sustainer G/Duration of 4.0/50 for the AIM-54CE gave me a top speed of 1139.44 M/S, or Mach 3.44. The range given by the tool was only 75277.88 Meters-BUT I was able to fire at 99+ miles with a 60% kill ratio. That seems pretty "realistic"-or at least pretty much in the ballpark of what we've been led to expect. Try this thing out, it's one cool tool.

 

For the AIM-54 I wa talking bout TK's Version that one goes just above mach 4 (although only tested two firings) Making its value as base(althugh some sources says the phoenix only reach mach 3.92, probably sea level launches) and just adding a low value on the sustainer at a lengthly duration to maintain constant speed and reach desired range as i think the missile can carry itself a long distance at sustained high speed (probably about mach3 and mach 2 for really max range) Just have to test it in game to see how it works

Edited January 27, 2013 by saisran

[orsin]

well, I assume the sonic speed means the actual 1Mach speed at altitude.

 

Here: http://combatace.com...ange-simulator/

 

To be honest, I don't really see how this thing works; what is sonic speed? What does the gravity entry signify? There are separate entries for Gs of acceleration. What about the IAS/TAS entry; is that a ratio or a constant? Some pointers would be appreciated.

 

I assume the sonic speed means the actual sound speed at simulation altitude, such that an entry will contrast with ground sound speed, and help achieve the IAS/TAS ratio.

 

As for the Gs, they could be acceleration Gs for booster and sustainer, etc.

 

 

With the help of this tools, we were able to adjust most of missiles. Even stock US/NATO missiles are way off accuracy.

 

After adjustment, we were able to see missile speed slow down to less than 1mach at max range. They struggle to catch up any air target in max range

[saisran]

well, I assume the sonic speed means the actual 1Mach speed at altitude.

 

 

 

I assume the sonic speed means the actual sound speed at simulation altitude, such that an entry will contrast with ground sound speed, and help achieve the IAS/TAS ratio.

 

As for the Gs, they could be acceleration Gs for booster and sustainer, etc.

 

 

With the help of this tools, we were able to adjust most of missiles. Even stock US/NATO missiles are way off accuracy.

 

After adjustment, we were able to see missile speed slow down to less than 1mach at max range. They struggle to catch up any air target in max range

 

Thanks this tool actually helps a lot. the Soviet missiles are actually more accurate now as they now have the room to manuver by flying in their designated speed and not flying at mach6,8 or 11.

[Fubar512]

Again, there are a lot of misconceptions in this thread. Mach numbers have nothing to do with altitude. There is no Mach level difference between sea level pressure and the air pressure at 50,000 feet. There is a difference in Mach level due to ambient temperature.

 

 

From the wikiedia:

 

The Mach number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At Standard Sea Level conditions (corresponding to a temperature of 15 degrees Celsius), the speed of sound is 340.3 m/s^[5] (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmospheric composition and largely independent of pressure. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at Mach 1 at 20°C or 68°F, at sea level, will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at -50°C or -58F, even though it is traveling at only 86% of its speed at higher temperature like 20°C or 68°F

 

Ergo, the best tool is one that expresses terminal velocity in meters/sec, and then you adjust the value to an average based on the intended operating altitude of that weapon.

Edited January 28, 2013 by Fubar512

[saisran]

Fubar that just makes it more complicated and my Literature inclined brain didn't understand any of that :). Anyway the tool Depicts the numbers in meter per second i just use an online converter to convert it to mach numbers and test it in game with the DeBug which displays mach number KIAS and m/s. It's just that TK and some missile models are a bit off for my taste and i find satisfaction in changing them to meet the published numbers found on the net. Having a missile goes 1834 right of the bat in my opinion is wrong. Mow those R-27 can now hit their target as i've been shot down more now than before as the missiles are not overspeeding.

Edited January 28, 2013 by saisran

[orsin]

Again, there are a lot of misconceptions in this thread. Mach numbers have nothing to do with altitude. There is no Mach level difference between sea level pressure and the air pressure at 50,000 feet. There is a difference in Mach level due to ambient temperature.

 

 

From the wikiedia:

 

The Mach number is commonly used both with objects traveling at high speed in a fluid, and with high-speed fluid flows inside channels such as nozzles, diffusers or wind tunnels. As it is defined as a ratio of two speeds, it is a dimensionless number. At Standard Sea Level conditions (corresponding to a temperature of 15 degrees Celsius), the speed of sound is 340.3 m/s^[5] (1225 km/h, or 761.2 mph, or 661.5 knots, or 1116 ft/s) in the Earth's atmosphere. The speed represented by Mach 1 is not a constant; for example, it is mostly dependent on temperature and atmospheric composition and largely independent of pressure. Since the speed of sound increases as the temperature increases, the actual speed of an object traveling at Mach 1 will depend on the fluid temperature around it. Mach number is useful because the fluid behaves in a similar way at the same Mach number. So, an aircraft traveling at Mach 1 at 20°C or 68°F, at sea level, will experience shock waves in much the same manner as when it is traveling at Mach 1 at 11,000 m (36,000 ft) at -50°C or -58F, even though it is traveling at only 86% of its speed at higher temperature like 20°C or 68°F

 

Ergo, the best tool is one that expresses terminal velocity in meters/sec, and then you adjust the value to an average based on the intended operating altitude of that weapon.

 

 

I am not the author of the software so my interpretation is just guesstimation.

 

From my understanding the software requires parameters such as missile acceleration at different stages, launch altitude and launch platform speed. so that it evaluates the range of the missile. The air friction varies at different altitude in game, and we found it is in general a linear function of speed given the altitude is constant.

 

Now, regarding the "sonic speed", I may have misinterpreted its use. however please not there are plenty of Mach-altitude tables online. Which are in contrast of your mention, but could have been deduced from experiential function of air density and average temperature....

[+SupGen]

Okee-dokee, given that math is not my strong suit, I had skipped working out the formula given in the first post and went straight for the tool; however I just tried the formula and it doesn't seem correct even for the example given. The formula is: Delta V=10xSpecific Impulsex(initial weight/final weight)in M/S. So plugging in the author's numbers we get: Delta V=10x245x(335/(335-132))in M/S. Or 10x245x(335/203)in M/S. Or, 10x245x1.65=4042.5 M/S, not 1227 as the author states. Am I screwing this up somehow, (wouldn't be the first time)? Any guidance would be appreciated. Thanks in advance.

[bwild]

Okee-dokee, given that math is not my strong suit, I had skipped working out the formula given in the first post and went straight for the tool; however I just tried the formula and it doesn't seem correct even for the example given. The formula is: Delta V=10xSpecific Impulsex(initial weight/final weight)in M/S. So plugging in the author's numbers we get: Delta V=10x245x(335/(335-132))in M/S. Or 10x245x(335/203)in M/S. Or, 10x245x1.65=4042.5 M/S, not 1227 as the author states. Am I screwing this up somehow, (wouldn't be the first time)? Any guidance would be appreciated. Thanks in advance.

 

you forgot to take the logarithm of the weight ratio, ln(1.65)=0.5

 

The equation is the first post is known as the Tsiolkovsky rocket equation, see http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

Edited January 31, 2013 by bwild

[+SupGen]

Thanks, buddy. I just pretty much found it all on Wiki, with some brain stretching, and an on-line calculator. (Eyes glazed over!) Natural logarithm. I cut out of three years of math classes to avoid this stuff, now I can't have fun without it. Life is just not fair!

[+SupGen]

Check out this missile performance tool I found online: http://www.ecf.utoronto.ca/~pavacic/missiles/minizap.zip This one kinda resembles the ThirdWire Weapons Editor. Very cool.

[saisran]

Nice find Supgen it even calculates Loft.