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Document title: Wing Loading on the Raptor - F-16.net - The Ultimate F-16 Reference
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Printed on: 18 November 2008

Forum: F-22A Raptor

Wing Loading on the Raptor



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Poll
Is wing loading truly a factor in the turning ability of a thrust vectoring fighter?
yes
85%
 85%  [ 17 ]
no
15%
 15%  [ 3 ]
Total Votes : 20


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Raptor_One
PostPosted: Jun 08, 2007 - 05:01 AM Reply with quote Back to top
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Crash,

If there actually was something to debate, I might agree with your analysis. Unfortunately, this was a simple case of one guy being VERY wrong about a huge number of things. When I, and then other more experienced/knowledgeable persons, tried to set bf-fly straight, he turned into an egomaniac. There is no excuse for that kind of bad behavior.
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johnwill
PostPosted: Jun 08, 2007 - 03:08 AM Reply with quote Back to top
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ATFS, let's hope so, I'd enjoy more "debate mode" with him. And thanks for your commentary, all good points to consider. What bothered me about him was his superiority complex - I'm a pilot, so I know everything. Well, I'm an engineer and I do not know everything and I'm willing to respectfully consider other points of view and maybe learn something.
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Tinito_16
PostPosted: Jun 08, 2007 - 03:51 AM Reply with quote Back to top
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You guys answered my question pretty much, and I want to thank you. Just remember, whomever you are, when there are discussions like this, it's best not to get too passionate or personal. This isn't life or death, this is one of my favorite message boards if not my outright favorite and when this kind of thing happens it just takes away from the discussion. Even if someone's wrong and doesn't want to admit it, we can learn from his/her point of view i.e. why that person believes in his opinion se fervently. Lets try to be nice to each other Cool
That said, it is a very interesting discussion we have going on.

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Raptor_One
PostPosted: Jun 08, 2007 - 04:35 AM Reply with quote Back to top
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Hehehe... the topic which sparked such heated debate is not really open to interpretation. There are no opinions on this one. There are the facts, and then there's everything else. I'm not sure what one can learn from reading misinformation. Something can be learned from reading the debunking of misinformation through cold, hard scientific fact, however. So in that sense, this thread was probably educational. I learned a few things myself.
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johnwill
PostPosted: Jun 08, 2007 - 05:09 AM Reply with quote Back to top
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Tinito, I tried my best to be respectful and courteous. If you can show me where I was otherwise, I will apologize. Of course, I agree with you completely about the need for civility. However, if you go back and read the posts, I think you will see who was courteous and who was not. bf-fly's childish rantings were not helpful to his position.

Idea Well, it finally dawned on me why bf-fly was so hung up on bank angle as THE controlling factor in turns. Commercial and light planes generally do not have g-meters and those pilots are taught to estimate load factor by observing bank angle in level turns. He doesn't know load factor effects in turn performance, because he doesn't know load factor.
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Afterburned
PostPosted: Jun 08, 2007 - 02:42 PM Reply with quote Back to top
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I didn't really want to get into this debate but I think i know that by-fly was saying when he mentioned the aircraft producing lift at 90 AoA. If you define lift as purely forces pointing in the axis away from the Earth's surface, what I believe by-fly's perception of lift was is the horizontal forces pointing parallel with the Earth's surface, but perpendicular to the plane of the wing created by the airfoil as it moves through the air. Kind of like how if you bank right 90 degrees... the immediate loss in lift causes the plane to fall from the sky, but the natural "lift" force created by the wing's airfoil will cause the plane to turn or move to the right even with no elevator input... sorry if this was already mentioned, didn't wanna sift through the drama back there haha
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Raptor_One
PostPosted: Jun 08, 2007 - 09:57 PM Reply with quote Back to top
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johnwill wrote:

Idea Well, it finally dawned on me why bf-fly was so hung up on bank angle as THE controlling factor in turns. Commercial and light planes generally do not have g-meters and those pilots are taught to estimate load factor by observing bank angle in level turns. He doesn't know load factor effects in turn performance, because he doesn't know load factor.


No kidding?! Are accelerometers that big of a deal to install? I can understand for light private planes, but commercial aircraft?
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Tinito_16
PostPosted: Jun 09, 2007 - 06:37 AM Reply with quote Back to top
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I just saw what bf-fly was saying in FSX, in the lessons. I think this whole thing is an incredible simplification, because it just doesn't make sense. Load factor or the g-force you feel has absolutely nothing to do with bank angle per-se, but with how fast you're turning with that bank angle. Because, if I'm in a plane on a constant heading and altitude, I'm feeling one g (toward the ground) regardless of the bank angle. I think the part about the stall speed increasing with bank angle is correct but I don't think a plane (much less a military one) will stall even on 90 degrees bank. A civilian plane, yes maybe, but even an F-4 has got to be able to get to extreme bank angles without stalling. The thing I guess bf-fly was getting at is that the bank angle affects lift against gravity, but I think with a little "up" rudder in an AF plane, this is counteracted. So... I don't know why instructors would teach that. How can the aircraft all of a sudden weigh twice as much at a certain bank angle (and worse, it said the pilot felt the g, too)? It's wrong...

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Raptor_One
PostPosted: Jun 09, 2007 - 07:12 AM Reply with quote Back to top
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Tinito_16 wrote:
I just saw what bf-fly was saying in FSX, in the lessons. I think this whole thing is an incredible simplification, because it just doesn't make sense. Load factor or the g-force you feel has absolutely nothing to do with bank angle per-se, but with how fast you're turning with that bank angle. Because, if I'm in a plane on a constant heading and altitude, I'm feeling one g (toward the ground) regardless of the bank angle. I think the part about the stall speed increasing with bank angle is correct but I don't think a plane (much less a military one) will stall even on 90 degrees bank. A civilian plane, yes maybe, but even an F-4 has got to be able to get to extreme bank angles without stalling. The thing I guess bf-fly was getting at is that the bank angle affects lift against gravity, but I think with a little "up" rudder in an AF plane, this is counteracted. So... I don't know why instructors would teach that. How can the aircraft all of a sudden weigh twice as much at a certain bank angle (and worse, it said the pilot felt the g, too)? It's wrong...


Do yourself a favor and pay NO attention to anything bf-fly posted. Look at the equations I wrote out and the posts by Jonhwill and Raptor_claw. Also keep in mind that wing loading, wing loading will not tell you how well the F-22 performs relative to other fighters unless you are given the aerodynamic coefficients for the F-22 and the fighters you are comparing it to. Specifically, you need to know the maximum lift coefficients. Forget about bank angle in the context of maximum performance flight maneuvers... please! Very Happy
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ATFS_Crash
PostPosted: Jun 09, 2007 - 12:43 PM Reply with quote Back to top
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Raptor_One wrote:
Crash,

If there actually was something to debate, I might agree with your analysis. Unfortunately, this was a simple case of one guy being VERY wrong about a huge number of things. When I, and then other more experienced/knowledgeable persons, tried to set bf-fly straight, he turned into an egomaniac. There is no excuse for that kind of bad behavior.


johnwill wrote:
ATFS, let's hope so, I'd enjoy more "debate mode" with him. And thanks for your commentary, all good points to consider. What bothered me about him was his superiority complex - I'm a pilot, so I know everything. Well, I'm an engineer and I do not know everything and I'm willing to respectfully consider other points of view and maybe learn something.


I'm sorry I really wasn't following the debate. I skimmed through it and really didn't understand what you were debating about, it didn't spark my interest enough spend the time to analyze deeper.

Part of my point is that most of us are pretty arrogant, take an honest look at your own posts.

Another part of my point is that no one is always right (except for maybe God or Jesus). So try to be forgiving of people if they are wrong.

Another part of my point is just because it is popular to think something is wrong, doesn't necessarily make it fact that it is wrong. If it wasn't for nuts like Galileo challenging convention that the world was the center of the solar system/universe, science would have not advanced. It takes people that are willing to break convention like Columbus saying the earth was round.

Sometimes in engineering particularly in the early stages or if they are having difficulties overcoming problems they will use something that resembles these weird “brainstorming sessions where people are encouraged to speak out with any idea no matter how stupid it may seem”. Sometimes people get stuck in a rut of thought, if you have a team of people that are trying to find a solution to a problem is less effective if they are all trying the same ideas the same ways that have been done before. Sometimes it takes a bit of foolishness and a willing to break from tradition to find solutions to problems.

Did Edison give up because he had failures? Did Edison give up because his ideas were scoffed at?

I have seen people argue with each other that actually were in agreement, but for some minor communication or comprehension problem they didn't realize that they were in agreement. I heard my slightly senile parents debating for a considerable time (hours) and seeming to go in circles, so I started to pay closer attention to their arguments and I realized that they both had the same position, they were just too hardheaded, forgetful and poor listeners that they didn't realize that they were in agreement. They usually were really good listeners and debaters and very intelligent, however everyone has their moments of foolishness.

Sometimes people in scientific community have extremely bizarre ways of looking at things and describing them. People often misinterpret what they are saying. Sometimes their concepts are so deep that people have a hard time understanding their concepts.

Most people that would meet Stephen Hawking (or Albert Einstein) on the street that did not know who he was would probably think he is an idiot.
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Raptor_One
PostPosted: Jun 10, 2007 - 03:57 AM Reply with quote Back to top
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ATFS_Crash wrote:

I'm sorry I really wasn't following the debate. I skimmed through it and really didn't understand what you were debating about, it didn't spark my interest enough spend the time to analyze deeper.

Part of my point is that most of us are pretty arrogant, take an honest look at your own posts.

Another part of my point is that no one is always right (except for maybe God or Jesus). So try to be forgiving of people if they are wrong.


If you only skimmed through this thread, you may have missed the parts where I, Raptor_Claw, and Johnwill tried to correct bf-fly on a number of points he made. It's not arrogant to correct someone on a subject you have a firm grasp of, whereas they do not. No one was railing on the guy for being wrong. He got railed on for insisting that he was right and all three of us were wrong. He first tried to use his 8000+ flight hours as a commercial airline pilot along with his numerous licenses to back up his incorrect statements. He then proceeded to link to unreliable internet sources that didn't prove a thing and were sometimes factually incorrect as well. Lastly, he tried to back up his arguments with ill-conceived examples that only served to reduce his credibility. Not for a second did he consider he might be wrong. He insisted that he was right and the three of us (myself, Raptor_claw, and Johnwill) were all wrong.

Quote:
Another part of my point is just because it is popular to think something is wrong, doesn't necessarily make it fact that it is wrong. If it wasn't for nuts like Galileo challenging convention that the world was the center of the solar system/universe, science would have not advanced. It takes people that are willing to break convention like Columbus saying the earth was round.


No one thought it was popular to think bf-fly was wrong. It was just he and I going at it for a while until Raptor_claw entered the debate. I'm pretty sure Raptor_claw has more aeronautical engineering knowledge and experience than I do. As for Johnwill, I have no doubt he is more knowledgeable in these subjects given the number of aircraft programs he's been involved with over the years. If you read certain posts in this thread carefully, you should understand exactly why bf-fly is wrong. The equations I wrote out for minimum turn radius and maximum turn rate should have been enough to set the guy straight, but he refused to pay them any mind.

With all due respect (and I do mean that), I thought your mention of Galileo and Columbus was utterly ridiculous and out of place. Are you implying that bf-fly is some misunderstood genius or a bold pioneer going against "conventional wisdom"? I assure you he's not. I can also assure you that the aeronautical engineering principles that were "debated" in this thread are well understood and fully accepted by the scientific community. bf-fly is not operating on a higher plane of understanding than everyone else here. If he can't clearly and coherently disprove the validity and/or relevance and/or applicability of a few simple aircraft performance equations, he's obviously in over his head.

Quote:
Sometimes in engineering particularly in the early stages or if they are having difficulties overcoming problems they will use something that resembles these weird “brainstorming sessions where people are encouraged to speak out with any idea no matter how stupid it may seem”. Sometimes people get stuck in a rut of thought, if you have a team of people that are trying to find a solution to a problem is less effective if they are all trying the same ideas the same ways that have been done before. Sometimes it takes a bit of foolishness and a willing to break from tradition to find solutions to problems.


Say what? What are these problems you're talking about? The only problem that existed in this thread was a forum member's lack of fundamental aeronautical engineering knowledge. You act as though this were a debate about M theory (aka string theory). It's not that complicated, trust me.

Quote:
Did Edison give up because he had failures? Did Edison give up because his ideas were scoffed at?


Oh my... this is rich. I'm sorry, but bf-fly was not presenting novel ideas. He was making factually incorrect statements that can, and were proved wrong... easily. This is not the year 1900 before Orville and Wilbur Wright made the first powered flight of an aircraft. We, as a civilization, have over 100 years of aeronautical engineering knowledge and experience. There are, of course, aspects of air and space flight we don't fully comprehend, but this thread hasn't touched on any of them.

Quote:
I have seen people argue with each other that actually were in agreement, but for some minor communication or comprehension problem they didn't realize that they were in agreement. I heard my slightly senile parents debating for a considerable time (hours) and seeming to go in circles, so I started to pay closer attention to their arguments and I realized that they both had the same position, they were just too hardheaded, forgetful and poor listeners that they didn't realize that they were in agreement. They usually were really good listeners and debaters and very intelligent, however everyone has their moments of foolishness.


Trust me, bf-fly CLEARLY has some fundamental misconceptions about aircraft flight from a scientific/engineering standpoint. At least three different people agreed upon this.

Quote:
Sometimes people in scientific community have extremely bizarre ways of looking at things and describing them. People often misinterpret what they are saying. Sometimes their concepts are so deep that people have a hard time understanding their concepts.


If simple algebraic equations are extremely bizarre ways of analyzing basic aircraft performance, I don't know what constitutes simple. Minimum turn radius, angle of attack, lift coefficient, and so on and so forth are NOT deep concepts. Students majoring in aerospace engineering begin with these concepts. It doesn't get any easier. Aircraft and spacecraft performance was the first aerospace engineering course I took in college. There weren't many prerequisites for the course either. Basic integral and differential calculus hardly even came into play.

Quote:

Most people that would meet Stephen Hawking (or Albert Einstein) on the street that did not know who he was would probably think he is an idiot.


Stephen Hawking is probably a bad example, but Albert Einstein? No, I don't think the average Joe would have thought Einstein the fool. Anyone with enough education to even grasp Einstein's theories would know the man was no fool. If someone doesn't know how to add, they shouldn't think someone the fool for telling them 2+2=4. Your arguments just don't hold water I'm afraid. Ignorance is not an excuse to attack knowledge. And yes, I'm talking about 2+2=4 knowledge. Not E=mc^2 knowledge.
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bf-fly
PostPosted: Jun 10, 2007 - 01:47 PM Reply with quote Back to top
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At no time did I intend nor desire to insult you guys, I only tried to explain something to you. I refrained on many occaision from getting personal, the reverse is not true.

Once more for old times sake, then you can resume insulting me if you wish. I only ask you you calmly read what I wrote, and cross reference it with material publically available, specifically bank angle/load factor charts and stall speed/load factor charts. For those of you who chose to turn a blind eye, I have no expectations. to the others I hope you will thoughtfully read what I have written. Thank you:


A aircraft in a level flight must always create enough lift to counter it's weight to maintain altitude. Too much and it climbs, too little an it decends. In constant speed, constant altitude turn, like you see at an airshow, it must somehow also counter the centrifical force that a turn creates. As the bank steepens, the centrifical force increases. This must be countered by increased lift since there is no other force or mechanism available (save TVC)(thrust compensates for drag). This countering force is called the horizonal component of lift. The vertical component maintains altitude as always. This additional lifting force required to counter centrifical force is also called load factor. There is a direct correlation between the angle of bank and the load factor. This is an unquestioned fact (except perhaps here).

The classical example is a 60 degree bank, (constant altitude turn, constant airspeed) because the math becomes easy. At 60 degrees, the weight of the a/c must be countered (as always) to maintain alttude, but also at 60 the centrifical generated also happens to equal the weight of the airplane. To compensate for both the vertical AND now horizonal (ie centrifical) component of lift, the plane must generate twice the normal amount of lift. If it did not, it would lose altitude. Ther are two way to increase lift, increase speed, or increase the Angle of Attack. If we are at a constant speed, then the only way to increase lift is to increase the angle of attack of the wing, AoA. This is why the stick or controls must be pulled back in a turn to maintain altitude, ie increase the AoA. If you simply banked but did not pull back, you will lose altitude.

These following examples work if all else is equal. We know in real life that is not true, especially comparing different types of aircraft.

Lets take the F-16 (or F-22 if you wish) and create articicial limits. This is for a constant altitude turn, constant speed turn:

One F-16 we will artificially limit it to 20 degrees AoA, the other to 15.

In a 15 degree bank, the one with the higher AoA limit can fly slower, therefore fly a smaller radius turn. That is a given.

Now lets do a 60 degree bank (2g's). As the turn steepens, so does the load factor increase (ie, centrifical force). As the load factor increases, so does the stalling speed because now in essense the plane weights twice as much. In a 60 degree bank constant altitude the stall speed increases about 1.5 times (actually about 1.41 if I recall correctly). This is an unquestioned fact. Google a load factor chart, use any source you like. An aircraft that stalls at 100 knots wings level will now stall at about 150 in a 60 degree bank. This is called an Accelerated Stall.

At 80 degrees of bank the stall speed increases 3 times, (check your chart) so that aircraft will now stall at 300 knots. The aircraft with the higher AoA limit, be that it's actual stall speed or it's artificial stall speed for the purpose of this example, will have a lower accelerated stalling speed. Hense, it can bank more steeply. Why? Plane A without limits may stall at say 80 (knots or MPH whichever you prefer), plane B is articicially limited so it will reach that limit at say 110. In a 80 degree bank, plane A will stall at 240 (3 times 80), plane B will stall at 330 (3 times 110).

What does that mean to radius? The plane with 20 AoA can both fly slower and bank more steeply. Not sure? look at it the opposite way. At 250, plane A can bank at or very close to 80 degrees, plane B can only bank at 250 about 75 degrees. (check you chart) This will create the smallest radius of turn. Same speed, different radius.

How do you get the smallest overall radius? The slowest speed at the greatest bank angle. I have provided this link on several occiasions. You can cross reference this charts provided with any source you like to verify them. I am providing this because it has all 3 charts, load factor, stall speed/bank and minimum radius turns per speed an bank angle. Use any source you like to verify these charts


http://www.aerospaceweb.org/question/pe ... 0146.shtml


Example 2, wing loading

Again, choose a F-16 or 22 or 747 is doesn't matter, just 2 identical planes except the weight different.

Plane A has no weapons, and half tanks of fuel. Plane B is 20% heavier (however way you wish to load it without additional external drag to complicate things).

Plane A is lighter, therefore has a lower wing loading. Plane B has a higher wing loading.

Plane A must maintain a positive AoA attack to maintain altitude (all planes do, save TVC)

Plane B is heavier so it must create more lift to counter that weight. Since it's identical to plane A it must have a higher AoA to create that extra lift (without getting into high lift devices)

Plane A, just for the sake of a baseline, lets say it needs 2 degrees AoA to maintain altitude. Plane B will need more, lets say 4 degrees. Each will stall about the same, when it reaches critical AoA (or CLmax if you wish), but at different speeds. Plane B is already at a disadvantage. Lets say each stall at 20 AoA. Plane A can fly slower since it is lifting less it's only used up 2 degrees of it's AoA limit to maintain alt. Plane B has used up 4. When plane A reaches a speed near it's stall speed, plane B has already stalled.

Wing loading and it's role in maneuverability really is only a guide or reference. There are too many additional variables to account for and wing loading is an oversimplification. However to say that a lower wing loading generally equals better maneuverability is mostly true.

The calculation provided of Rmin is also one of these comparisions. It creates a roughly equal condition so an inference can be drawn. But it does not calculate the actual min radius of turn for a specifice a/c since it only assigns 1 load factor. If you change the value of n, then you are changing the load factor, ie, changing the bank angle.


I said a while ago the following

all things being equal:

The plane with the highest AoA limit will have the smallest radius turn

The plane able to bank the steepest will have the smallest radius turn

The plane with the lowest load factor will have the smallest radius turn.

And lastly, a 90 degree bank turn is not possible (save TVC) because the is no vertical component of lift. From 85 to 90 the G loads get so great that it becomes an imposibility due to structural or pilot limitations. If you look closely at any fighter doing a min radius 7-9g turn at an airshow, they are in the 75-80 degree bank range.

The chart I provided gives the speed and bank angle reference as it pertains to minium radius of turn. It is correct.

To whoever took the time to read this and the source material, Thank you. You'll note I'm the only one who backed up what they said with sources.

To be sure, I expect more insults. So be it.

BYE.
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Raptor_One
PostPosted: Jun 10, 2007 - 04:38 PM Reply with quote Back to top
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Well, well, well... look who couldn't stay away. Didn't you say you were done with this forum? Do you want someone to correct you yet again, or are you hoping that people will eventually get tired of arguing with you and give you the last (wrong) word? I know you think that everything you're stating is correct. Most of what you say is not correct, however. Let me just address a single thing you've gotten wrong...

Quote:

The calculation provided of Rmin is also one of these comparisions. It creates a roughly equal condition so an inference can be drawn. But it does not calculate the actual min radius of turn for a specifice a/c since it only assigns 1 load factor. If you change the value of n, then you are changing the load factor, ie, changing the bank angle.


I assume when you say "1 load factor" you mean a single load factor and not a load factor, n=1. If you substituted n=1 into the equation below you'd get an undefined (i.e. infinite) Rmin.

**correction
Rmin = Vsturn^2 / {g * (n^2 - 1)^0.5}

If n=1, the term (n^2-1)^0.5 will be...

(1^2-1)^0.5 --> (1-1)^0.5 --> (0)^0.5 ---> 0

So the denominator becomes {g * 0} .... which equals 0.

Moving on...

What you fail to understand about the variable n in the Rmin equation is that it's not a predetermined value. It's whatever value minimizes Rmin. The value of Vsturn is given by...

*** correction

Vsturn = { (2*n*W) / (rho * S * CLmax) }^0.5.

You can simply substitute this into the equation for Rmin. When you do that, you'll find that the only unknown variable on the right side of the Rmin equation is the load factor, n. To find where Rmin is at its minimum value, you just graph the Rmin equation for all conceivable values of n. In other words, if the aircraft is limited to 9.0g, you would graph the Rmin equation for values of n in the range 1.0 to 9.0. As I noted before, 1.0 gives an undefined/infinite turn radius, so you wouldn't actually include this lower limit. You'd start at 1.01 or 1.1 or something like that. Anyway, the graph of Rmin as a function of n will have a minimum value somewhere. The value of n that corresponds to this minimum Rmin value is the load factor that gives you your absolute minimum turn radius. Once you graphically determine this load factor for minimum turning radius, you can figure out what your bank angle will be. You won't know the bank angle ahead of time.

Like I said, your bank angle is what it is. If I, as the engineer, tell a pilot that his aircraft's minimum turning radius at 1,000 ft occurs at a speed of 100 KIAS for an aircraft gross weight of X lbs, the pilot will enter a level turn around that airspeed and pull as much g as possible such that the aircraft does not lose altitude and maintains a constant speed. The pilot will adjust the bank angle as necessary to maintain the steady, level turn giving minimum turn radius. If I tell the pilot what his load factor will be during this minimum radius turn, that will just give him yet another reference point.


Last edited by Raptor_One on Jun 11, 2007 - 02:17 AM; edited 2 times in total
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johnwill
PostPosted: Jun 10, 2007 - 07:56 PM Reply with quote Back to top
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bf-fly,

Welcome back, sincerely. There was never any intention to insult you (although it was a temptation). I am sorry you felt insulted. You had to take some fairly intense criticism, not because of mistakes in your arguments, but because of your "I'm a pilot, so I know all about airplanes" attitude. The aerodynamic and flight mechanics principles you talk about are all basics that I learned when you were in diapers. If we could sit down at a table with a pad of paper, I am sure we could get this topic settled.

Although you might not admit it, I think you have learned something from the discussions. You learned what normal load factor is for one thing. Your last post was your best, so there is hope. And study the difference between load factor and acceleration. Many aero engineers have never stopped to think through the relationship, but they are different most of the time.
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Raptor_One
PostPosted: Jun 11, 2007 - 02:02 AM Reply with quote Back to top
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bf-fly wrote:

In a 15 degree bank, the one with the higher AoA limit can fly slower, therefore fly a smaller radius turn. That is a given.


Here's another example of where you keep getting it wrong despite being told again and again that it's NOT angle of attack limit (or critical angle of attack) that determines an aircraft's turn radius. Aside from the aircraft's weight which is always a factor in just about any aspect of fundamental aircraft flight performance, it's the maximum load factor an aircraft can generate in a level turn at the aircraft's level turn stall speed given by

***correction

Vsturn = { (2*n*W) / (rho * S * CLmax) }^0.5.

If you're not talking about the aircraft's stall speed, then you just use whatever velocity you want in the Rmin equation, which makes it simply a turning radius equation...

R = V^2 / {g * (n^2 - 1)^0.5}

Once you decide the turn velocity, how much load factor you choose to apply in the turn determines your turning radius. And yes, to the extent that the equations relating bank angle and load factor are accurate (they're not exact relations as Raptor_claw pointed out a while back), the amount of bank you apply will determine your load factor assuming you, as the pilot, don't allow the aircraft to change altitude during the turn.

Now where did "angle of attack limit" come into play here? Nowhere. Where did critical angle of attack come into play here? Nowhere except indirectly in terms of the minimum radius turn equation in which you turn at your stall speed, Vsturn. Since Vsturn depends on the aircraft's CLmax, and since CLmax happens at the critical or limit AoA, Rmin is indirectly tied to critical/limit AoA... but only in the sense that CLmax occurs at this AoA. There's no equation relating an aircraft's angle of attack and it's load factor. Nor is there any equation relating angle of attack and turning radius. To relate angle of attack and turning radius, you'd have to draw a graph of turning radius R, vs. AoA from the aircraft's CL vs. AoA curve. This, of course, is nonsensical, but that's what you'd have to do. CL is related to AoA by... a graph. You determine CL vs. AoA experimentally. Do you now understand why your statement above is incorrect? If not, what is your confusion?
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