CLmax as f(cg), equation for estimation
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CLmax as f(cg), equation for estimation
Hello All - My first post.
Can anyone point me to the equation which corrects CLmax for a change in CG? In other words, if CLmax is measured to be 2.4 at 20% CG, what would the equation be to estimate CLmax at say 7% fwd CG? (all else being constant, e.g. weight + alt).
I recall seeing this from years ago; any help appreciated -
Can anyone point me to the equation which corrects CLmax for a change in CG? In other words, if CLmax is measured to be 2.4 at 20% CG, what would the equation be to estimate CLmax at say 7% fwd CG? (all else being constant, e.g. weight + alt).
I recall seeing this from years ago; any help appreciated -
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It's a matter of taking moments about the reference point to determine the extra tail load.
If the cg is X forward of the reference point and the tailplane is L aft of the reference point, then the extra download on the tail, T, due to the CG being forward is:
T = W * (X/L)
So the total "effective weight" is the actual weight plus the extra download, i.e. W+T
So that means that when CLmax was deteremined with the CG at the reference, for a weight W0, now that lift has to support W+T
W0 = W+T = W + W * (X/L) = W * { 1 + (X/L) }
reversing the equation, we get
W = W0 / { 1 + (X/L) }
But our reference CLmax0 came from the W0 case, so our new CLmax would be reduced by the same factor, i.e.
CLmax = CLmax0 / { 1 + (X/L) }
So the further forward the CG moves, the more CLmax appears to reduce. the longer the tail arm is, the less a given CG shift affects it.
Note that X and L both have to be in the same units - if L is in metres, so is X - you can't just use the %CG shift, you need to multiply it by the reference dimension (the mean chord, usually). Also, typically the reference CG is 25% of the chord, and the reference tail arm measured from the 25% wing chord position to the 25% tail chord position.
If the cg is X forward of the reference point and the tailplane is L aft of the reference point, then the extra download on the tail, T, due to the CG being forward is:
T = W * (X/L)
So the total "effective weight" is the actual weight plus the extra download, i.e. W+T
So that means that when CLmax was deteremined with the CG at the reference, for a weight W0, now that lift has to support W+T
W0 = W+T = W + W * (X/L) = W * { 1 + (X/L) }
reversing the equation, we get
W = W0 / { 1 + (X/L) }
But our reference CLmax0 came from the W0 case, so our new CLmax would be reduced by the same factor, i.e.
CLmax = CLmax0 / { 1 + (X/L) }
So the further forward the CG moves, the more CLmax appears to reduce. the longer the tail arm is, the less a given CG shift affects it.
Note that X and L both have to be in the same units - if L is in metres, so is X - you can't just use the %CG shift, you need to multiply it by the reference dimension (the mean chord, usually). Also, typically the reference CG is 25% of the chord, and the reference tail arm measured from the 25% wing chord position to the 25% tail chord position.
Of course that does assume the aeroplane has a tailplane at the back (which wasn't explicitly stated in the question) - if it's a canard you can get an answer for the apparent Cl(max) increase by making the moment length a negative number, and it all gets rather complicated for tail-less aeroplanes.
I'm obviously missing some thing here. I thought Cl(max) was a function of the particular airfoil eg NACA 2410 a little over 2.4 at a Reynolds 6X10^6. I thought angle of attack was the sole key, forgetting about the rate of change of angle of attack and it's effect on Cl(max). My Abbott & Doenhoff seemingly makes no mention of CoG having any effect. Confused. 
What are the aerodynamic reasons behind the phenomena? Thanks folks.
What are the aerodynamic reasons behind the phenomena? Thanks folks.
No, it's just a rather dubious* way of representing the overal "net lift" after accounting for tailplane download and the way tailplane download is a function of static margin. The idea is that if you looked at the maximum available lift using Cl(max) and the wing area as the reference area it would yeild a number that was significantly higher than the "real" maximum lift, because the "real" maximum lift would have to take into account the tailplane download. So they produce this numerical method which is based on the concept of calculating the tailplane download/static margin relationship, and then producing a function showing this as a Cl(max) to static magin relationship for calculation purposes.
The *actual* Cl(max) and the actual lift slope would remain exactly where the polars said they were, which is one of the reasons I don't like this simplification. Another is that it continues to reinforce the myth that the behaviour of 3-dimensional wings can be accurately described using just the 2-dimensional section data. Which is another way of pointing out that there are a few dozen other parameters which could just as validly come into play here rather than just CG position.
* in that while it may be numerically arguable it does not describe the physical reality
The *actual* Cl(max) and the actual lift slope would remain exactly where the polars said they were, which is one of the reasons I don't like this simplification. Another is that it continues to reinforce the myth that the behaviour of 3-dimensional wings can be accurately described using just the 2-dimensional section data. Which is another way of pointing out that there are a few dozen other parameters which could just as validly come into play here rather than just CG position.
* in that while it may be numerically arguable it does not describe the physical reality
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There's nothing "dubious" about correcting a value measured at one CG to the equivalent for a different CG. Indeed, it's REQUIRED that this be done when determining the performance data for an aircraft, which must be shown for the critical (usually most forward) CG. That's the context of the original question - correcting measured data from one reference to another.
It's got nothing to do with trying to analytically correct some theoretical "CLmax" to some other theoretical value. It's to do with the practical question of correcting measured test data.
And, speaking from that same practical point of view, i couldn't care less what the notional CLmax is of an aerofoil, or even of a wing for that matter. What I need to be able to do is present stall speeds in a flight manual, and to do that i need a means of determining that speed, which it is convenient to base on the concept of a maximum lift coefficient, corrected, as required by regulation, to the CG limit.
see AC25-7C, 29.d(5)(d) (on page 131 of the AC) for an "official" description of the correction
It's got nothing to do with trying to analytically correct some theoretical "CLmax" to some other theoretical value. It's to do with the practical question of correcting measured test data.
And, speaking from that same practical point of view, i couldn't care less what the notional CLmax is of an aerofoil, or even of a wing for that matter. What I need to be able to do is present stall speeds in a flight manual, and to do that i need a means of determining that speed, which it is convenient to base on the concept of a maximum lift coefficient, corrected, as required by regulation, to the CG limit.
see AC25-7C, 29.d(5)(d) (on page 131 of the AC) for an "official" description of the correction
Last edited by Mad (Flt) Scientist; 4th Jan 2018 at 16:48. Reason: Added reference to AC25-7C
But Cl(max) remains the same regardless of CG position - that's the point. Similarly the stalling angle-of-attack (alpha) also remains the same regardless of CG position. The speed at which the stalling angle is achieved in S&L flight will increase slightly as static margin increases (for a tailplane-at-the-back aeroplane), but this is not because Cl(max) has changed because it doesn't.
But for calculation purposes it can be useful to model it as a Cl(max) shift as a convenient conceit, in the same way that when analysing material properties it is a convenient conceit to assume that atoms solid, uniform spheres packed so close together that they touch their neighbours in a lattice, or when analysing cyclic motions it can be convenient to assume that -1 has a square root and it governs the behaviour. None of these are literally true - they are just convenient way of looking at or modelling things in specific contexts.
[soap-box mode]
As for stalling speeds in flight manuals - this is something I have always argued against because they are dangerously misleading. Stall speeds vary with weight, bank angle, CG, air density, phase of the moon and ambient mean grandmother's age (I may have made some of those up).
I have ALWAYS felt that all aircraft should simply be equipped with a good alpha gauge, because stalling alpha only really varies with density altitude (and even then not dramatically) so it would be extremely simple to have a single instrument which tells you directly and accurately just how close you are to stalling - it could even incorporate air density correction (mechanically - no bug-prone safety-critical software needed). Fast jet pilots love 'em, they're no more expensive to make than (say) a decent turn & slip indicator, and there's no earthly reason why this wonderful safety-enhancing instrument couldn't be a mandatory inclusion in everything from VLA upwards.
[end of soapbox mode]
But for calculation purposes it can be useful to model it as a Cl(max) shift as a convenient conceit, in the same way that when analysing material properties it is a convenient conceit to assume that atoms solid, uniform spheres packed so close together that they touch their neighbours in a lattice, or when analysing cyclic motions it can be convenient to assume that -1 has a square root and it governs the behaviour. None of these are literally true - they are just convenient way of looking at or modelling things in specific contexts.
[soap-box mode]
As for stalling speeds in flight manuals - this is something I have always argued against because they are dangerously misleading. Stall speeds vary with weight, bank angle, CG, air density, phase of the moon and ambient mean grandmother's age (I may have made some of those up).
I have ALWAYS felt that all aircraft should simply be equipped with a good alpha gauge, because stalling alpha only really varies with density altitude (and even then not dramatically) so it would be extremely simple to have a single instrument which tells you directly and accurately just how close you are to stalling - it could even incorporate air density correction (mechanically - no bug-prone safety-critical software needed). Fast jet pilots love 'em, they're no more expensive to make than (say) a decent turn & slip indicator, and there's no earthly reason why this wonderful safety-enhancing instrument couldn't be a mandatory inclusion in everything from VLA upwards.
[end of soapbox mode]
see AC25-7C, 29.d(5)(d) (on page 131 of the AC) for an "official" description of the correction
The correction factors you are talking about are those which are then applied to attempt to remove the sources of error in that test method - like the tailplane download adding to weight, the vertical component of thrust from the engine reducing weight and mach number corrections.
So it is using the variation in stall speed with CG position to calculate the Cl(max), not measuring a variation in Cl(max) with CG position. To clarify this point I'll go back to your original question:
The answer would be that if the method works the Cl(max) calculated for the 7% fwd CG would also be 2.4, but the airspeeds at which the stalls occurred (the test parameter being measured) would be different. The algorithms given in AC25-7C, 29.d(5)(d) would then (hopefully) yield the same Cl(max) from the two different airspeeds.
Last edited by PDR1; 4th Jan 2018 at 19:36.
