CG--RUDDER EFFECTIVENESS AND Vmc

[Mad (Flt) Scientist]

OK, WAG time, because we don't actually test minimum control speeds at other than aft c.g., that being the critical case, so I can't just look it up.

Let's assume that the ideal is to have the theoretical Vmca somewhere near stall speed, since that should ensure that the various takeoff speed limits are driven by e.g. 1.2Vs and not by 1.1 Vmca.

Therefore the lift coefficient in "steady flight at the stall" (yes, I know there's no such thing, but this is theoretically) is CLmax. If I bank the aircraft through 5 degrees, then the weight of the aircraft now acts 85degrees in the lift plane and 5 degrees in the sideforce plane. So the amount of sideforce I need to generate is:

Y=mgsin(5deg)

Since the lift and sideforce coefficients are nondimensionalised by the same factor, qS, I can make the links:

mg=qS*CLmax

so

qS*CY=qS*CLmax*sin(5deg)

so to prevent the aircraft accelerating in the sideways direction, we'll need a sideforce coefficient roughly

CY= CLmax * sin(5) ~0.08CLmax.

If we assume a CLmax of about 2, that gives us a CY of about 0.16.

As we move the c.g. back and forward, the couple due to the sideforce will change - the aerodynamic force is defined as acting at 25%cg (standard reference point for most aircraft) while the weight component of course acts at the cg. So the (dimensional) yawing moment is:

weight component * [cg-to-ref-point distance]

We've already defined the weight/sideforce as being about 0.16 CLmax. To get a non dimensional distance for yawing moment we need to refer the distance offset to the wing span.

non-dim offset = [cg-to-ref-point distance]/b
= (cg% - 0.25)*mac/b

b/mac is, of course, the aspect ratio. So the non-dimensional yawing moment due to the cg is:

weight component * [cg-to-ref-point distance] = 0.08CLmax * (cg% - 0.25)/AR

To make my sums easier I'm going to assume an aspect ratio of 8.

Therefore the yawing moment coefficient generated by the weight effect at 5 degrees bank will be

0.08 * 2.0 * (cg%-0.25) /8 = 0.02 * (cg%-0.25)

Therefore the delta yawing moment caused by moving the cg is:

deltaCG * 0.02

If one assumes a cg range of 20%, that means the yawing moment variation between forward and aft c.g would be approximately 0.004. That sounds pretty damned small, but to convert it to something useful, rudder angle, we need to consider how the rudder will generate a corresponding moment:

Cn(due to rudder) = Cn(rudder)*rudder.

Typical values for Cn(rudder) are of the order of 0.001 per degree (I don't have any exact numbers to hand). That would imply that some 4 degrees of rudder would be additionally needed at aft c.g. compared to forward. That's not insignificant.

For an aircraft which was rudder limited with perhaps 30 degrees of maximum rudder, in order to generate the equivalent of 4 degrees more rudder (or about 10% more yawing moment) would require flight about 5% faster - meaning a five knot or so penalty on Vmca, 6 knots or so on V2, etc. No performance engineer will lightly give up that kind of number.

Just to note that, of course, the effect of bank itself, rather than cg is MUCH more powerful than the numbers in my previous post. It's very easy to alter the rudder required to trim at a given speed by banking into and out of the live engine, which is of course why there has to be a bank angle limit in the regs.

PS I plan on checking the maths and assumptions at work tomorrow, just in case I dropped an order of magnitude somewhere by mistake.
OK, the main assumption I was worried about was the rudder derivative, 0.001 per degree; that checks about right on one of our aircraft (it's a function of angle of attack, but that's a good enough number for high alpha)