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Old 21st Sep 2004, 15:31
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Mad (Flt) Scientist
 
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The axis for considering forces in the horizontal is the lateral centre of pressure unless manoeuvre is also considered.
Technically, it does not matter whether one considers the equations of motion to be solved about any particular reference point; what is important is that all the relevant forces acting upon the aircraft be considered to be acting at their appropriate locations. How you choose to resolve the sum of the forces and moments is irrelevant.

If one considers the simplified Vmca case of wings-level flight, where the yaw control system only generates a couple, then the moment balance about the yaw axis is simple: the engine generates a yawing moment arising from the laterally displaced force axis, and the yaw control must generate a moment equal and opposite to the engine-induced moment. In this case the longitudinal cg position is irrelevant, and the aircraft is found to be stable at a zero sideslip case.

However, there is no yaw control on an aircraft which generates a pure couple; instead, the rudder will generate a sideforce which is removed from the centre of the aircraft; this therefore generates a moment.

If we now seek to trim the aircraft in a wings level condition, we quickly find that a solution at zero sideslip does not exist, as the sideforce exerted by the rudder cannot be balanced. Therefore, in order to trim a Vmca-like condition with zero bank, we must have a (small) angle of sideslip on the aircraft.

In order to determine the trim conditions (steady state solution for the equation of motion) we must obtain solutions to two simultaneous equations, requiring that both sideforce and yawing moment be balanced. (Again, our case is simplified by omitting the roll axis).

CY(total) = CY(beta)*beta + CY(rudder)*rudder = 0

Cn(total) = Cn(beta)*beta + Cn(rudder)*rudder + Cn(OEI) = 0

The aerodynamic derivatives in the above equations are not defined about the c.g., but rather an arbitrary reference point; c.g. has no bearing on these values. The Cn(OEI) engine yawing moment term is not affected by longitudinal c.g. position.

Therefore there is a single solution for beta and rudder which is not dependent upon c.g. position (or mass, or weight).

If now we adopt a slightly more representative Vmca condition and permit a little bank angle to be used, there is now a component of the aircraft weight acting at the cg (of course) in the sideforce axis. This provides a fixed offset to the CY(total) equation, and provides an offset to the Cn(total) equation which is a function of the cg position. (It's the weight component in the sideforce axis multiplied by the distance between the c.g. and the reference point used for Cn, all normalised appropriately).

Note that "lift" doesn't actually enter into this at all - the force and moment balances we're dealing with are in the Y axis direction and about the Y axis.

Adding the roll axis moment balance to the considerations doesn't materially affect the conclusion. The lift, drag and pitching moment don't matter at all, unless we get an indirect effect on derivatives due to changes in e.g. alpha, which again we're neglecting.

So I have to say I'm not convinced that cg is of no effect on Vmca.

Consequently when you use any bank to assist in asymmetric control the small resolved portion of lift acts from the centre of total lift most often with its centre above the cg. The effect into the horizontal plane is so insignificant that for practical purposes it can be ignored.
Bank is a very powerful effect on rudder required to trim, because its affecting the weight component and hence the yawing moment generated by the weight about the reference point. I hope that statement isn't implying that bank isn't important for Vmca.
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