Well if empirical flight test results demonstrate otherwise, the mathematics requires amendement. One should ensure the facts work with the hypothesis not vice versa!
I make the following comments.
I see this as essentially a lateral/directional stability problem. Normally these calculations are resolved about the CG.
It is the load on the fin that counteracts the component of the lift along the inertial y-axis, thereby allowing straight flight inspite of the bank angle.
Lets assume an a/c with C_l_\beta ~= 0 and where the No 1 eng fails. In this case the roll DOF is not important - we can assume wings level. Assuming the rudder has sufficient authority to counteract the assymmetric yaw, an equilibrium position will develop where the a/c is sideslipped away from the live engine. Unfortunately relative to inertial space, the a/c cannot maintain the runway centreline because resolving the thrust vector and fin load, they both in the same direction.
In order to arrest the azimuthal drift, you need to reverse the sense of the sideslip. This is done via judicious use of the rudder.
However, in reversing the sense of the sideslip, the rudder is now working hard. Not only must it balance the component of thrust resolved in the relevant axis, but its natural propensity to generate a load in the same direction courtesy of the sideslip.
Now assume by this judicious use of rudder, whereupon I'm able to change the flight path vector in azimuth, I've regained the runway centreline.
One way of off-loading the rudder is merely to incline the lift vector in order to generate a component in the direction of the inertial y-axis i.e., bank.
If I do this without reducing the sideslip, i.e., the load on the rudder, the aircraft will now drift in azimuth away from the runway centreline in the direction of the live engine.
Reduce the sideslip and it should be possible to balance the forces again all the way until you have a zero sideslip condition.
You are merely balancing the thrust component, the lift component and the fin sideload along the inertial y-axis via a combination of sideslip and roll angle.
That said, I don't believe you can have a nil sideslip, nil rudder force, angle of bank /= 0 condition and maintain the runway heading because the fin is symmetric. In this case, the only way to counter the tendency of the a/c to drift in the direction of the bank is to counter with rudder. This effectively cambers the rudder whereupon you get a sideload.
As for V_mca....
The way I understand it, you can change the available moment necessary to counteract asymmetric yaw due to an ENG OUT situation via modification of the force or the distance that the opposing aerodynamic surface generates.
Whereupon my conclusion that the CG location is intimately acquainted with the concept....
