\tan(\alpha)=u/V & \sin(\beta)=v/V
where
V = u
i + v
j + w
k &
V =\sqrt[u^2+v^2+w^2]
The way I see it is once you've got an assymetric condition you've got two options once you've noticed it, whereupon by definition you've got an angle of sideslip, and depending on the stability derivative C_l_\beta (& C_l_r), some roll angle. Which one you use is a matter of preference in most cases but in critical cases where you've let the angle get large and configuration and performance considerations come into play you best choose one not the other:
1) Jump on the appropriate rudder to arrest the yaw. C_n_\beta i.e., static directional stability requirements tend to ensure the contribution of the fin is to oppose any increasing sideslip, and with the effective fin angle of attack increased by the rudder deflection, a flight path with a characteristic \beta will develop where the forces and moments are in balance. Its a dirty flight condition and unless C_l_\beta ~=0, you need an initial amount of aileron to re-gain wings level. The actual value of sideslip that develops is merely a function of how much load you choose to apply to the rudder.
However, if you want to continue in
straight flight, the lateral loads on the aircraft must also sum to zero whereupon the value of \beta will probably be fixed and unique.
2) Accomplish the above, but now you've got an aft CG condition and you're at MTOW and you've let \beta get large (on Sector 4, Day 6, 0200Z perhaps?) at 200' when a donk lets go off R/W32 @ AGP, 32C and 975mb. This is approaching a V_mca condition.
JAR 25.149 requires no more than 150lbs/667N of force be applied to the rudder controls in order to restore directional control in this condition and continue in
straight flight. Neither may thrust be reduced.
The proviso for no more than 5 degrees angle of bank is presumably because if one supposes that full rudder deflection is required to counteract the asymmetric yawing tendency, you have lost the ability to control the path of the aircraft in azimuth through space via expedient use of the rudder. Hence the allowance for use of bank. Since the lateral component of lift due to bank is proportional to \sin(\gamma) you now have ~=9% of the aircraft lift to aid the process.
The aerodynamic advantages of the use of bank are numerous, and again, because the vertical component of lift whilst banked is proportional to \cos(\gamma) you now still have ~=99.5% of the original lift available for performance at 5 degrees of bank.
My concern is that for the purposes of calculating RTOW's at speeds approaching V_mca (or for that matter any speed), the performance will clearly be affected by whether or not you choose to employ some bank angle. Oddly, there is no performance requirement to be met associated with V_mca.
For presumably 99% of the time, you needn't concern yourself with this matter, but in the interests of discussion, if the only way you can meet a climb gradient requirement is by flying in a particular configuration, it becomes paramount to ensure you do so.
Now if only I could fly within +/- 5 degrees of bank.
PS: V_mca certainly
is affected by CG location.
From JAR 25.149:
(c) VMC may not exceed 1·13 VSR with –
(1) Maximum available take-off power or
thrust on the engines;
(2) The most unfavourable centre of
gravity;
(3) The aeroplane trimmed for take-off;
(4) The maximum sea-level take-off
weight (or any lesser weight necessary to show
VMC);
(5) The aeroplane in the most critical
take-off configuration existing along the flight
path after the aeroplane becomes airborne, except
with the landing gear retracted;
(6) The aeroplane airborne and the
ground effect negligible; and
(7) If applicable, the propeller of the
inoperative engine –
(i) Windmilling;
(ii) In the most probable position
for the specific design of the propeller
control; or
(iii) Feathered, if the aeroplane has
an automatic feathering device acceptable
for showing compliance with the climb
requirements of JAR 25.121.
It affects the length of the tail moment arm, a crucial component towards directional stability.