Let's go through the calculations. Actually, this will answer markfer20's original question on how to do the calculations.
Using Centuri TIR-33;
http://www.rockets4schools.org/images/Calculating.pdf
Using the Javelin rocket example on Pages 21-23 of TIR-33, the Javelin Nose and Fin fb (Fin in the Presence of the Body) CNalpha and CP are:
The Total Rocket CP is 11.3 inches.
We will assume that the Nose is bent up 2.0 degrees, so, as an example, when the rocket is at an angle of attack of 2.0 deg, the local angle of attack for the Nose will be 4.0 deg. The local angle of attack for the Fins will be 2.0 deg.
The Moment Reference will be about the Center of Gravity (CG). Assuming 1.0 Caliber Stabilit Margin, TIR-33 has the Javelin CG at:
The Moments are balanced about the CG, using the Nose Contribution and the Fin Contribution. The moment arms are the distances from the component CP's to the Rocket CG:
Note, that as will be seen, we can't balance the CNalpha's, we have to balance the Normal Forces.
The Nose and Fin Normal Forces are:
Note that 1/2 rho V^2 (dynamic pressure) and Aref (the Reference Area) will be on both sides of the equation, and will drop off.
With a positive angle of attack, the moment from the Nose tries to rotate the rocket clockwise, the moment from the Fins tries to rotate the rocket counterclockwise. If the two moments are equal, there is no rotation of the rocket. This will occur at the trim angle of attack.
Balancing the moments:
(CNalphaNose x (aoa + 2 deg)) x (Xcg - NoseCP) = (CNalphaFin)fb x aoa x (FinCP-Xcg)
Putting in the numbers, converting 2 deg to radians (the Barrowman CNalpha's are per radian), 2 deg = 0.03491 radians,
(2.0 x (aoa + 0.03491)) x (10.55 - 1.68) = (33.9 x aoa) x (11.86 - 10.55)
(17.74 x aoa) + 0.6193 = (44.409 x aoa)
0.6193 = 26.669 aoa
aoa = 0.02322 radians = 1.33 deg
So the trim angle of attack will be at an angle of attack of 1.33 deg.
Note that TIR-33 has an Appendix (Appendix 12 on Page 35) providing a proof on why we normally can use CNalpha to replace Normal Force when doing these calculations. Of course this time, since the bent Nose had a different local angle of attack than the Fins, we had to use the actual Normal Force, not the Cnalpha's.
Again, in summary, the trim angle of attack is 1.33 deg. Absent any other disturbing forces, the rocket will fly at an angle of attack of 1.33 deg. Because of the continuous Normal Force, lift is being generated. The rocket will fly a slowing arcing trajectory because of the continuous (small) lift.
Now addressing Thrust misalignment. The Thrust is now misaligned by only 1.33 deg (the trim angle of attack). The normal component of the misaligned thrust will only be 2.3% of the total thrust. One could argue also how much the nozzle might be misaligned anyway, a 1.33 deg misalignment might not be so large compared to the expected variation of the rocket nozzle alignment in our rocket motors.
In summary, the rocket will fly, it will just fly a gently arcing trajectory. I could actually predict the trajectory using Normal Force from the trim angle of attack, but again a 1.33 deg trim angle of attack is very small.
In TIR-33 on Page 36 a projected "Banana Rocket" is included.
Using the 1/2 deg (0.5 deg) deflection of the tail, and balancing the moments again, we get (0.5 deg converted to 0.0087266 radians)
(CNalphaNose x (aoa + 2 deg)) x (Xcg - NoseCP) = ((CNalphaFin)fb x (aoa - 0.5 deg)) x (FinCP-Xcg)
(CNalphaNose x (aoa + 0.03491)) x (Xcg - NoseCP) = ((CNalphaFin)fb x (aoa - 0.0087266)) x (FinCP-Xcg)
Note that the bent Nose gets 2.0 deg added to the angle of attack, the bent Tail (Fins) gets 0.5 deg subtracted from the angle of attack.
Continuing, again using the Javelin numbers:
(2.0 x (aoa + 0.03491)) x (10.55 - 1.68) = (33.9 x (aoa - 0.0087266)) x (11.86 - 10.55)
(17.74 x aoa) + 0.6193 = (44.409 x aoa) - 0.3875396
1.00684 = 26.669 aoa
aoa = 0.037753 radians = 2.163 deg
So for the "Banana Rocket" the trim angle of attack will be at an angle of attack of 2.163 deg. Still a pretty small trim angle of attack.
Charles E. (Chuck) Rogers