# In a non-symmetric turn, how do the side slip angle and thrust side slip angle affect the equations of motion?

I'm doing a problem on Flight Mechanics in which a banked turn is performed. The center of gravity of the aircraft is always within an horizontal plane.

In order to write the dynamic equations, I have to assume some simplifications and conditions. One of them is the following:

The thrust’s sideslip angle is equal and with opposite sign than the aeroplane’s sideslip angle, that is: ($\nu = −\beta$).

Side slip angle ($\beta$) is defined as the angle made by the velocity vector to longitudinal axis of the vehicle at the center of gravity in an instantaneous frame.

Thrust side slip angle ($\nu$) is defined as the angle made by the thrust vector to longitudinal axis of the vehicle at the center of gravity in an instantaneous frame. https://en.wikipedia.org/wiki/Slip_(aerodynamics)#Sideslip_angle

Then, I have to write the dynamic equations for the symmetric flight and for the non-symmetric flight.

My questions are:

• In my opinion, the condition written above should not be accounted for the symmetric flight, so $(\beta = \nu = 0)$. Is this affirmation correct?
• For the non-symmetric case, I have drawn the following scheme of velocity and thrust. But I'm not sure if it's correct because it doesn't make sense, despite that is what the condition says: $(\nu=-\beta)$. Is the scheme correct according to the condition? (The scheme is the top view of the aircraft.)

• In your graphic, what do subscripts b and I mean? Usually b is body and I is inertial. But then why shouldn't thrust be parallel to the body x-axis? – DeltaLima Sep 26 '15 at 14:48
• Yes! It means that. The reason why thrust is not parallel to the axis is because the statement says that the thrust has a sideslip angle $\nu$ and I think this angle is measured between thrust and X-axis. Please correct me if I'm wrong. @DeltaLima – Airman01 Sep 26 '15 at 15:14
• If I understand the situation correctly, the aircraft is flying with the body axis aligned with the inertial axis and slipping to the right. E.g. the aircraft is on close final, aligned with the runway and correcting for a crosswind from the right. – DeltaLima Sep 26 '15 at 15:39
• Can you state the definition of thrust's sideslip angle and airplane's sideslip angle? Between which two vectors/axis each is measured? – Jan Hudec Sep 26 '15 at 19:52
• @Airman01, the diagram appears to be correct according to that definition, but with engines fixed to the airframe I have to wonder what could cause the change of thus defined $\nu$. – Jan Hudec Sep 26 '15 at 20:45