How does bank angle differ from roll angle with respect to stability in fixed-wing aircraft? I would like a clarification on bank angle and how its different from roll angle with respect to to fixed wing aircraft. It is my understanding that the bank angle is a result of rotating the aircraft body to the stability frame, implying that if the angle of attack $\alpha$ and the side slip angle $\beta$ are zero then, and only then the bank and angle and roll angle are the same. Based on Stevens and Lewis Aircraft Control and Simulation (which doesn't define bank angle) the rotation from body to stability(wind) frame is given by
$$
C_{w-b} = \begin{bmatrix} &amp;\cos\alpha \cos\beta &amp; \sin\beta &amp; \sin\alpha cos\beta \\ &amp;-\cos\alpha\sin\beta &amp; \cos\beta &amp; -\sin\alpha\sin\beta\\ &amp;-\sin\alpha &amp; 0 &amp; \cos\alpha  \end{bmatrix}
$$

Therefore, the bank angle $\mu$ I think is given by:
$$\mu \triangleq \begin{bmatrix} &amp;\cos\alpha \cos\beta &amp; \sin\beta &amp; \sin\alpha \cos\beta \end{bmatrix} \begin{bmatrix} \phi \\ \theta \\ \psi\end{bmatrix} $$ 

where $\phi,\, \theta,\, \psi$ are standard aerospace Euler angles defined based on a 3-2-1 rotation.
My specific questions are:

1) is my understanding and calculation of bank angle correct? If not can someone point me to a good resource for this. I was surprised to be able to find a clear definition and formula quickly.

2) by multiplying the second and third row of $C_{w-b}$ with Euler angles we obtain a set of other two angles relative to stability frame. Do these angles have any names or specific role in aerospace dynamics/control?
In my experience, "bank angle" and "roll angle" mean exactly the same thing. I've only ever heard of one meaning of either term. 

For completeness, here are some "plain English" definitions of all three Euler angles:

Pitch is the angle that the nose (the longitudinal axis) forms with the horizon. If the nose is pointed at the horizon, the pitch is 0°; if it's pointed straight up, the pitch is 90°; and if it's pointed straight down, the pitch is -90°.

Heading (sometimes called "yaw angle") is the horizontal direction that the nose is pointing, measured clockwise from true north.

Bank angle (sometimes called "roll angle") is the amount that the aircraft would have to roll (rotate around its longitudinal axis) in order to bring the wings level (which is to say, bring the lateral axis parallel with the horizon), with the top side of the aircraft facing above rather than below the horizon.

Pitch ranges from -90 to 90. The other two angles have a range consisting of a full circle; I think bank is usually expressed as a number from -180 to 180, and heading as a number from 0 to 360.