# Aircraft turn displacement from non-linear turn rate (instantaneous roll rate)

I've been trying to figure out an aircraft kinematics problem to estimate the x and y offset relative to current position after completing a turn. The turn is a specific change in heading, finishing in level flight but not necessarily starting in level flight.

Assumptions:

• Instantaneous maximum roll rate
• Constant velocity
• No wind
• End turn in straight level flight
• Begin turn at any roll angle (not necessarily level)

Provided inputs/constants:

• Current velocity
• Max roll angle
• Max roll rate
• Gravity
• Initial roll angle

What I tried:

• Split problem into 3 parts: (a) transition from current roll angle to max roll angle (b) sustained max roll angle (c) transition from max roll angle to level.
• For smaller maneuvers, max bank angle won't be reached and phase b will be skipped. It should (I hope) be fairly straightforward to derive the max roll angle reached, substitute for max roll angle, and solve.
• I found the turn angle by integrating turn rate as a function of time w=g tan(b)/v.
• My hope was to use the arc equation (x,y)=(R cos(T),R sin(T) and integrate wrt time but it was pointed out that this is only true for a constant radius arc and is not applicable in this scenario (plus the integral was insane - Wolfram Alpha couldn't solve it).

I believe this problem would be classified as non-uniform angular acceleration in a spiral with constant radial velocity but might have that wrong?

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
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Commented Apr 19, 2022 at 2:05