Pitching up with constant bank -- how to compute final attitude, from initial attitude and degrees of rotation about pitch axis? (Specific example)

Question

(Note: highly related, but different: How to compute final aircraft attitude, if we know starting attitude and degrees of rotation around each axis? (Specific example))

I'm flying due north (000 degrees heading), with bank angle of 0 degrees and a pitch attitude of 0 degrees.

I roll the airplane 30 degrees to the right, rotating only around the aircraft's roll axis (longitudinal axis).

Then I pitch the aircraft up through 45 degrees of rotation around the aircraft's pitch axis (lateral axis), with no rotation around the airplane's yaw axis (sometimes called the "directional" axis), but rotating as needed around the aircraft's roll axis (longitudinal axis) to hold the bank angle constant.

At the instant the pitch rotation is finished, what is the airplane's heading and pitch attitude? And what are the relevant formulae?

Note that -- just as in the related question linked above -- no information is given about the instantaneous direction of the flight path at any point in the maneuver. We could imagine that the angle-of-attack remains constant, in which case large variations in thrust and in airspeed would be required to fly the maneuver. Or we could imagine that the maneuver involves a very large change in angle-of-attack and minimal change in the direction of the flight path, at least as seen in side view, resembling some sort of modified version of the famous "Cobra" maneuver. Or we could imagine that the maneuver is being flown by a hovering "Harrier"-type VTOL airplane, or a helicopter, with zero airspeed. Or we could imagine the maneuver being flown in some hypothetical way that would never be possible in reality (at least in a fixed-wing aircraft)-- e.g. the flight path is completely horizontal and linear throughout the maneuver, with substantial airspeed, and during the maneuver the aircraft develops an extreme sideslip ("yaw") angle as well as an extreme angle-of-attack. Those details shouldn't affect the answer -- it appears that all these cases boil down to the same problem in spherical geometry.

In fact, it might not actually be possible to fly this maneuver at all in any conventional fixed-wing aircraft in a way that exactly meets all the constraints of the problem, including zero rotation about the yaw axis. It appears that it would be possible to fly the maneuver in some helicopters, or a Harrier-type VTOL aircraft operating with zero airspeed, or in some as-yet-unbuilt future derivations of such aircraft. The point being, this is not a question about the flight dynamics of any particular aircraft. It's a question about how to compute the final bank angle, final pitch attitude, and delta heading, from a known initial attitude and a prescribed series of rotations about the various axes of the aircraft.

Answer 1

Just as in the related answer How to compute final aircraft attitude, if we know starting attitude and degrees of rotation around each axis? (Specific example), this answer will use a navigational computation to solve this problem in spherical geometry.

We'll imagine the aircraft to be centered in the middle of an imaginary globe with the nose pointing at 0 degrees latitude and 0 degrees longitude. We'll set the specified total degrees of rotation about the aircraft's pitch axis (45 degrees), equal to the total angular path length that would be followed by a vessel starting at 0 degrees N 0 degrees E, and travelling along a rhumb line, which is a line of constant heading.

The aircraft's bank angle-- 30 degrees--will be represented by the vessel's heading. In this problem the bank angle is constrained to remain constant-- that's why we are modelling the problem with a rhumb line calculation rather than a great circle calculation. Note that maintaining a constant bank angle requires some rotation about the aircraft's roll axis --there must be a non-zero rate of roll toward the high wingtip.

The final latitude of the vessel will represent the aircraft's final pitch attitude, and the final longitude of the vessel will represent the aircraft's total heading change.

The particular rhumb line calculator used to solve this problem gives the option to use a "normal sphere" for the shape of the earth, in which case the expected input for distance travelled (path length) is in "geographical miles". A geographical mile is very nearly equal to a nautical mile, but is defined to be equal to exactly 1 minute of longitude at the equator, so for the distance travelled, we'll multiply the specified degrees of rotation about the aircraft's pitch axis (45) by 60 minutes / degree to get the total angular path length in units of minutes of equatorial longitude, which is also the total path length in "geographical miles". (The value is 2700.)

The calculator outputs a final position of 38 degrees 58.269 min N, 24 degrees 19.530 minutes E.

Converted to decimal degrees, this means that the aircraft's final pitch attitude is 38.97 degrees, and the aircraft's change in heading is 24.33 degrees.

Note that this is about 2.2 degrees greater change in pitch attitude, and 2.2 degrees less change in heading, than in the related problem where we had the same 45 degrees of rotation about the aircraft's pitch axis, but rather than constraining the bank angle to remain constant (which requires some rotation about the aircraft's roll axis), we specified that there was no rotation about the aircraft's roll axis (which led to a 9.1 degree increase in bank angle during the course of the pitch rotation.)

It's perhaps a little surprising that there wasn't a greater difference in final results between the two related problems. Pitch rotation values of 60 degrees or more would likely yield a much greater difference in final results between the two related problems.