Attitude heading computers generate both a turn rate and a yaw rate. How are these different from each other? Is Euler yaw rate the same as turn rate?
Rate of turn is easy to define: it is the rate at which the direction of aircraft motion changes with respect to the ground (rate of change of track). If the aircraft is not slipping and there is no change in wind, then it is equivalent to the rate of change of heading.
Yaw rate, on the other hand, is slightly messier. Technically, yaw rate of the rate of change of the z-axis Euler angle. As an industry standard, the Euler angle in the aviation industry is defined by the z-y-x rotation, which means that the yaw occurs first, followed by pitch, followed by bank. Therefore, yaw rate is technically exactly equal to the rate of change of heading.
In many applications and loose technical speaks, however, yaw rate may also refer to the third component of the angular velocity. Angular velocity is the instantaneous rate of rotation about its axis of rotation. If the aircraft is rotating about the vertical axis only, then the third component of the angular velocity is, again, exactly equal to the rate of change of heading.
In a coordinated turn, however, the aircraft is pitched and banked, and therefore it is not rotating purely about the vertical axis. In this definition, then, yaw rate is not equal to turn rate.
As an aside, it's mathematically incorrect to call any component of the angular velocity a "rate", since angular velocity is not a derivative of any quantity, even though angular velocity can be transformed into Euler rates if you know the current Euler attitudes.
Turn rate is the rate at which an aircraft's direction of motion changes.
Yaw rate is:
- If the aircraft is wings-level: the rate at which an aircraft's heading (the direction in which the aircraft's longitudinal axis points) changes.
- If the aircraft is not wings-level: the rate at which the aircraft's heading would be changing if the aircraft were wings-level. (To visualise this, roll your head until the aircraft appears to be wings-level, and then measure the rate at which its apparent heading1 changes.)
Ideally, the aircraft's turn rate and yaw rate are equal, so that the aircraft's heading stays aligned with its direction of motion through the air (this is good because the aircraft's profile area - the cross-sectional area that it presents to the oncoming air - is smallest when its direction of motion is aligned with its longitudinal axis, and the amount of pressure drag2 experienced by an aircraft is roughly proportional to its profile area; thus, aligning the aircraft's longitudinal axis and direction of motion minimises drag). If the turn and yaw rates are equal and nonzero, this is called a coordinated turn.
It is also possible for the aircraft's turn rate to be different from its yaw rate:
- For instance, if you step hard on the right rudder pedal while holding the yoke neutral, the aircraft will immediately yaw (change its heading) hard to the right, but will not start to turn (change its direction of motion) significantly until a large yaw angle has already developed; for the first few seconds, the aircraft has a large yaw rate but a negligible turn rate.
- On the flip side of things, if you turn the yoke hard to the right while applying just enough left rudder to hold your heading (initially) constant, the aircraft will roll to the right and start slipping sideways through the air; this causes its direction of motion to change (becoming more rightward), introducing a nonzero turn rate, but, thanks to your left rudder input, the aircraft's heading remains constant, and the aircraft's yaw rate remains zero (until the aircraft's sideslip angle becomes large enough that you run out of rudder authority and the large right-yawing moment imposed by the vertical stabiliser forces the aircraft to yaw to the right).
1: The heading it would have if we redefined "up" and "down" to be aligned with the vertical axis of your head.
2: The drag that results from the aircraft having to push air out of its way as it moves through said air. Also known as profile drag.
This is a comparison of two frames of reference: turn rate is earth referenced on a horizon based yaw plane, whereas aircraft yaw rate is based on the aircraft yaw plane.
In a skidding turn with no bank, they would be equal. In a 90 degree "bank and yank" or "knife-edge", the rate of turn is 100% in the pitch plane, with 0 yaw.
A coordinated turn is a combination of pitch and yaw, producing a change in heading at a given rate.
To put it simply, turn rate is a change in aircraft heading, (over time) while yaw is rotation about the vertical axis of the airplane.
So, if the aircraft yaws the aircraft heading will also change. However, the reverse is not true: In a coordinated turn at a constant angle of bank the aircraft heading will be changing, but it will not be rotating about the vertical axis (unless you introduce a slip or skid) therefore the yaw rate should be zero.