Is there a specific relation between sideslip angle and yaw rate OR how sideslip angle depend on yaw rate?

Question

Since, yaw rate is controlled via rudder and sideslip angle can also be controlled by rudder, I want to know if there is any dependence of sideslip angle on yaw rate in fly by wire aircraft as I am working on the SAS design of a/c lateral/directional dynamics?

Answer 1

Since, yaw rate is controlled via rudder

No, yaw is controller by many things, and rudder is there mainly to compensate for them to keep the plane flying nose-forward.

The usual way to make a control system is that you control (with a PID controller):

• Move elevator to target vertical (plane coordinates) acceleration.
• Move ailerons to target roll rate.
• Move rudder to target zero lateral (plane coordinates) acceleration. Lateral acceleration is measured by the ball in traditional cockpit.

Note that each axis is basically independent and rudder is only used to eliminate side-slip, which is proportional to the lateral acceleration unless you have asymmetric thrust or drag—and I have not heard of a FBW that would automatically correct for those conditions.

When flying with auto-pilot, the auto-pilot sets vertical speed target from altitude error and vertical acceleration from vertical speed error for the pitch and it sets roll from heading error and roll rate from roll error for roll axis. Autopilots don't set lateral acceleation targets at all—that is always zero.

That is why auto-land has much lower cross-wind limit—autopilot does not handle rudder, so it can't de-crab.

Answer 2

Sideslip angle β is the "sideways" angle of attack of the aircraft body. In a given time span, sideslip angle is the integrated signal of yaw rate, which is the integrated signal of yaw acceleration, which follows from the yawing moment and the inertia. This is the case in any aircraft, fly-by-wire or conventional control.

$r_{dot}$ = $M_β$ / $I_Z$

r = $∫$ $r_{dot}$ dt

$β$ = $∫$ r dt

Sideslip is an aerodynamic entity defined relative to air velocity (body axes), while inertial entities are defined in earth axes. A transformation matrix describes the relationship of the two axes systems. For flight behaviour the body axes form the frame of reference: a sudden side wind changes initial side slip angle at the start of our time interval, plus the moments acting upon the aircraft.

Answer 3

No, there is no specific or dependent relationship between sideslip angle and yaw rate.

A momentary yaw rate can result in a slip, but a sustained sideslip maneuver will involve a zero yaw rate. Note that a yaw rate is a measurement of rotational velocity while a slip angle is a measurement of relative position.

Consider a momentary yaw rate brought about by rudder deflection with the wings held level. That momentary yaw rate will result in a slip of some angle (which angle will depend on a number of factors including but not limited to amplitude and duration of yaw rate), but to sustain a sideslip will necessarily require a zero yaw rate. That is to say, to fly in a sustained sideslip maneuver—as opposed to a slipping turn—the angular difference between the relative wind and aircraft's longitudinal axis must be maintained with a zero yaw rate.

While yaw rate or angle can be controlled by rudder, there are several other forces or control inputs that can affect or control yaw rate or angle. As one example, consider an asymmetrical thrust scenario involving a sustained sideslip and zero yaw rate. In that scenario, rudder input opposite the direction of the sideslip may be required to maintain the sideslip.