Do aerodynamic forces and moments change aircraft pitch and yaw in the same way?

Several questions on the Aviation SE site have dealt with the question: can the rudder alone turn the aircraft:

• This one, which was marked a duplicate of
• this one, which was itself marked as a duplicate of
• this question, with a skilful and detailed answer.

So we know in triplicate that the rudder alone can turn an aircraft. Not very comfortably due to the side forces, wrong way lean, different airspeeds on inside & outside wing etc, that is why the normal way to turn is to bank.

However, the following has been posted in the comments (partial quotes only due to length):

Yaw does not cause aircraft turn. It only causes yaw. Yaw is not turn.

Lift, pointed in a direction other than perpendicular to the horizon, is what creates turn. Nothing else.

With no other yaw force on the aircraft, if you have right rudder, that does create a leftwards force, but it also creates a right yaw (sideslip) which itself creates a rightwards force that opposes the leftwards force from the rudder.

It works in exactly the same way as it does in pitch. The elevator pulls the tail down and, because it's behind the CG, creates a nose up pitching moment (torque). Lift on the wings lifts the aircraft up, and because it's also (generally) behind the CG it produces a Nose down pitching moment. Because the moment arm of the elevator is so much longer the torques balance, but the Up force from wings is much larger than the down force from the elevator and the aircraft is able to fly. The force from the rudder, and the force from the sideslip angle of the fuselage act in the same way.

The nose of the aircraft does not "turn" away from the "direction the pilot ... " it only "yaws" away, and it only yaws away until the sideslip angle reaches the point where equilibrium occurs.

The gist of all this is that the aircraft would behave in yaw in the same way as in pitch: rudder deflection only causes an angle of sideslip, not a continuous turn in heading. That would conflict with the earlier questions posted above.

So the question is: does yaw really behave exactly like pitch? Does deflecting the rudder only cause a greater angle of sideslip but no flight path change?

• What are the assumptions required for the answer? are we applying only rudder and are we leaving the bank angle free to increase or are we applying aileron to keep the wings level? are we increasing thrust or are we leaving the aircraft free to decelerate due to the increase in drag? – Federico Dec 19 '17 at 9:36
• @Federico good questions that will lead to a good answer I reckon. – Koyovis Dec 19 '17 at 9:42
• Should we expect that changing the aircraft's vector in a direction parallel to the lift vector (i.e. pitch) will work the same as changing it in a direction perpendicular to that vector (i.e. yaw)? – Ralph J Dec 19 '17 at 9:47
• No, I won't answer before knowing what are the assumptions required. – Federico Dec 19 '17 at 9:49
• The additional lift on the vertical stabilizer, which is caused by deflecting the rudder, mainly creates a moment so that the aircraft turns the way you want to go. This repoints the thrust vector and also causes a side force due to the sideslip angle which works your way. In order to be more general, one should have a look at the additional forces and moments created by sideslip angle, which probably depend on details of the aircraft's configuration. – user7241 Dec 19 '17 at 10:14

You could also take into consideration the secondary effect of yaw, which is roll. As the aircraft yaws one wing moves forward in the airflow while the other moves back. This increases the amount of lift generated by each with the wing moving forward creating more lift and the trailing wing creating less. The aircraft therefore, will roll in the direction on yaw. This can be countered by applying opposite aileron.

• Yes, however since the rudder sticks out above the fuselage, deflecting rudder creates an immediate rolling moment to outboard, only later compensated for by the effect you describe. – Koyovis Dec 21 '17 at 12:28
• Assuming that the center of pressure for the rudder airfoil lies above the flight path of the aircraft CG. If the center of pressure happens to be below the extension of the flight path from the CG, then the aircraft will roll right from right rudder. This can be the case because of unusual aircraft design, or from very high AOA. Also, in swept wing aircraft, yaw produces roll due to decreasing the sweep angle on the wing moving forward, and increasing the sweep angle on the opposite wing. – Charles Bretana Dec 23 '17 at 16:15
• Finally, in high performance aircraft, roll at high AOA itself produces a pitch moment, increasing AOA. This effect, now called Inertial roll coupling, caused loss of control and the loss of an F-16 in the mid 80s (I believe). It comes because the aircraft is rolling about its flight path, not its fuselage, and its kinda like mounting a barbell on a steel rod that is offset from the bar by some angle, and then spinning it on the steel rod. The two ends of the barbell will describe circles around the steel bar rotational axis, and exert a centrifugal force that pulls them further away. – Charles Bretana Dec 23 '17 at 16:22

Of course will yaw also change direction if we define this as a change in the flight path. But it does so much less efficiently than banking.

Yaw creates a side force on the fuselage and fins which accelerates the aircraft sideways. This does change the resulting flight path, but the amount of side force is small due to the low aspect ratio of the fuselage. The lift-to-drag ratio of the fuselage is poor, so another change to the aircraft's vector is a deceleration if the increased drag is not compensated by more thrust.

Banking also creates a side force (when seen in the geocentric reference system), but now the wing is responsible for that force. Its lift-to-drag ratio is much better than that of the fuselage, therefore, the amount of side force is much higher and less drag is created. By first banking and then pulling more than 1g, the pilot changes the flight path much more effectively while maintaining altitude.

So the answer is yes, yaw behaves the same way as pitch. Only the wing and elevator are replaced by the fuselage and fins, with all changes to the magnitude of the forces this causes. The only difference is that pitch is needed to adjust lift, and changing the pitch attitude will change the speed of the aircraft or the load factor. Flying in a sideslip with no adjustment for the side force by a corresponding bank angle will continuously accelerate the aircraft sideways and will result in a uncoordinated turn. Now the sideways load factor (which is zero in coordinated flight) will be nonzero while in the longitudinal motion the normal size of the load factor is one (and be different from one when pitch is changed).

EDIT:

@TomMcW has contributed a practical example in the comments. Consider a crosswind landing with a glider or motor glider where crabbing is your only option to fly a straight approach over ground. Right before touchdown you should then step on the rudder to align the aircraft with the runway so the tires can touch down without moving sideways. Now consider that you stepped on the rudder too early and the aircraft floats along the runway with a sideslip angle. You will drift leewards, right? That is exactly the change in flight path I am talking about.

In crabbed flight the aircraft flies without sideslip. The fuselage with its forward center of pressure position destabilises the aircraft in yaw and the vertical tail adds the required stability. Both create no side force in crabbed flight. Now you want to align the fuselage with the runway and step on the rudder: The tail swings the aircraft around into a sideslip until the fuselage points into the desired direction. Now opposite rudder must be applied briefly to stop the yawing motion and then returned to a position between neutral and the initial deflection in order to trim the new sideslip. Note that this works exactly in the same way as the elevator when a new angle of attack is trimmed!

In the sideslip condition both the fuselage and the vertical tail will create a side force in the same direction. The side force of the fuselage is higher because its center of pressure is closer to the center of gravity than that of the tail, and in order to reach an equilibrium, both must point to the same side. This combined side force accelerates the aircraft sideways such that a speed component is added which reduces the sideslip condition. Since the aircraft is trimmed for a specific sideslip, the changing balance of moments with a change in sideslip will yaw the aircraft further so that the trimmed sideslip angle is restored.

Why is that? Remember that the rudder is deflected. This creates a camber in the vertical tail which moves its zero-sideforce angle of sideslip away from the sideslip angle of zero to a negative angle (for example, right rudder deflection will shift the zero-sideforce sideslip angle on the tail to a left sideslip value). Any change in sideslip will, therefore, bring a relatively larger change in side force at the tail which changes the balance of yawing moments such that the old sideslip is restored. Note again how similar this is to static longitudinal stability.

This means that the aircraft must maintain a slow yawing motion in order to keep the sideslip constant. The side force causes a reduction in sideslip and the yawing motion is a result of the sideslip-maintaining tendency of the vertical tail.

And, no, yaw is not a moment. Please let's all use an agreed version of English.

• Does deflecting the rudder only cause a greater angle of sideslip but no flight path change? – Koyovis Dec 19 '17 at 9:39
• @Koyovis: No, in addition to yaw the flight path does change, too. – Peter Kämpf Dec 19 '17 at 9:40
• @Koyovis In a crosswind landing you can use either bank or yaw to compensate. If yaw didn't affect flight path you couldn't do a crabbed landing. – TomMcW Dec 20 '17 at 20:53
• @TomMcW: This is a little different. In a crabbed landing you only fly with a sideslip right before touchdown (if performed right). You want to keep the sideslip part of the landing as short as possible precisely because the side force involved will accelerate the aircraft sideways and makes it leave the runway to the leeward side. But that is precisely what my answer is about, so your example is actually very helpful. – Peter Kämpf Dec 20 '17 at 20:59
• It's a strangely worded question. – TomMcW Dec 20 '17 at 21:37

The flight path of the aircraft CoG is defined relative to earth axes. In the six degrees of freedom of a rigid body, the three angular DoFs are defined as:

• Pitch is the nose up/down angle relative to the gravity field of the earth.
• Roll is the wingtip up/down angle relative to the gravity field of the earth. The angle relative to gravity is also referred to as the bank angle.
• Yaw is the angle relative to a suitable earth reference, usually magnetic north or geographical north. Note that yaw is defined as an angle with reference to earth field of gravity, writing it with a capital does not re-define it as a torque about the aircraft Z-axis.

There is additional information here. Aircraft pitch, roll and yaw are defined with reference to earth field of gravity, while aerodynamic forces use the aircraft axes, described in more detail below. Aircraft axes are defined relative to the free airstream - to avoid confusion, nose-up angle relative to free airstream is defined as Angle of Attack, and nose sideways angle as Angle of Sideslip. Confusingly, aircraft pitch, roll and yaw is sometimes used relative to free stream, which is only valid if the aircraft flies at constant altitude in the direction chosen for yaw.

The aircraft axes and sign definitions used for aerodynamic force considerations are defined as follows:

Aircraft motion and forces about the angle of attack axis (X-axis) is also referred to as symmetrical flight in aircraft stability and control: equilibrium of forces and moments in the plane of symmetry of the aircraft. For a-symmetrical flight, we look at forces, moments and deflections around the Z- and the X-axis, and forces in direction of Y-axis.

A rigid body has six degrees of freedom: three linear displacements, and three angular ones. Stability and Control of aircraft generally divides aircraft behaviour into:

• Symmetrical behaviour: forces, moments and displacements in the aircraft field of symmetry of the aircraft (nose up/down). Note that the gravity vector is then broken into components relative to aircraft axes.
• A-symmetrical behaviour: forces, moments and displacements not in the aircraft field of symmetry.

The three equations that determine the state of stationary, a-symmetrical flight are the forces in Y-direction, and the torques about the X-axis and the Z-axis. Within these three equations, there are five variables:

• rudder deflection;
• aileron deflection;
• sideslip angle;
• roll angle;
• yaw angular velocity.

A resulting angular velocity $r$ can therefore be the result of an infinite number of combinations of the remaining four variables. Only if one of the variables is fixed, is the resulting $r$ a function of the remaining three variables. For instance keep the ailerons at zero, and a rudder deflection results in a sideslip angle, a bank angle, and a nose sideways angular velocity. Different combinations for different aircraft. A much better option is of course to choose sideslip angle zero: a combination of ailerons, roll and rudder then results in a co-ordinated turn.

Aircraft heading change is the result of a sideways force relative to the velocity vector and to the earth field of gravity: a force in this direction acts as a centripetal force. There is an infinitely large combination of aircraft states that can cause heading change and turn the aircraft.

Deflection of rudder only is one of them: it is the back-up mechanical mode of the A-320, and is the way that Anthony Fokker controlled his flying birds nest, the "Spin", which had no ailerons. This way of turning is uncomfortable due to rolling the wrong way when the rudder sticks out on top, creating the rolling moment; and due to cross-effects with increasing sideslip drag and a-symmetrical lift. Pilots are taught to turn the aircraft using the ailerons, and only use the rudder to keep the nose in the wind, for a very good reason!

Does aircraft yaw behave like aircraft pitch?

• Yes of course, in the way that it responds to moments and forces around the CoG. Basic physics.
• No, in the way that there are more degrees of freedom for the aircraft to change heading. In pitch, elevator deflection changes AoA changes wing lift: one degree of freedom. in yaw, either rudder deflection or aileron deflection creates the combination of yaw and/or bank angle resulting in the centripetal force that changes flight path.
• So, are you concluding that yaw does or does not behave the same way as pitch? Basically, if you boiled this answer down to yes or no, which would it be? – TomMcW Dec 22 '17 at 21:18
• @Koyovis, +1 for first sentence in your last paragraph - spot on correct. TomMcW, As in most things technical, it depends on the context. In this context, "the way it affects aircraft trim about the corresponding axis of rotation", the answer is yes, it behaves exactly the same as pitch. If you want to include everything including which axis of rotation it affects, then obviously the answer would be no. How should he "boil" that down? – Charles Bretana Dec 23 '17 at 15:48
• @Koyovis, for second sentence in last paragraph, "No, in the way that there are more degrees of freedom for the aircraft change heading.", How are there more degrees of freedom? In pitch, Once the aircraft pitch AOA is in equilibrium (and AOA is static) the aircraft will either maintain pitch flight path (in level flight or constant climb or dive angle), or it will increase or decrease flight path pitch angle. These latter two conditions are both "turns" in the vertical. – Charles Bretana Dec 23 '17 at 15:55

First of all, by Yaw, I mean the torque or moment created by the lift produced by the vertical stabilizer and rudder when at a sideslip angle, acting on the aircraft through a moment arm between the CG and the rudder.
This Yaw, itself, exclusive of analyzing the total sum of all other aerodynamic forces on the total surface of the aircraft, does NOT cause the aircraft heading to continuously change.

F=ma !!!!

Unless the total lateral force (parallel with the wings) is non- zero, the aircraft will remain on a constant heading. Yaw, (by definition) is nothing more than the condition that the aircraft Heading is not aligned with the relative wind. PERIOD.

Depressing the rudder, does cause a Yaw. But it also causes Lateral Lift to the side. It is this Lift which can potentially cause the aircraft to turn. But it will ONLY cause a turn if the sum of all other lateral Lift is insufficient to balance out and compensate for the Lift created by the rudder deflection. The sideslip generated by the rudder (THIS IS THE YAW) also generates Lift, and it is in the opposite direction from the Lift generate by the rudder Lift. Which direction the aircraft turns in is determined by the total Lateral Lift produced by not just the rudder, but by all aircraft surfaces, including the side of the Fuselage as a result of the sideslip.

The point is that it is not the Yaw which causes the turn, as Yaw is a torque. It is the Lift. Lift Is a force

Yaw is torque, and can only produce rotational acceleration, which can only change attitude, and the equilibrium sideslip angle. Absent a non-zero total lateral force, it cannot change direction of motion. It will just change the angle of the fuselage to the relative wind, (sideslip). As the sideslip increases, the torque from the rudder deflection decreases, and the opposing torque frm the sideslip increases, until they reach equilibrium. At that point, the sideslip angle stops increasing.

IF at this point of equilibrium, the total lateral (sideways) force is non-zero, then the aircraft will turn.

As a thought experiment, imagine a side thruster on the nose of the aircraft pointing to the left, that puts out 100 pounds of thrust, let's say, 5 feet forward of the aircraft CG. This would produce the exact same right Yaw as a rightwards rudder deflection at the tail, that produced 50 pounds of leftwards force on the tail, 10 feet behind the CG. Clearly, the aircraft is not going to turn in the same direction at the same rate of turn in these two scenarios. Why Not? you might say - the yaw is the same! Both are producing an identical amount of Yaw (500 ft-lbs) in the same direction (clockwise)! But obviously, they do not. This is because, absent consideration of the forces produced by the fuselage due to the sideslip, the rocket thruster is pushing the aircraft to the right, and it will turn right, the rudder is pushing the aircraft to the left, and, (Absent consideration of sideslip forces), it would turn left.

It is difficult to understand this because any aircraft has a fuselage, with significant sideslip forces, and these distort the effect of the issue, but if you imagine an aerodynamic body that consists of a flat plate and a rudder, so that there is no sideslip force, then imagine what would happen if you deflect the rudder. Why, the entire body would just spin around the yaw axis until the rudder is once again streamlined in the relative wind, and there would be no turn created at all.

• Yaw would be a rotation. Yaw moment or yawing moment would be the torque. – user7241 Dec 21 '17 at 19:10
• Yes, but the rotation (the angle between fuselage and relative wind), is the result of the moment. The moment, and the force that produces that moment are the two abstractions we are discussing. When I use the term Yaw I am referring to the moment, not the sideslip that results from it. And you need to be careful about using the term rotation. In this context it does not mean continuous rotation, it is just the static angle between the two vectors. – Charles Bretana Dec 21 '17 at 20:03
• You are empatically stating that deflecting the rudder alone does not change the aircraft heading, is that correct? – Koyovis Dec 22 '17 at 3:36
• @Koyovis, No, What I am emphatically stating is that the yaw, or yaw moment from the rudder deflection does not turn the aircraft. Lift generated by the vertical stabilizer and rudder, creates a yaw moment (which causes sideslip), and a horizontal force opposite the rudder. This horizontal force can turn the aircraft, but it will turn the aircraft only if the sum of all other forces (from fuselage sideslip, asymmetric thrust, etc.) is non-zero and the turn generated will be in the direction of the total lateral force, which may or may not be in the same direction as the rudder force. – Charles Bretana Dec 22 '17 at 3:49
• @Koyovis, If you imagine a rudder, by itself, moving through an air mass. Now apply a force (How doesn't matter) that twists the rudder by some angle to the right. What would happen? would the rudder start to turn continuously? of course not. Now put the rudder on a stick, which is of minimal thickness, but measurable mass, (so that the CG is forward of the "rudder", and repeat. What will happen? The entire body will twist laterally until the rudder is again aligned with the relative wind, and then it will remain in that stable position. – Charles Bretana Dec 22 '17 at 4:01