Rudder Control and CG Position

[Milt]

Most Text Books Are Wrong

Most text books lead you to conclude that one of the significant factors affecting steady state symmetric or asymmetric directional stability of an aircraft is the position of the centre of gravity.

Why do they perpetuate this misconception and con most of you aviators into such a belief?

[FlyingForFun]

Why do you believe that this is wrong? You can not say that something is wrong without justifying that point of view.

My own thoughts would be that the effectiveness of the vertical stabiliser would depend on its distance from the centre of gravity - more distance equals more moment, which equals a greater stabilising effect. Therefore, the centre of gravity does have an effect. I'm not an expert, and I don't have any text books to hand to confirm this, but it seems to make sense to me. Can you please explain why this is not the case, if that's what you believe?

FFF
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[bookworm]

Why do they perpetuate this misconception and con most of you aviators into such a belief?

Because we aviators are so talented that we even "raise the dead"? :)

[Milt]

FlyingForFun

Have I only found one so far who has been conned?

Consider further.

The CG represents the location of the point through which gravity acts VERTICALLY on the mass of an aircraft.

Gravity cannot be resolved to the horizontal and have any value except ZERO.

With steady state directional stability and yaw control one should only consider forces in the horizontal plane.

The rudder generates predominantly horizontal forces. Now ponder what other HORIZONTAL forces come into play and you will see that the CG plays NO part.

The qualification of steady state simplifies very complex interactions which are introduced when manoeuvre is introduced.

Keep it simple at this stage to avoid further confusion

[bookworm]

With steady state directional stability and yaw control one should only consider forces in the horizontal plane.

You're missing something.

Conventionally in an engine out situation we apply up to 5 degrees of bank for both control and performance. That allows us to use a horizontal component of lift to minimize the sideslip force.

Where does that horizontal component of lift act? At the same point as the rest of the lift, at the centre of gravity (otherwise the aircraft would start to pitch).

So in summary, if you fly wings level in asymmetric flight, you're right, the C of G plays no part. But if you bank for no slip ("raise the dead" [engine]), then the position of the C of G matters for yaw control.

[Tony_EM]

Hang on a moment, you are forgetting inertia. Just because this balance point is called the centre of gravity, dpoes not mean it is only related to the earth's gravitational pull. The CG position WILL determine the behaviour and stability of the aircraft in pitch as well as yaw. Having said that, yaw performance is hardly affected because the CG range will never come close to producing more side area infront of the CG than behind it, or even affecting the ration to a significant degree.

Lift acts at the centre of pressure (CP), not the CG. The relative position of the CG and CP will produce a pitch over effect which is balanced by the negative angle of the horizontal stab. These rules change with FBW aircraft, especially the unstable ones.

Yaw is different in that it is not counteracting a constant force (gravity), but maintaining directional stability. Now, if Milt's theory was correct, and CG played no part in the equation, why isn't the V/stab in the centre of the plane? Having the V/stab in the tail ensures that the lateral CP is well behind the CG (much more than the vertical, or lift CP). FlyingForFun is correct when he states that the distance of the V/stab from the CG determines its moment arm and therefore effectiveness in terms of yaw stability and authority. Taking the 747sp as an example; when the fuselage was shortened, the size of the V/stab had to be increased to maintain its effectiveness because of the shorter distance from the CG and reduced moment arm.

There is no question that changuing the CG will affect the effectiveness of the V/stab in terms of stability and rudder in terms of authority. However, because of the large distance of the lateral CP from the CG, this effect will, in practice, be negligable compared to other factors. I seriously doubt that any pilot can detect the difference in yaw stability and authority when the CG moves within its permitted range on most aircraft, especially large commercial transports. The differences in question can be natural yaw oscillation period, rudder authority, susceptibility to PIO and extreme attitude recovery amongst others. Certification is supposed to ensure that these differences remain negligable as long as the CG remains within the envelope. Any aircraft that does demonstrate significant differences in yaw stability/authority would be, IMO, a bad design.

Where the CG can have an effect on yaw stability/authority is in marginal control situations, such as spin entry and recovery. While rearward CG condition does have serious implications on pitch authority and therefore makes spin entry much easier and recovery much harder, this is mainly due to pitch characteristics, but, is also true because of the reduced effectiveness/authority of the V/stab. That is why certification relies on comprehensive flight testing throughout the CG range. My only concern here is that both large passenger jet manufacturers rely heavily on computer simulations and extrapolation of flight test conditions to determine the performance in extreme and marginal conditions.

I haven't read any operational text books which conclude that CG is a "significant" factor wrt yaw stability, but that is because the location and size of the V/stab has been designed to perform as advertised as long as the CG stays within the permitted envelope. However, at the design stage, the exact size of the V/stab is mostly dictated by its location relative to the CG.

So, to say that CG has no effect on yaw stability is technically incorrect. However, in practise, it's hardly worth worrying about compared to the effects that CG has on pitch.

[bookworm]

Tony_EM and studi

You're ignoring a very important part of what Milt wrote. His claim was about the steady state directional stability. So there is no acceleration, no angular acceleration (d omega/dt = 0) and inertia plays no role in the equilibrium.

So this is statics. What theoretical mechanics actually says about a rigid body in such circumstances, is that there must be no net force and no net moment on the body. Zero net moment is zero net moment about any point -- there's nothing special about the C of G.

All that matters is where the forces act in relation to each other. That means that when the equilibrium is maintained by:

* the asymmetric thrust of an engine

* the force applied by the rudder and fin

* the side force on the fuselage that's slipping

then it's the positions at which those forces act that matter.

For example, what matters in the wings level case is not that the V/stab is well behind the CG, but that it is well behind the lateral centre of pressure of the aircraft at which a side force is produced when the aircraft slips.

Finally

Lift acts at the centre of pressure (CP), not the CG. The relative position of the CG and CP will produce a pitch over effect which is balanced by the negative angle of the horizontal stab.

If you consider lift to be the sum of the lift from the mainplane plus the lift from the tailplane (and any other sources of aerodynamic lift), then it must net out to acting at the CG. If it didn't, the aircraft would have a pitch-axis angular acceleration as there would be an unbalanced couple with the weight.

[cwatters]

Milt,

You are so wrong.

Try throwing a dart backwards. It will turn round so that it's center of vertical area is behind the center of gravity every time.

[Milt]

bookworm and others

For simplicity let's keep wings level for now and without manoeuvre. We can complicate things later if you want to followm through on this theme.

But let me correct 'bookworm ' on one important point first.

An aircraft's wing and body lift ONLY act in close proximity of the CG for a CCV - Control Configured Vehicle - which then must have artificial longitudinal stability to be flyable. eg F16 and some/most fly by wire aircraft.

For naturally longitudinally stable aircraft the lift acts through a point aft of the CG. A down force from the tail balances the lift/weight pitching moment.

A heavy 747 on the cruise has a down force of about 30 tons on the tail. The wings have to overcome this down force. So what price natural stability?

Back to the horizontal.

With the asymmetric aircraft straight and level we have horizontal forces which can be conveniently broken down as horizontal components of engine thrust, rudder force and drag.

There is still no weight component so NO CG position effect.

Single engine thrust will be acting close to the prop boss or jet engine centre line. Rudder force will be acting sideways close to the rudder hinge line as there is a contribution from the fin because of sideslip.

Drag will act through "the lateral/sideways centre of pressure" of the fuselage. Oddly I have never heard of a better name for it.

The combination of these three forces balance to result in horizontal untidy progress with a sideslip angle as a result of the combined relative values and directions of each force.

Absolutely no relationship to CG!!

Oh what have I started here? But it is all very interesting to hear the differing points of view. Keep them coming.

[bookworm]

Steady state directional stability" is nonsense. If the aircraft would in reality be in a steady state, there would not be any need for a vertical fin, since all acting forces and moments would be zero. But since there are forces and moments unequal zero, directional stability is needed. Only the absence of a steady state creates the need for stability measures. A steady state is intrinsic stable.

Steady state simply means that the forces sum to zero, so there is no acceleration. There are still forces, and still moments of force. Stability is about the response to small displacements from that steady state. Just because there is no moment at the equilibrium position doesn't mean that the equilibrium is stable.

Nevertheless, I was wrong to make that claim about stability, even static stability. I was actually thinking about the case of yaw control, which is what Milt wrote in the subject line of the thread. In that case, it is statics, because the net force is and remains zero.

Stability is different because the forces in response to a perturbation do not sum to zero, so the same logic does not apply.

Wrong. A moment is a product of a force and a length. So it matters around which point you consider the moment.

Not if the net force is zero. Consider a rigid body with forces Fi acting at points ri. Then the total moment about the origin is:

Sum_over_i(Fi x ri)

Now change the point about which we take moments to R.

The total moment is now

Sum_over_i(Fi x (ri - R)) = Sum_over_i(Fi x ri) - Sum_over_i(Fi) x R

If Sum_over_i(Fi) = 0, the moment remains the same. So it doesn't matter what point you take moments about, if the net force on the body is zero.

In the example that you give, there is a net force on the object of 20 units westbound. If you try again with a net force of zero (e.g. reverse one of the forces and make it eastbound) you'll find that the moment is the same, regardless of the point about which you take moments.

The implication of that is that CG position does not have a role to play in determining the minimum control speed with asymmetric power if the wings are kept level. That is a steady state situation.

The stability case is rather different, and, though I was wrong before, I'm not actually convinced that it's as simple as the usual books make out.

The directional stability should be determined by the tendency for slip angle to be restored in response to a perturbation in slip angle. The derivative that's normally used for this is dN/dbeta, where N is the net yawing moment (yes, about the C of G :O) and beta is the slip angle. While that seems sensible at first sight, there's a problem. N determines the angular acceleration but beta is also affected by the direction of motion of the aircraft, and therefore by the linear acceleration that comes in response to the perturbation. Perhaps that term is small by comparison, but I'd need convincing that it can be ignored.

For naturally longitudinally stable aircraft the lift acts through a point aft of the CG. A down force from the tail balances the lift/weight pitching moment.

That's all lift in my books, Milt. The net moment of the vertical aerodynamic forces and the weight has to be zero in steady state, which means that the net effect of all the vertical aerodynamic forces must be to act through the C of G.

[Tony_EM]

What's the point in talking about steady states and statics when a flying aircraft is about the most dynamic thing ever invented?

Lets take a single moment in the life of a pencil balancing on its tip; no resultant forces, so is it a stable system? Of course, but only for that single moment in time.

I'm ignoring the steady state directional stability because it has nothing to do with how a flying aircraft achieves dynamic stability in flight.

[bookworm]

I agree with what you write about behaviour in a steady state. Very nice how you show that the point around which you consider the moments is not relevant in case that the forces sum up to zero. I guess Fi, ri and R are vectors and that "x" stands for the vector product (also known as cross product).

Yes, it can be, or it can be a good old fashioned multiplication in one dimension (provided the Fs and the rs are perpendicular).

BUT: "steady state directional stability" doesn't exist. Either it is steady state, then you don't have any disturbance and stability is not necessary, or then you have disturbances (hence it is not steady state anymore) which are controlled by stability.

I agree. One way of thinking about it is to look at the result of changing all those Fs in response to a change in a parameter (e.g. slip angle), i.e. to differentiate.

So assuming the ri distances don't change:

Moment about R = Moment about the origin - Sum_over_i(Fi) x R

becomes

d(Moment about R) = d(Moment about the origin) - Sum_over_i(d(Fi)) x R

So if Sum_over_i(d(Fi)) is non-zero, i.e. if the change causes a force which causes an acceleration of the body, then it does matter where you take moments about. There are cases in mechanics where the net force remains zero, but directional stability of an aircraft (i.e. response to a change in slip angle) is not one. My bad.

In this dynamic case, the C.G. has definitely an influence. And flying is definitely a permanent dynamic process.

Do you fly helicopters then? :)

Seriously, Milt has a point with yaw control (authority, if you like). That's not dynamics, it's simple statics.

John Denker explains it (http://www.av8n.com/how/htm/multi.html#sec-1out-cm-effect) rather better than I can.

However, static stability, despite its name, appears to be about the acceleration (angular or otherwise) response of a body to a perturbation, and that is necessarily about the dynamics of rigid bodies, and that necessarily involves the centre of mass.