Longitudinal Dihedral

[Mad (Flt) Scientist]

No, no, no.

I'm sorry, but changing the tailplane incidence on an all-moving tail aircraft WITHOUT changing the CG, which is possible by countering with elevator - is NOT going to change the stability of the aircraft.

You cannot assume that the same change in AoA occurs at tail and at the wing; the downwash gradiet has a very significant input into this calculation.

Plus the centre of pressure is not the correct thing to be talking about when dealing with this - it's the aerodynamic centre and the neutral point which matter for this kind of work.

I'll try to put a more point-by-point response together....

ok, Beagle's post with italic comments

Draw a diagram with the wing centre of pressure at distance a ahead of the centre of mass and the tailplane centre of pressure at distance b behind the centre of mass. Fairly obviously, for things to be in balance, then the lift generated by the wing, Lw and the lift generated by the tail, Lt, must be such that:

Lw x a = Lt x b.

Not true - this is assuming that there is no pitching moment about the CP, which is by no means guaranteed. Or, rather, it's assuming the CP doesn't move when the disturbance is introduced later in the analysis

Or, in other words:

CLw x ½ rho V² Sw x a = CLt x ½ rho V² St x b

Which can be re-arranged to CLw/CLt = St x b/Sw x a

Now, imagine our original diagram is given a small disturbance. Fairly obviously, to stop the whole thing tipping on its arse, then the increase in lift from the tail, dLt at the tail centre of pressure must be greater than the increase in lift from the wing, dLw, at the wing centre of pressure. That is:

actually, the pitching moment restoring the static situation (which generally comes from the tail) must be greater than the destabilishing moment from the wing...

dLt x b > dLw x a Or, in other words:

dCLt x ½ rho V² St x b > dCLw x ½ rho V² Sw x a

Which simplifies to:

dCLt x St x b > dCLw x Sw x a

Now, for aerofoils in the cruise range concerned, the lift curve slope may be assumed to be linear and the same for both wing and tail aerofoils, hence dCLt = dCLw.

You cannot RPT cannot assume that the same alpha disturbance occurs on both wing and tail - due to downwash. So you cannot make the leap that the delta CL on both surfaces is the same, regardless of what you assume about the lift-curve slopes

So that:

St x b > Sw x a

Going back a few steps, we already showed that CLw/CLt = St x b/Sw x a. So it is clear that, for stability, CLw > CLt and this is achieved by ensuring that the wing is rigged at a greater incidence than the tailplane.

OK – a bit simplistic but far more algebraically correct than the so-called ‘CFS’ explanation which is utter hogwash!

And a tip for you boffins - K.I.S.S

Unless your tail and/or wing are operating in the non-linear range then changing the setting angle of either will NOT affect the stability, because stability is determined by:

the lift curve slope and pitch moment slopes of the wing WITH alpha
the tailplane lift curve slope
the downwash gradient (again, alpha)
and the relative position of these surfaces in the fore/aft sense, plus their relative sizes.

as long as the various slopes are not impacted, I can change the tail setting angle and stability will not be in ANY way affected

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OK, let\'s keep it simple, as suggested.

An aircraft with wing and tail located as previously is trimmed in steady level flight. Ignoring the pedantics of CP and pitching moment reference points, let\'s say that

Lw * a = Lt * b

Let us now disturb the aircraft from the trimmed position by introducing an instantaneous change in alpha, da.

The effect of an increase in wing aoa of da is to change the wing downwash by a corresponding amount, de (where de=da * downwash_gradient)

Therefore, the change in angle of attack seen by the tailplane is : d(at)=da-de

Assuming now that the lift-curve slopes of the two airfoils are identical, and equal \'k\', we get:

the change in wing lift: dCL=k * da

the change in tail lift: dCLt=k * d(at) = k * (da-de)

Since de and da are related by the downwash gradient, if we denote that as d_g we get:

dCLt = k * ( da - da*d_g) = k * da * (1-d_g)

Therefore the total change in pitching moment about the cg would be:

+dCL * a - dCLt * b

= k * da * a - k * da * (1-d_g) * b

= k * da * [a - b *(1-d_g) ]


Notice that nowhere in that calculation of the resulting pitching moment does the actual absolute AoA on either surface play a role - what matters is the gradient with alpha of the aerodynamic characteristics.

Additionally, this is hideously simplified; to do it properly would mean starting from using something other than CPs AND would need some decent greek symbols!