ongitudinal Stati Stability Some definitions C m M V S pithing moment without dimensions (so without influene of ρ, V and S) it is a shape parameter whih varies with the angle of attak. Note the hord in the denominator beause of the unit Nm! C V S For the wing+airraft we use the surfae area of the wing S! C V S For the tail we use the surfae of the tail: S! Definition of aerodynami enter of a wing: The aerodynami enter (a..) is the point around whih the moment does not hange when the angle of attak hanges. e an therefore use Cm a as a onstant moment for all angles of attak. The aerodynami enter usually lies around a quarter hord from the leading edge.
Criterium for longitudinal stati stability (see also Anderson 7.5): e will look at the onsequenes of the position of the enter of gravity, the wing and the tail for longitudinal stati stability. For stability, we need a negative hange of the pithing moment if there is a positive hange of the angle of attak (and vie versa), so: Cm 0 Cm 0 0 0 Graphially this means C m (α) has to be desending: For small hanges we write: dc m d 0 e also write this as: Cm 0
hen C m (α) is desending, the Cm 0 has to be positive to have a trim point where C m = 0 and there is an equilibrium: So two onditions for stability: ) Cm 0 > 0; if lift = 0; pithing moment has to be positive (nose up) dc ) m 0 d ( or C 0 m ); pithing moment has to beome more negative when the angle of attak inreases Condition is easy to hek. But what is the onsequene of ondition? For this we have to study what happens when the angle of attak hanges. Therefore we have to look at the derivatives to the angle of attak and then use this to predit what the hange in pithing moment will be. For this we will first look at the tail and then look at the effet on the whole onfiguration. 3
The horizontal tail surfae The angle of attak of the horizontal tail α : The wing has pushed air down and dereased the angle of attak with ε, the downwash angle. The relation between α and α now beomes: i So the hange in α due to a hange in angle of attak α now an be alulated: d d d ( i ) d d d The term d basially means: the hange in downwash due to the hange in angle of d attak. Typial values are around 0.0 for tails that do not have a T-onfiguration. 4
Calulate hange in pithing moment due to a hange in angle of attak α: the Cm α In this figure is drawn upward. In reality it ould just as well be pointed downwards, but the sign onvention is that the lift is positive upward, and therefore we draw it like this. Moment around enter of gravity: Pithing moment: ith: M M M l ( l l ) a g g M a = Moment of wing around aerodynami enter, so onstant for all α! l = ing (and fuselage) lift fore times arm relative to.g., positive (lokwise) g - ( l l ) = Moment of tail lift fore rel. to.g., is negative (ounter lokwise) g e an simplify the moment equation by using the total lift fore : = + M M l l l a g g M ( ) l l a g M l l a g 5
Now make this pithing moment dimensionless with : V S M M l l a g V S V S V S V S e an now simplify this enormously by using the definitions in the start (Note how the differene in moments and fores all work out alright). One ompliation however is that the lift oeffiient of the tail surfae is defined using the area of the tail surfae, so S instead of S: C V S Using all this transforms the moment equation into its dimensionless form: l g C m Cm C a C V S l V S l S l C C C C S g m m a This we now all V Sl S l C C C C V g m m a The ratio V (the tail area times the arm divided by the wing area times the hord) is alled the tail volume V (even though it is dimensionless) : V S l S 6
e want to know dcm ; so differentiate to α : d dc dc m m dc l dc d d d d a w g V A number of observations an be made: dc m a w d 0 The moment around the aerodynami enter does not hange when the angle of attak α hanges. So this term is zero by definition and disappears. Note how the tail volume V is independent of the angle of attak, and so it an be treated as onstant. : hat we re left with is this: dcm dc l dc d d d g V The dc d is simply a harateristi of the airraft shape: the steepness of the C -α urve: e normally should have similar data for the tail airfoil, however then the the angle of dc dc attak of the tail surfae α is on the x-axis. So we know the and not the d d. But we have seen: i and therefore: d d d d So: dc dc d dc d d d d d d 7
So we substitute Both dc d and dc d with dc d d d : dc d dcm dc l g dc d V d d d d are onstants for any given shape, and indiate the steepness of the C -α urve. They are also written as a and a t, where the index t refers to the tail. hen also writing dcm d as Cm we an write the last equation above as follows: lg d Cm a at V d And we onluded for stati stability that this will be stable if: lg d a atv 0 d Cm should be less than zero, so the airraft So for the tail this means: d lg atv a d From this relation we an not only onlude the following: - A larger tail will ontribute to stati stability - A longer distane between tail and wing will ontribute to stability - A enter of gravity that is just after the wing or even before the wing ontributes to stability (forward g => more stable, aft.g. less stable) 8
From this equation we an, for a given airraft onfiguration, alulate what.g. position is just on the edge of stability. This point is alled the neutral point. If the.g. is before this point the airraft will be stable, if the.g. is after this point the airraft will be unstable. e an alulate this by solving the borderline ase between stability and instability. So at neutral point: l g l andc 0 : np m lnp d a at V 0 d lnp d a at V d The neutral point then is: l a d V with V S l a d S np t The distane between the neutral point and the enter of gravity is alled the stati margin: Exerise A similar analysis an be done for a anard plane. ould a forward.g. there also be a benefit or would everything reverse? e leave that as an exerise for the reader. Normal onfiguration Canard onfiguration (Beeh 99) (Beeh Rutan Starship 000) 9