Pitching moment

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The pitching moment of an airfoil, in aerodynamics, is a moment produced by a vertical force applied at a distance forward or aft from the aerodynamic center of the airfoil, causing the aircraft to pitch up or down^[1].

1 Longitudinal static sability
- 1.1 Trim
- 1.2 Static stability
2 References
3 See also

[edit] Longitudinal static sability

The aerodynamic pitching moment is important for the study of the stability of aircraft and missiles.

Near the cruise condition most of the lift is generated by the wings, with ideally only a small amount generated by the fuselage and tail. We may analyse the longitudinal static stability by considering the aircraft in equilibrium under wing and tail lift, and weight. The moment equilibrium condition is called trim, and we are generally interested in the directional stability of the aircraft about this trim condition.

Equating forces in the vertical direction:

W = L w + L t

where W is the weight, $L w$ is the wing lift and $L t$ is the tail lift.

For a symmetrical aerofoil at low incidence, the wing lift is proportional to the incidence angle:

$L_w=qS_w\frac{\partial C_L}{\partial \alpha} (\alpha-\alpha_0)$

where $S w$ is the wing area $C L$ is the (wing) lift coefficient, : $α$ is the incidence angle. The term $α 0$ is included to account for camber, which results in lift at zero incidence. Finally q is the dynamic pressure:

$q=\frac{1}{2}\rho U^2$

where $ρ$ is the air density and U is the speed.

[edit] Trim

The tailplane is usually a symmetrical aerofoil, so its lift is proportional to incidence, but in general, there will also be an elevator deflection to maintain moment equilibrium (trim). In addition, the tail is located in the flow field of the main wing, and consequently experiences a downwash, reducing the incidence at the tailplane.

For a statically stable aircraft the tailplane lift acts in the opposite sense to that of the main wing, whilst it acts in the same sense for unstable aircraft. This is exploited in high performance combat aircraft, which overcome the inherent aerodynamic instability by using feedback control (artificial stability).

Similarly, the trim lift is in the same direction as the wing lift for a stable canard configuration, although the downwash from the canards reduces the main wing lift.

The tail lift is, therefore:

$L_t=q S_t\left(\frac{\partial C_l}{\partial \alpha}\left(\alpha-\frac{\partial \epsilon}{\partial \alpha}\alpha\right)+\frac{\partial C_l}{\partial \eta}\eta\right)$

where $S t$ is the tail area, $C l$ is the tail lift coefficient, $η$ is the elevator deflection, and $ε$ is the downwash angle.

Note that for a tail configuration, the tail incidence is usually less than that of the mainplane, so the main wing should stall before the tail, ensuring sufficient control remains for early recovery. Loss of main wing lift whilst maintaining tail lift lowers the nose, reducing incidence and tending to recover from the stall.

There are a few classical cases where this favourable response was not achieved, notably the Boeing 727 when it first entered service. This aircraft had a high T-tail, which in the event of a stall immersed the tailplane in the separated flow from the main wings, causing loss of tail lift, increasing the incidence angle still further; a phenomenon known as 'deep stall'.

Taking moments about the centre of gravity, the net nose-up moment is:

M = L w x g - (l t - x g) L t

where $x g$ is the location of the centre of gravity behind the aerodynamic center of the main wing, $l t$ is the tail moment arm. For trim, this moment must be zero. For a given maximum elevator deflection, there is a corresponding limit on centre of gravity position at which the aircraft can be kept in equilibrium. When limited by control deflection this is known as a 'trim limit'. In principle trim limits could determine the permissible forwards and rearwards shift of the centre of gravity, but usually it is only the forward cg limit which is determined by the available control, the aft limit is usually dictated by stability.

In a missile context 'trim limit' more usually refers to the maximum angle of attack, and hence lateral acceleration which can be generated.

[edit] Static stability

The nature of stability may be examined by considering the increment in pitching moment with incidence at the trim condition. If this is nose up, the aircraft is directionally unstable if nose down it is stable. Differentiating the moment equation with respect to $α$ :

$\frac{\partial M}{\partial \alpha}=x_g\frac{\partial L_w}{\partial \alpha}-(l_t-x_g)\frac{\partial L_t}{\partial \alpha}$

Note: $\frac{\partial M}{\partial \alpha}$ is a stability derivative.

It is convenient to treat total lift as acting at a distance h ahead of the centre of gravity, so that the moment equation may be written:

M = h (L w + L t)

Applying the increment in incidence:

$\frac{\partial M}{\partial \alpha}=h(\frac{\partial L_w}{\partial \alpha}+\frac{\partial L_t}{\partial \alpha})$

Equating the two expressions for moment increment:

$h=x_g-l_t\frac {\frac {\partial L_t}{\partial \alpha}}{\frac{\partial L_w}{\partial \alpha}+\frac{\partial L_t}{\partial \alpha}}$

The denominator of the second term is dominated by the wing lift, so the tail lift term can be ignored, yielding:

$h=x_g-c(1-\frac{\partial \epsilon}{\partial \alpha})\frac{\frac{\partial C_l}{\partial \alpha}}{\frac{\partial C_L}{\partial \alpha}}\frac{l_t S_t}{c S_w}$

Where c is the mean aerodynamic chord of the main wing. The term:

$V=\frac{l_t S_t}{c S_w}$

is known as the tail volume ratio, its rather complicated coefficient, according to Piercy, has values in the range 0.5 to 0.65 for typical configurations, hence the expression for h may be written more compactly, though somewhat approximately, as:

h = x g - 0.5 c V

h is known as the static margin. For stability it must be negative.

For a tailess aircraft V=0, so the condition of static stability is for the centre of gravity to lie ahead of the aerodynamic centre.

[edit] References

^ Preston, Ray (2006). Main Wing Stability. Aerodynamics Text. Selkirk College. Retrieved on 2006-04-01.

Piercy N A V - Aerodynamics. English Universities Press. London. 1943. pp384-386.

Category: Aerodynamics

Pitching moment

From Wikipedia, the free encyclopedia

Contents

[edit] Longitudinal static sability

[edit] Trim

[edit] Static stability

[edit] References

[edit] See also

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