Pitching

So far we've been talking a lot about the forces that act on a fixed wing aircraft. That's because the translational acceleration of an object is given by the following equation:

F = ma

But there are also moments which cause rotational accelerations. We will use the letters L, M, and N to denote the rotational moment about the vehicle's x, y, and z axes.

For longitudinal motion, we only care about the pitching moment M. The pitching moment is given by the following equation:

M = C_M \bar{q} S \bar{c}

Most of the interesting physics is wrapped up in the dimensionless coefficient C_M. Generally, this coefficient will be a function of the angle of attack, elevator deflection, and pitching rate, so we can say:

C_M = f(\alpha, \delta E, q)

We generally assume the pitching rate is close enough to zero that it can be ignored. We model this coefficient with the following equation:

C_M = C_{M_0} + C_{M_\alpha}\alpha + C_{M_{\delta E}} \delta E

This coefficient C_{M_\alpha} is important. When it's negative, the vehicle will be "statically stable in pitch". So if the vehicle has some small positive angle of attack, this term will lead to a negative pitching moment which will bring the angle of attack back towards zero.

Why does M depend on \alpha?

It should be clear that elevator deflection will cause the vehicle to pitch. But why does the pitching moment depend on the angle of attack? Consider an airplane viewed from the side:

The center of gravity for this plane is located somewhere in the plane around where the wing meets the fuselage.

The red arrows show the effect of lift acting everywhere on the wing.

We can define something called the center of pressure and treat this distributed lift force as if it were acting at one point on the wing.

In this case (above), the center of pressure is directly below the center of gravity, so this will not produce any pitching moment clockwise or counter-clockwise.

But imagine if the center of pressure were shifted slightly forward on the wing:

Now the upwards pressure will cause a rotation about the center of gravity.

And in fact, this is exactly what happens in real aircraft. As the aircraft's angle of attack (\alpha) changes, the center of pressure will tend to move forwards and backwards along the wing.

It’s this motion of the center of pressure that explains why the pitching moment depends on the vehicle’s angle of attack.

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