Derivatives 21 Part 1a
To better understand what a static stability derivative may show, let us focus on a particular entry. Let us look at first column (Zw, Xw and Mw). The column describes what happens when 'w' (downward velocity) is increased from zero (level flight) to a small positive value (descent). Let us in particular look at Mw, the tendency for the airplane to pitch up (or down) as 'w' is increases. Mw is a good example because it is an important factor in aerodynamic stability, and at the same time it is one of the easier derivatives to get an intuitive mathematical understanding of. In normal cruise 'w' and Mw are both zero, i.e. level flight and steady pitch attitude. That means that an increased 'w' is the same thing as a positive 'w', an increased Mw is the same as a positive Mw and decreased Mw is the same as negative Mw.
Imagine our airplane cruising happily along in still air and then enter an area with rising air. Presumably the balance of the stable cruise flight (in the still air) is disturbed by the sudden change to the relative velocity of the airplane (relative to the surrounding air mass). As the airplane enters the rising air, from the airplane's perspective, it feels like the airplane is no longer in level flight, but now in a descent.
Several possible aerodynamic behaviours might be reasonably expected as the airplane enters rising air. I think most of us will agree that it is reasonable to expect the updraft to somehow push the airplane upwards. Indeed, that particular expectation is so reasonable that it is included in the FAR stability derivatives tool tip for Zw; the negative Zw value means that, all else being equal, the vertical velocity will tend back towards zero. It is less obvious what happens to the forward speed of the airplane when the airplane enters rising air, and in fact it is possible to design airplanes with positive or negative Xw. Finally, what we really wanted to know is, when the airplane enters rising air, will its nose pitch up or down?
We characterise a stability derivative as positive static stability if the initial resulting trend of the disturbance is in the direction we consider beneficial. If the trend is in the opposite direction it is characterised as negative static stability. If there is little or no trend we call it neutral static stability. Positive static stability is highlighted with a green font, and negative static stability is highlighted with a red font. The FAR analysis will not identify neutral static stability, and will instead simply use a green or red font based on the sign of the derivative, regardless of how close to zero the number is.
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Now that we understand Mw, let us look at Zw and Xw too.
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The final 'updraft derivative' to look at is Xw.
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We do not have any particular reason to aim for a negative Xw, but if we want to see if we can create such a design just for the fun of it, then we should design it so that the increased drag portion exceeds the forward lift portion of the Xw-effect. The first thing that comes to mind, is that we need more drag to do that. Technically we want a great amount of 'increased drag' as a function of angle of attack, but in practical terms, in order to produce a big drag increase, we need a lot of drag to start with. One way to solve that problem is to install brake fins, like the curious but potent brake fins on this modified airplane.
When fully active, the brake fins will create a great amount of drag. The fully extended brake fins will at least quadruple the drag of the airplane, and in some landing-like configurations it is more like a ten fold increase in drag.
With the brake fins in their travel setting the stability derivative Xw for the modified airplane shows 0.083 (1 km, mach 0.3). With fully deployed brake fins Xw shows -0.123. The airplane is easier to land with brake fins deployed, but I doubt that it is the Xw sign change that makes the difference. Still, the point is that we can modify our airplane design to manipulate Xw if we want.