How do the linearised equations of motion work?

Question

I am reading this book: "Flight Dynamic Principles" by Cook, second edition. You can read it here: https://aerocastle.files.wordpress.com/2012/04/flightdynamicsprinciples.pdf

First, go to the page 76, read the sentence:

Further, the only significant higher order derivative terms commonly encountered are those involving $\dot w$.

Why is the only $\dot w$ involved here? the $\dot u$ and $\dot v$ have the same roles, huh??

Second, same page 76, section 4.2.3, the sentence:

The assumptions applied to the aerodynamic terms are also applied to the control terms thus, for example, the pitching moment due to aerodynamic controls may be expressed.

And you can see the equation (4.32)

it is totally different from aerodynamics terms equation..why is that?. Why are moments not dependent u, v, w, and p, q, r like aerodynamics terms.

Lastly, look at the example 4.2 at page 80, what is the meaning of this example? where do the values of dimensionless longitudinal derivatives at page 81 come from? What are they? The book said seeing the appendix 2 at page 413. Now go to the page 413, where do these multipliers come from?

Answer 1

Um..it's a bit of a counterintuitive and hard to visualise field of aeronautics. There are six degrees of freedom (if we assume that the aircraft is rigid) that need to be transformed from aircraft axes into earth axes. And since we're looking at aerodynamic stability, we need to consider wind axes to introduce disturbances: if the aircraft moves as an effect of the disturbance, the relative wind changes.

Since we're working with transformation matrices, we can only solve these analytically if there are simple multipliers in the matrix, as soon as there are trigonometric terms such as sines & cosines introduced, we'll need to use numerical methods. So we linearise by looking at small disturbances only, and set $cos(\phi)$ = 1 and $sin(\phi)$ = $\phi$. Usually OK from a body axis perspective, since stall occurs at 15 degree which is a small enough angle. But..

..to make it all even worse, it is one of the few remaining bits of engineering where the dark art of quaternions are used, since when the nose is pointing straight up some of the Euler angles are undefined and we need a fourth dimension to provide continuity.

I reckon if you're interested in this book, the only prior knowledge you need to have is the realisation that it is going to be complex. What has always worked for me is just taking the results at face value, start to work with it to gain experience, then look it up again so that the context is much clearer.

The book does indeed seem to be a bit laborious to read. What they're saying with the statement on $\dot{w}$ is that in the end result, the linearisations and simplifications are justified, in that no big inaccuracies occur if effects of $\dot{u}$ and $\dot{v}$ are neglected. Effects of $\dot{w}$ are significant though and do need to be considered, which the authors do in the rest of the text.

Equation 4.32 is the most generic statement there is on moments - what is says is: the total moment is (how the moment changes with a variable) times the variable. It does not yet consider what the cause of the moment change is.