Aircraft stabilty and control analysis extensively uses so-called linear state space models. The basic equation in this type of model is :
| x • = A . x + B . u | (1) |
We will assume your familiarity with the concept. If not, please follow the link above.
The motions of an airplane can be divided into symmetrical ( up/down ) and asymmetrical ( left/right ).
This section discusses the symmetrical part.
We will not derive the physical model of a generic aircraft here. It is a long story, and it is best studied from a thorough textbook like Etkin's. One outcome is that the small motions of the aircraft around straight and level flight can be cleanly separated into the "symmetrical" and the "asymmetrical" motions.
The symmetrical motions ( fore-aft, up-down, and rotation in pitch ) can be completely described by a linear state space model with these four state elements :
|
u forward velocity // short for Δu, an extra on the airspeed V. v vertical ( down ) velocity. // in body fixed frame, see below. q pitch rate. // nose up. θ pitch attitude. // nose up. | (2) |
It is not immediately obvious that these four variables are necessary and sufficient to describe the motion, but they are. However, the state elements themselves are not as obvious as they may seem.
The problem lies in the fact that the equations of motion are formulated in a body fixed reference frame. In particular, the airplane velocity is defined relative to the X body axis fixed in the airplane ( pointing forward ) and the Z body axis ( down in the airplane ).
The great theoretical advantage is that in the body frame of reference, the inertial properties do not change. Having to take care of the change in inertial properties would complicate the equations of motion no end.
This is the conventional point of view. For a critique, see ****
The downside is that the body is not an inertial reference frame, because the airplane itself shifts and rotates. And Newton's equations of motion only hold in an inertial frame. After some subtle math, it turns out that in horizontal flight, Newton's law for the vertical direction in body axes becomes :
| (3) |
Here, Z is the sum of all aerodynamic forces in the vertical direction ( body fixed, positive down ) after subtracting the steady state lift needed to keep the aircraft flying on a horizontal flight path. We could strictly call this force ΔZ instead of Z, but we won't.
What this says is that the change in vertical (down) body-fixed velocity comes first of all from the sum of the aerodynamic vertical forces as expected, but it also comes from a purely geometrical term. When the airplane rotates backward in straight and level flight, the airspeed V remains horizontal in the inertial world, but it points more downward relative to the airplane. In body axes, the velocity V gets a "vertical" down component.
The geometrical transformation from the inertial frame to the moving one contains all the other cross terms between the horizongtal and vertical velocities and the rate of rotation q as well, but the products between the small perturbations u, w and q are an order smaller than the perturbations themselves and can be neglected when linearizing. The only product term that is not an order smaller is the ( V . w ) one, because V is not zero in the steady state.
By the same subtle derivation as for (3), the moment equation around the pitch axis is unchanged relative to the inertial case :
| (4) |
A vertical down velocity ( in the airplane's own body axes ) is the equivalent of an angle of attack. The air comes in from below because the aircraft is moving down by a velocity w, and forward by ( nearly ) its flying velocity V. The equation is :
| (4) |
Note again that these velocities are defined as vertical and forward as measured in the airplane's body axes. This can happen in either of two ways, or by a combination of both :
- the airplane attitude is horizontal, ( θ = 0 ), but in this flat attitude it sinks with a non-zero vertical speed.
We will shortly see that this gives a descending flight.
- the airplane has a non-zero pitch attitude θ, but it follows a horizontal flight path.
The air will come in from below at the angle θ, which means that α = θ.
The somewhat unexpected result is that the climb angle, more properly called the flight path angle γ, is :
| (4) |
The flight path angle at zero angle of attack ( zero vertical velocity v in the body axes ) is equal to the pitch attitude θ. At first this does not feel right, because a sudden change in pitch angle would not cause a sudden change in flight path angle. But this is made up for by the kinematics of the motion, which comes from the same subtle derivations as before. In fact, we already met the kinematic rate of change of w with q in equation (3). The rate of change of θ with q is purely kinematic :
Non-dimensional parameters and scaling.
| (5) |
A pitch rate q has two simultaneous effects : it increases θ, and it increases α by the same amount.
By (4), a pitch rate by itself does not change the flight path angle.
- make more formal that α = w/V, and mention that alpha has an aerodynamic effect(and theta does not). Briefly touch on gravity
- mention here, or refer to, the Ax and Az output specific forces.
- mention that q *does* have an effect on the aerodynamic force.
- find a derivation for F = m . ( v_dot + omega x v ) in rigid body dynamics.
In the matrix formulation of the section on state space, the state vector and input vectors are :
| (6) |
TODO Explain x and h.
TBW.