Similar to the symmetrical motions ( up- down ), the asymmetrical motions ( left-right ) can be completely described by a linear state space model with only four state elements :
|
v sideways velocity // in body fixed frame. φ roll angle. p roll rate. r yaw rate. | (1) |
All motions are positive "to the right" TBC.
Like in the symmetrical case, 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 ( forward in the airplane ) and the Z body axis ( down in the airplane ).
With the geometrical terms similar to the symmetrical case, Newton's law for the sideways direction in body axes becomes :
| (2) |
Here, Y is the sum of all aerodynamic forces in the sideways direction ( body fixed, force to the right TBC ).
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 moment equations are once again unchanged relative to the non-body-fixed, 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 :
| (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 ψ and y.
TBW.