Systems of Axes and Notation
2.2 AIRCRAFT BODY FIXED AXES .1 Generalised body axes
It is usual practice to define a right handed orthogonal axis system fixed in the aircraft and constrained to move with it. Thus when the aircraft is disturbed from its initial flight condition the axes move with the airframe and the motion is quantified in terms of perturbation variables referred to the moving axes. The way in which the axes may be fixed in the airframe is arbitrary although it is preferable to use an accepted standard orientation. The most general axis system is known as a body axis system (oxbybzb) which is fixed in the aircraft as shown in Fig. 2.2. The (oxbzb)
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o xb
zb zw yb, yw
xw V0
ae
ae
Figure 2.2 Moving axes systems.
plane defines the plane of symmetry of the aircraft and it is convenient to arrange the oxb axis such that it is parallel to the geometrical horizontal fuselage datum.
Thus in normal flight attitudes the oybaxis is directed to starboard and the ozbaxis is directed “downwards’’. The origin o of the axes is fixed at a convenient reference point in the airframe which is usually, but not necessarily, coincident with the centre of gravity (cg).
2.2.2 Aerodynamic, wind or stability axes
It is often convenient to define a set of aircraft fixed axes such that the ox axis is parallel to the total velocity vector V0 as shown in Fig. 2.2. Such axes are called aerodynamic, wind or stability axes. In steady symmetric flight wind axes (oxwywzw) are just a particular version of body axes which are rotated about the oybaxis through the steady body incidence angle αeuntil the oxwaxis aligns with the velocity vector.
Thus the plane (oxwzw) remains the plane of symmetry of the aircraft and the oyw and the oybaxes are coincident. Now there is a unique value of body incidence αefor every flight condition, therefore the wind axes orientation in the airframe is different for every flight condition. However, for any given flight condition the wind axes orientation is defined and fixed in the aircraft at the outset and is constrained to move with it in subsequent disturbed flight. Typically the body incidence might vary in the range−10◦ ≤ αe ≤ 20◦over a normal flight envelope.
2.2.3 Perturbation variables
The motion of the aircraft is described in terms of force, moment, linear and angular velocities and attitude resolved into components with respect to the chosen aircraft fixed axis system. For convenience it is preferable to assume a generalised body axis system in the first instance. Thus initially, the aircraft is assumed to be in steady rectilinear, but not necessarily level, flight when the body incidence is αeand the steady velocity V0resolves into components Ue, Veand Weas indicated in Fig. 2.3.
In steady non-accelerating flight the aircraft is in equilibrium and the forces and
L, p, f N, r,y
M, q, q
o
Z, We, W, w
X, Ue, U, u
Pitch
Yaw
Roll
Y, Ve, V, v
z
x
y
Figure 2.3 Motion variables notation.
Table 2.1 Summary of motion variables
Trimmed equilibrium Perturbed
Aircraft axis ox oy oz ox oy oz
Force 0 0 0 X Y Z
Moment 0 0 0 L M N
Linear velocity Ue Ve We U V W
Angular velocity 0 0 0 p q r
Attitude 0 θe 0 φ θ ψ
moments acting on the airframe are in balance and sum to zero. This initial condition is usually referred to as trimmed equilibrium.
Whenever the aircraft is disturbed from equilibrium the force and moment balance is upset and the resulting transient motion is quantified in terms of the perturbation variables. The perturbation variables are shown in Fig. 2.3 and summarised in Table 2.1.
The positive sense of the variables is determined by the choice of a right handed axis system. Components of linear quantities, force, velocity, etc., are positive when their direction of action is the same as the direction of the axis to which they relate.
The positive sense of the components of rotary quantities, moment, velocity, attitude, etc. is a right handed rotation and may be determined as follows. Positive roll about the ox axis is such that the oy axis moves towards the oz axis, positive pitch about the oy axis is such that the oz axis moves towards the ox axis and positive yaw about the oz axis is such that the ox axis moves towards the oy axis. Therefore, positive roll is right wing down, positive pitch is nose up and positive yaw is nose to the right as seen by the pilot.
A simple description of the perturbation variables is given in Table 2.2. The inten-tion is to provide some insight into the physical meaning of the many variables used in the model. Note that the components of the total linear velocity perturbations
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Table 2.2 The perturbation variables
X Axial “drag’’ force Sum of the components of
Y Side force aerodynamic, thrust and
Z Normal “lift’’ force weight forces
L Rolling moment Sum of the components of
M Pitching moment aerodynamic, thrust and
N Yawing moment weight moments
p Roll rate Components of angular
q Pitch rate velocity
r Yaw rate
U Axial velocity Total linear velocity
V Lateral velocity components of the cg
W Normal velocity
V0 ae
qe q
xb U
Ue
We W
zb o
Horizon Perturbed
body axes
Equilibrium body axes
ge
Figure 2.4 Generalised body axes in symmetric flight.
(U , V , W ) are given by the sum of the steady equilibrium components and the transient perturbation components (u, v, w) thus,
U = Ue+ u V = Ve+ v
W = We+ w (2.1)
2.2.4 Angular relationships in symmetric flight
Since it is assumed that the aircraft is in steady rectilinear, but not necessarily level flight, and that the axes fixed in the aircraft are body axes then it is useful to relate the steady and perturbed angles as shown in Fig. 2.4.
With reference to Fig. 2.4, the steady velocity vector V0defines the flight path and γeis the steady flight path angle. As before, αeis the steady body incidence and θeis the steady pitch attitude of the aircraft. The relative angular change in a perturbation is also shown in Fig. 2.4 where it is implied that the axes have moved with the airframe
and the motion is viewed at some instant during the disturbance. Thus the steady flight path angle is given by
γe= θe− αe (2.2)
In the case when the aircraft fixed axes are wind axes rather than body axes then,
ae= 0 (2.3)
and in the special case when the axes are wind axes and when the initial condition is level flight,
αe= θe= 0 (2.4)
It is also useful to note that the perturbation in pitch attitude θ and the perturbation in body incidence α are the same thus, it is convenient to write,
tan (αe+ θ) ≡ tan (αe+ α) = W
U ≡ We+ w
Ue+ u (2.5)
2.2.5 Choice of axes
Having reviewed the definition of aircraft fixed axis systems an obvious question must be: when is it appropriate to use wind axes and when is it appropriate to use body axes? The answer to this question depends on the use to which the equations of motion are to be put. The best choice of axes simply facilitates the analysis of the equations of motion. When starting from first principles it is preferable to use generalised body axes since the resulting equations can cater for most applications.
It is then reasonably straightforward to simplify the equations to a wind axis form if the application warrants it. On the other hand, to extend wind axis based equations to cater for the more general case is not as easy.
When dealing with numerical data for an existing aircraft it is not always obvious which axis system has been used in the derivation of the model. However, by reference to equation (2.3) or (2.4) and the quoted values of αeand θeit should become obvious which axis system has been used.
When it is necessary to make experimental measurements in an actual aircraft, or in a model, which are to be used subsequently in the equations of motion it is preferable to use a generalised body axis system. Since the measuring equipment is installed in the aircraft its location is precisely known in terms of body axis coordinates which, there-fore, determines the best choice of axis system. In a similar way, most aerodynamic measurements and computations are referenced to the free stream velocity vector.
For example, in wind tunnel work the obvious reference is the tunnel axis which is coincident with the velocity vector. Thus, for aerodynamic investigations involving the equations of motion a wind axis reference is to be preferred. Traditionally all aerodynamic data for use in the equations of motion are referenced to wind axes.
Thus, to summarise, it is not particularly important which axis system is chosen provided it models the flight condition to be investigated, the end result does not depend on the choice of axis system. However, when compiling data for use in the equations of motion of an aircraft it is quite common for some data to be referred
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q f y y
q
o f x0
x1 x2, x3
y0
y1, y2
y3
z3
z2 z0, z1 Figure 2.5 The Euler angles.
to wind axes and for some data to be referred to body axes. It therefore becomes necessary to have available the mathematical tools for transforming data between different reference axes.