THE LINEAR AIR REACTIONS

At the heart of the subject of atmospheric flight mechanics lies the problem of deter­mining and describing the aerodynamic forces and moments that act on a given body in arbitrary motion. It is primarily this aerodynamic ingredient that distinguishes it from other branches of mechanics. Aerodynamic forces and moments are strictly speaking functionals of the state variables. Consider for example the time-dependent lift L(t) on a wing with variable angle of attack a(t). Because the wing leaves behind it a vortex wake that in general generates an induced velocity field at the wing, and because hysteresis is present in flow separation processes, the aerodynamic field that fixes the lift at any given moment is actually dependent not only on the instantaneous value of a but strictly speaking on its entire past history. This functional relation is expressed by

L(t) = Т[а(т)] —oo < r<t (4.9,11)

When a(r) can be expressed as a convergent Taylor series around t, i. e.

a(f) = a{t) + (t — t)a(t) + {t — t)[9] [10]a(t) + ••• (4.9,12)

then the infinite series a(t), a(t), a(t) ••• can replace a(r) in (4.9,11), i. e.

L{t) = L(a, a, a ■ ■ ■) (4.9,13)

where a, a ■ ■ ■ are values at time t. Thus the lift at time t is in this case determined by a and all its derivatives at time t. A further series expansion of the right-hand side of (4.9,13) around t = 0 yields

A L(t) = LaAa + L(jAa)2 + – + La Да + hLaa(Aa)2 + – (4.9,14)

in which all the products and powers of Act, A a ••• appear, and where

dL

La = — etc. (4.9,15)

• da a=0

The classical assumption of linear aerodynamic theory, due to Bryan (1911) is to ac­cept the linear reduction of (4.9,14) as a representation of the aerodynamic force, even when Aa(t) is not an analytic function as implied by (4.9,12), i. e.

AL(t) = LaAa + Ьы Ad + L& Да + ••• (4.9,16)

Derivatives such as La in (4.9,16) are known as the stability derivatives, or more gen­erally as aerodynamic derivatives. For most forces and state variables, only the first term of (4.9,16) is kept, but in some cases, terms up to the second derivative must be retained for sufficient accuracy. This assumption has been found to work extremely well over a wide range of practical applications. Occasionally the addition of nonlin­ear terms such as Laa(Aa)2 = Ьаг(Аа)2 can extend the useful range considerably. Another way of including nonlinear effects is to treat the derivatives as functions of the variables, for example, La = La(a).

A major fraction of the total effort in aerodynamic research in the past has been devoted to the determination, by theoretical and experimental means, of the aerody­namic derivatives needed for application to flight mechanics. A great mass of infor­mation about these parameters has now been accumulated and Chap. 5 is devoted to this topic.

For a truly symmetric configuration, it is evident that the side force Y, the rolling moment L, and the yawing moment N will all be exactly zero in any condition of symmetric flight, that is, when the plane of symmetry remains in a fixed vertical plane. In that case, /3, p, г, ф, ф, and yE are all identically zero. Thus the derivatives of the asymmetric or lateral forces and moments, Y, L, N with respect to the symmet­ric or longitudinal motion variables u, w, q are zero. In writing out the complete lin­ear expression for the aerodynamic forces and moments, we use this fact, and in ad­dition make the further approximations:

3. The derivative Xq is also negligibly small.

4. The density of the atmosphere is assumed not to vary with altitude (see Sec. 6.5).

It should be emphasized that none of these assumptions is basically necessary for the solution of airplane dynamics problems. They are made as a matter of experience and convenience. When it appears necessary to do so, any of the terms dropped can be restored into the equations. With these assumptions, however, the linear forces and moments are:

ДА = XUA и + Xww + AXC	(a)
AY = Yvv + YpP + Y, r + A Yc	ib)
A Z = ZuAu + Zww + ZH, w + Zqq + A Zc A L = Lvv + Lpp + Lrr + A Lc	(c) w> <4’9>17)
AM = MuAu + Mww + M^w + Mqq + Д Mc	(e)
AN = Nvu + Npp + Nrr + ANc	(/)
In the above equations, the terms on the right with subscript c are control forces and moments that result from the control vector c. Explicit forms for the controls will be introduced as they are needed in the following.

Aerodynamic Transfer Functions

The preceding equations are subject to the theoretical objection (not of great practical importance) that the Bryan formulation for the aerodynamics is not quite sound even within the restriction of linearity. This is readily illustrated by considering the lift on a wing following a step change in angle of attack. Let Да be given by Дa(t) = a0l(f) where a0 is a constant. For t > 0, the Bryan formula (4.9,16) gives

A L(t) = Laa0 = const

whereas in fact the lift undergoes a transient approach to the asymptote Laa0, the de­tails of which depend on the wing shape and the Mach number. Equation (4.9,16) fails in this case because Да is not an analytic function, having a discontinuity at t = 0. Now the transient process is often well approximated as a linear one, and as such is subject to exact representation by linear mathematics in the form of an indi – cial function (Tobak, 1954), or an aerodynamic transfer function (Etkin, 1956). The implementation of these alternative representations of aerodynamic force is described in Etkin (1972, Sec. 5.11).