The input to the airplane is the set of incremental aerodynamic forces and moments that derive from the turbulence—six associated with the rigid-body degrees of freedom and others with the elastic degrees of freedom. All of these inputs are, of course, random functions of a single variable, time, and are described statistically by the methods previously given. Once they are known, the problem of calculating system response is relatively routine. Let us illustrate the structure of the problem with a linear/invariant aerodynamic model. Let g be a vector (g for “gust”) that somehow defines the atmospheric velocity field (specific forms for g are given below), let f be the associated aerodynamic force vector, and let T be a matrix of “gust transfer functions” that relates them:

f(s) = T(s)g(s) (13.3,1)

The determination of the input then consists of two parts—defining g and finding the elements of T. When both of these are known, (13.3,1) yields the force vector, which can then be incorporated into the vehicle system equations in a more or less straightforward manner. The details of the process depend very much on the degree of idealization used and the assumptions made; examples are given below.

One approximation that is almost always made is to ignore the departure of the airplane from rectilinear flight, i. e. to assume it samples a frozen field on a straight line. The input statistics can then be derived quite readily from those of the turbulence given in frame FA. Thus let FA have axes parallel to Fw, and zero time be chosen so that the coordinates of the airplane mass center relative to FA are (Vet, 0, 0). The connection between (x, у, z), the coordinates of a point in Fw, and (xv x2, x3) the coordinates of a point in Fa is then

xi = + *, *2 = 2/. *з = * (13.3,2)

We now change notation for the turbulent velocities, to emphasize that they are parallel to the axes of ■*W> denoting them (ug, vg, wg). Being functions of (aq, x2, x3) they become functions of (x, y, z, t) via (13.3,2)—or for a fixed point of the airplane, functions of f only. The spectral component (see after

544 Dynamics of atmospheric flight 13.2,14) is then a velocity field of the form

exp i[Q1(Fe< + x) + D.2y + Q. sz] = еІПіГ ^еі(ПіХ+П2У+Пз! і) (13.3,3)

or for the two-dimensional ease, the above with the z term absent. It is seen to consist of a time-periodic velocity at any fixed point (x, y, z) of the vehicle.

Even when the system is linear, it is not in general true (as was erroneously stated on p. 321 of Dynamics of Flight—Stability and Control) that the response to turbulence can be constructed of a superposition of the three separate responses to ug, vg, and wg. This is the case only when there are no cross-correlations between elements of the input vector associated with different components of the turbulence. Equations (3.4,48 and 49) make it clear that there are contributions to response power and cross spectra that derive from cross spectra of the input components. Such cross spectra exist even in isotropic turbulence if variations over the vehicle are allowed for, as illustrated in Eig. 13.2 for the points A and C of a wing-fin combination. In spite of the above theoretical condition, practical calculations of gust response are often made for one input component at a time. There is no assurance, however, that significant errors of omission will not occur when that is done.