The method proposed by Ribner (ref. 13.4) does not fit the pattern of the foregoing ones in that no function equivalent to g(t) is explicitly defined. Instead the response is found as a superposition of responses to individual spectral components like that pictured in Fig. 13.5. Thus let the wg component of a single wave be described by [cf. (13.3,21a)]
This time-periodic velocity field induces periodic incremental pressure distributions that integrate to periodic incremental forces and moments, of which for example the lift is described by
The relationship between the lift and the velocity is given by an aerodynamic transfer function, Г(£11; Q,), i. e.
dL = I^Qj, Q2) dW
(note that 1c = Qjc/2). The mean-square incremental lift produced by the whole turbulent field is then given by the basic response theorem [(3.4,51)
extended to two dimensions]
X2 =JJ 02)<Ш1(Ш2
and the one-dimensional spectrum for lift is
Фіі(^і) = f |Г(^і> £i2)|2xF33(01, £12)<Ш2
Any vehicle response variable such as angle of attack or load factor is treated like the lift above, but the transfer function is of course different. .
The heart of this approach is the availability of aerodynamic transfer functions like Г, of which a whole matrix is in general required for all the generalized forces and moments associated with rigid and elastic degrees of freedom, and with ug, vg, and wg inputs. There are methods available for calculating some of these transfer functions for some wing shapes (refs. 7.16, 13.12), and for propellers (ref. 13.13).
In view of the fact pointed out previously, that the spectra of vehicle responses to ug, vg, wg cannot in general be simply superposed (owing to the nonvanishing of certain cross-correlations or cross spectra), the three velocity components should, strictly speaking, be considered simultaneously. Ribner’s method has not yet been explicitly extended to cover this case.