THE “PANEL” METHOD (ref. 13.5)

In this method the principle aerodynamic surfaces are divided into N panels, as illustrated in Fig. 13.9. At a reference point of the wth panel the

turbulent velocities are [ugJt), vgJt), wgJt)], and the gust vector g is the 3N column of all these components. The force vector is then

f=Tg (13.3,15)

where f is an (M x 1) vector, M being dependent on the problem, and T == is an (M x 3N) matrix of aerodynamic coefficients. To carry out the analysis of f and subsequently of the spectra of vehicle response one must first evaluate all the 3MN transfer functions ti}(s) and then apply the

input/output theorem (3.4,48). The latter includes all the cross-spectral densities of the components of g, which do not all vanish.

When the method is applied for one velocity component only, say wg, and for a relatively small number of panels, the matrix T is not excessive in size. The time functions for the input elements are obtained by using

(13.3,2) , and the relevant cross spectra are derived from them (note that in this formulation each input and output quantity is a function of time only). Consider for example the wg components at the mth and nth panels, wgJt) and wgJt). The cross-correlation is

в’тп(т) = worSt)’ w„Sl + T)

which by using Pig. 13.96 we can identify as

R’mn(T) = – йзз(£і + VeT, i2, 0)

where £x and f2 are as shown, and R33 is obtained from (13.2,6) as

Д33 = [1]V{(f1 + Kr)2 + £22}-

g(£) for the von Karman model is given by (13.2,18). The Fourier integral of Ii’mn(r) is then the required one-dimensional input spectrum

Ф’шпЫ = ± Г R’mne~^ dr.


For further details of the panel method the reader is referred to ref. 13.5 and the literature cited therein.