# THE POWERS SERIES METHOD

A method proposed by the author (refs. 13.9, 13.10) is a “natural” ex­tension of the point approximation to higher order. In it the velocity field of the airplane is expanded in a Taylor series around the C. G. Thus a typical component such as wg would be described by

«’»(*> У’ 0 = WM + wgJf)x + wg(t)y + wgJt)x‘l —————— (13.3,16)

in which wg(t), wgJf) • • • denote values of wg, dwgldx • • • at the C. G. Since the velocity field is now completely fixed by the coefficients of series like

(13.3,16) , the vector describing the gust field is the column of all these coefficients. In ref. 13.10 the elements of the vector are separated into those that produce longitudinal and lateral forces, i. e.

where only the coefficients of the linear terms in the Taylor series expansions have been listed. The number of terms retained fixes both the domain of validity in wave number space and the complexity of the analysis.

We consider now the limits of validity of the first-order Taylor expansion corresponding to (13.3,17). The method of ref. 13.9 [Eqs. (9.1) et seq.] when applied to the linear part only of the velocity field yields values of CL and Cm in good agreement with the exact Sears function for к < .5. Thus the

552 Dynamics of atmospheric flight limit on £2j is given by [cf. (13.3,12)]

Oj – < .5
2

^i>6

A large gain in the valid range of (one decade) is obtained relative to the point approximation, but only at the cost of using transfer functions for unsteady oscillatory motion to represent the aerodynamics. If quasi-steady aerodynamics is used (e. g. GLa = CLx etc.) then (3.3,12) still holds. It may well be questioned why the power-series method should be used at all with unsteady aerodynamics—why not preferably go directly to the exact two­dimensional transfer functions for gust penetration (see Ribner’s method below)? The advantage, if any, of this method rests in the availability, or ease of obtaining, results for oscillatory translation and rotation of the vehicle. f The theoretical and experimental problems posed by the oscillatory boundary condition have proved more tractable in the past than that of the “running wave” characteristic of gust penetration; solutions for oscillatory motions have been vigorously pursued in connection with flutter analyses, and measurements of oscillatory transfer functions, although by no means easy, are much simpler than those for gust penetration.

The limit on Q2 is assessed from a consideration of the rolling moment acting on the wing. An argument based on symmetry considerations (only antisymmetric distributions of velocity produce rolling moments) shows that an expansion in wave number would be of the form

Filotas’ approximate solution for rolling moment can in fact be expressed in this form, with к = Now the linear power series approximation, as we show below, is equivalent to retaining only the first term in (13.3,19). Hence the error can be assessed from the Q23 term, leading to the limit for about 10% error, 026 < 2 which is the same as (13.3,13) for the lift in the point approximation.

In summary then, the first-order power series method, with quasi-steady aerodynamics, has the effect of extending the point approximation to embrace lateral responses, with the limitations

Aj/c > 60 ?..Jb > it

If unsteady oscillatory aerodynamics are used, the A2 limitation is relaxed to Aj/c > 6.

f The method was presented at a time when no was available.

We turn now to the gust transfer function for the power series method. As an example let us consider the equations for rigid-body response, and use the first-order series. Then from (13.3,17) we get [ef. (13.3,7)—for sim­plicity of notation, the subscript g has been omitted here]

 GTu A GTw A. GTux GTw, x GTvv GDu GDw GDux A GDwx GDvv &Lu gLw GLux GLwx A ™mu Gmv> A ‘~rmux A A ^mvy

with an obviously similar matrix for T2. In the quasi-steady approximation, some of these matrix elements would be neglected, and the remaining ones would be expressed as aerodynamic derivatives. We have already discussed the aerodynamic forces associated with (ug, vg, wg), i. e. the elements of the first two columns above, which are identical with (13.3,7). The remaining elements describe the effect of “gust-gradients” on the airplane. The gradient terms wx, wy correspond to linearly varying downwash over the airplane surface, which provides boundary conditions on relative motion precisely equivalent to rigid-body pitch and roll rotations of the vehicle—see Fig. 7.13, ates the wx case. The equivalent rates of pitch and roll are readily found for an upwash wave of unit amplitude given by [see (13.3,3)]

(a)

= Qi – і = ki
2

Associated with these velocity-gradient terms are aerodynamic forces and moments exemplified by

ЛСг = СгЛ = -°lfwav

= Gmfwgx

etc.

The x and у gradients of ug and vg that appear in (13.3,17) do not have correspondingly elegant general interpretations. For example, the influence

of ug on unswept wings of large aspect ratio is clearly like that of yaw rate, with equivalent value

= = і0.гид (13.3,22)

However, for small aspect ratio or swept wings the situation is not so simple. For a further discussion of the gradient terms, see ref. 13.10.

The zero elements arise from isotropy (ref. 13.10).

Formulae and graphs of the above spectrum functions associated with the Dryden model of the turbulence are given in ref. 13.10. (No corresponding information is available for the von Karman spectrum, although it can readily be derived.)