# THE AIRPLANE AS A POINT

The simplest approximation is that in which the variations of (ug, vg, wg) over the vehicle are neglected. The airplane is in effect treated as a point traversing the aq axis, with coordinates (Vet, 0, 0). The input vector is then clearly

(13.3,4)

w„

Furthermore, the usual aerodynamic assumptions that lead to decoupling of the system equations into lateral and longitudinal sets make it possible to separate the response problem into two parts—the longitudinal response to

(13.3,6)

and the lateral response to

The associated force vectors and gust transfer functions are

(13.3,7)

(13.3,8)

where [ugvgwg] = [мл%] – r – Ve.

The disturbing forces AGT^ etc. can be incorporated in (5.13, 18 to 20) or (5.14,1 to 3) by adding them to the associated control term, i. e. by replacing AGT by (AGTc + AGTJ, etc. Now in the point approximation there is no difference between aerodynamic forces associated with relative translation of the airplane w. r.t. the air whether it is the air that moves or the airplane, and in linear approximation vg = /?„, vog — ctg. Thus the eleven transfer functions above are recognized as being identical to those previously used to relate airplane motion to aerodynamic forces, as follows (note the minus signs, ug reduces the relative velocity, etc.):

GTv |
GTg |
_ A _ |
||

@DV |
gd« |
Gyl) A |
||

Ti = — |
®LV |
&La |
; Tt=- |
в» PnK |

®mV |
К |

The adoption of the point approximation means that the airplane is assumed to be vanishingly small with respect to the wavelengths of all significant spectral components (e. g. A >> span in Fig. 13.3). The nondimensional frequency parameter used in the Theodorsen and Sears functions for unsteady flow effects is 1c = wc/2 Ve, which we can relate to by (13.3,3). It gives со = OxFc, whence

k=Q. l- = ‘u – (13.3,10)

2

Thus (c/Aj) —»■ 0 implies QjC -> 0 and к -* 0. Hence it is consistent in this

approximation to use the quasistatic aerodynamic representation by aerodynamic derivatives. Finally then the gust transfer functions are

RANGE OF VALIDITY OF THE POINT APPROXIMATION

It should he observed at the outset that the only excitation of the lateral modes that can exist in this approximation is that provided by vg. In fact comparable inputs may arise from the span wise gradients in wg and ug, which are explicitly excluded in this approximation. It must therefore he considered of limited usefulness for calculating lateral response.

In considering the validity for longitudinal response, we must ascertain for what limiting values of (Qlt 02) or (Av A2) the airplane of Fig. 13.5 can be considered to be vanishingly small. We consider the limits on Qx and £l2 separately.

For £lx we use the criterion that the complex amplitude of the lift on a finite wing flying through a sinusoidal inclined wave of upwash shall not depart too far from its value at к = 0. This problem has been solved by Filotas (ref. 7.16), and from his results we may take as a reasonable upper limit к = .05. It follows that the range of validity is

£1 – < .05

x2

(13.3,12)

^ > — = 60 c .05

For an airplane with mean chord of 20 ft, this yields Qx < .005, and as shown on Fig. 13.6 for large-scale turbulence a small part of the turbulent energy is contained in the spectral components of wavelength shorter than this. This fraction increases rapidly, however, with decrease in L or increase in chord.

For the limit on Q2 we again use Filotas’ result for finite wings. He finds that the effect of spanwise variation is given by the factor

aj> 2!

where J1 denotes a Bessel function of the first kind and b is the wing span. This factor is unity when ff2 = 0, and decreases by roughly 10 % at Q26/2 = 1.

We therefore take this value as the upper limit for 02 in the point approximation, i. e.

Q,6 , –

~T<X

x (13.3,13)

T>7r

For an airplane of span 100 ft, the upper limit on 02 is 2 x 10~2. Its effect is not immediately apparent, however, as was the case with the Qx limit. To evaluate it, we must calculate the truncated one-dimensional spectra

ФЛ(Оі) = Г ¥„(0* 02) dQa (13.3,14)

j-n;

in which the integration excludes those wave numbers that exceed the valid limit. These truncated spectra cannot he evaluated explicitly in terms of elementary functions for the von Karman spectra, but can he for the Dryden spectra. Formulae and graphs of the latter are given in ref. 13.10. To show the effect of truncation, Фз^О^ has been evaluated numerically for the von Karman spectrum, with L = 5000 ft, 6 = 200 ft, and £1′ = 2/6. The result is shown on Fig. 13.6. It is seen to be quite close to the basic spectrum Ф33 for these values of scale and span, the difference being confined to the high wave numbers. The areas under Ф33 and Ф*3 differ by only a few percent. For smaller scale of turbulence the difference increases.

In summary we may conclude that for many cases, especially for large-scale turbulence and small airplanes, the point approximation can give useful results of good accuracy for the longitudinal rigid-body responses. It is probably better, and certainly simpler, to use the basic (not truncated) onedimensional spectra, on the grounds that including the small contribution from the short-wavelength components of the spectrum with an inaccurate theory is better than leaving them out altogether. On the other hand, no such general statement can be made about the responses in the structural or lateral rigid-body modes.

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