Category Dynamics of. Atmospheric Flight

The term gust alleviation interpreted in its broadest sense can mean the reduction of any response variable associated with the turbulence. If in these responses we include structural stresses and vehicle accelerations as well as attitude and trajectory variables it may well be that reducing one response increases another. For example, if pitch attitude is controlled to try to keep the lift constant, then a reduction in ДL2 would be associated with increases in AO2 and q2. The term gust alleviation is sometimes used in a more restricted sense, applied to the load factor only.

When one tries to control load factor by a feedback control to the elevator, the inherent time lag associated with pitching motion is usually such as to make this approach not highly effective (ref. 13.15). When a wing flap control is simultaneously used, however, to control the wing lift almost instantaneously in response to aircraft normal acceleration, pitch rate, and pitch attitude, reductions of an order of magnitude in An2 can be achieved (ref. 13.16).

This illustrates the direction in which we must go in striving for ideal gust alleviation (no doubt unachievable in practice). That is, the per­turbations in all forces and moments produced by the gust field should be just cancelled by automatic fast-acting aerodynamic devices, such as flaps and spoilers, circulation control, etc. The ideal result would be a vehicle that would have the same motion and structural stresses in rough air as in smooth—i. e. rectilinear translation and unity load factor, but with its various automatic gust alleviation devices being very active indeed. To be successful
such a system would probably need gust field sensors (perhaps angle of attack and sideslip vanes much like those used in the measurement of turbulence by aircraft) located at strategic points such as wing tips and tail. With suitable input-rate terms incorporated, sufficient lead time for actu­ating the aerodynamic devices might be obtained. There does not appear to be any fundamental technological impediment to achieving very sub­stantial reductions in gust response by this approach. Considerations of weight, cost, and reliability, however, may present serious economic and operational impediments.

[1] = K> v • • • > wo/’

a (9 x 1) vector. The method is therefore similar to the panel method in that the gust vector is defined by the turbulent velocities at a discrete set of points. To compute the aerodynamic transfer functions Skelton assumes

We shall calculate the longitudinal response of the jet transport used in previous examples, cruising at 30,000 ft through turbulence of 5000 ft scale. In this situation the point approximation is valid, and (13.3,7 and 11) give the needed input function. For the aerodynamic derivatives we use the same numerical values as in Sec. 9.1. The system equations are used in Laplace transfrom form, i. e. (5.14,2) with cj, T and ye both zero.

We noted in Sec. 11.2 that the phugoid oscillation could be suppressed by the pilot by a simple feedback of pitch-attitude to the elevator deflection. We provide for this in the following equations by including the control equation Д<5в = —Кв. On combining the equations we get:

A(s)y(«) = B(s)

where (йд, wg) are Laplace transforms of the nondimensional gust velocities (ug, wg), and

(13.5,3)

(13.5,4)

The required frequency-response functions are found by substituting s = ilc( — ifljFgt*) and solving the resulting complex algebraic equation for the ratios Уі(ік)Ійд and «/г(г&)/й>9.

A response variable of interest not directly included in the above is the load factor. It is defined by An = ALjW. The lift increment AL is taken

as the sum of two parts, that due to aircraft motion (Да, AF) and that due to atmospheric motion (ug, wg). The result obtained is

(«)

(13.5,5)

(b)

Some of the more interesting transfer functions and output spectra are plotted in Figs. 13.10 and 13.11. In Fig. 13.10 we show the squares of the moduli of the transfer functions for speed, angle of attack, pitch attitude, and load factor for vertical gust input. Both stick-fixed and controlled motion are shown. All the motion responses fall off rapidly at high wave number (or high frequency), hut the load factor response tends to the constant value associated with flight on a rectilinear path at constant speed (i. e. no motion response). At wave numbers above 10~2 the load factors are pro­gressively more approximate because of the neglect of unsteady aerodyna­mics. Much more accurate values could he obtained by the simple expedient of multiplying these by a reduction factor for finite wings in sinusoidal gusts—obtained from the generalized Sears function as given by Filotas (ref. 7.16). [The appropriate factor is actually Filotas’ S(kv A)2.]

The effect of the simple elevator-control law (the simple gain is not, of course, the optimum control law for reducing gust response) is seen, as expected, to eliminate the phugoid peaks and substantially to reduce the pitch response for all frequencies lower than that of the short-period mode. With respect to a response, the airplane is seen to act like a low-pass filter, with cut-off frequency at the short-period mode.

On combining these transfer functions with the input spectrum, we get the output spectra, e. g. for speed response

etc. These are shown on Fig. 13.11. (Note that these are two-sided spectra— twice the area gives the mean square.) It is seen that the point approximation is quite adequate in this example for giving the responses in the motion variables (AF, Да, в) but is less satisfactory for the load factor, for which a substantial fraction of the mean-square value is contributed by frequencies above the limit of validity of this approximation. The use of a corrected transfer function as noted above would improve the accuracy of this result appreciably.

If the gust input vector were extended to include the gust-gradient term

(b)

Fig. 13.10 Transfer functions for response to wg input. Jet transport cruising at 30,000 ft and 500 mph with pitch feedback.

(b)

Fie. 13.11 Power spectra of response to wg input. Jet transport cruising at 30,000 ft and 500 mph with pitch feedback.

dwjdx = qg, then the right-hand side of (13.5,1) would read

where qg is the Laplace transform of qa = qgt*, and B'(.s) would he like B(s) but with the additional column [0 — CL — CM 0 0]T. In this case the general response theorem (3.4,49) would hare to be used to calculate outputs, since the cross spectra of both ug and wg with qg are not zero (ref. 13.10). This would entail the calculation not only of the moduli of the transfer functions, but of their real and imaginary parts. [An alternative but equivalent method for this case was given in Dynamics of Flight—Stability and Control (Sec. 10.6), that does not use the input cross spectra.]

When the system equations are nonlinear or the input is nonstationary the foregoing methods of analysis all fail. In such situations one approach is to construct an appropriate mathematical model of the system—analog or digital—and feed in random inputs representing the turbulence. The statisti­cal properties of the output can then be determined by analogue or digital analysis techniques.

In this connection mention should be made of a possible experimental technique that does not appear to have been applied yet. It would consist of exposing a rigid model of the vehicle to a wind-tunnel flow simulating the real turbulence. Force transducers could then produce time records of the actual input forces, thus bypassing the whole problem represented by the second box of Fig. 13.1. The measured forces and moments provide directly the required inputs to the mathematical model, which could be connected on line. In transient situations an ensemble of records could supply the appropriate statistics.

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)]

dWeiUei^x^y’)

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

dLeikl

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]

00

X2 =JJ 02)<Ш1(Ш2

—oo

and the one-dimensional spectrum for lift is

Фіі(^і) = f |Г(^і> £i2)|2xF33(01, £12)<Ш2

J—oo

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.

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]

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.)

A method proposed by Skelton (ref. 13.11) for a study of a VTOL airplane is in some respects similar to both the preceding and following methods, yet different from each. In it three points on the vehicle, for example two wing tips and the tail, are used to identify nine inputs—three gust components at each of these three control points, thus
that the disturbance velocity field is linear in both x and y, so that a change in wgi for example implies a change in wg over the whole vehicle of an amount proportional to the perpendicular distance from the line passing through points 2 and 3. In making an assumption about the whole velocity field associated with each input it resembles the following method. The complete gust matrix T in this formulation of the analysis, for six degrees of freedom would be a (6 X 9) matrix. However it would with the usual assumptions separate into two smaller matrices for lateral and longitudinal subsystems.

For further details the reader is referred to ref. 13.11.

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.

£ttJ—oq

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

The finite extent of the airplane is seen to be important when significant variations of gust velocity can occur between one point and other—e. g. between right and left wing tips, or between wing and tail. An example of these effects for a wing is seen in the experimental results of Nettleton (ref. 13.14), a sample of which is shown in Fig. 13.8. This is a rather extreme case in that the scale of the turbulence L is about equal to the wing chord. The aspect ratio is effectively infinite. Here w is the upwash measured a short

distance in front of the wing, L is the lift measured on a small strip of wing, Ay is the spanwise separation of two lift strips, or of one strip and the upwash probe, and t* is the time delay for which JtwIi(Ay, r) is a maximum. A rel­atively small correlation length for w is seen to lead to a larger correlation length for (w, L) and a still larger one for (L, L).

To allow for such effects, i. e. to remove altogether or in part the limitations we found above on wave number, naturally entails some cost in additional complexity of analysis or experiment. We outline below the principles of five methods of doing this that seem adequately to span the spectrum of possible approaches, although they are not all-inclusive. In all the analysis methods the approximation is made that the airplane has no significant z dimension, i. e. that variations of the gust field with z are negligible. The tur­bulence is then characterized by a two-dimensional spectrum function TXQj, Q2) or its associated correlation function. Each of the methods has advantages and limitations, and the choice for any particular study will re­flect the problem itself, the kind and extent of aerodynamic information and computing machinery available, and the tastes of the analyst.

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.):

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.

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.