RESPONSE OF AN AIRPLANE TO A GUST OF SPECIFIED PROFILE

One of the critical design conditions for airplane structures is the gust loading, which the airplane encounters when flying through a turbulent atmosphere. It is customary to assume that the nonuniformity in the flow consists of small disturbances superimposed on a uniform steady flow. Generally, only the component of the disturbing velocity normal to the flight path is considered. Such normal disturbances are called gusts.

To study the response of an airplane to a gust, let us make the following assumptions

1. The airplane is rigid.

2. The disturbed motion is symmetrical with respect to the airplane’s longitudinal plane of symmetry, but the pitching motion can be neglected.

3. The airplane is initially in horizontal flight at constant velocity U.

4. The gust is normal to the flight path, and is uniform in the spanwise direction.

5. The variation of the forward speed of the airplane can be neglected.

6. The quasi-steady lift coefficient may be used, and the chordwise distribution of the gust velocity may be regarded as constant at any instant and equal to the gust at the mid-chord point.

The disturbed motion of the airplane, consequently, has only the degree of freedom the vertical displacement z (measured at the airplane’s center

of mass, positive downward). According to Newton’s law, the equation of motion is

mz — — L (1)

where m is the total mass of the airplane, L is the total lift (positive upward), and a dot indicates a differentiation with respect to time. To derive an expression for the lift L, the gust profile must be specified. Evidently, it is the gust distribution relative to the airplane that is of significance. Hence, no generality is lost by regarding the gust speed, w(t), as a function of time. Then, according to assumption 6, the lift can be written as

where p is the density of the air, and 8 is the wing area. Using Eq. 2 and introducing a parameter A, of physical dimension [Г-1],

(3,

2m ax

we can write Eq. 1 as

z = — A(w + z) (4)

This equation is to be solved for the initial conditions

z = z = 0 when t — 0 (5)

An integrating factor of Eq. 4 is easily seen to be eu. Equation 4 may be written as

j

— (zeu) = — hv{t) eu at

Integrating, and using Eqs. 5, we obtain

z(t) = — Ае_Д( J w{x) dx

A second integration gives

z(t) — — А Г е“Дт dr f w(x) elx dx Jo Jo

Changing the order of integration, we have

z(t) = — A f w(x) elx dx f e~x~ dr Jo Jx

i-e.,

If w(x) is a step function, the so-called sharp-edged gust, so that ii’(a;) is equal to a constant w0 for x > 0 and vanishes for x < 0 (Fig. 8.5), then an integration of Eq. 6 leads to

<t) = j w0(l – e~u) – иу (7)

and

z(t) = — Xw0 e~u (8)

The acceleration reaches the maximum when / = 0.

2wn

Dividing zmax by the gravitational acceleration g, we obtain the sharp – edged gust formula

_ Anax _ ^ ^0__ pU Sw, dCL g g 2 mg d*

(10)

where An denotes the increment of load factor. The product of An and the weight of the structures gives the acting inertia force.

Equation 7 is the indicial admittance of the displacement z for a sharp – edged gust. Equation 6 is the Duhamel integral for an arbitrary gust

w0

Sharp-edged gust

Fig. 8.5. Sharp-edged gust.

profile. According to § 8.1, the same problem can be as easily solved by the method of mechanical impedance, in which the response to a sinu­soidal gust, w0 sin mt, is first obtained. The response to an arbitrary gust can then be obtained by a Fourier integral. It is easy to show that the results obtained by these two methods agree with each other.

The sharp-edged gust formula is derived under the six simplifying assumptions named above and the idealized gust profile of Fig. 8.5. In reality, none of these assumptions can be fulfilled. Nevertheless, the formula is convenient for use in airplane design. If the gust speed is based on an “effective” value which is derived by reducing the experi­mental acceleration data according to Eq. 10, the result can be used to predict the gust load factor on similar airplanes. The effective gust speed Wo, however, would have to be determined for each type of airplane, because the effects of the simplifying assumptions are different for different airplane size, geometry, flexibility, center-of-gravity location, dynamic stability characteristics, and flight Mach number.

Much work has been done in the direction of relaxing one or another of the assumptions made in deriving the sharp-edged gust formula. Using aerodynamics of an unsteady incompressible flow, KUssner8,20 obtained in 1931 the response of a rigid airplane, restrained against pitch, to a gust with a finite velocity gradient. KUssner also extended his analysis to take into account the elasticity of the wing in bending, but assumed the deflection mode to be of the same form as the static deflection curve under a uniformly distributed load. He concluded that the stresses in an elastic wing may be considerably higher than that in a rigid wing. In addition, he showed that, for a gust of given intensity, the load factor reaches a maximum when the gust is inclined at 65° to 70° to the flight path, but the load factor due to a normal gust (gust velocity perpendicular to the flight path) differs from this maximum by less than 10 per cent. The response of a rigid airplane free to pitch is treated by Bryant and Jones in 19328,5 under the assumptions of quasi-steady lift, and in 19368,6 for a semirigid wing including the unsteady-flow characteristics as given by Wagner. Similar extensions were made by Williams and Hanson8-36 in 1937, Sears and Sparks8,32 in 1941, and Pierce8-24 and Putnam8,27 in

1947.

More extensive investigation on the effect of elastic deformation was made by Goland, Luke, and Kahn8,10 in 1947. Jenkins and Pancu8,16 in

1948, Bisplinghoff, Isakson, Pian, Flomenhoft, and O’Brien8,3’8,4 in 1949. In these studies the bending and torsion deflections of the wing are approximated by a number of deflection modes.

A comprehensive analysis of the effect of the gust gradient and the airplane pitching was made by Greidanus and van de Vooren8,11 (1948). The allowance of the pitching degree of freedom introduces great compli­cation into the analysis because of the phase lag in the downwash between the wing and the tail.

Further extensions were made by Bisplinghoff and his associates, Mazelsky, Diederich, Houbolt, and others, to account for aerodynamic forces in a flow of a compressible fluid.

Since transient problems can be best treated by the method of Laplace transformation, details of some of these extensions will be discussed in Chapter 10. In the remaining sections of the present chapter, let us turn to the statistical aspects of the dynamic-stress problem.