THE METHOD OF QUASISTATIC DEFLECTIONS

Many of the important effects of distortion can be accounted for simply by altering the aerodynamic derivatives. The assumption is made that the changes in aerody­namic loading take place so slowly that the structure is at all times in static equilib­rium. (This is equivalent to assuming that the natural frequencies of vibration of the structure are much higher than the frequencies of the rigid-body motions.) Thus a change in load produces a proportional change in the shape of the vehicle, which in turn influences the load. Examples of this kind of analysis are given in Sec. 3.5 (ef­fect of fuselage bending on the location of the neutral point), and Secs. 5.3 and 5.10.

THE METHOD OF NORMAL MODES

When the separation in frequency between the elastic degrees of freedom and the rigid-body motions is not large, then significant inertial coupling can occur between the two. In that case a dynamic analysis is required, which takes account of the time dependence of the elastic motions.

The method that is described here for accomplishing this purpose is based upon the representation of the deformation of the elastic vehicle in terms of its normal modes of free vibration. Imagine that the vehicle is at rest under the action of no ex­ternal forces, aerodynamic, gravity, or other, and that a frame of reference with origin at the mass center, but otherwise arbitrary, is attached to it. The position of mass ele­ment dm is then (x0, y0, z0). Now let the structure be deformed by a self-equilibrating set of external forces and couples, so that it takes a new form, stationary with respect to the coordinate system. Upon instantaneous release of this force system, a free vi­bration ensues, that is, one in which external forces play no part, and in which the po­sition of Sm at time t is (x, y, z). Since there is zero net force, and zero net moment, the linear and angular momenta of the elastic motion must vanish, whatever the ini­tial distorted shape. In particular this is true for each and every undamped normal mode of free vibration. Any small arbitrary elastic motion of the vehicle can, there­fore, relative to the chosen axes (transients as well as steady oscillations), be repre­sented by a superposition of free undamped normal modes as follows:

oo

x'{t) = X /„(*о. Jo, Zo)e„(t)

1

oo

y’it) = X 8n(*o> Уо, z0)e„(0

1

00

z'(t) = X hn(x0, y0, Zo)€„(t)

where (x’, y’, z!) are the elastic displacements, (x — x0) etc., (/„, gn, h„) are the mode shape functions,7 and e„(f) are the generalized coordinates giving the magnitudes of the modal displacements.

We have specified idealized undamped modes, as opposed to the true modes of a real physical structure with internal and external damping, because the latter may not be “simple” modes with fixed nodes, describable by a single set of three functions. More generally they each consist of a superposition of two “submodes” 90° out of phase. Because of this, the equations of motion for the elastic degrees of freedom of the real structure are not perfectly uncoupled from one another, but contain intercou­pling damping terms that would usually be negligible in practical applications.

The use of the free undamped normal modes is seen to ensure that the linear and angular momenta of the distortional motion vanishes. Consequently the elastic mo­tions have no inertial coupling with the rigid-body motions except through the mo­ments and products of inertia. However, it can be shown that this coupling is second – order and negligible in the small-perturbation theory. The determination of the shapes and frequencies con of the normal modes is a major task, and the methods for finding them are beyond the scope of this text. For treatises on this subject the reader should refer to (Bisplinghoff et al., 1955; Fung, 1955). As indicated in (4.12,1), there are ac­tually an infinite number of normal modes of vehicle structures. In practice, of course, only a finite number N of those at the low-frequency end of the set need be retained, and the summations in (4.12,1) are approximated by finite series of N terms. Some judgment and experience is needed to decide just how many modes are needed in any application, but a general rule that is helpful is to discard those whose frequen­cies are substantially higher than the significant ones present in the spectral represen­tation of inputs arising from control action or atmospheric turbulence.