# CHOICE OF THE GENERALIZED COORDINATES

Representing the deformation of a continuous elastic structure by a finite number of generalized coordinates is equivalent to imposing certain constraints on the elastic body. The structure is no longer elastic, but “semirigid.”

A choice of the generalized coordinates in describing the elastic displacements is a choice of the semirigid modes. When the generalized coordinates are chosen, the equations of motion can be derived according to Lagrange’s equations. Explicit use of the differential equations of the theory of elasticity is avoided. The elastic properties of the structure are summed up completely in the expression of the elastic strain energy.

The crucial question remains whether such a simplified model can yield sufficiently accurate results for engineering purposes. The answer is affirmative provided that the semirigid modes were properly chosen. Guidance to this choice has to be sought from model tests and mathematical experiments.

Some of the consequences of the semirigid approximation can be clarified at once. To be concrete, consider the particular representation of a cantilever wing as adopted in Eqs. 1 of § 7.1. This representation implies (1) that the wing-bending deformation can only assume the form f(y), the torsion ф(у), the aileron g(y), for all airstream speed, density, and direction of flow; (2) if f ф, g are real-valued functions, no phase shift occurs across the span in bending, torsion, or aileron deflection. Now it is known that a wing is capable of oscillating in many different modes when immersed in a flow of a given speed and density. But most of these multitudes of modes are irrelevant to the flutter problem, since at a critical speed the flutter mode assumes a definite form. At, or near, the critical flutter speed, the other oscillation modes, if accidentally excited, are relatively heavily damped and quickly die out. Thus, if the assumed semirigid mode closely approximates the actual displacements which occur in flutter at the critical speed, little error will result from neglecting other possible modes.

As for the phase shift across the span, examples show that generally considerable phase shift exists in the bending displacement, but not in the torsional motion. In many examples the bending phase shift across the span is unimportant. Duncan7-25 offers the following explanation: In flutter the motion is much larger near the wing tip than inboard. Thus, at half the wing span from the wing root, the amplitude is of the order of one quarter of that at the tip, which implies that the dynamical importance of the motion at half-span is only of the order of one sixteenth of that at the tip. Thus, phase differences are only of importance when they occur in the region near the tip, and within this region they are in fact very small for both bending and torsion. This argument, of course, fails when a wing carries large isolated masses, for then a motion of small amplitude may be dynamically important.

Duncan and his associates6,6,5-24,5,25 have shown that, for a cantilever wing of uniform rectangular planform and uniform cross sections along the span, of either (1) a two-spar construction which derives its torsional stiffness entirely from the differential bending, or (2) a monocoque structure with bending stiffness negligibly small in comparison with its torsional stiffness, the semirigid assumption is exact—meaning that

{a) The distribution of the bending and torsion displacements along the span are independent of air speed for the mode of oscillation which develops into flutter at the critical speed, and

(b) For this mode, the bending oscillations at all parts of the span are in phase, and likewise for the torsional oscillations.

Generally speaking, the application of the semirigid concept has been extremely successful. Disagreement between the calculated flutter speed and the experimental value can usually be attributed to other causes than the semirigidity approximations. In general, the calculated flutter speed is remarkably insensitive to the exact form of the semirigid modes, but it is rather sensitive to the aerodynamic assumptions. Thus, in the past, efforts have been made repeatedly to replace one or a few of the aerodynamic coefficients according to the experimental evidence. Such partial tampering with aerodynamics often leads to inferior results. Better results are obtained by using either the whole set of coefficients exactly as given by the linearized aerodynamic theory or the whole set of experimental derivatives.

There are two ways of representing the deformation of a structure: (1) by a series expansion in terms of a set of continuous functions, (2) by recording displacements at a number of points on the structure. The first type leads to generalized coordinates such as those used in §7.1, whereas the second type leads to the so-called lumped-mass method. In the lumped-mass method, a wing is divided into a number of strips, each of which is supposed to move as a unit. For certain types of electrical analog computors, this is the most convenient method. A full description of the lumped-mass method can be found in Myklested’s book6,17.

If the strip assumption for aerodynamic force is relaxed and the finite – span effect is to be estimated, the method of generalized coordinates is preferred, because the deformation pattern of the whole wing is fixed for each coordinate, and the aerodynamic problem can be solved. On the other hand, in the lumped-mass method the deformation pattern across the span is not known until the displacements at all sections are determined; thus finite-span effect cannot be easily accounted for.

The idea of Rauscher’s “station-function” method7-53 stems from a desire to adopt the simplicity of the lumped-mass method to the calculation of the aerodynamic finite-span effect. Consider the deflection of a wing as an example. The deflections Z1( Z2, • • •, Z„ at a series of stations Уі> Уї> ‘ • ‘,Уп are chosen to describe the wing deformation. The deflection curve of the entire wing, expressed as a continuous functionf(y), may be approximated by

M = zjm + • • • + zjm

where/iQ/), • • – yfniy) are certain functions of y, independent of Z1( • • •, Zn. Each of the reference deflections Z1( • ■ •, Zn thus makes its own contribution to f(y): As f{y) must satisfy the boundary conditions regardless of the values of Zlt • ■ •, Z„, the component functions f^(y), • • -,fn(y) must individually satisfy all boundary conditions. In order that f(y) may have the value Zx at the station yv irrespective of the values Z2, • • •, Z„, it is necessary that f^) = 1, and/2(г/,) = • • • =/п(уг) = 0. In a like manner, for all /, j — 1, 2, • • •> tv, we must have ft{y^ = 1, and ffyjj) = 0 if і Ф j. The functions ffy) so defined are called “station functions.” Their application to the flutter problem is similar to the generalized coordinates used in § 7.1.

A question of paramount importance is the minimum number of degrees of freedom (i. e., the number of generalized coordinates) that should be allowed in each particular problem to insure reasonable accuracy of the final result. For example, for a multi-engined airplane wing, or for a wing having a large fuel tank attached to the outer span, a simple semirigid representation, with one pure bending and one pure torsion mode of deflection, is, in general, inadequate. Better results can be obtained by using the first few normal modes of free oscillation of the wing as generalized coordinates, or by numerical integrations of the differential equations of motion (see Runyan and Watkins7-97). In some cases the flexibility of the engine mount and the fore and aft motion of the wing are important, and must be accounted for.

The most complicated case is probably the tail flutter. A large number of distinct kinds of deformation of the tail unit and the fuselage are possible, and the effect of the freedom of the airplane as a whole is serious.

To reduce the complexity of the analysis, one may first distinguish the symmetrical and antisymmetrical types which are independent of each other. In the former, the motions occur symmetrically about the fuselage centerline; in the latter, antisymmetrically. The number of degrees of freedom for the general case of each type is so great that it is impractical to make routine calculations of critical speeds without introducing simplifying assumptions, which must be guided by experimental results (see Duncan7,25).

Unless there are reasons to believe otherwise, at least the following degrees of freedom should be considered.7,25

For symmetrical tail flutter:

1. Elevator rotation about the hinge line (elevators treated as a single rigid unit).

2. Vertical bending of the fuselage.

3. Bending of the horizontal tail surface.

4. Pitching of the airplane as a whole.

For antisymmetrical tail flutter:

(a) Elevator flutter:

1. Rotation of elevators about their hinge line, in opposition.

2. Torsion of the fuselage.

3. Bending of the horizontal tail surface.

(b) Rudder flutter:

1. Rotation of the rudder about its hinge line.

2. Lateral bending of fuselage.

3. Fuselage torsion or bending of the fin.

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