Co

Here lw, lg, mw, and nig are defined in a manner similar to the quantities on the right-hand side of Eqs. (4.35) with the loads from Theodorsen theory,

and the fundamental bending and torsion frequencies are

2 V mb2r2l2

Finally, the constant An = 0.958641. It is clear that these equations are in the same form as the ones solved earlier for the typical section and that the influence of wing flexibility for this simplest two-mode case only enters in a minor way, namely, to adjust the coupling terms by a factor of less than 5%.

The main purpose of this example is to demonstrate how the tools already presented can be used to conduct a flutter analysis of a flexible wing. Addition of higher modes can certainly affect the results, as can such things as spanwise variations in the mass and stiffness properties and concentrated masses and inertias along the wing. Incorporation of these additional features into the analysis would serve to make the analysis more suitable for realistic flutter calculations. However, to fully capture the realism afforded by these and other important considerations, such as aircraft with delta-wing configurations or very low aspect ratio wings, a full finite element analysis would be necessary. Even in such cases it is typical that flutter analyses based on assumed modes give the analyst a reasonably good idea of the mechanisms of instability. Moreover, the full finite element method can be used to obtain a realistic set of assumed modes that could, in turn, be used in a Ritz analysis similar to that above.

4.7 Flutter Boundary Characteristics

The preceding sections have described procedures for the determination of the flutter boundary in terms of altitude, speed, and Mach number. For a standard atmosphere any two of these conditions is sufficient to describe the flight condition. The final flutter boundary is usually presented in terms of a dimensionless flutter speed as UF/(boje). This parameter is similar to the reciprocal of reduced frequency and is called the reduced velocity. A useful presentation of this reduced flutter speed as a function of the mass ratio, // = m/(Ttp-^b1), is illustrated in Fig. 4.13. It is immediately apparent that the flutter speed increases in a nearly linear fashion with increasing mass ratio. This result can be interpreted in either of two ways. For a given configuration, variations in [i would correspond to changes in atmospheric density and thus altitude. In such a case the mass ratio increases with increasing altitude.

Uf ba>e

Uf ba>e Figure 4.13 Plot of dimensionless flutter speed versus mass ratio for the case a = l/x/To, r = 1 /2, xg = 0, and a = —3/10.

This implies that any flight vehicle is more susceptible to aeroelastic flutter at low altitudes than it is at higher ones.

A second interpretation of the mass ratio is related to its numerical value for any fixed altitude. The value of [i will depend on the type of flight vehicle as reflected by its mass per unit span, m. Table 4.2 gives some vehicle configurations and typical mass ratio values for atmospheric densities between sea level and 10,000 ft.

The flutter boundary is very sensitive to the dimensionless parameters. In Fig. 4.14, for example, one sees a dramatic change in the flutter speed versus the frequency ratio a — o>h/o),,. The significant drop in the flutter speed for Xg — 0.2 around a — 1.4 is of utmost practical importance. There are certain frequency ratios at which the flutter speed becomes very small, depending on the values of the other parameters. This dip will be observed in the plot of flutter speed versus frequency ratio for the wings of most highperformance aircraft, which have relatively large mass ratios and positive static unbalances. The chordwise offsets also have a strong influence on the flutter speed, as seen in Fig. 4.15. Indeed, a small change in the mass center location can lead to a large increase in the flutter speed. The mass center location, e, cannot be changed without simultaneously changing the dimensionless radius of gyration, r, but the relative change in the flutter speed for a small percentage change in the former is more than for a similar percentage change in the latter. These facts have led to a concept of mass balancing wings to alleviate flutter, similar to the way control surfaces are mass balanced. If the center of mass is moved forward of the reference point, the flutter speed is generally relatively high. Unfortunately, however, this

Table 4.2. Variation of Mass Ratio fin-Typical Vehicle Types

-rtpxb-

Gliders and Ultralights 5-15

General Aviation 10-20

Commercial Transports 15-30

Attack Aircraft 25-55