# FUNDAMENTALS OF FLUTTER ANALYSIS

This chapter is essentially divided into two parts. In §§ 6.1-6.6 the flutter of a cantilever wing having a straight elastic axis is treated under the assumption of quasi-steady aerodynamic derivatives. In §§ 6.7-12, the quasi-steady assumption is removed, and the analysis is based on the linearized thin-airfoil theory. This division is necessary because the results of the unsteady airfoil theory are so complicated that a great deal of numerical work is required in the solution, and the main analytical features are masked by the calculative complications. On the other hand, the quasi-steady assumption introduces such great simplifications that one will have no difficulty in carrying through a detailed analysis explicitly. Furthermore, the unsteady airfoil theory, as outlined in Chapters 13-15, is quite elaborate. Therefore it seems advantageous to use the quasi­steady theory as an introduction.

Although the quasi-steady assumption is used here chiefly as an intro­duction, the results so obtained may find practical applications for low – speed airplanes. To present a greater variety of methods of attack, Galerkin’s method, as applied to flutter analysis by Grossman,613 is used in the first part, whereas the method of generalized coordinates is used in the second part. The advantage of Galerkin’s method is its direct rela­tionship with the partial differential equations of motion. It is a natural extension of the approach used in § 1.10 for torsion-flexure oscillation of a cantilever beam. The main mathematical feature of the method, par­ticularly in the first approximation, is very similar to the method of generalized coordinates to be explained later in §7.1. It is easy to see how the analysis can be improved by successive approximation, or general­ized so that the results of more accurate aerodynamic theory can be incorporated. However, these refinements will not be discussed. Here, after a review of the thin-airfoil theory in a two-dimensional steady incompressible flow in § 6.1, the quasi-steady aerodynamic coefficients are derived in § 6.2. Then, in § 6.3, the partial differential equations of motion of a straight cantilever wing in a flow are derived, and the possi­bility of flutter and divergence is discussed. In § 6.4 Galerkin’s method for calculating the critical flutter speed is given. This is followed by § 6.5 treating the stability of the wing as the speed of flow varies. In § 6.6 some
condusions regarding the effect of changing several structural parameters on the critical torsion-flexure flutter speed are drawn. 

To obtain more accurate answers, the unsteady airfoil theory may be used. To simplify the calculation, we introduce not only the linearization of the hydrodynamic equations, but also the “strip” assumption regarding the finite-span effect: that the aerodynamic force at any chordwise section is the same as if that section were situated in a two-dimensional flow. Information regarding the finite-span effect is still incomplete, and the inclusion of known results in the analysis will introduce tremendous complications in the numerical work. The linearity and strip assumptions will be discussed further in Chapter 7.

The unsteady aerodynamic forces in an incompressible fluid are sum­marized in § 6.7 and are used to study the forced oscillation of a two – dimensional wing in a flow (§ 6.9). The existence of a critical flutter speed is again demonstrated, thus confirming a result obtained earlier in § 6.3. Using the linearized airfoil theory with compressibility effects properly included, the flutter of a two-dimensional wing is analyzed in §6.10. Methods of solving the flutter determinant are discussed in § 6.11. The general problem of determining the critical speed is then discussed in § 6.12. Further remarks about the practical engineering flutter analysis are reserved for the next chapter.