Classical Flutter Analysis
Until at least the late 1970s, the aircraft industry performed most lifting-surface flutter analyses based on what is commonly called “classical flutter analysis” based on the flutter determinant. The objective of such an analysis is to determine the flight conditions that correspond to the flutter boundary. It was previously noted that the flutter boundary corresponds to conditions for which one of the modes of motion has a simple harmonic time dependency. Because this is considered to be a stability boundary, it is implied that all modes of motion are convergent (i. e., stable) for less critical flight conditions (i. e., lower airspeed). Moreover, all modes other than the critical one are convergent at the flutter boundary.
The method of analysis is not based on solving the generalized equations of motion as described in Section 5.1. Rather, it is presumed that the solution involves simple harmonic motion. With such a solution specified, the equations of motion are then solved for the flight condition(s) that yields such a solution. Whereas in the p method we determine the eigenvalues for a set flight condition—the real parts of which provide the modal damping—it is apparent that classical flutter analysis cannot provide the modal damping for an arbitrary flight condition. Thus, it cannot provide any definitive measure of flutter stability other than the location of the stability boundary. Although this is the primary weakness of such a method, its primary strength is that it needs only the unsteady airloads for simple harmonic motion of the surface, which for a given level of accuracy are derived more easily than those for arbitrary motion.
To illustrate classical flutter analysis, it is necessary to consider an appropriate representation of unsteady airloads for simple harmonic motion of a lifting surface. Because these oscillatory motions are relatively small in amplitude, it is sufficient to use a linear-aerodynamic theory for the computation of these loads. These aerodynamic theories usually are based on linear potential-flow theory for thin airfoils, which presumes that the motion and thickness of the wing structure create a small disturbance in the flow field and that perturbations in the flow velocity are small relative to the freestream speed. For purposes of demonstration, it suffices to reconsider the typical section of a two-dimensional lifting surface that is experiencing simultaneous translational and rotational motions, as illustrated in Fig. 5.2. The motion is simple harmonic; thus, h and в are represented as
h(t) = h exp(i at) в (t) = в exp(i at)
where a is the frequency of the motion. Although the h and в motions are of the same frequency, they are not necessarily in phase. This can be taken into account
mathematically by representing the amplitude в as a real number and h as a complex number. Because a linear aerodynamic theory is to be used, the resulting lift, L, and the pitching moment about P, denoted by M, where
M = Mi + b( 1 + dL (5.37)
also are simple harmonic with frequency rn, so that
L(t) = L exp(i rnt) M(t) = M exp(i rnt)
The amplitudes of these airloads can be computed as complex linear functions of the amplitudes of motion as
mh(k, M») ь + me (к, M»)e
Here, the freestream air density is represented as рж and the four complex functions contained in the square brackets represent the dimensionless aerodynamic coefficients for the lift and moment resulting from plunging and pitching. These coefficients in general, are, functions of the two parameters к and MTO, where
As in the case of steady airloads, compressibility effects are reflected here by the dependence of the coefficients on M(Xl. The reduced frequency к is unique to unsteady flows. This dimensionless frequency parameter is a measure of the unsteadiness of the flow and normally has a value between zero and unity for conventional flight vehicles. Also note that for any specified values of к and MTO, each coefficient can be written as a complex number. As in the case of h relative to в, the fact that lift and pitching moment are complex quantities reflects their phase relationships relative to the pitch angle (where we can regard в as a real number, for convenience). The speed at which flutter occurs corresponds to specific values of к and MTO and must be found by iteration. Examples of how this process can be carried out for one – and two-degree-of-freedom systems are given in the following subsections.