Aeroelasticity Airfoil

Divergent Oscillations

Unstable

< 0

= 0

Continuous Convergence

Stable

= 0

= 0

Time Independent

Stability Boundary

> 0

= 0

Continuous Divergence

Unstable

Recall from Eq. (4.11) that the total displacement is a sum of all modal contributions. It is therefore necessary to consider all possible combinations of r* and where F/, can be <0, = 0, or >0 and Q/, can be = 0 or / 0. The corresponding type of motion and stability characteristics are indicated in Table 4.1 for various combinations of r* and Q/,. Although our primary concern here is with regard to the dynamic instability of flutter for which Q-k / 0, Table 4.1 shows that the generalized equations of motion can also provide solutions to the static aeroelastic problem of divergence. This phenomenon is indicated by the unstable condition for — 0, and the divergence boundary occurs when H = Q/, = 0.

In many published works on flutter analysis, the method outlined in this section is known as the p method, named for the reduced complex eigenvalue p — bv/U, which appears in Eq. (4.9) and in terms of which the eigenvalue problem can be posed instead of in terms of v as we have done. To provide accurate prediction of flutter characteristics the p method must use an aerodynamic theory that accurately represents the loads for transient motion of the lifting surface. Such theories may involve aerodynamic states, in addition to the structural generalized coordinates and their first time derivatives; see, for example, the theory outlined in Section 4.5.2. These additional states do not affect the meaning of the real and imaginary parts of the eigenvalues as discussed here. We now apply the p method to a simple configuration.

4.2 Aeroelastic Analysis of a Typical Section

In this section we will demonstrate the flutter analysis of a linear aeroelastic system. To do this a simple model is needed. In the older literature of aeroelasticity, flutter analyses were often performed using simple, spring-restrained, rigid-wing models such as the one shown in Fig. 4.2. These were called typical section models and are still very appealing Figure 4.2 Schematic showing geometry of the wing section with pitch and plunge spring restraints.

because of their physical simplicity. This configuration could represent the case of a rigid, two-dimensional wind tunnel model that is elastically mounted in a wind tunnel test section, or it could correspond to a typical airfoil section along a finite wing. In the latter case the discrete springs would reflect the wing structural bending and torsional stiffnesses, and the reference point would represent the elastic axis.

Of interest in such models are points P, C, Q, and T, which refer, respectively, to the reference point (i.e., where the plunge displacement h is measured), the center of mass, the aerodynamic center (presumed to be the quarter-chord in thin-airfoil theory), and the three-quarter-chord (an important chordwise location in thin-airfoil theory). The dimensionless parameters e and a (— 1 < e < 1 and — 1 < a < 1) determine the locations of the points C and P; when these parameters are zero, the points lie on the mid-chord, and when they are positive (negative), the points lie toward the trailing (leading) edge. In the literature, the chordwise offset of the center of mass from the reference point, rather than e, often appears in the equations of motion. It is typically made dimensionless by the airfoil semichord b and denoted by x„ — e — a. This so-called static unbalance parameter is positive when the center of mass is toward the trailing edge from the reference point. The rigid plunging and pitching of the model is restrained by light, linear springs with spring constants ki, and kg.

It is convenient to formulate the equations of motion from Lagrange's equations. To do this, one needs kinetic and potential energies as well as the generalized forces resulting from aerodynamic loading. One can immediately write the potential energy as

To deduce the kinetic energy, one needs the velocity of the mass center C, which can be found as vc = vp + 0b3 x b [(1 + a) - (1 + e)] bi, (4.14) where the inertial velocity of the reference point P is yP — -hi2, (4.15)

and thus

Vc = —hb + bO(a — e)b2- (4.16) The kinetic energy is then given by

where Ic is the moment of inertia about C. By virtue of the relationship between bi and the inertially fixed unit vectors ii and ¡2, assuming 6 to be small, one finds that

= -m(h2 + 2b x gild) + -IP02, where IP — Ic + mb2xj.

The generalized forces associated with the degrees of freedom h and 0 are easily derived from the work done by the aerodynamic lift through a virtual displacement of the point Q

and by the aerodynamic pitching moment about Q through a virtual rotation of the model. The velocity of Q is vQ = -h\2 + be(^-+a^\>2. (4.19)

The virtual displacement of the point Q can be obtained simply by replacing the dot over each unknown in Eq. (4.19) with a S in front of it, that is,

where <5pg is the virtual displacement at Q. The angular velocity of the wing is //b ;, so that the virtual rotation of the wing is simply <50l>3. Therefore, the virtual work of the aerodynamic forces is

and the generalized forces become Qh = -L,

It is clear that the generalized force associated with h is the negative of the lift, whereas the one associated with 0 is the pitching moment about the reference point P.

Lagrange's equations (as found in the Appendix, Eqs. A.35) are here specialized for the case in which the kinetic energy K depends only on q\, q2,..., and so d /dK\ dP

Here n — 2,q\ — /;, and q2 — 6 and the equations of motion become m(h + bxgO) + k/,h — — L,

For the aerodynamics, the steady-flow theory we used in the previous chapter gives L = 2n prxjbU16,

+3 -2