Theodorsen Function
where the generalized forces are given in Eqs. (4.22). The function C(k) is a complexvalued function of the reduced frequency k, given by
where W,"(k) are Hankel functions of the second kind, which can be expressed in terms of
Bessel functions of the first and second kind, respectively, as Hf\k) = J„(k)  iY„(k).
The function C(k) is called Theodorsen's function and is plotted in Fig. 4.9. Note that C(k) is real and equal to unity for the steady case (i.e., for k — 0). As k increases, one finds that the imaginary part increases in magnitude while the real part decreases. As k tends to infinity, C(k) approaches 1 /2. However, for practical situations k does not exceed values of the order of unity. Hence, the plot in Fig. 4.9 only extends to k — 1. When any harmonic function is multiplied by C(k), its magnitude is reduced and a phase lag is introduced.
A few things are noteworthy concerning Eqs. (4.78). First, in Theodorsen's theory the liftcurve slope is equal to lit. Thus, the first of the two terms in the lift is the circulatory
lift without the effect of shed vortices multiplied by C(k). The multiplication by C(k) is a consequence of the theory having taken into account the effect of shed vorticity. The second term in the lift as well as the pitching moment are noncirculatory, depending on the acceleration and angular acceleration of the airfoil. The circulatory lift is the more significant of the two terms in the lift.
For steady flow the circulatory lift is linear in the angle of attack, but for unsteady flow there is no single angle of attack since the flow direction varies along the chordline as the result of the induced flow. However, just so we can discuss the concept for unsteady flow, it is possible to introduce a socalled effective angle of attack. For simple harmonic motion it can be inferred from Theodorsen's theory that an effective angle of attack is a = C(k)
As we will show in Section 4.5.2 by comparison with the finitestate aerodynamic model introduced therein, a is the angle of attack measured at the threequarter chord based on an averaged value of the induced flow. Recall that in steadyflow aerodynamics, the angle of attack is the pitch angle 0. Here, however, a depends on 0 as well as on /;, 0, and k. Because of these additional terms and because of the behavior of C(k), we expect changes in magnitude and phase between 0 and a carrying over into changes in the magnitude and phase of the lift and pitching moment relative to that of 0. Indeed, the function C(k) is sometimes called the liftdeficiency function because it reduces the magnitude of unsteady lift relative to steady lift. It also introduces an important phase shift between the peak values of pitching oscillations and corresponding oscillations in lift and pitching moment. An approximation of Theodorsen theory in which C(k) is set equal to unity is called a "quasisteady" thinairfoil theory. Such an approximation has value only for cases in which k is restricted to be very small.
Theodorsen's theory may be used in classical flutter analysis. There the reduced frequency of flutter is not known a priori. One can find k at the flutter condition using the method described in Section 4.3. Theodorsen's theory may also be used in the k and pk methods, as described in Sections 4.4.1 and 4.4.2, respectively.
4.5.2 FiniteState Unsteady ThinAirfoil Theory of Peters et al.
Although the Theodorsen theory is an excellent choice for classical flutter analysis, there are situations in which an alternative approach is needed. First, frequently one needs to calculate the modal damping in subcritical flight conditions. Second, there is a growing interest in the active control of flutter, and design of controllers requires that the system be represented in statespace form. To meet these requirements, one needs to represent the actual aerodynamic loads (which are in the frequency domain in Theodorsen's theory) in terms of timedomain differential equations. Finitestate theories approximate the actual infinitestate aerodynamic model to within engineering accuracy. One such approach is the finitestate, inducedflow theory for inviscid, incompressible flow of Peters et al.
Consider a typical section of a rigid, symmetric wing, shown in Fig. 4.2, and the additional vectorial directions defined in Fig. 4.10. To begin the presentation of this theory, we first relate the three sets of unit vectors:
1. a set fixed in the inertial frame, ii and ij, such that the air is flowing at velocity
2. a set fixed in the wing, bi and b2, with bi directed along the zerolift line toward the leading edge and bj perpendicular to bi,
3. a set ai and kn associated with the local relative wind vector at the threequarter chord, such that aj is along the relative wind vector and a2 is perpendicular to it, in the assumed direction of the lift.
The relationships among these unit vectors can be simply stated as and bi b2
Inducedflow theories approximate the effects of shed vortices based on changes they cause in the flow field near the airfoil. Thus, the velocity field near the airfoil consists of the
^ lift direction
^ lift direction zerolift line direction of the velocity of T relative to wind
Figure 4.10 Schematic showing geometry of the zerolift line, relative wind, and lift directions.
zerolift line direction of the velocity of T relative to wind
Figure 4.10 Schematic showing geometry of the zerolift line, relative wind, and lift directions.
freestream velocity plus an additional component to account for the induced flow. Although the induced flow varies throughout the flow field, we will approximate its value near the airfoil as an average value along the chordline. Thus, the local inertial wind velocity is written approximately as — Ui\ + /.obi, where /,« is the average induced flow (perpendicular to the airfoil zerolift line). According to classical thinairfoil theory, one should calculate the angle of attack using the instantaneous relative wind velocity vector as calculated at T. To represent the relative wind velocity vector at T, one can write the relative wind vector (i.e., the velocity of the wing with respect to the air) as Wai and set it equal to the inertial velocity of T minus the inertial air velocity, that is,
Wai = \'T ~ (Uii + A0b2) = Yp + Uii  A0i>2, where vr is the inertial velocity of the threequarter chord, given by vT — vp + Ob'i x rPT, and rpx is the position vector from P to T. From Fig. 4.2 one finds that r pt —
Thus,
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