## Deficiency Lift Function Theodorsen

Figure 4.7 Comparison between p and p-k methods of flutter analysis for a twin-jet transport airplane. From Hassig (1971) Fig. 2, used by permission.

one can compute [A(ifei)]. Using this new matrix in Eq. (4.75) leads to another set of ps, so that k2 - |3(p)|, Y2 - -7^. (4.77)

Continual updating of the aerodynamic matrix in this way provides an iterative scheme that is convergent for each of the roots, with negative y being a measure of the modal damping. The earliest presentation of this technique was offered by Irwin and Guyett in 1965.

Hassig applied the p-k method to the configuration of Fig. 4.6. As illustrated by Fig. 4.7 (which is his Fig. 2), the p-k method appears to yield approximately the same result as the p method. This, of course, simply validates the convergence of the scheme. Its greatest advantage is that it can utilize airloads that have been formulated for simple harmonic motion. Another comparison offered by Hassig was between the widely used k method and the p—k method for a horizontal stabilizer/elevator configuration. This example of a strongly coupled system provided the results given in Fig. 4.8 (which is his Fig. 3). Here again, as in the k versus p comparison of Fig. 4.6, widely differing conclusions can be drawn regarding the modal coupling. In addition to the easily interpreted frequency and damping plots versus airspeed for strongly coupled systems, a second advantage is offered by the p-k method regarding computational effort. The k method requires numerous computer runs at constant density to ensure matching the Mach number with airspeed and altitude. The p—k method does not have this requirement.

The accuracy of the p-k method depends on the level of damping in any particular mode. It is left as an exercise to the reader (see Problem 13) to show that the p-k method damping is only a good approximation for the damping in lightly damped modes. Fortunately, these are the modes about which we care the most.

Figure 4.8 Comparison between p—k and k methods of flutter analysis for a horizontal stabilizer with elevator. From Hassig (1971) Fig. 3, used by permission.

Figure 4.8 Comparison between p—k and k methods of flutter analysis for a horizontal stabilizer with elevator. From Hassig (1971) Fig. 3, used by permission.

### 4.5 Unsteady Aerodynamics

In Section 4.2, flutter analysis was conducted using an aerodynamic theory for steady flow. The lift and pitching moment used were functions only of the instantaneous pitch angle, 0. Fung (1955) suggested a simple experiment to demonstrate that things are not that simple: Attempt to rapidly move a stick in a straight line through water and notice the results. In the wake of the stick there is a vortex pattern, with vortices being shed alternately from each side of the stick. This shedding of vortices induces a periodic force perpendicular to the stick's line of motion, causing the stick to tend to wobble back and forth in your hand. A similar phenomenon happens with the motion of a lifting surface through a fluid and must be accounted for in unsteady aerodynamic theories.

In a more complete unsteady aerodynamic theory, the lift and pitching moment consist of two parts from two physically different phenomena: noncirculatory and circulatory effects. Noncirculatory effects, also called apparent mass and inertia effects, are generated when the wing motion has a nonzero acceleration. It has to then carry with it a part of the air surrounding it. That air has finite mass, which leads to inertial forces opposing its acceleration.

Circulatory effects are generally more important for aircraft wings. Indeed, in steady flight it is the circulatory lift that keeps the aircraft aloft. Vortices are an integral part of the process of generation of circulatory lift. Basically, there is a difference in the velocities on the upper and lower surfaces of an airfoil. Such a velocity profile can be represented as a constant velocity flow plus a vortex. In a dynamic situation, the strength ofthe vortex (i.e., the circulation) is changing with time. However, the circulatory forces of steady-flow theories do not include the effects of the vortices shed into the wake. Restricting our discussion to two dimensions and potential flow, we recall an implication of the Helmholtz theorem: The total vorticity will always vanish within any closed curve surrounding a particular set of fluid particles. Thus, if some clockwise vorticity develops about the airfoil, a counterclockwise vortex of the same strength has to be shed into the flow. As it moves along this shed vortex changes the flow field by inducing an unsteady flow back onto the airfoil. This behavior is a function of the strength of the shed vortex and its distance away from the airfoil. Thus, accounting for the effect of shed vorticity is in general a very complex undertaking and would necessitate knowledge of each and every vortex shed in the flow. However, if one assumes that the vortices shed in the flow move with the flow, then one can estimate the effect of these vortices.

In this section we present two types of unsteady aerodynamic theories, both of which are based on potential flow theory and take into account the effects of shed vorticity. The simpler theory is appropriate for classical flutter analysis as well as for the k and p—k methods. The other is a finite-state theory cast in the time domain, appropriate for time-domain analysis as well as for eigen-analysis in the form of the p method.

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