A method to predict the onset of a limit cycle for an aeroelastic testbed has been developed. The prediction is based on wavelet processing of measurement data that have been recorded under various flight conditions. The method has been considered for only a small testbed; however, the concepts may lead to techniques that could assist in prediction of the aeroelastic behaviors of aircraft during flight testing.

Some background information is prerequisite to an explanation of this development. The term "wavelet" denotes a signal that is nonzero for a short time. The relevant wavelet in the present case is the Morlet wavelet, which is essentially a windowed sinusoidal signal of finite length. The Morlet wavelet is a function of parameters, called scale and position, that affect, respectively, the period of the sinusoidal component and the time upon which the nonzero component is centered. Wavelet processing involves computation of the magnitudes of correlation between a measured signal and wavelets of different scales and positions. The wavelets with the highest correlation magnitudes represent dominant features in the measurement data.

Wavelet analysis is useful for extracting features in data that relate to nonlinear dynamics. These features can be used to indicate the nature of nonlinearities in an aeroelastic testbed, as reported, for example, in "Characterizing Nonlinear Dynamics by Use of Wavelet Analysis" (DRC-98-42), NASA Tech Briefs, Vol. 23, No. 9 (September 1999), page 70. These features are also the main mathematical tools used in the present method for predicting limit cycles.

In a typical application that involves aeroelasticity, for the purpose of the present method, limit cycles are behaviors that basically consist of steady oscillations of parts of an aircraft. These oscillations are usually stable in the sense that they are of a fixed amplitude; however, they are highly dangerous in flight because of their effect on the pilot and the stress caused to the airframe. It is particularly difficult to predict the onset of limit cycles because they are associated with nonlinearities that are often poorly represented in analytical models.

In the present method, responses from an aeroelastic testbed are analyzed at a series of increasing airspeeds in order to develop an ability to predict the onset of a limit cycle. The testbed is a standard wing section that undergoes pitching and plunging motions. The wing is also equipped with a nonlinear spring that affects the pitching motion and provides the dynamics for the limit cycle. This limit cycle occurs at an airspeed of 9.8 m/s.

Figure 1 shows the maps that result from wavelet processing of pitch measurements at different airspeeds. The color varies from white to black to indicate low to high magnitude of correlation between the signal and the wavelets. Of particular interest is the peak level of correlation at each point in time. This feature is indicated by the solid curved lines through the maps.

The feature associated with the peak level of correlation seems to be related to the airspeed. Notably, the responses at low airspeeds show a sudden and large change in scale of peak correlation after a time interval of 2 seconds, whereas at high airspeeds, the change is more gradual and occurs at later times.

A prediction function for the limit cycle can be computed as a ratio between (1) the magnitude of the maximum change in scale and (2) the time when this change occurs. A plot of this function (see Figure 2) clearly demonstrates a trend toward zero as the airspeed increases. A linear extrapolation of the function becomes zero near 9.8 m/s; thus, the wavelet approach appears to enable correct prediction (through extrapolation) of the onset of the limit cycle.

This work was done by Rick Lind and Martin Brenner of Dryden Flight Research Center. For further information, access the Technical Support Package (TSP) free on-line at www.nasatech.com/tsp  under the Information Sciences category.

DRC-01-10

##### NASA Tech Briefs Magazine

This article first appeared in the March, 2001 issue of NASA Tech Briefs Magazine.

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