A method to detect and to characterize the type of nonlinear dynamics in an aeroelastic system involves the utilization of information from wavelet processing of measurement data. This method is intended to assist in identifying unmodeled dynamics of aircraft during flight testing.

Some background information is prerequisite to an explanation of this development. The term "wavelet" as used here 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 when 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 inherently extracts time-varying features of a signal because of the short duration of the nonzero portion of the wavelet. The wavelets at a position in time determine only the features of the signal near that position. Thus, the changes in wavelets that correlate highly with the signal at different times indicate the changes in features of the signal as time progresses.

In the present method, responses from several configurations of an aeroelastic test bed are analyzed to determine the nature of the nonlinear dynamics that affect an aeroelastic system. The various configurations include various springs associated with pitch movement of a wing assembly. The forces generated by these springs can vary linearly with pitch angle or can vary nonlinearly, as in the cases of hardening or softening springs. Pitch angles are measured during free decay of oscillations of the system in response to an initial pitch placement.

The figure presents maps that result from wavelet processing of pitch responses for different system configurations. These maps can be regarded as three-dimensional in the sense that they depict magnitudes of correlation as functions of wavelet scales and positions: The color at each point in a map indicates the magnitude of correlation; specifically, white represents low correlation, the color changes gradually to darker levels of gray as the correlation increases, and the color becomes blue for particularly high correlation.

The dominant features of each map are extracted by identifying correlation peaks. Because these peaks are sometimes difficult to identify visually from the colored two-dimensional representations of the three-dimensional maps, a curve is drawn in each map to indicate the scale associated with the dominant feature at each position in time.

The dominant features extracted from the wavelet maps clearly indicate the presence of a nonlinearity in the aeroelastic dynamics. The curve is level and shows no change in scale for the dominant feature of the response from a linear system, but the curves vary, showing changes in scale for the dominant features of the responses from the nonlinear systems. This result is expected because (a) the response from a linear system should be a decaying sinusoid that has a constant frequency, while (b) the response from a nonlinear system should be a decaying sinusoid with a changing frequency. Thus, it is straightforward to detect the presence of nonlinearity from the variation in scale shown by the curve through the correlation peaks in a map.

The type of nonlinearity can be characterized by the type of change in the curve. The response from a hardening spring features a decreasing frequency, so its map should show an increasing scale. Conversely, the response from a softening spring features an increasing frequency, so its map should show a decreasing scale. The curves indicate these behaviors, demonstrating that wavelet analysis provides means for both detection and characterization of nonlinear dynamics.

*This work was done by Rick Lind and Martin Brenner of *Dryden Flight Research Center* and Kyle Snyder of the University of Tennessee Space Institute. **DRC-98-42*

##### NASA Tech Briefs Magazine

This article first appeared in the September, 1999 issue of *NASA Tech Briefs* Magazine.

Read more articles from the archives here.