Complex Modal Extraction for Estimating Wave Parameters in One-Dimensional Media

Abstract

A method of complex orthogonal decomposition is applied to the extraction of modes from simulation data of multi-modal traveling waves in one-dimensional continua. The decomposition of a transient wave is performed on a nondispersive pulse. Complex wave modes are then extracted from a two-harmonic simulation of a dispersive medium. The wave frequencies and wave numbers are obtained by looking at the whirl of the complex modal coordinate, and the complex modal function, respectively, in the complex plane. From the frequencies and wave numbers, the wave speeds are then estimated, as well as the group velocity associated with the two waves. The group velocity is also extracted directly from a decomposition of the traveling envelope of the waveform. The observations from the first two examples are used to help interpret the decomposition of a simulation of the traveling waves produced by a Gaussian initial displacement profile in an Euler-Bernoulli beam. While such a disturbance produces a continuous spectrum of wave components, the sampling conditions limit the range of wave components (i.e. mode shapes and modal coordinates) to be extracted. Within this working range, the wave numbers and frequencies are obtained from the extraction, and compared to theory. The frequency distribution is then approximated. The results are robust to random noise.

Complex Modal Decomposition for Estimating Wave Properties in One-Dimensional Media

“…The extracted mode shapes (not shown, see [29]) resembled the harmonic building blocks of the periodic pulse, with some distortion, particularly near the spatial end points. Likewise, the modal coordinates resembled harmonic responses with distortion near the temporal end points.…”

Complex Modal Analysis of the Swimming Motion of a Whiting

“…The method we use to analyze the motion of the fish is the complex orthogonal decomposition (COD), developed for structures [7] and since applied to the movements of worms [23] and waves in beams [24]. COD is a generalization of the well known proper orthogonal decomposition (POD).…”