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

Abstract

A method of complex orthogonal decomposition is summarized for the time-domain, and then formulated and justified for application in the frequency-domain. The method is then applied to the extraction of modes from simulation data of sampled multimodal traveling waves for estimating wave parameters in one-dimensional continua. The decomposition is first performed on a transient 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 decomposition is finally applied to 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 modal 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. Modal signal energies are also quantified. The results are robust to random noise.

Characterizing Wave Behavior in a Beam Experiment by Using Complex Orthogonal Decomposition

“…Since the higher frequencies travel faster through the beam, the higher frequency modal coordinates have shorter durations of nearly harmonic oscillation. This trend was also observed in decompositions of numerically simulated waves [25]. In the experiment, the modes are not pure harmonics, and there is noise and modeling error that contribute to some modal pollution apparent as a lower amplitude, lower frequency oscillation after the strong harmonic has left the measurement zone.…”

“…v is a complex orthogonal mode (COM) and λ is the corresponding complex orthogonal value. Indeed, the correlation matrix can also be formed in the frequency domain as R = Z Z H /N, to produce the same COMs [25].…”

“…In the previous study [25], the COD was applied to a simulated response to an initial Gaussian displacement distribution starting at rest [26]. The extracted traveling wave modes could be related to a form that resembled Eq.…”

“…Thus far COD has been applied to nematode posturing [23], whiting fish locomotion [24], and discerning traveling and standing modal waves [1]. COD has also been applied to extract the dispersion relation from a simulated beam [25].…”

Complex Orthogonal Decomposition Applied to Nematode Posturing

“…The COD fills this void, as complex modes can be used to describe nonstanding and traveling waves [33]. Complex modal motions also occur in mechanical vibration systems with gyroscopic terms, general damping [34,35], and asymmetric stiffness matrices (flutter and friction).…”