Complex modal analysis of carangiform swimming kinematics

The overall purpose of this study is to do a complex modal analysis of the kinematic data of carangiform swimming. The research considers the midline motion in itself, leaving out the causes to which the initiation of motion is ascribed. Today, motion analysis is commonly based on the FFT (fast Fourier transform). Instead, this research examines a more recent technique, COD (complex orthogonal decomposition), to perform complex modal analysis. This analysis describes the main modes of the lateral displacement, and the associated frequency and wavelength. Body center lines were used for the kinematic analysis of the swimming motion. We acquired raw data of midline lateral movements from the resources available in the literature for three carangiform fish: whiting, mackerel, and saithe. The midline motion data for these fish was originally available in photo form, so it had to be digitized in order to be used as input to COD. Using COD, the total motion was decomposed into the main modes (eigenvectors) with the corresponding contribution (eigenvalues). Each mode, in addition, was decomposed to its standing and traveling parts. COD was used also to do accurate measurement of amplitude, wavelength and frequency (as functions of location or time). The main focus of this analysis is to compare the traveling wave model with the true motion (raw data). In this regard, a traveling wave model was also built based on amplitude profile, average wavelength, and frequency of raw data, and the same analysis that was applied to the raw data was applied to this model. The results for both cases (raw data and the traveling wave model) were then compared. The results show that in both cases, there is only one main mode which incorporates more than 99% of the energy. The other finding is that if we consider the main mode of carangiform motion in the context of amplitude(x)×cos(ωt - cx), then the coefficients ω and c in the argument of harmonic function are not constant, but functions of time and location, respectively. Also, this analysis shows that the standing part for both cases has non-zero amplitude, which challenges the notion of a pure traveling wave suggested in nearly all literature for the kinematics of fish swimming. COD can be performed with the goal of isolating a single mode of interest, and then representing the mode by the real and imaginary parts of the extracted complex mode, quantifying the motion parameters based on the complex mode and modal coordinate, enabling visualization and computation of additional quantities of interest, and “purifying” or isolating the motion in terms of the extracted mode.