Complex Orthogonal Decomposition Research Articles

Generation of quasi-traveling waves in a finite rectangular membrane with two internal viscoelastic line supports

In this work we study the forced vibration of a finite rectangular membrane driven by arbitrarily distributed boundary motion on two adjacent edges and internally coupled with multiple viscoelastic line supports, as an extension of the analysis of coexisting vibrations and waves in one-dimensional elastic strings and ducts. Using the static solution for the membrane displacement in the absence of interior supports, the response of the nonclassically damped system is obtained through a normal-mode expansion. A traveling-wave index based on complex orthogonal decomposition is utilized to identify and optimize the traveling waves in the rectangular membrane. A vibration localization condition in a semi-infinite membrane is analytically derived, from which the complex boundary excitations and constraints in a finite membrane are specified to effectively realize pure traveling waves and confine the vibration to a restricted area. Then the membrane is edge-driven by a simpler time-periodic, spatially sinusoidal excitation for practical realization, with the line-support parameters optimized using the traveling-wave index function and a derivative-free optimization routine with lower-bound constraints. Analytical results show that quasi-traveling waves are produced for ranges of line-support parameters (location, stiffness and damping) following optimization using the traveling-wave index. The developed wave patterns are examined with a power-flow analysis and numerically verified through finite element analysis. Wave localization is demonstrated to be valid in a finite membrane with specific boundary conditions and multiple viscoelastic line supports. The proposed method shows the predictive ability of the orthogonally stiffened membrane configuration as the basic unit cell for the design of periodic structures with directional wave propagation as well as vibration localization dynamics.

Fluidic undulation effects on carangiform swimmers propelled by internal active bending moments

With different shapes and material properties, fish all achieve undulatory swimming gait under the action of internal active muscle stimulation and external fluid forces. Such locomotion can be decomposed into deformation affected by internal and external forces in the body frame and overall translation and rotation solely determined by fluid forces. In order to revisit the undulatory swimming gait, we investigate the hydrodynamic performance of two-dimensional flexible carangiform swimmers with varying stiffnesses and thicknesses, which are driven by the active internal bending moments, and employ the complex orthogonal decomposition and Fourier decomposition methods to quantitatively measure and analyze the proportion of undulation. It is found that standing wave deformation characteristics are prominently observed along fish-like bodies with high stiffness, whereas traveling wave characteristics are more evident in bodies with lower stiffness. The self-propelled fish body demonstrates lateral oscillation and rotation around its center of mass, namely, the heaving and pitching movement, particularly in specimens with high stiffness. The present analysis shows that the heaving and pitching locomotion induced by the fluid significantly increase the traveling wave proportion by modulating the amplitude and phase of the left and right traveling waves viewed in forward frame. We called it fluidic undulation effects (FUE), which is different from the undulation of body deformation. This effect is more pronounced for large stiffnesses and thin airfoils. The standing wave deformation observed with a large stiffness transforms into a traveling wave propulsion pattern, with its traveling wave index even slightly surpassing that of a small-stiffness pattern. Although the efficiency of the standing wave deformation is low, it facilitates a faster forward speed (body lengths per stroke). The positive impact of the FUE on the swimming performance is also confirmed by restricting the recoil motions of the lateral translation and rotation of the body. Furthermore, we observe that there is no undulatory swimming gait that has both the highest energy efficiency and the highest speed.

Complex Modal Characteristic Analysis of a Tensegrity Robotic Fish's Body Waves.

A bionic robotic fish based on compliant structure can excite the natural modes of vibration, thereby mimicking the body waves of real fish to generate thrust and realize undulate propulsion. The fish body wave is a result of the fish body's mechanical characteristics interacting with the surrounding fluid. Thoroughly analyzing the complex modal characteristics in such robotic fish contributes to a better understanding of the locomotion behavior, consequently enhancing the swimming performance. Therefore, the complex orthogonal decomposition (COD) method is used in this article. The traveling index is used to quantitatively describe the difference between the real and imaginary modes of the fish body wave. It is defined as the reciprocal of the condition number between the real and imaginary components. After introducing the BCF (body and/or caudal fin) the fish's body wave curves and the COD method, the structural design and parameter configuration of the tensegrity robotic fish are introduced. The complex modal characteristics of the tensegrity robotic fish and real fish are analyzed. The results show that their traveling indexes are close, with two similar complex mode shapes. Subsequently, the relationship between the traveling index and swimming performance is expressed using indicators reflecting linear correlation (correlation coefficient (Rc) and p value). Based on this correlation, a preliminary optimization strategy for the traveling index is proposed, with the potential to improve the swimming performance of the robotic fish.

Computational Study of Stiffness-Tuning Strategies in Anguilliform Fish.

Biological evidence demonstrates that fish can tune their body stiffness to improve thrust and efficiency during swimming locomotion. However, the stiffness-tuning strategies that maximize swimming speed or efficiency are still unclear. In the present study, a musculo-skeletal model of anguilliform fish is developed to study the properties of variable stiffness, in which the planar serial-parallel mechanism is used to model the body structure. The calcium ion model is adopted to simulate muscular activities and generate muscle force. Further, the relations among the forward speed, the swimming efficiency, and Young's modulus of the fish body are investigated. The results show that for certain body stiffness, the swimming speed and efficiency are increased with the tail-beat frequency until reaching the maximum value and then decreased. The peak speed and efficiency are also increased with the amplitude of muscle actuation. Anguilliform fish tend to vary their body stiffness to improve the swimming speed and efficiency at a high tail-beat frequency or small amplitude of muscle actuation. Furthermore, the midline motions of anguilliform fish are analyzed by the complex orthogonal decomposition (COD) method, and the discussions of fish motions associated with the variable body stiffness and the tail-beat frequency are also presented. Overall, the optimal swimming performance of anguilliform fish benefits from the matching relationships among the muscle actuation, the body stiffness, and the tail-beat frequency.

Regulation of the swimming kinematics of lampreys Petromyzon marinus across changes in viscosity.

The bodies of most swimming fishes are very flexible and deform as result of both external fluid dynamic forces and internal musculoskeletal forces. If fluid forces change, the body motion will also change unless the fish senses the change and alters its muscle activity to compensate. Lampreys and other fishes have mechanosensory cells in their spinal cords that allow them to sense how their body is bending. We hypothesized that lampreys (Petromyzon marinus) actively regulate body curvature to maintain a fairly constant swimming waveform even as swimming speed and fluid dynamic forces change. To test this hypothesis, we measured the steady swimming kinematics of lampreys swimming in normal water, and water in which the viscosity was increased by 10 or 20 times by adding methylcellulose. Increasing the viscosity over this range increases the drag coefficient, potentially increasing fluid forces up to 40%. Previous computational results suggested that if lampreys did not compensate for these forces, the swimming speed would drop by about 52%, the amplitude would drop by 39%, and posterior body curvature would increase by about 31%, while tail beat frequency would remain the same. Five juvenile sea lampreys were filmed swimming through still water, and midlines were digitized using standard techniques. Although swimming speed dropped by 44% from 1× to 10× viscosity, amplitude only decreased by 4%, and curvature increased by 7%, a much smaller change than the amount we estimated if there was no compensation. To examine the waveform overall, we performed a complex orthogonal decomposition and found that the first mode of the swimming waveform (the primary swimming pattern) did not change substantially, even at 20× viscosity. Thus, it appears that lampreys are compensating, at least partially, for the changes in viscosity, which in turn suggests that sensory feedback is involved in regulating the body waveform.

Complex modal analysis of the movements of swimming fish propelled by body and/or caudal fin

Most fish generate thrusts by undulating their body and/or caudal fin (BCF). The movement of a fish body can typically be described as a propulsive wave with an increasing amplitude. In this study, we decompose the midline motions of swimming fish into travelling and standing components based on the complex orthogonal decomposition (COD) analysis. The travelling index, i.e., the relative ratio between the magnitudes of the travelling and standing components, is employed to analyze the typical locomotion of BCF fish. Using existing biological data, we find that anguilliform swimmers prefer wave motions with high travelling index (0.74∼0.90), while the motions of thunniform fish are dominated by a standing wave with the travelling index around 0.36∼0.64. For subcarangiform and carangiform fish, their motions are mixed by the travelling and standing waves with a medium travelling index (0.52∼0.78). The locomotion of BCF fish can be broadly classified into three categories by the travelling index: the standing-wave form, the mixture wave form, and the travelling-wave form. The new criterion is firstly applied to classify the locomotion of BCF fish, different from the traditional classification based on the fraction of fish body with lateral displacement. Moreover, the travelling index can also be used to evaluate the swimming performance, and to analyze the fish swimming dynamics.

Experimental Study on Complex Modes of an End-Damped Continuous Beam

The complex modes of an end-damped cantilevered beam are studied as an experimental example of a nonmodally damped continuous system. An eddy-current damper is applied, for its noncontact and linear properties, to the end of the beam, and is then characterized to obtain the effective damping coefficient. The state-variable modal decomposition (SVMD) is applied to extract the modes from the impact responses in the cantilevered beam experiments. Characteristics of the mode shapes and modal damping are examined for various values of the end-damper damping coefficient. The modal frequencies and mode shapes obtained from the experiments have a good consistency with the results of the finite element model. The variation of the modal damping ratio and modal nonsynchronicity with varying end-damper damping coefficient also follow the prediction of the model. Over the range of damping coefficients studied in the experiments, we observe a maximum damping ratio in the lowest underdamped mode, which correlates with the maximum modal nonsynchronicity. Complex orthogonal decomposition (COD) is applied in comparison to the modal identification results obtained from SVMD.

CFD Studies of the Effects of Waveform on Swimming Performance of Carangiform Fish

Carangiform fish, like mackerel, saithe and bass, swim forward by rhythmically passing body waves from the head to the tail. In this paper, the undulating motions are decomposed into the travelling part and the standing part by complex orthogonal decomposition (COD), and the ratio between these two parts, i.e., the travelling index, is proposed to analyse the waveform of fish-like movements. To further study the relative influences of the waveform on swimming performance, a self-propelled model of carangiform fish is developed by the level set/immersed boundary (LS-IB) method, and the in-house code is tested by two cases of flow past a sphere and an oscillating cylinder, respectively. In this study, the travelling index is varied in ranges up to 50% larger or smaller than the biological data. The results show that carangiform fish seem to favour a fast and efficient swimming motion with a travelling index of around 0.6. Meanwhile, we study several numerical cases with different amplitude coefficients (0.5~1.1) and tail-beat frequency (2 Hz~5 Hz), and then compare their swimming performance with each other. We found that the forward speed is closely related to the travelling index and tail-beat frequency, while the swimming efficiency is increased with the tail-beat frequency and amplitude coefficient. These results are also consistent with biological observations, and they might provide beneficial guidance with respect to the future design of robotic fish.

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

Abstract Complex orthogonal decomposition (COD) is applied to an experimental beam to extract the dispersive wave properties from response measurements. The beam is made of steel and is rectangular with a constant cross section. One end of the beam is free and is hung by a soft elastic cord. An impulse is applied to the free-end. The other end is buried in sand to absorb the wave as it travels from the impact site on the free-end; this effectively prevents reflections of the wave off the buried end and emulates a semi-infinite beam. The beam response is measured with an array of accelerometers, whose signals are integrated to obtain an ensemble of displacement signals. Acceleration responses are also compared in the frequency domain to predictions from the Euler–Bernoulli model. COD is applied to the displacement ensemble to obtain complex modal vectors and associated complex modal coordinates (COCs). The spatial whirl rates of nearly harmonic modal vectors are used to extract the modal wave numbers, and the temporal whirl rates of the modal coordinates are used to estimate the modal frequencies. The dispersion relationship between the frequencies and wave numbers compare favorably to those of the theoretical infinite Euler–Bernoulli beam.

Complex Orthogonal Decomposition Applied to Nematode Posturing

The complex orthogonal decomposition (COD), a process of extracting complex modes from complex ensemble data, is summarized, as is the use of complex modal coordinates. A brief assessment is made on how small levels of noise affect the decomposition. The decomposition is applied to the posturing of Caenorhabditis elegans, an intensively studied nematode. The decomposition indicates that the worm has a multimodal posturing behavior, involving a dominant forward locomotion mode, a secondary, steering mode, and likely a mode for reverse motion. The locomotion mode is closer to a pure traveling waveform than the steering mode. The characteristic wavelength of the primary mode is estimated in the complex plane. The frequency is obtained from the complex modal coordinate's complex whirl rate of the complex modal coordinate, and from its fast Fourier transform. Short-time decompositions indicate the variation of the wavelength and frequency through the time record.

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.

Complex Modal Analysis of the Swimming Motion of a Whiting

The kinematics of the transverse motion of a swimming fish are analyzed using a complex modal decomposition. Cinematographic images of a swimming whiting (Gadus merlangus) were obtained from the work of Sir James Gray (1933, “Studies in Animal Locomotion III. The Propulsive Mechanism of the Whiting (Gadus merlangus),” J. Exp. Biol., 10, pp. 391–402). The position of the midline for each image was determined and used to produce planar positions of virtual markers distributed along the midline of the fish. Transverse deflections of each virtual marker were then used for the complex orthogonal decomposition of modes. This method was applied to images of a whiting before and after amputation in a Newtonian frame of reference and an “anterior-body-fixed” frame as well. The fish motions were well represented by a single complex mode, which was then used as a modal filter. The extracted mode and modal coordinate were used to estimate the frequency, wavelength, and wave speed. The amputated fish was compared to the nonamputated fish, and the amount of traveling in the respective waveforms was quantified. The dominant complex mode shape and the estimated modal frequency were employed to reanimate the fish motion.

A complex orthogonal decomposition for wave motion analysis

A method is presented for decomposing wave motion into its principle components. The basic idea is a complex generalization of proper orthogonal decomposition. The method involves the representation of real oscillatory signals as complex analytic signals. The relationship between complex modes and wave motion is explored. From an ensemble of complex signals, a complex correlation matrix is formed, and its complex eigensolution is the basis of the decomposition (like a complex singular value decomposition). The complex eigenvectors contain standing and traveling characteristics. A traveling index is proposed to quantify the relative degree of traveling and standing in a waveform. A method of dissecting a wave mode into its traveling and standing parts is also proposed. From the complex modes and modal coordinates, frequencies, wavelengths, and characteristic wave speeds can be obtained. The method is applied to traveling and standing-wave examples.