This paper provides analytical insights into the hypothesis that fish exploit resonance to reduce the mechanical cost of swimming. A simple body-fluid fish model, representing carangiform locomotion, is developed. Steady swimming at various speeds is analysed using optimal gait theory by minimizing bending moment over tail movements and stiffness, and the results are shown to match with data from observed swimming. Our analysis indicates the following: thrust-drag balance leads to the Strouhal number being predetermined based on the drag coefficient and the ratio of wetted body area to cross-sectional area of accelerated fluid. Muscle tension is reduced when undulation frequency matches resonance frequency, which maximizes the ratio of tail-tip velocity to bending moment. Finally, hydrodynamic resonance determines tail-beat frequency, whereas muscle stiffness is actively adjusted, so that overall body-fluid resonance is exploited.
This paper develops a design method for the interconnections of a network of Andronov-Hopf oscillators such that the system exhibits a desired strange attractor. Because of the structure of the oscillators, the desired behavior can be achieved via weak linear coupling, which destabilizes the oscillators' phase difference. First, a set of sufficient conditions are established that result in phase destabilization, and thus instability, of a desired periodic solution. Then, an additional condition is determined to ensure that all harmonic periodic orbits will be unstable. Finally, additional numerical properties are assessed, where tuning of a small parameter can result in chaos.
Abstract-Animal locomotion can be viewed as mechanical rectification due to the dynamics that convert periodic body movements to a positive average thrust, resulting in a steady locomotion velocity. This paper considers a general multi-body mechanical rectifier under continuous interactions with the environment, with full rotation and translation in three dimensional space. The equations of motion are developed with respect to body coordinates to allow for direct analysis of maneuvering dynamics. The paper then formulates and solves an optimal turning gait problem for a mechanical rectifier traveling along a curved path, with propulsive forces generated by periodic body deformation (gait). In particular, the gait is optimized to minimize a quadratic cost function, subject to constraints on average locomotion velocity and average angular velocity. The problem is proven to reduce to two separate, tractable minimization problems solvable for globally optimal solutions. The first problem solves for the optimal shape offset that results in turning, while the other solves for the optimal gait that results in locomotion along a straight path. A case study of a locomotor in a fluid environment is presented to demonstrate the utility of the method for robotic locomotor design.
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