Modelling a Swimming Fish as a Series of Linked Harmonic Oscillators

  • Andrew Liszewski

Student thesis: Masters ThesisMPhil


Researchers have studied the behaviours of generalist swimmers for decades with aim to develop more agile and efficient robotic swimmers. The high efficiency of subcarangiform swimming motion in particular is postulated to be a result of the ability of the muscles and other elastic tissues to passively transmit an undulatory wave, generated in the pre-caudal region, posteriorly through the caudal region (Blight, 1977; Wassersug and Von Seckendorff Hoff, 1985). Wassersug and Von Seckendorff Hoff (1985) and Liao (2004) have shown that even the bodies of dead fish are able to produce thrust in certain circumstances because of the stiffness properties inherent in their bodies. This passive transmission of an undulatory wave through natural energy storage mechanisms suggests that the high efficiencies exhibited in biological swimmers may be both internal forces and external forces using calculated values of the drag and added mass based on the literature. Physical and theoretical models with the same lengths, masses, stiffnesses, damping and drag characteristics are tested at the same angular amplitude and frequency of oscillations. Comparisons between the performances of the theoretical and analogous physical models are used to validate the theoretical models. The theoretical model without fluid resistance is proven accurate in its ability to predict the undulatory amplitude (UA) of its physical analogue within ±5%. The performance of the model with fluid resistance shows that the effect of fluid resistance is overestimated. Fluid resistance in the physical experiments reduced the undulatory amplitude by 35.7% on average. In the numerical simulations, fluid resistance was calculated to reduce the undulatory amplitude 59.4%. However the calculated relative response at resonance was observed to increase dramatically, underscoring the importance of tuning the oscillatory properties in the presence of fluid resistance. An abridged system identification process is adopted to explore the numerical relationships of certain parameters to the movement of the body. This reveals the likely causes for the overestimation: the added mass is currently underestimated; the estimated drag force is currently overestimated; the stiffness value is underestimated in water. Changing these values would increase the calculated TBA with fluid resistance. One could carry out the full system identification process on the model to determine a set of parameters which replicate this set of experiments but there is no value in doing so until the stiffness and dissipation coefficients are more accurately accounted for and the flow around the physical model is incorporated into the model. In summary, this thesis has derived and validated a theoretical model which accurately predicts the motion of linked segment models without fluid resistance as a series of linked harmonic oscillators. The incorporation of fluid dynamics into the model caused noticeable changes in the frequency response of the system as well as greatly decreased the calculated undulatory amplitude. Based on numerical analysis, likely causes for the misrepresentation of the fluid dynamics are identified. Recommendations are made for improvements of the model with fluid resistance which will make it a valuable tool in the design of AUVs. achieved in man-made propulsion mechanisms by tuning their stiffness properties. The literature review of this study outlines findings from previous research into the modes of swimming, swimming behaviours and the forces present on the body of a fish as it propels itself. From this, it is shown that the stiffness properties of the bodies of real fish are highly specialised swimming modes. Previous studies in robotics have focused on creating biomimetic autonomous underwater vehicles (AUVs) with anteriorly placed actuators which generate a wave to be transmitted through the rest of the tail passively. The current studies identify flexible continuous beam models and linked segment structures as two leading biomimetic designs for such robots. In the literature review it is shown that linked segment structures hold a considerable advantage in imitating kinematics and are thus a promising avenue for progressing the understanding of how fish achieve such high levels of efficiency. It is shown that the undulatory wave travelling along the swimming fish body can be approximated as a series of finite heaving linked segments (Lighthill, 1969) connected by flexible joints (Hebrank, 1982; Bowtell and Williams, 1991). This study builds and validates two theoretical models, derived from kinematics and hydrodynamic principles outlined in the literature review, for tuning the oscillatory properties of linked segment systems. The first theoretical model accounts for only the internal forces of the mechanism by neglecting fluid resistance. The second theoretical model incorporates
Date of Award26 Feb 2013
Original languageEnglish
Awarding Institution
  • University of Bath
SupervisorJames Cunningham (Supervisor)


  • filose
  • biometrics
  • fish-like swimming

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