The European Physical Journal Special Topics

, Volume 224, Issue 17–18, pp 3379–3392 | Cite as

Free vibration analysis of a robotic fish based on a continuous and non-uniform flexible backbone with distributed masses

  • W. CoralEmail author
  • C. Rossi
  • O.M. Curet
Regular Article Applied Physics and Robotics
Part of the following topical collections:
  1. Dynamics of Animal Systems


This paper presents a Differential Quadrature Element Method for free transverse vibration of a robotic fish based on a continuous and non-uniform flexible backbone with distributed masses (fish ribs). The proposed method is based on the theory of a Timoshenko cantilever beam. The effects of the masses (number, magnitude and position) on the value of natural frequencies are investigated. Governing equations, compatibility and boundary conditions are formulated according to the Differential Quadrature rules. The convergence, efficiency and accuracy are compared to other analytical solution proposed in the literature. Moreover, the proposed method has been validate against the physical prototype of a flexible fish backbone. The main advantages of this method, compared to the exact solutions available in the literature are twofold: first, smaller computational cost and second, it allows analysing the free vibration in beams whose section is an arbitrary function, which is normally difficult or even impossible with other analytical methods.


Free Vibration European Physical Journal Special Topic Timoshenko Beam Ionic Polymer Metal Composite Slenderness Ratio 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    C.W. Bert, M. Malik, Appl. Mech. Rev. 49, 1 (1996)ADSCrossRefGoogle Scholar
  2. 2.
    Y. Cha, M. Verotti, H. Walcott, S. Peterson, M. Porfiri, Bioinspir. Biomimet. 8, 3 (2013)CrossRefGoogle Scholar
  3. 3.
    Y. Chen, J. Appl. Mech. 30, 310 (1963)ADSCrossRefGoogle Scholar
  4. 4.
    W. Coral, et al., Smart Actuation and Sensing Systems – Recent Advances and Future Challenges. Chapter 3, SMA-Based Muscle-Like Actuation in Biologically Inspired Robots: A State of the Art Review (INTECH, 2012), p. 53Google Scholar
  5. 5.
    M.A. De Rosa, C. Franciosi, M.J. Maurizi, Comp. Struct. 58, 1145 (1955)CrossRefGoogle Scholar
  6. 6.
    H. Du, M.K. Lim, N.R. Lin, Inter. J. Numer. Meth. Eng. 37, 1881 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    H. Du, M.K. Lim, N.R. Lin, J. Sound Vibr. 181, 279 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    M. Gurgoze, J. Sound Vibr. 96, 461 (1984)ADSCrossRefGoogle Scholar
  9. 9.
    M. Gurgoze, J. Sound Vibr. 100, 588 (1985)ADSCrossRefGoogle Scholar
  10. 10.
    T. Kaneko, J. Phys. D: Appl. Phys. 8, 1928 (1975)ADSCrossRefGoogle Scholar
  11. 11.
    G. Karami, P. Malekzadeh, Comp. Meth. Appl. Mech. Eng. 191, 3509 (2002)CrossRefGoogle Scholar
  12. 12.
    P. Laura, M.J. Maurizi, J.L. Pombo, J. Sound Vibr. 41, 397 (1975)ADSCrossRefGoogle Scholar
  13. 13.
    P. Laura, P.L. Verniere de Irassar, G.M. Ficcadenti, J. Sound Vibr. 86, 279 (1983)ADSCrossRefGoogle Scholar
  14. 14.
    S.Y. Lee, S.M. Lin, J. Sound Vibr. 183, 403 (1995)ADSCrossRefGoogle Scholar
  15. 15.
    R.M. Lin, M.K. Lim, H. Du, Comput. Struct. 53, 993 (1994)CrossRefGoogle Scholar
  16. 16.
    W.H. Liu, J.R. Wu, C.C. Huang, J. Sound Vibr. 122, 193 (1988)ADSCrossRefGoogle Scholar
  17. 17.
    G.V. Rao, K.M. Saheb, G.R. Janardhan, J. Sound Vibr. 298, 221 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    C. Rossi, W. Coral, et al., Bioinspir. Biomimet. 6, 15 (2011)CrossRefGoogle Scholar
  19. 19.
    C. Rossi, W. Coral, et al., A Motor-less and Gear-less Bio-mimetic Robotic Fish Design, 2011 IEEE International Conference on Robotics and Automation (2011)Google Scholar
  20. 20.
    S. Timoshenko, D.H. Young, W. Weaver, Vibration problems in engineering (Wiley, New York, 1974)Google Scholar
  21. 21.
    M. Aureli, V. Kopman, M. Porfiri, Free-locomotion of underwater vehicles actuated by ionic polymer metal composites. IEEE/ASME Transactions on 15(4), 603 (2010)Google Scholar
  22. 22.
    P. Phamduy, R. LeGrand, M. Porfiri, Robotics & Automation Magazine, IEEE 22(1), 86 (2015)CrossRefGoogle Scholar
  23. 23.
    Tracker, Video Analysis and Modelling Tool, (accessed September 10, 2015)
  24. 24.
    W.-H. Chu, Technical Report No. 2, DTMB, Contract NObs-86396(X), Southwest Research Institute (San Antonio, Texas, 1963)Google Scholar
  25. 25.
    U.S. Lindholm, D.D. Kana, W.-H. Chu, H.N. Abramson, J. Ship. Res. 9, 11 (1965)Google Scholar

Copyright information

© EDP Sciences and Springer 2015

Authors and Affiliations

  1. 1.Universidad Politécnica de MadridMadridSpain
  2. 2.Centre for Automation and Robotics (CAR) UPM-CSICMadridSpain
  3. 3.Department of Ocean and Mechanical EngineeringFlorida Atlantic University (FAU)Boca RatonUSA

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