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Floating rigid bodies: a note on the conservativeness of the hydrostatic effects

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Abstract

Within the framework of Lagrangian mechanics, the conservativeness of the hydrostatic forces acting on a floating rigid body is proved. The representation of the associated hydrostatic potential is explicitly worked out. The invariance of the resulting Lagrangian with respect to surge, sway and yaw motions is used in connection with the Routh procedure in order to convert the original dynamical problem into a reduced one, in three independent variables. This allows to put on rational grounds the study of hydrostatic equilibrium, introducing the concept of pseudo-stability, meant as stability with respect to the reduced problem. The small oscillations of the system around a pseudo-stable equilibrium configuration are discussed.

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Notes

  1. Needless to say, the expression (4) is identical to \(\pi ^{ \text {(B)}}=\underline{F}^{ \text {(B)}}\cdot \underline{v}_G+\underline{M}^{ \text {(B)}}_{G} \cdot \underline{\omega }\).

  2. By properly choosing the origin \({\varOmega }\) and the axes \(\underline{k}_1, \underline{k}_2\), every equilibrium configuration can always be represented in the stated form. This reflects once again the invariance of the algorithm under the subgroup \({\mathfrak {E}}_2\subset {\mathfrak {E}}_3\) of rigid motions preserving the plane \(\zeta =0\).

References

  1. Lewandowski EM (2004) The dynamics of marine craft. World Scientific Publishing Co., Singapore

    Book  Google Scholar 

  2. Fossen TI (2002) Marine control systems. Marine Cybernetics, Trondheim

    Google Scholar 

  3. Lewis EV (ed) (1988) Principles of naval architecture, vol. 1: stability and strength. Society of Naval Architects and Marine Engineers, Jersey City

    Google Scholar 

  4. Loney SL (1956) Elements of hydrostatics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Biran A, Lopez Pulido R (2014) Ship hydrostatics and stability, 2nd edn. Butterworth-Heinemann, Kidlington

    Google Scholar 

  6. Goldstein H (1959) Classical mechanics. Addison-Wesley, Reading

    MATH  Google Scholar 

  7. Gantmacher FR (1975) Lectures in analytical mechanics. Mir Publishers, Moscow

    Google Scholar 

  8. Vignolo S (2003) A geometric approach to Routh procedure. Addendum: “Legendre transformation and analytical mechanics: a geometric approach”. J Math Phys 44:3141

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. DIME, Sez. Metodi e Modelli Matematici, Università di Genova, Piazzale Kennedy, Pad. D, 16129, Genova, Italy

    Enrico Massa & Stefano Vignolo

Authors
  1. Enrico Massa
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  2. Stefano Vignolo
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Correspondence to Enrico Massa.

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Massa, E., Vignolo, S. Floating rigid bodies: a note on the conservativeness of the hydrostatic effects. Meccanica 52, 2491–2497 (2017). https://doi.org/10.1007/s11012-016-0598-5

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  • DOI: https://doi.org/10.1007/s11012-016-0598-5

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