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Asymptotic analysis and numerical solutions for the rigid body containing a viscous liquid in cavity in the presence of gyrostatic moment

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Abstract

This work touches upon the dynamical motion of a symmetric rigid body around a principal axis. It is assumed that the body has a cavity of spherical form filled with a viscous liquid and a movable mass connected with double elastically with a point lying on the axis of dynamical symmetry that exerts a viscous friction. This body is acted upon by a gyrostatic moment, in which the first two components are selected to be null and the third one is different from zero. The combined effect of both the viscous liquid and the movable mass on the suggested motion is studied. The achieved results reveal that in the existence of internal dissipation, the motion of the system approaches to a steady rotation around the axis of greatest inertia moment when time tends to infinity. The attained results are plotted to detect the good impact of the body parameter on the motion. Numerical results of the governing system are obtained applying the fourth-order Runge–Kutta method and represented in other plots. The significance of this work is focused on the great applications in the field of submarines and gyroscopes industries.

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References

  1. Chernous’ko, F.L.: On the motion of a satellite about its centre of mass under the action of gravitational torques. J. Appl. Math. Mech. 27(3), 208–722 (1963)

    MathSciNet  Google Scholar 

  2. Chernous’ko, F.L.: Motion of a rigid body with cavities filled with viscous liquid at small Reynolds numbers. Zh. Vychisl. Mat. Fiz. 5(6), 1049–1070 (1965)

    Google Scholar 

  3. Beletskii, V.V., Grushevskii, A.V.: The evolution of the rotational motion of a satellite under the action of a dissipative aerodynamics moment. J. Appl. Math. Mech. 58(1), 11–19 (1992)

    Article  Google Scholar 

  4. Akulenko, L.D., Leshchenko, D.D., Chernous’ko, F.L.: The rapid motion of a heavy rigid body about a fixed point in a dragging medium. Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela 3, 5–13 (1982)

    Google Scholar 

  5. Kuznetsova, EYu., Sazonov, V.V., Chebukov, SYu.: Evolution of the satellite rapid rotation under the action of gravitational and aerodynamic torques. Mech. Solids 35(2), 1–10 (2000)

    Google Scholar 

  6. Leshchenko, D.D., Rachinskaya, A.L.: Motion of a satellite relative to the centre of mass under the action of light pressure torque in a resistive medium. Visn. Odes. Derzh. Univ. Ser. Fiz.-Mat. Nauk. 12(7), 85–98 (2007)

    MATH  Google Scholar 

  7. Akulenko, L.D., Leshchenko, D.D., Rachinskaya, A.L.: Evolution of the satellite fast rotation due to the gravitational torque in a dragging medium. Mech. Solids 43(2), 173–184 (2008)

    Article  Google Scholar 

  8. Akulenko, L.D., Kovaleva, A.S.: Estimation of time perturbed Lagrangian system stays in an assigned region. J. Appl. Math. Mech. 73(2), 115–123 (2009)

    Article  MathSciNet  Google Scholar 

  9. Amer, T.S.: On the rotational motion of a gyrostat about a fixed point with mass distribution. Nonlinear Dyn 54, 189–198 (2008)

    Article  MathSciNet  Google Scholar 

  10. Amer, T.S.: The rotational motion of the electromagnetic symmetric rigid body. Appl. Math. Inf. Sci. 10(4), 1453–1464 (2016)

    Article  Google Scholar 

  11. Ismail, A.I., Amer, T.S., El Banna, S.A., El-Ameen, M.A.: Electromagnetic gyroscopic motion. J. Appl. Math. 2012, 1–14 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Panayotounakos, D.E., Theocaris, P.S.: On the decoupling and the solutions of the Euler dynamic equations governing the motion of a gyro. ZAMM 70(11), 489–500 (1990)

    Article  Google Scholar 

  13. Amer, T.S., Abady, I.M.: On the solutions of the Euler’s dynamic equations for the motion of a rigid body. J. Aero. Eng. 30(4), 04017021 (2017)

    Article  Google Scholar 

  14. Nayfeh, A.H.: Perturbations Methods. WILEY, Weinheim (2004)

    Google Scholar 

  15. Malkin I. G., Some problems in the theory of nonlinear oscillations, United States Atomic Energy Commission, Technical Information Service, ABC-tr-3766 (1959)

  16. Ismail, A.I., Amer, T.S.: The fast spinning motion of a rigid body in the presence of a gyrostatic momentum. Acta Mech. 154, 31–46 (2002)

    Article  Google Scholar 

  17. Amer, T.S.: Motion of a rigid body analogous to the case of Euler and Poinsot. Analysis 24, 305–315 (2004)

    Article  MathSciNet  Google Scholar 

  18. Amer, T.S., Amer, W.S.: On the motion of a gyro close to Kovalevskaya’s case. J. Comput. Theor. Nanosci. 14, 1009–1015 (2017)

    Article  Google Scholar 

  19. Elfimov, V.S.: Existence of periodic solutions of equations of motion of a solid body similar to the lagrange gyroscope. J. Appl. Math. Mech. 42(2), 262–269 (1978)

    Article  MathSciNet  Google Scholar 

  20. Amer, T.S.: On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector. Nonlinear Dyn 54, 249–262 (2008)

    Article  MathSciNet  Google Scholar 

  21. Amer, T.S.: On the dynamical motion of a gyro in the presence of external forces. Adv. Mech. Eng. 9(2), 1–13 (2017)

    Article  Google Scholar 

  22. Amer, T.S., Galal, A.A., Abady, I.M., El-Kafly, H.F.: The dynamical motion of a gyrostat for the irrational frequency case. Appl. Math. Model. 89, 1235–1267 (2021)

    Article  MathSciNet  Google Scholar 

  23. Amer, T.S., Abady, I.M.: On the application of KBM method for the 3-D motion of asymmetric rigid body. Nonlinear Dyn 89, 1591–1609 (2017)

    Article  MathSciNet  Google Scholar 

  24. Amer, T., Ismail, A., Amer, W.: Application of the Krylov-Bogoliubov-Mitropolski technique for a rotating heavy solid under the influence of a Gyrostatic Moment. J. Aerosp. Eng. 25(3), 421–430 (2012)

    Article  Google Scholar 

  25. Amer, T.S., Farag, A.M., Amer, W.S.: The dynamical motion of a rigid body for the case of ellipsoid inertia close to ellipsoid of rotation. Mech. Res. Commu. 108, 103583 (2020)

    Article  Google Scholar 

  26. Karapetyan, A.V.: The stability of regular precession of a symmetric rigid body with an ellipsoidal cavity. Vestnik Mosk. Gos. Univ., Ser. 1: Mathematika Mehkanika 6, 122–125 (1972)

    Google Scholar 

  27. Markeyev, A.P.: The stability of rotation of a top with a cavity filled with liquid. Izv. Akad. Kauk SSSR. MTT 3, 19–26 (1985)

    Google Scholar 

  28. Karapetyan, A.V., Prokonina, O.V.: The stability of permanent rotations of a top with a cavity filled with liquid on a plane with friction. PMM 64(1), 85–91 (2000)

    MATH  Google Scholar 

  29. Rudenko, T.V.: The stability of the steady motion of a gyrostat with a liquid in a cavity. J. Appl. Math. Mech. 66(2), 171–178 (2002)

    Article  MathSciNet  Google Scholar 

  30. Leshchenko D. D.,: Motion of a rigid body with movable point mass, Izv. AN SSSR, Mekh. Tverd. Tela. 11(3), 37–40 (1976)

  31. Akulenko L. D., Roshchin Yu. R.,: Response optimal braking of the rotation of a solid by controls limited by an ellipsoid, Izv. AN SSSR, MTT 1 (1977)

  32. Akulenko L. D., Roshchin Yu. R.,: Optimal control of the rotation of a solid by a low thrust pivoted motor, Izv. AN SSSR, MTT, 5 (1977)

  33. Smol’nikov, B.A.: Generalization of Euler case of a solid. PMM 31(4), 735–736 (1967)

    MATH  Google Scholar 

  34. Chernous’ko F. L.,: Motion of a rigid body with moving internal masses, Izv. AN. SSSR, MTT 4, 33–44 (1973)

  35. Akulenko L. D., Leshchenko D. D.,: Some problems on the motion of a rigid body with a movable mass, Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela 13 (5), 29–34 (1978)

  36. Leshchenko, D.D., Sallam, N.S.: Some problems on the motion of a rigid body with internal degrees of freedom. Int. Appl. Mech. 28(8), 524–528 (1992)

    Article  Google Scholar 

  37. Akulenko, L.D., Zinkevich Ya, S., Leshchenko, D.D., Rachinskaya, A.L.: (2011) Rapid rotations of a satellite with a cavity filled with viscous fluid under the action of moments of gravity and light pressure forces. Cosmic Res. 49(5), 440–451 (2011)

    Article  Google Scholar 

  38. Galal, A.A., Amer, T.S., El-Kafly, H., Amer, W.S.: The asymptotic solutions of the governing system of a charged symmetric body under the influence of external torques. Results Phys. 18, 103160 (2020)

    Article  Google Scholar 

  39. Disser, K., Galdi, G.P., Mazzone, G., Zunino, P.: Inertial motions of a rigid body with a cavity filled with a viscous liquid. Arch. Rational Mech. Anal. 221, 487–526 (2016)

    Article  MathSciNet  Google Scholar 

  40. Chernousko, F.L., Akulenko, L.D., Leshchenko, D.D.: Evolution of motions of a rigid body about its center of mass. Springer, AG (2017)

    Book  Google Scholar 

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This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El-Koom,Menoufia, 32511, Egypt

    W. S. Amer

  2. Mathematics Department, Faculty of Science, Tanta University, Tanta, 31527, Egypt

    A. M. Farag

  3. Mathematics and Computer Science Department, Faculty of Science, Suez University, Suez, 43511, Egypt

    I. M. Abady

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  1. W. S. Amer
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  2. A. M. Farag
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Correspondence to W. S. Amer.

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Amer, W.S., Farag, A.M. & Abady, I.M. Asymptotic analysis and numerical solutions for the rigid body containing a viscous liquid in cavity in the presence of gyrostatic moment. Arch Appl Mech 91, 3889–3902 (2021). https://doi.org/10.1007/s00419-021-01983-5

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