Acta Applicandae Mathematicae

, Volume 130, Issue 1, pp 81–113

Global Dynamics of Froude-Type Oscillators with Superlinear Damping Terms

Article

Abstract

This paper deals with the damped superlinear oscillator
$$x'' + a(t)\phi_p\bigl(x' \bigr) + b(t)\phi_q\bigl(x'\bigr)+ \omega^2x = 0, $$
where a(t) and b(t) are continuous and nonnegative for t≥0; p and q are real numbers greater than or equal to 2; ϕr(x′)=|x′|r−2x′. This equation is a generalization of nonlinear ship rolling motion with Froude’s expression, which is very familiar in marine engineering, ocean engineering and so on. Our concern is to establish a necessary and sufficient condition for the equilibrium to be globally asymptotically stable. In particular, the effect of the damping coefficients a(t), b(t) and the nonlinear functions ϕp(x′), ϕq(x′) on the global asymptotic stability is discussed. The obtained criterion is judged by whether the integral of a particular solution of the first-order nonlinear differential equation
$$u' + \omega^{p-2}a(t)\phi_p(u) + \omega^{q-2}b(t)\phi_q(u) + 1 = 0 $$
is divergent or convergent. In addition, explicit sufficient conditions and explicit necessary conditions are given for the equilibrium of the damped superlinear oscillator to be globally attractive. Moreover, some examples are included to illustrate our results. Finally, our results are extended to be applied to a more complicated model.

Keywords

Damped oscillator Superlinear differential equations Global asymptotic stability Free rolling motion Froude’s expression 

Mathematics Subject Classification (2000)

34D05 34D23 34D45 37B25 

References

  1. 1.
    Artstein, Z., Infante, E.F.: On the asymptotic stability of oscillators with unbounded damping. Q. Appl. Math. 34, 195–199 (1976/77) MathSciNetGoogle Scholar
  2. 2.
    Bacciotti, A., Rosier, L.: Lyapunov Functions and Stability Control Theory. Springer, Berlin (2005) Google Scholar
  3. 3.
    Ballieu, R.J., Peiffer, K.: Attractivity of the origin for the equation \(\ddot{x} + f(t,\dot{x},\ddot{x})|\dot{x}|^{\alpha}\dot{x} + g(x) = 0\). J. Math. Anal. Appl. 65, 321–332 (1978) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bass, D.W., Haddara, M.R.: Nonlinear models of ship roll damping. Int. Shipbuild. Prog. 35, 5–24 (1988) Google Scholar
  5. 5.
    Bass, D.W., Haddara, M.R.: Roll and sway-roll damping for three small fishing vessels. Int. Shipbuild. Prog. 38, 51–71 (1991) Google Scholar
  6. 6.
    Berg, M.: A three-dimensional airspring model with friction and orifice damping. Veh. Syst. Dyn. 33, 528–539 (1999) Google Scholar
  7. 7.
    Brauer, F., Nohel, J.: The Qualitative Theory of Ordinary Differential Equations. Benjamin, New York (1969). Dover, New York (1989) (revised) MATHGoogle Scholar
  8. 8.
    Bulian, G.: Approximate analytical response curve for a parametrically excited highly nonlinear 1-DOF system with an application to ship roll motion prediction. Nonlinear Anal., Real World Appl. 5, 725–748 (2004) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cardo, A., Francescutto, A., Nabergoj, R.: On damping models in free and forced rolling motion. Ocean Eng. 9, 171–179 (1982) CrossRefGoogle Scholar
  10. 10.
    Cesari, L.: Asymptotic behavior and stability problems. In: Ordinary Differential Equations. Springer, Berlin (1959). Springer, Berlin (1963) (2nd ed.) Google Scholar
  11. 11.
    Chan, H.S.Y., Xu, Z., Huang, W.L.: Estimation of nonlinear damping coefficients from large-amplitude ship rolling motions. Appl. Ocean Res. 17, 217–224 (1995) CrossRefGoogle Scholar
  12. 12.
    Chun, H.H., Chun, S.H., Kim, S.Y.: Roll damping characteristics of a small fishing vessel with a central wing. Ocean Eng. 28, 1601–1619 (2001) CrossRefGoogle Scholar
  13. 13.
    Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. Heath, Boston (1965) MATHGoogle Scholar
  14. 14.
    Dalzell, J.F.: A note on the form of ship roll damping. J. Ship Res. 22, 178–185 (1978) Google Scholar
  15. 15.
    Duc, L.H., Ilchmann, A., Siegmund, S., Taraba, P.: On stability of linear time-varying second-order differential equations. Q. Appl. Math. 64, 137–151 (2006) MATHMathSciNetGoogle Scholar
  16. 16.
    Grace, I.M., Ibrahim, R.A., Pilipchuk, V.N.: Inelastic impact dynamics of ships with one-sided barriers. Part I: analytical and numerical investigations. Nonlinear Dyn. 66, 589–607 (2011) CrossRefGoogle Scholar
  17. 17.
    Haddara, M.R.: On the stability of ship motion in regular oblique waves. Int. Shipbuild. Prog. 18, 416–434 (1971) Google Scholar
  18. 18.
    Haddara, M.R., Bass, D.W.: On the form of roll damping moment for small fishing vessels. Ocean Eng. 17, 525–539 (1990) CrossRefGoogle Scholar
  19. 19.
    Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York (1966) MATHGoogle Scholar
  20. 20.
    Hale, J.K.: Ordinary Differential Equations. Wiley, New York (1969). Krieger, Malabar (1980) (revised) MATHGoogle Scholar
  21. 21.
    Hatvani, L.: On the asymptotic stability for a two-dimensional linear nonautonomous differential system. Nonlinear Anal. 25, 991–1002 (1995) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hatvani, L.: Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. Proc. Am. Math. Soc. 124, 415–422 (1996) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Hatvani, L., Totik, V.: Asymptotic stability of the equilibrium of the damped oscillator. Differ. Integral Equ. 6, 835–848 (1993) MATHMathSciNetGoogle Scholar
  24. 24.
    Hatvani, L., Krisztin, T., Totik, V.: A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. Differ. Equ. 119, 209–223 (1995) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Hinemo, Y.: Prediction of ship roll damping-state of art. Dept. of Naval Architecture and Marine Engineering, The University of Michigan, Report No. 239 (1981) Google Scholar
  26. 26.
    Ibrahim, R.A., Grace, I.M.: Modeling of ship roll dynamics and its coupling with heave and pitch. Math. Probl. Eng. 2010, 1–32 (2010). Article ID 934714 CrossRefGoogle Scholar
  27. 27.
    Ignatyev, A.O.: Stability of a linear oscillator with variable parameters. Electron. J. Differ. Equ. 1997(17), 1–6 (1997) MathSciNetGoogle Scholar
  28. 28.
    Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Stability Analysis of Nonlinear Systems. Dekker, New York (1989) MATHGoogle Scholar
  29. 29.
    Levin, J.J., Nohel, J.A.: Global asymptotic stability for nonlinear systems of differential equations and application to reactor dynamics. Arch. Ration. Mech. Anal. 5, 194–211 (1960) CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Michel, A.N., Hou, L., Liu, D.: Stability Dynamical Systems: Continuous, Discontinuous, and Discrete Systems. Birkhäuser, Boston (2008) Google Scholar
  31. 31.
    Neves, M.A.S., Pérez, N.A., Valerio, L.: Stability of small fishing vessels in longitudinal waves. Ocean Eng. 26, 1389–1419 (1999) CrossRefGoogle Scholar
  32. 32.
    Pucci, P., Serrin, J.: Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math. 170, 275–307 (1993) CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Pucci, P., Serrin, J.: Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. Anal. 25, 815–835 (1994) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Rouche, N., Habets, P., Laloy, M.: Stability Theory by Lyapunov’s Direct Method. Applied Mathematical Sciences, vol. 22. Springer, New York (1977) CrossRefGoogle Scholar
  35. 35.
    Senjanović, I., Parunov, J., Ciprić, G.: Safety analysis of ship rolling in rough sea. Chaos Solitons Fractals 8, 659–680 (1997) CrossRefMATHGoogle Scholar
  36. 36.
    Shimozawa, K., Tohtake, T.: An air spring model with non-linear damping for vertical motion. Q. Rep. RTRI 49, 209–214 (2008) CrossRefGoogle Scholar
  37. 37.
    Smith, R.A.: Asymptotic stability of x″+a(t)x′+x=0. Q. J. Math. 12, 123–126 (1961) CrossRefMATHGoogle Scholar
  38. 38.
    Sugie, J.: Global asymptotic stability for damped half-linear oscillators. Nonlinear Anal. 74, 7151–7167 (2011) CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Sugie, J.: Smith-type criterion for the asymptotic stability of a pendulum with time-dependent damping. Proc. Am. Math. Soc. 141, 2419–2427 (2013) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Sugie, J., Hata, S.: Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients. Monatshefte Math. 166, 255–280 (2012) CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Sugie, J., Shimadu, T., Yamasaki, T.: Global asymptotic stability for oscillators with superlinear damping. J. Dyn. Differ. Equ. 24, 777–802 (2012) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Taylan, M.: The effect of nonlinear damping and restoring in ship rolling. Ocean Eng. 27, 921–932 (2000) CrossRefGoogle Scholar
  43. 43.
    Üçer, E.: Examination of the stability of trawlers in beam seas by using safe basins. Ocean Eng. 38, 1908–1915 (2011) CrossRefGoogle Scholar
  44. 44.
    Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Applied Mathematical Sciences, vol. 14. Springer, New York (1975) CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsShimane UniversityMatsueJapan

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