ANALYTICAL APPROACH OF TWO GENERAL NONLINEAR SHIP MOTIONS IN TIME DOMAIN

S. REKHA, D. GOWTHAMAN, P. BALAGANESAN, B. ACHOUR

Abstract


Two nonlinear models of the roll motion of ships are discussed and analytically solved. Both models are second-order differential equations with nonlinear restoring and damping moments. A new approach to the homotopy perturbation method is applied to derive analytical expressions for the roll angle, velocity, acceleration and restoring and damping moments. The analytical results are validated by direct comparison with the fourth-order Runge‒Kutta method. The analytical approach of this paper can be efficiently extended to various vibrating dynamical models arising in mechanical systems.


Keywords


Mathematical modeling, ship dynamics, capsizing, restoring and damping moments, analytical solution.

Full Text:

PDF

References


ABUALRUB T., ABUKHALED M. (2015). Wavelets approach for optimal boundary control of cellular uptake in tissue engineering, International Journal of Computer Mathematics, Vol. 92, Issue 7, pp. 1402-1412.

ABUKHALED M., KHURI S., SAYFY A. (2011). A numerical approach for solving a class of singular boundary value problems arising in physiology, International Journal of Numerical Analysis and Modeling, Vol. 8, Issue 2, pp. 353-363.

ABUKHALED M., KHURI S. (2020). Efficient numerical treatment of a conductive-radiative fin with temperature dependent thermal conductivity and surface emissivity, International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 21, Issue 4, pp. 159-168.

ABUKHALED M. (2013). Variational iteration method for nonlinear singular two-point boundary value problems arising in human physiology, Journal of Mathematics, Article 720134.

BASS DW., HADDARA MR. (1988). Nonlinear models of ship roll damping. International Shipbuilding Progress, Vol. 35, Issue 401, pp. 5-24.

BHRAWY A. (2013). A Jacobi-Gauss-Lobatto collocation method for solving generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Applied Mathematics and Computation, Vol. 222, Issue 1, pp. 255-264.

BIAZAR J., GHAZVINI H. (2008). Homotopy perturbation method for solving hyperbolic partial differential equations, Computers and Mathematics Applications Vol. 56, Issue 2, pp. 453-458.

DASHTIMANESH A., ENSHAEI H., TAVAKOLI S. (2019). Oblique-asymmetric 2D1T model to compute hydrodynamic forces and moments in coupled sway, roll, and yaw motions of planning hulls, Journal of Ship Research, Vol. 63, Issue 1, pp. 1-15.

DUAN J-S., RACH R., WAZWAZ A.M. (2015). A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, Vol. 5, Issue 1, pp. 59-74.

FALZARANO J., SHAW S., TROESCH A. (1992). Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. International Journal of Bifurcation and Chaos, Vol. 2, Issue 1, pp. 101-115.

GHAMARI I., GRECO M., FALTINSEN O., LUGNI C. (2020). Numerical and experimental study on the parametric roll resonance for a fishing vessel with and without forward speed, Applied Ocean Research, Vol. 101, Article 102272.

HAMAMOTO M., PANJAITAN J. (1996). A critical situation leading to capsize of ships in astern seas, Journal of the Society of Naval Architects of Japan, Vol. 1996, Issue 180, pp. 215-221.

HARIHARAN G., RAJARAMAN R., SATHIYASEELAN D. (2016). Wavelet based spectral algorithm for nonlinear dynamical systems arising in ship dynamics, Ocean Engineering, Vol. 126, pp. 321-328.

HASSAN I. (2008). Application to differential transformation method for solving systems of differential equations, Applied Mathematical Modelling, Vol. 32, Issue 12, pp. 2552-2559.

HE J.H. (1999). Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, Vol. 178, Issue 3-4, pp. 257-262.

HE J.H. (2007). Variational iteration method-some recent results and new interpretations, Journal of Computational and Applied Mathematics, Vol. 207, Issue 1, pp. 3-17.

IBRAHIM R.A., GRACE I.M. (2010). Modeling of ship roll dynamics and its coupling with heave and pitch, Mathematical Problems in Engineering, Vol. 2010, Article ID 934714, pp. 1-31.

JANG T.S., KWON S.H., LEE. J.H. (2010). Recovering the functional form of the nonlinear roll damping of ships from a free-roll decay experiment: An inverse formulism, Ocean Engineering, Vol. 37, Issue 13-14, pp. 1337-1344.

KAWAHARA Y., MAEKAWA K., IKEDA Y. (2011). A simple prediction formula of roll damping of conventional Cargo ships on the basis of Ikeda's method and its limitation, Fluid Mechanics and Its Applications, Vol. 97, pp. 465–486.

LIAO S.J. (2012). Homotopy analysis method in nonlinear differential equations, Springer and Higher Education Press, Heidelberg, Shanghai, China.

MITRA R.K., BANIK A.K., DATTA T.K., CHATTERJEE S. (2018). Nonlinear roll oscillation of semisubmersible system and its control, International Journal of Non-Linear Mechanics, Vol. 107, pp. 42-55.

MUNIF A., UMEDA N. (2000). Modeling extreme roll motions and capsizing of a moderate-speed ship in astern waves, Journal of the Society of Naval Architects of Japan, Vol. 187, 2000, pp. 51-58.

SUN S., SHAO M. (2019). Estimation of nonlinear roll damping by analytical approximation of experimental free-decay amplitudes, Journal of Ocean University of China, Vol. 18, pp. 812-822.

TIRMIZI I., TWIZELL E. (2002). Higher-order finite-difference methods for nonlinear second-order two-point boundary-value problems, Applied Mathematics Letters, Vol. 15, Issue 7, pp. 897-902.

WASSERMANN S., FEDER D.F., ABDEL-MAKSOUD M. (2016). Estimation of ship roll damping-A comparison of the decay and the harmonic excited roll motion technique for a post panamax container ship, Ocean Engineering, Vol. 120, pp. 371-382.

WAZWAZ A.M. (2013). A reliable iterative method for solving the time-dependent singular Emden-Fowler equations, Central European Journal of Engineering, Vol. 3, Issue 1, pp. 99-105.

WILSON P., CARRICA P., STERN F. (2006). Unsteady RANS method for ship motions with application to roll for a surface combatant, Computers and Fluids, Vol. 35, Issue 5, pp. 501-524.

XU H., HASSANI V., GUEDES SOARES C. (2019). Uncertainty analysis of the hydrodynamic coefficient’s estimation of a nonlinear maneuvering model based on planar motion mechanism tests, Ocean Engineering, Vol. 173, Issue 3, pp. 450-459.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.