Skip to main content
SpringerLink
Log in

Mathematical modeling of sway, roll and yaw motions: order-wise analysis to determine coupled characteristics and numerical simulation for restoring moment’s sensitivity analysis

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

This paper investigates sway, roll and yaw motions of a floating body with the aim to determine coupled motion characteristics based on the order-wise analysis of hydrodynamic coefficients. To compute the hydrodynamic coefficients and wave force exerted on the floating body, we employ speed-dependent strip theory. The governing equations are solved analytically for linear restoring moment. For nonlinear restoring moment which is expressed as an odd-order polynomial of fifth degree in roll angle, we apply the Runge–Kutta–Gill method to solve the coupled equations. To investigate the effect of initial disturbances on sway, roll, yaw and speed of the body, numerical experiments have been carried out for a Panamax Container ship under the action of a sinusoidal wave of periodicity 11.2 s with varying wave height and speed. For the linear restoring moment, we first derive associated motion equations for various cases based on the relative magnitude of the hydrodynamic coefficients. The order-wise analysis leads to the classification of coupled characteristics exhibiting the nature of coupling. For the nonlinear restoring moment, we notice that the initial disturbance plays an important role in the ship’s stability. The effects of forward speed and variation in wave heights are illustrated through typical numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A ij :

Added-mass cross coupling coefficients for the jth mode coupled into the ith mode of motion

\({A_{ij}^0}\) :

Speed-independent part of A ij

B ij :

Damping cross coupling coefficients for the jth mode coupled with the ith mode of motion

\({B_{ij}^0}\) :

Speed-independent part of B ij

\({B_{44}^\ast}\) :

Viscous damping in roll

C ij :

Hydrostatic restoring cross coupling coefficients for the jth mode coupled with the ith mode

F j :

Exciting force and moment due to waves

\({\bar{F}_j}\) :

Amplitude of wave exciting force/moment

\({\overline{GM}}\) :

Metacentric height

I j :

Moment of inertia in jth mode

I jk :

Product of inertia

M :

Mass of the body

M jk :

Generalized mass matrix for the body

U :

Speed of the body

a ij :

two-dimensional sectional added mass coefficient

\({a_{ij}^A}\) :

a ij for aftermost section

b ij :

two-dimensional sectional damping coefficient

\({b_{ij}^A}\) :

b ij for aftermost section

f j :

Sectional Froude–Kryloff force

g :

Gravitational acceleration

h j :

Sectional diffraction force

\({h_j^A}\) :

h j for aftermost section

i, j, k:

Subscript (i, j, k =  1, 2, . . . , 6)

t :

Time variable

x, y, z:

Coordinate system

z c :

z-Coordinate of center of gravity

Δ:

Displacement of vessel

α :

Incident wave amplitude

β :

Undamped natural frequency of the damped system

η :

Displacement

\({\dot{\eta}}\) :

First order derivative of η with respect to time

\({\ddot{\eta}}\) :

Second order derivative of η with respect to time

θ :

Phase angle

ρ :

Mass density of water

ς :

Damping factor

ω :

Wave encountering frequency

References

  1. Froude W.: On the rolling ships. Instit. Naval Archit. Trans. 2, 180–229 (1861)

    Google Scholar 

  2. Kerwin J.E.: Notes on rolling in longitudinal waves. Int. Shipbuil. Prog. 2, 597–614 (1955)

    Google Scholar 

  3. Tasai F.: On the swaying, yawing and rolling of ships in oblique waves. Int Shipbuil. Prog 14, 216–228 (1967)

    Google Scholar 

  4. Salvesen N., Tuck E.O., Faltinsen O.M.: Ship motions and sea loads. Trans. Soc. Naval Archit. Mar. Eng. 78, 250–287 (1970)

    Google Scholar 

  5. Haddara M.R.: On the stability of ship motion in regular oblique waves. Int. Shipbuil. Prog. 18, 416–434 (1971)

    Google Scholar 

  6. Schmitke R.T.: Ship sway, roll, and yaw motions in oblique seas. Trans. Soc. Naval Archit. Mar. Eng. 86, 25–46 (1978)

    Google Scholar 

  7. Fang M.C., Chen S.Y.: On the stability and wave profile of a ship in oblique waves. Int. Shipbuil. Prog. 43, 359–380 (1996)

    Google Scholar 

  8. Cardo A., Francescutto A., Nabergoj R.: On damping models in free and forced rolling motion. Ocean Eng. 9, 171–179 (1982)

    Article  Google Scholar 

  9. Roberts J.B.: A stochastic theory for non-linear ship rolling in irregular seas. J. Ship Res. 26, 229–245 (1982)

    Google Scholar 

  10. Taylan M.: Solution of nonlinear roll model by a generalized asymptotic method. Ocean Eng. 26(11), 1169–1181 (1999)

    Article  Google Scholar 

  11. Taylan M.: Static and dynamic aspects of a capsize phenomenon. Ocean Eng. 30(3), 331–350 (2003)

    Article  Google Scholar 

  12. Das S.N., Das S.K.: Determination of coupled sway, roll and yaw motions of a floating body in regular waves. Int. J. Math. Math. Sci. 41, 2181–2197 (2004)

    Article  Google Scholar 

  13. Das S.K., Das S.N.: Note on sway, roll and yaw motions of a ship with forward speed: analytical study. Acta Mech. 186, 221–227 (2006)

    Article  MATH  Google Scholar 

  14. Das S.K., Das S.N.: Determination of roll motion for a floating body in regular waves. Proc. IMechE. 220, 41–48 (2006)

    Google Scholar 

  15. Das S.K., Das S.N., Sahoo P.K.: Investigation of coupled sway, roll and yaw motions of a floating body: numerical modelling for non-linear roll restoring. Ships Offshore Struct. 3(1), 49–56 (2008)

    Google Scholar 

  16. Tick L.J.: Differential equations with frequency-dependent coefficients. J. Ship Res. 3, 45–46 (1959)

    Google Scholar 

  17. Cummins W.E.: The impulse response function and ship motions. Schiffstechnik 9, 101–109 (1962)

    Google Scholar 

  18. Korvin-Kroukovsky B.V., Jacobs W.R.: Pitching and heaving motions of a ship in regular waves. Trans. Soc. Naval Archit. Mar. Eng. 65, 590–632 (1957)

    Google Scholar 

  19. Gerritsma, J., Beukelman, W.: The distribution of the hydrodynamic forces on a heaving and pitching ship model in still water. In: Proceedings 5th Symposium on Naval Hydrodynamics, pp. 219–251 (1964)

  20. Ogilvie, T.F., Tuck, E.O.: A rational strip theory for ship motions, Part 1. Report No. 013, Dept. of Naval Arch. and Marine Engineering, University of Michigan, Ann Arbor (1969)

  21. Wang, Z.H.: Hydroelastic analysis of high-speed ship. Ph.D. Thesis, Department Naval Architecture and Offshore Engineering, Technical University of Denmark, Lyngby (2000)

  22. Vugts, J.H.: The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. Report 194, Shipbuilding Lab., Delft University of Technology, Delft (1968)

  23. Frank, W., Salvesen, N.: The Frank Close-Fit Ship-Motion Computer Program, Report 3289, NSRDC, Washington, D.C. (1970)

  24. Krishnamurthy E.V., Sen S.K.: Numerical Algorithms, Computations in Science and Engineering. Affiliated East-West Press, New Delhi (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Central Water and Power Research Station, Khadakwasla, Pune, 411024, India

    S. N. Das

  2. Hokkaido Institute of Technology, 15-4-1, Maeda, 7-jou, Sapporo, Hokkaido, 006-8585, Japan

    S. Shiraishi

  3. International Institute of Information Technology, Pune Infotech Park, Hinjewadi, Pune, 411057, India

    S. K. Das

Authors
  1. S. N. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Shiraishi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. K. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. K. Das.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Das, S.N., Shiraishi, S. & Das, S.K. Mathematical modeling of sway, roll and yaw motions: order-wise analysis to determine coupled characteristics and numerical simulation for restoring moment’s sensitivity analysis. Acta Mech 213, 305–322 (2010). https://doi.org/10.1007/s00707-009-0278-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-009-0278-9

Keywords

Search

Navigation