Contemporary Ideas on Ship Stability pp 91-115 | Cite as

# Improved Maneuvering-Based Mathematical Model for Free-Running Ship Motions in Following Waves Using High-Fidelity CFD Results and System-Identification Technique

## Abstract

Predicting maneuverability and stability of a free running ship in following and quartering waves are one of the most important topics to prevent broaching; however current mathematical models show quantitative errors with the experimental data while high-fidelity CFD simulations show quantitative agreement, which provides the opportunity to improve the mathematical models for free running ship dynamics in waves. In this study, both maneuvering coefficients and wave model in the mathematical model are improved utilizing system identification technique and CFD free running outputs. From turning circle and zigzag calm water CFD free running data, the maneuvering coefficients are estimated. The wave correction parameters are introduced to improve the wave model, which are found from a few forced and free running CFD simulations in waves. The mathematical model with the improved parameters shows much better agreement with experiments in both calm water and waves than the original mathematical model. The original mathematical model was based on the maneuvering coefficients estimated from several captive tests and wave forces calculated from linear Froude-Krylov forces and diffraction forces based on a slender ship theory.

## Keywords

System identification Constrained least square method Maneuvering coefficients Wave model Free-running ship## List of Symbols

- \(a_{1,2,3,4}\)
Tuning parameter for wave forces amplitude in surge

- \(a_{H}\)
Rudder and hull hydrodynamic interaction coefficient in sway

- \(B\)
Ship breadth

- \(b_{1,2,3,4}\)
Tuning parameter for wave forces amplitude in sway

- \(c_{1,2,3,4}\)
Tuning parameter for wave forces amplitude in roll

- \(C_{X}\)
Nondimensionalized drift wave force in surge

- \(C_{Y}\)
Nondimensionalized drift wave force in sway

- \(C_{N}\)
Nondimensionalized drift wave moment in yaw

- \(d\)
Ship draft

- \(d_{1,2,3,4}\)
Tuning parameter for wave forces amplitude in yaw

- \(Fr\)
Froude number

- \(g\)
Gravitational acceleration

- \(GZ\)
Restoring arm in roll

- \(I_{xx}\)
Moment of inertia in roll

- \(I_{zz}\)
Moment of inertia in yaw

- \(J_{xx}\)
Added moment of inertia in roll

- \(J_{zz}\)
Added moment of inertia in yaw

- \(k\)
Wave number

- \(K_{p}\)
Derivative of roll moment with roll rate

- \(K_{r}\)
Derivative of roll moment with yaw rate

- \(K_{R}\)
Rudder force in roll

- \(K_{s}\)
Rotational index in nonlinear first order Nomoto’s model

- \(K_{v}\)
Derivative of roll moment with sway velocity

- \(K_{w}\)
Wave moment in roll

- \(K_{w}^{Dif}\)
Diffraction wave moment in roll

- \(K_{w}^{FK}\)
Froude-Krylov wave moment in roll

- \(K_{rrr}\)
Derivative of roll moment with cubed yaw rate

- \(K_{rrv}\)
Derivative of roll moment with squared yaw rate and sway velocity

- \(K_{rvv}\)
Derivative of roll moment with squared sway velocity and yaw rate

- \(K_{vvv}\)
Derivative of roll moment with cubed sway velocity

- \(K_{\phi }\)
Derivative of roll moment with roll angle

- \(L\)
Ship length

- \(l_{R}\)
Longitudinal position of rudder center from center of ship gravity

- \(m\)
Ship mass

- \(m_{WD}\)
Shape parameter of Weibull distribution for wave drift forces/moment

- \(m_{x}\)
Added mass in surge

- \(m_{y}\)
Added mass in sway

- \(N_{r}\)
Derivative of yaw moment with yaw rate

- \(N_{R}\)
Rudder force in yaw

- \(N_{s}\)
Nonlinear index in nonlinear first order Nomoto’s model

- \(N_{v}\)
Derivative of yaw moment with sway velocity

- \(N_{w}\)
Wave moment in yaw

- \(N_{w}^{Dif}\)
Diffraction wave moment in yaw

- \(N_{w}^{FK}\)
Froude-Krylov wave moment in yaw

- \(N_{rrr}\)
Derivative of yaw moment with cubed yaw rate

- \(N_{rrv}\)
Derivative of yaw moment with squared yaw rate and sway velocity

- \(N_{rvv}\)
Derivative of yaw moment with squared sway velocity and yaw rate

- \(N_{vvv}\)
Derivative of yaw moment with cubed sway velocity

- \(N_{\phi }\)
Derivative of yaw moment with roll angle

- \(p\)
Roll rate

- \(r\)
Yaw rate

- \(R\)
Ship resistance

- \(T\)
Propeller thrust in surge

- \(t_{R}\)
Rudder and hull hydrodynamic interaction coefficient in surge

- \(T_{s}\)
Time constant index in nonlinear first order Nomoto’s model

- \(T_{w}\)
Wave period

- \(u\)
Surge velocity

- \(u_{w}\)
Wave particle velocity in surge

- \(v\)
Sway velocity

- \(v_{w}\)
Wave particle velocity in sway

- \(W_{CN}\)
Drift wave moment in yaw

- \(W_{CX}\)
Drift wave force in surge

- \(W_{CY}\)
Drift wave force in sway

- \(x_{H}\)
Rudder and hull hydrodynamic interaction coefficient in yaw

- \(X_{R}\)
Rudder force in surge

- \(X_{rr}\)
Derivative of surge force with squared yaw rate

- \(X_{rv}\)
Derivative of surge force with yaw rate and sway velocity

- \(X_{rr}\)
Derivative of surge force with squared sway velocity

- \(X_{w}\)
Wave force in surge

- \(X_{w}^{Dif}\)
Diffraction wave force in surge

- \(X_{w}^{FK}\)
Froude-Krylov wave force in surge

- \(Y_{r}\)
Derivative of sway force with yaw rate

- \(Y_{R}\)
Rudder force in sway

- \(Y_{v}\)
Derivative of sway force with sway velocity

- \(Y_{w}\)
Wave force in sway

- \(Y_{w}^{FK}\)
Froude-Krylov wave force in sway

- \(Y_{w}^{Dif}\)
Diffraction wave force in sway

- \(Y_{rrr}\)
Derivative of sway force with cubed yaw rate

- \(Y_{rrv}\)
Derivative of sway force with squared yaw rate and sway velocity

- \(Y_{rvv}\)
Derivative of sway force with squared sway velocity and yaw rate

- \(Y_{vvv}\)
Derivative of sway force with cubed sway velocity

- \(Y_{\phi }\)
Derivative of sway force with roll angle

- \(z_{H}\)
Height of hydrodynamic sway force application point from center of ship gravity

- \(z_{HR}\)
Height of rudder force application point in roll

- \(\alpha_{X}\)
Tuning parameter for drift wave force in surge

- \(\alpha_{Y}\)
Tuning parameter for drift wave force in sway

- \(\alpha_{N}\)
Tuning parameter for drift wave moment in yaw

- \(\beta_{1}\)
Tuning parameter for wave particle velocity in surge

- \(\beta_{2}\)
Tuning parameter for wave particle velocity in sway

- \(\gamma_{R}\)
Flow straitening coefficient

- \(\delta\)
Rudder angle

- \(\varepsilon\)
Rudder effectiveness coefficient

- \(\varepsilon_{a1,2,3,4}\)
Tuning parameter for wave forces phase lag in surge

- \(\varepsilon_{b1,2,3,4}\)
Tuning parameter for wave forces phase lag in sway

- \(\varepsilon_{c1,2,3,4}\)
Tuning parameter for wave forces phase lag in roll

- \(\varepsilon_{d1,2,3,4}\)
Tuning parameter for wave forces phase lag in yaw

- \(\eta_{WD}\)
Scale parameter of Weibull distribution for wave drift forces/moment

- \(\zeta_{w}\)
Wave amplitudes

- \(\lambda\)
Wave length

- \(\xi_{G}\)
Longitudinal position of center of ship gravity from a wave trough

- \(\rho\)
Water density

- \(\phi\)
Roll angle

- \(\psi\)
Yaw angle

- \(\psi_{0}\)
Tuning parameter for wave drift wave force phase lag in surge

- \(\omega\)
Wave frequency

## Notes

### Acknowledgements

This research was sponsored by the Office of Naval Research Grant N000141-21-05-6-8 and NICOP Grant N00014-09-1-1089 under the administration Dr. Patrick Purtell. The CFD simulations were conducted utilizing DoD HPC. The authors are grateful to Mr. K. Tanimoto and Ms. K. Takagi of Osaka University and Mr. A. Hanaoka of The University of Iowa, IIHR for assistance conducting the experiments.

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