# Prediction of ship steering capabilities with a fully nonlinear ship motion model. Part 1: Maneuvering in calm water

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## Abstract

This paper introduces a new method for the prediction of ship maneuvering capabilities. The new method is added to a nonlinear six-degrees-of-freedom ship motion model named the digital, self-consistent ship experimental laboratory (DiSSEL). Based on the first principles of physics, when the ship is steered, the additional surge and sway forces and the yaw moment from the deflected rudder are computed. The rudder forces and moments are computed using rudder parameters such as the rudder area and the local flow velocity at the rudder, which includes contributions from the ship velocity and the propeller slipstream. The rudder forces and moments are added to the forces and moments on the hull, which are used to predict the motion of the ship in DiSSEL. The resulting motions of the ship influence the inflow into the rudder and thereby influence the force and moment on the rudder at each time step. The roll moment and resulting heel angle on the ship as it maneuvers are also predicted. Calm water turning circle predictions are presented and correlated with model test data for NSWCCD model 5514, a pre-contract DDG-51 hull form. Good correlations are shown for both the turning circle track and the heel angle of the model during the turn. The prediction for a ship maneuvering in incident waves will be presented in Part 2. DiSSEL can be applied for any arbitrary hull geometry. No empirical parameterization is used, except for the influence of the propeller slipstream on the rudder, which is included using a flow acceleration factor.

### Keywords

Ship steering capabilities Maneuvering Seakeeping Rudder angle Rudder force Roll motion Comparison of the numerical solution and experimental data### List of symbols

*x*_{r,}*y*_{r,}*z*_{r}Location of the center of pressure of the rudder

**x**_{p}The location of a point, p, on the ship surface

**I**Moment vector of inertia about the ship

*I*_{(i,i)}Moments of inertia about the ship’s center of mass in the

*i*th direction**F**Force vector acting at the ship’s center of mass in the ship-fixed frame

**Γ**Moment vector acting at the ship’s center of mass in the ship-fixed reference frame

**F**^{p},**F**^{q}Exciting force and restoring force acting at the ship’s center of mass in the ship-fixed frame

*x*_{c},*y*_{c},*z*_{c}Center of mass in the ship-fixed coordinate system

**x**Unit vector in the ship-fixed coordinate system

**x**_{s},**y**_{s},**z**_{s}Three components of unit vectors in the ship-fixed coordinate

- \( \hat{x},\hat{y},\hat{z} \)
Unit vectors in the Earth-fixed coordinate system

**X**Translational motion of the ship

*X*_{i}Translational motion of the ship in the

*i*direction**u**_{s}Ship speed vector

**u**_{r}Effective inflow velocity into the rudder

**u**_{t}Total velocity at a point on the ship hull including all ship motions

**v**_{s}Translational velocity of the ship’s center of mass

**v**_{h}Horizontal component vector of the translational velocity

**n**Normal vector of the ship’s surface

*n*_{x},*n*_{y},*n*_{z}Three components of the normal vector of the ship’s surface

*β*Drift angle of the ship

*δ*Effective rudder deflection angle

*ρ*Density of the water in which the rudder is operating

*P*Pressure

**θ**Rotational angle of the ship

*θ*_{i}Rotational angle of the ship in the

*i*th direction**Ω**Angular velocity vector of the ship

**Ω**_{h}Horizontal angular vector (roll and pitch) of the ship

- Ω
_{i} Angular velocity of the ship in the

*i*direction*θ**Total heading angle, equal to

*θ*_{3}+*β***Ω***Total heading angular velocity, equal to \( \frac{{\text{d}}\theta^{*}}{{\text{d}}t} \)

*η*Free surface elevation

*η*_{e}Environmental free surface elevation

*η*_{s}Free surface deformation due to the ship’s motion

- \( \bar{\eta }_{\text{e}} \)
The over bar indicates the average (this is the average of the environmental free surface elevation)

*φ*Velocity potential

*φ*_{e}Velocity potential of the environment, including incident waves

*m*_{ship}Total ship mass

*v*Dissipation coefficient due wave breaking

*A*_{0}Surface area of the rudder planform

*R*_{3}Moment arm for the rudder force yaw moment

*R*_{1}Moment arm for the roll moment

**R**Ship maneuvering radius vector

*i*stepNumber of the time step

**F**_{ship}Force on the ship’s surface, except on the rudder

*F*_{l}Lift force in the

**z**_{s}coordinate**F**_{rudder}Force on the rudder

*F*_{x}Force on the rudder in the

**x**_{s}direction*F*_{y}Force on the rudder in the

**y**_{s}direction**Γ**_{ship}Moment on the ship, except on the rudder

**Γ**_{rudder}Moment on the rudder

**Γ**_{rudder(3)}Yaw moment on the rudder

- Γ
_{c(1)} Roll moment due to the direction of ship speed change

*D*_{trans}Dissipation of translational motion due to wave breaking [1]

*D*_{rotat}Dissipation of rotational motion due to wave breaking and bilge keels

*H*Water depth

*V*Volume

- d
*V* Element of the volume

**x**_{track}Track in Earth coordinates

*x*_{track}Track in the

*x*direction in Earth coordinates*y*_{track}Track in the

*y*direction in Earth coordinates

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