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Ship Motion Mathematical Model

а) waterline plane

b) frame plane (bow view)

с) central plane (starboard view)

Fig. 4. Coordinates Systems

Chapter 1. General.

19

Ship Motion Mathematical Model

The fixed axes origin lies in fixed point Og. Og Xg – axis and OgYg – axis lie in the plane parallel to the calm free water surface. Og Zg – axis is perpendicular to the plane. The axes directions are as follows: Og Xg – axis directs polewards, OgYg axis directs eastwards, Og Zg axis directs downwards.

The body axes origin is in the ship center of gravity. OX – axis and OY – axis are parallel to the base plane. OZ – axis is perpendicular to it. The axes directions are as follows: OX – axis-forward , OY axis-starboard, OZ – downwards.

The ship inclination angles are positive when ship turns clockwise when watching from the axis end.

The equations describing the ship motion are as follows:

(m + λ11)	dVx						+ (m + λ22 )Vyω z + (m + λ33 )Vzω y = ∑Fx + ∑Fx(M ),
					dt
(m + λ22 )			dVy					+ (m + λ11)Vzω x − (m + λ33 )Vyω x = ∑Fy + ∑Fy(M ),
					dt
(m + λ33 )			dVz					− (m + λ11)Vzω y − (m + λ22 )VyVz = ∑Fz + ∑Fz(M ),
					dt
(Jx + λ44 )					dω x					+ [(Jz + λ66 ) − (Jy + λ55 )]ω yω z + (λ33 − λ22 )VyVz = ∑Mx + ∑Mx(M ),
					dt
(Jy + λ55 )					dω y					+ [(Jx + λ44 ) − (Jz + λ66 )]ω xω z + (λ11 − λ33 )VxVz = ∑My + ∑My(M ),
					dt					+ [(Jy + λ55 ) − (Jx + λ55 )]ω zω y + (λ22 − λ11)VyVz = ∑Mz + ∑Mz(M ),
(Jz + λ66 )				dω x
					dt

x&g = Vx cosϕ cosψ + Vy (sinθ cosϕ sinψ − cosθ sinϕ ) + ω (cosθ cosϕ sinψ + sinθ sinϕ ), y&g = Vx sinϕ cosψ + Vy (sinθ sinϕ sinψ + cosθ cosϕ ) + ω (cosθ sinϕ sinψ − sinθ cosϕ ), z&g = −Vx sinψ + Vy sinθ cosψ + ω cosθ cosψ ,

ωx = θ& − ϕ& sinϕ ,

ωy = ψ& cosθ + ϕ&cosψ sinθ ,

ωz = ϕ&cosψ cosθ − ψ& sinθ ,

where m is the ship mass ( m = ρ LBT CB , ρ – water density; L, B, T – length, breadth and draught at midship, CB – block coefficient);

Jx ,Jy , Jz – moments of ship inertia in body axis;

λ11,λ22 K λ66 – added masses;

Vx ,Vy , Vz – ship velocity components in body axis;

ω x ,ωy , ωz – ship angular velocity components in body axis;

∑Fx , ∑Fy , ∑Fz , ∑Mx, ∑My , ∑Mz – total force components and total moment components due to water and wind influence;

20 NAVI-TRAINER 4000. Mathematical Models. Technical Description.

∑Fx(M ), ∑Fy(M ), ∑Fz(M ) , ∑Mx(M ), ∑My(M

force and total mechanical moment components.

Ship Motion Mathematical Model

), ∑Mz(M ) – total mechanical

The total forces and moments components on the ship are defined by the following equations:

Total force components and total moment components due to water and wind influence longitudinal component of total force

∑Fx = (FxBH + ∑FxR + Fx P ) + FxA + FxC + FxW + Fx bank + F x bot + Fx ship ,

lateral force component

∑Fy = (FyBH + ∑FyR + Fy P + FyTHR ) + FyA + FyC + FyW + Fy bank + Fy ship ,

vertical force component

∑Fz = (FzBH + Fz ST + Fzg + Fzz ) + FzW + Fz bank + F z bot ,

roll moment

∑Mx = (MxBH + ∑MxR + MxTHR + Mx ST + Mxx ) + MxA + MxW + Mx bank

trim moment

∑My = (MyBH + My ST + Myy ) + MyW + Mx bank + M x bot + Mx ship ,

yaw moment

∑Mz = (MzBH + ∑MzR + MzP + MzTHR ) + MzC + MzA + MzW + Mz bank +

The nomenclature is as following:

– forces and moments, on bare hull

+ Mx ship

M z bot + Mzship ,

FxBH ,FyBH , FzBH ,

MxBH ,MyBH , MzBH

Fz ST , Fzz , M x ST , M y ST , M xx , M yy

Hydrodynamic force and hydrodynamic

moment components at bare hull;

Buoyancy force and restoring and damping moments for roll and trim;

– forces and moments out of steering device influence

∑FxR , ∑FyR , ∑MxR , ∑MzR

Fx P ,Fy P ,Fz P ,Mx P ,My P , Mz P

FyTHR , MxTHR , MzTHR

– external forces and moments

force and moment on the rudder (including the interaction forces);

thrust, lateral and vertical force on propellers (total value) and their moments;

thruster axial force and its moments relative to OX-axis and OZ-axis;

FxA,FyA, MxA, MzA

aerodynamic force and aerodynamic moment;

Chapter 1. General.

21

Ship Motion Mathematical Model

FxC ,FyC , MxC , MzC

FxW ,FyW , FzW ,

MxW ,MyW , MzW

Fx bot ,Fy bot , Fz bot ,

Mx bot ,My bot , Mz bot

Fx bank ,Fy bank , Fz bank ,

Mx bank ,My bank , Mz bank

Fx ship ,Fy ship, Mz ship

additional force and moments due to current influence;

additional force and moments due to wave influence;

additional force and moments due to shallow waters influence;

additional force and moments due to channel geometry influence

additional force and moments due to another ship influence.

The Total Mechanical Forces and Moments Components

The total mechanical force and moment components values depend on the nature of the performed operation. It is possible to consider one, two, or several components listed below:

Mechanical force longitudinal component

∑Fx(M ) = F x (M )wall + F x (M )ship ;+ ∑F x (M )rope + ∑F x (M )anch ;

Mechanical force lateral component

∑Fy(M ) = Fy (M )wall + Fy (M )ship + ∑Fy (M )rope + ∑Fy (M )anch ;

Mechanical force vertical component

∑Fz(M ) = Fz (M )wall + Fz (M )ship + ∑Fz (M )rope + ∑Fz (M )anch ;

Mechanical force roll moment

∑Mx(M ) = M x (M )wall + M x (M )ship + ∑M x (M )rope + ∑M x (M )anch ;

Mechanical force trim moment

∑My(M ) = M y (M )wall + M y (M )ship + ∑M y (M )rope + ∑M y (M )anch ;

Mechanical force yaw moment

∑Mz(M ) = M z (M )wall + M z (M )ship

where

Fx (M )wall ,Fy (M )wall Fz(M )wall ,

Mx (M )wall ,My (M )wall , Mz(M )wall

Fx (M )ship ,Fy (M )ship , Fz (M )ship ,

Mx (M )ship ,My (M )ship , Mz (M )ship

+ ∑M z (M )rope + ∑M z (M )anch ;

Mechanical force and moment components at wall collision,

Mechanical force and moment components at ships collision,

22 NAVI-TRAINER 4000. Mathematical Models. Technical Description.

	Ship Motion Mathematical Model
Fx (M )rope,Fy (M )rope, Fz (M )rope,	Mechanical force and moment
Mx (M )rope,My (M )rope, Mz (M )rope	components due to rope tension,
Fx (M )anch,Fy (M )anch, Fz (M )anch,	Mechanical force and moment
Mx (M )anch,My (M )anch, Mz (M )anch	components on an anchored vessel.

Chapter 1. General.

23

CHAPTER 2

Ship Motion in Calm Deep

Water

Copyright Transas Marine Ltd. 2003

Ship Motion Mathematical Model

A ship moving in calm deep water is effected by hydrodynamic forces at bare hull, buoyancy forces, stability forces, inertia forces and forces on ship’s propellers and steering arrangement (see Fig. 5).

At the same time, forces on ship’s propellers and steering gears depend on the ship control system parameters. The ship mathematical model description for calm deep water is given below. Forces effecting the ship are shown on Fig. 5.

b) frame plane (bow view)

a) water plane (top view)

Fig. 5. Forces on Ship Hull. Calm Deep Water

Chapter 2. Ship Motion in Calm Deep Water.

27

Hydrodynamic Forces and Moments

HYDRODYNAMIC FORCES AND MOMENTS

Hydrodynamic forces and moments on the ship are usually defined as the result of ship model experiment. Measurements are usually preformed in the body axes at given values of kinematics parameters, such as drift angle β, rudder angle β and

path curvature ω .

Drift Angle β. Drift angle is the angle between model centreline and velocity vector at the gravity center. The relative water ship velocity is used to obtain the drift angle value. It is calculated using the formula β = arctg(−Vy lq Vx lq ) , where Vx lq and Vy lq

are the relative ship water velocities. Drift angle is positive, when the ship turns starboard.

Path Curvature ω . Path curvature relative value is defined as a value inverse to the path curvature radius ω = L / R (R – curvature radius). Path curvature depends on the angular velocity value as follows: ω = (L V ) ω z . Path curvature is positive, when the ship turns clockwise, top view.

The hydrodynamic ship characteristics obtained in the following ranges of kinematics parameters:

Drift angle	− 180o ≤ β ≤ 180o ,
Relative path curvature	−∞ ≤	ω	≤ ∞ ,
Roll angle	− 30o ≤ θ ≤ 30o

Different accuracy requires mathematical description of hydrodynamic characteristics in different ranges. The highest accuracy requires description of the ship with ordinary rudder system (rudder, nozzle, etc.) manoeuvring in calm deep water. These ranges of kinematic parameters values are as follows:

Drift angle	− 25o ≤ β ≤ 25o ,
Relative path curvature	−1.0 ≤	ω	≤ 1.0 ,
Roll angle	− 10o ≤ θ ≤ 10o

So, the hydrodynamic characteristics were obtained with high accuracy to calculate the ship manoeuvring in calm deep water. The values of hydrodynamic characteristics in the remaining domain are defined by extrapolating the initial values of hydrodynamic characteristics.

The equations for bare hull hydrodynamic characteristics and for steering gears hydrodynamic characteristics are the combined functions of kinematic parameters. They are described as follows:

longitudinal force

	F (β ,ω )			(ω	min	≤ ω ≤ ω	max	), (−25o ≤ β ≤ 25o )
	x
Fx =	~	(β , 1	ω )	(− ∞ ≤ ω < ωmin , ωmax < ω < ∞ ),
	Fx
						(− 180o ≤ β < −25o ,25o < β < 180o )

28 NAVI-TRAINER 4000. Mathematical Models. Technical Description.