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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

MATHEMATICAL DESCRIPTION OF FORCES DUE

TO ANCHOR CHAINS, MOORS AND TOW LINES

The mathematical model used in the simulator allows to calculate the ship low speed or mechanically with locking engines when berthing, mooring or towing. The model brief description would be done below.

Anchor Gear Model

The force on ship hull due to the anchor gear is defined by the anchor winch, anchor chain and anchor influence. In ship mathematical model the description of one or two anchor gears can be done.

The following anchor gears parameters are used:

•chain gauge;

•number of shackles;

•anchor weight and type;

•anchor winch type and maximal chain haul speed;

•hawse position.

The forces on the ship hull due to anchor gear influence are shown on Fig. 1.

Mathematical model of each anchor gear includes:

•anchor mathematical model that can move on the bottom until it catches on;

•anchor chain mathematical model that transmits the holding force from anchor onto the hull;

•anchor winch mathematical model that allows to set the speed and length of the chain running out.

Anchor mathematical model is described by equations:

(m

ANCH

+ m

(l

1

))V&

= F

x ANCH

+ F

+ F

(V

x ANCH

) ,

CHN

x ANCH

x HOLD

x HD

(m

ANCH

+ m

(l

1

))V&

= F

y ANCH

+ F

+ F

(V

x ANCH

) ,

CHN

y ANCH

y HOLD

y HD

where m – the anchor mass, mchn (l1) – the mass of a chain segment of length l1 that is lying on the bottom, Vx anch – the anchor traverse speed, F x ANCH ,F y ANCH –

the chain tension at the chain attachment point to the anchor in fixed axis;

Fx HOLD,Fy HOLD – the holding anchor force including the holding force of the chain segment lying on the bottom, Fx HD,Fy HD – the complete force of the anchor and anchor chain hydrodynamic resistance.

Model described two variants of anchor gear influence: the influence of movable anchor and the influence of immovable anchor. Conditions defining the anchor movement are:

immovable anchor

(F ANCH + FHOLD ) < 0 ,

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

anchor is pulled along the bottom

(F ANCH + FHOLD ) ≥ 0 ,

a) when anchor catches on the bottom

b) when anchor is hauled

Fig. 22. Forces Due To Anchor-Handling Gear Influence

Anchor holding force is calculated as follows:

when anchor catches the soil (immovable anchor)

FHOLD = kHOLD mANCH g,

when anchor does not catch the soil (movable anchor)

FHOLD = μ HOLD mANCH g ,

where kHOLD – the anchor catch coefficient that depends on the soil type and anchor type, μHOLD – the dry friction coefficient (the pair anchor and soil).

The holding force of the anchor chain is defined by the equation:

FHOLDCHN = kHOLDCHN mCHN (l1)g

where kHOLDCHN – the holding force coefficient for the chain segment lying on the bottom.

Anchor is considered not to be catching the soil, if it is being dragged in horizontal plane in one direction over a distance exceeding the limiting length l , or if chain pluck angle exceeds the limiting value α .

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

Limiting length value and limiting angle value depend on the anchor type and sizes as well as the soil type.

FCHN (s) is the tension distribution in the chain, where s is the coordinate of point on

the chain line. It is defined using the catenary line equations defined by the equations of flexible heavy thread. The boundary conditions are the anchor coordinates and hawse coordinates.

The chain projection onto the horizontal plane supposed to be a straight line. When the chain overlaps the hull, changes shape of the chain line and of the force direction are taken into account.

The length of the pay away anchor chain is defined by the winch operation (capstan and windlass).

The winch is automatically controlled by commands “Drop”, “Stop”, “Up”. At “Drop” command, brakes are released, and moor or anchor chain are released at set length. At “Stop” command winch brakes are engaged. At “Up” command the chain is hauled with the set speed.

Winch operation is described by equation

mCHN (LCHN )V&CHN = FWINCH (LCHN ,LCHN ) + FF + FBRAKE + FCHN HAWSE ,

where mCHN – the pay away chain mass, FWINCH (LCHN ,LCHN ) – the effort from the winch, FF – the chain dry friction force in hawse, FBRAKE – the breaks effort,

FCHN HAWSE – the effort in hawse, VCHN is the chain pay away speed.

Mooring Gear Model

The mooring gear consists of the mooring lines and docking winches. The following parameters of the mooring gear are taking into account:

•mooring line diameter and type;

•winch type;

•knights positions.

A mooring line is modeled as a weightless stretchable thread without considering its special configuration.

The model describes the influence of a mooring lines from the winch to the another ship’s knight, bollard or mooring ring (see Fig. 23).

Two winch operation conditions are examined: constant length conditions (“Stop” conditions) and constant tension condition. The last condition suppose to have constant value of the force at ship’s end of tow-line. The force is considered to be directed along the tow-line.

1.Constant Length condition The force calculation formula is

FROPE = kROPE ε ,

where kROPE – the coefficient that depends on the tow-line material and type; ε

– the unit tensile elongation calculated by formula	ε = d − l
	l

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

(d – the distance between the tow-line fixing points, l – the length of the pay away line length).

Fig. 23. Mooring Forces.

The line is not stretched, if ε ≤ 0 . When ε > 0 , the line is stretched. In the last case, effort in the line is calculated by the formula given above.

A possibility of rope break is considered as the condition:

F > Fmax , where Fmax is the breaking force.

2.Constant Tension Mode

A possibility for the winch to support automatically constant effort F paying away or veer in the anchor chain is modeled. The line length is defined by the equation:

l =	d	, where ε =	F	.
	ε − l		kROPE

The line forces applied to the ship ( Fx(M )ROPE , Fy(M )ROPE , Fz(M )ROPE ) and their moments ( Mx(M )ROPE , My(M )ROPE , Mz(M )ROPE ) are obtained in body axis due to the forces values and directions and the mooring line attachment point position.

Mooring Buoy Model

The mathematical model of a ship mooring to a buoy considers the tension forces in the mooring line between the ship and buoy and the tension forces in the anchor chain between the buoy and anchor.

When calculating the efforts, the mooring line and anchor chain are considered to be flexible heavy long threads, and anchor – to be stationary fixed on the bottom.

Two types of mooring buoys are modelled: single-anchor buoy ( (SALM Single Anchor Leg Mooring) and multi-anchored buoy (CALM Catenary Anchor Leg Mooring). The equations of the buoy movement when effected by the external forces are as follows:

mMBV&y MB = F y WAVE MB+ F y C MB+ F y ROPE MB+ F y ANCH MB ,

mMBV&z MB = F zWAVE MB+ F z ROPE MB+ F z ANCH MB ,

where mMB – the buoy mass, Vx MB, Vy MB, Vz MB – the buoy speed components,

F xWAVE MB, F y WAVE MB, F zWAVE MB – the components of the wave force effecting the buoy, F x C MB, F y C MB – the components of the current force effecting the buoy,

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

F x ROPE MB, F y ROPE MB, F z ROPE MB – the components of the tow-line force effecting

the buoy, F	, F	, F	z ANCH MB	– the components of the anchor force
	x ANCH MB	y ANCH MB

effecting the buoy.

Fig. 24. Moor Forces Due To Mooring Buoy Influence

The forces due to the waves and current are calculated in the same way as for the forces on the ship hull taking into account the object shape. Non-dimensional characteristics necessary for the calculations are stored in the database. Anchor and mooring-line forces effecting the buoy are similar to the forces developed on the ordinary anchored systems.

Tow-Line Forces at Hull

Towing gear consists of towing lines, towing winch and hook.

The mathematical model considers

•Tow-line type and diameter;

•Towing winch type;

•Maximal allowed force at hook.

Forces effecting the ship at towing are illustrated on Fig. 25.

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

Fig. 25. Forces at Towing

Towing line equation is the same as in 0, page 71 and is calculated using formula

FROPE = kROPE ε ,

where kROPE – the coefficient depending on the towing line material and type, ε – the unit tensile elongation.

See section Ship Collision with Other Objects below for ship towing by pusher tugs.

Ship Collision with Other Objects

The mathematical model allows to consider the possibility of ship collision with the quay wall, bottom or other objects by examining the objects interaction at collision as partially elastic collision of two objects.

In case of collision of two ships or between ship and quay wall, fender mathematical model is considered.

Fenders operation is characterised by the force applied to the contact point

(see Fig. 26). This force depends on the fenders parameters and is proportional to the collision area as well as the ship velocity and displacement values and directions.

In case of two ships collision the force components are calculated according to formulas:

Force normal to the contact surface

Fn = kss1,2 + kVnVn1,2 ,

Force tangent to the contact surface

Fτ = kVτ Vτ 1,2 ,

where s1,2 is the contact surface area defined with the consideration of ship hull configuration shape at the contact area, Vn1,2 and Vτ 1,2 are normal and tangent

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Mathematical Description of Forces due to Anchor Chains, Moors and Tow Lines

components of the relative ship speed, fenders rigidity.

ks, kVn, kVτ

coefficients describing the

a) two ships collision	b) ship and quay wall collision
Fig. 26. Forces At Collision
Forces Fx(M )ship,Fy(M )ship, Fz(M )ship	are defined by recalculating forces Fn, Fτ in the

body axis. Moments Mx(M )ship,My(M )ship, Mz(M )ship are calculated according to body axis.

Forces and moments effecting the ship at collision with the quay wall are calculated in the same way.

When modeling the ship movement due to push-boat, tug-related forces and moments are calculated as in case of mechanical contact at two ships collision. In case of several tugs participating in towing, collision forces and moments are calculated with the consideration of ships masses and inertia moments.

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75

CHAPTER 5

Automatic Vessel

Movement Control Systems

Copyright Transas Marine Ltd. 2003