Auto-oscillations of anchored vessels under the action of wind and current

\yg1 i7

ARCH1EF

7 e-'e

/7L

D.CUMENTATIE _I:

c'4..72/t9

DATUM: 9 OKf l93

Ondrafdjn

AUTO-OSCILIMtTIONS OP ANCHORED VESSELS UNDER THE ACTION OP WIND AND CURRENT

by

A.Y.Gerassimov, RY.Pershitz, N.N.Rakhmniii

Introduction

It is known from practice that in the presence of wind an

anchored vessel swings from side to side with respect to wind

direction line. It performs angular

(yaw)

and translational (drift) oscillations in the horizontal plane. As is

shown by

full-scale observations, the intensity of oscillations depends

on the wind force, and their amplitude values may reach 900 to

1000 for the law while for the drift they may be equal to the depth at anchorage or even more than that. The depex3dence of

the oscillatory period on the wind force is weak. When

lying

at anchor is a working condition for a vessel, such oscillations

may prove to be extremely undesirable.

Yawing and drifting of anchored vessels are

auto-oscillat-ory in nature as they may be caused even by the wind of constant

direction and force. Such character of motion is due to

non-linear relationships inherent in an oscillatory system formed

by the anchored vessel. The most

significant

of these n1fest themselves in the nonlinear relationship between the horizontal

component of the anchor chain tension and the shifting of the

hawse-hole with respect to the sea bed, as well as in the

non-linear relationships between the aerodynamic and hydrodynamic

forcea

and, the

kinematics of

the ship's motion.

lab. v.

Scheepsbouwkunde

Technische Hogeschool

Hydrodynamics

Deift

E1

__.-For the purpose of making a detailed analysis of yawing

and drifting of anchored vessels this paper deals with the

discussion of forces acting on the vessel in the oircumstanca8,

and the derivation of relevant differential equations of motion0

In the derivation of thee equations 'eat attention was given

to determining the tension of the anchor chain as dependent on

the shifting of hawse-hole.

The aerodynamic and hydrodynamic forces are defined in

accordance with the Iown results [i], [2], [3j. The general

equatiOfl8 of motion Obtained for an anchored. ship are used for

finding her equilibrium positions and analyzing stability of

the same. It is shown that the main reason inducing the ship

to yaw is instability of her equilibrium position due to wind.

Consideration is given to conditions in which stability of

equilibrium is ensured for anchored veesela while periodic

yawing and drifting is ruled out.

1. Coordinate systems and Nomenclature

To solve the problem Under review, four coordinate systems

are used. Two of them are applied for the description of ship's

motion in the horizontal plane, viz., the fixed coordinate

system XOY with OX-axis directed oppositely to the wind and the

origin 0 which coincides with the center of gravity (CG) of a

non-diverted vessel., andthe body axis system with

axis directed forward and the origin in CG. The -axis is

directed to port side.

Fig.l shows the directions of coordinate axes and positive

denotes a point of the anchor chain breaking away from the

ground, = initial position

of

the hawse-hole, current position of the sane.

Two more coordinate systems (Pig.2) are required for the

description of the anchor chain positioning in space. One of

these, the Z'

fix

system is situated in the plane of the

anchor chain sagging. The origin A is made coincident with the

anchor lying on the ground. The -axis is directed

vertical-ly, while the horizontal axis is coincident with the ground

plane and directed to the hawse-hole 1,o.. The other system of

coordinates is characterized by the fact that the vertical

axis OIL always passes through the

point

B Where the anchor

chain breaks away from the

ground,

and that the origin 0 is at

a distance of

_T

(1)

below the ground level. Here

I

denotes the horizontal

component of the anchor chain tension and Z/ is the weight per

wait length of the chain submerged in water.

Besides, the following designations are also used in this

paper:

and

p

= mass density of water and

air,

= acceleration due to gravity,

L

= length between perpendiculars,

P.

= lateral area of the underwater body,

Q = sail area,

177 = own mass of ship,

. siiip's mass moment of inertia for central

vertical axis,

added mass mont of inertia for the same axis,

= wind velocity,

= flow velocity,

= angle btween wind and flow directions,

47 = center of sail area,

= abscissa of

Cñ,

Thba

hawee-hole coordinates in the body

coordinate sy8tem

0, 2

i aerodynamic force and its projections

on the body axis,

P4.

,P

P

= hydrodynamic force and its projections

'ST'

'2 on the body axis,

projections of horizontal component of the

-

anchor chain tension on the body axis,

Ma

= aerodynamic moment about the central vertical axis, = moment of resistance to ship'

8

rotating about the central vertical axis,

= moment of anchor chain ten8ion about the central vertical axis,

fiangle

between wind direction and ship's longitudinal

axis; the amplitude value of the same angle,

fi°

same angle at static equilibrium,

and = projections of CG velocity on the body axis,

= lateral (normal to the wind) displacement of CG,

= displacement of CG towardi the wind,

= displacement of hawse-hole towards the wind,

= lateral displacement of hawse-hole,

projection of absolute displacement of hawse-hole onto the anchor chain 8agging plane,

- angle between wind direction and the anchor chain sagging plane,

depth of sea at anchorage,

1/

hawse-hole elevation over the sea bed.

2. Basic assumptions

The discussion of yaw and drift problem for the.anchored

ships is based on the fol2owing assumptions:

It is assumed that the coupled pitching and heaving

motions do not affect the ship' s movement in the horizontal

plane.

The magnitude of hydrodynainic forces is taken as

in-dependent of athwartship inclinations.

In the estimation of inertial fcu'cea the vessel is

considered to be symmetric not only about the centerplane but

also about the athwartship plane, and the center of gravity

to be located in the athwartship plane.

In predicting the noninertial forces and moments

acting on the vessel use is made of steadiness hypothesis.

It is also assumed that the ship's movement is so slow that

the anchor chain inertia forces can be neglected when deter.

mining the tension of the chain.

Differential equations of motion

According to

Li]

the differential equations of the ship' s

horizontal motion in the body coordinate system can

(m

+ 2ff)

(/77

* 4)

₌

(m +,,)j.

(i7 +i,,) ;;

(y

-

2,,))

-

(2,i-1z,,) Lj.

_if

(2)

The right-hand

aide

of equations (2) could moat

convenient-lybewrittenastheaUma_

/r

4r&r

4.'°

Fe

%er'?'

(3)

;=

The terms inluded in the expressions

(3)

are detex"xnined by

the

aerodynamic

forces acting upon

the

above-water body in the

presence of wind, the

anchor

chain tension,

and

the noninertial

hydrodynTTrtc forces generated on the underwater body

during

its

motion.

The inertial forces considered in

this

problem are

taken into account in the left-hand side of

equations (2).

When

defining the

signs

of

formulae (3)

it was thought that the

forces

and

moments were calculated

for the positive shifts.

In equations (2) provision is made for taking account of

the constant current in

the vicinity

of anchorage.

For

this

purpose you need only to represent th CG velocity projections

with respect to Water in the form of the following obvious

expressions (Pig.l):

=±ZCoi(c6+J),

=

-

zr

5/?

(o t1).

(4)

In the absence of current t and are equal to zero. Thus three unknown values can be derived directly from

equations (2): yaw

angle ) and

projections '

and

of the CG velocity. In the fixed coordinate

systems

these projections

will have, the form

r Co,,s

i$a),

c==

3'z+ / 'os'J

(5)

By integrating expressions

(5)

time functions .

( _),an

J(

)can

be

found

which

determine

the position of CG in space. The position of the hawse-hole

can

be

found

from the

following Obvious relationships:

44 (

/

(6)

Along with the relationship for the functions of

full idea of

the

yawing and drifting of an anchored vessel under the action of wind

and

current.

4. Estimation of aerodynamic forces

Projections of aerodynamic forces on the

axis

of the

body

system of coordinates

are

defined by expressions

ar

0a°r

=

CQ°e

and

the moment about the central vertical axis by

/,

-''

'7Z

Q

ma

2

In the latter expression

and

give rather a

(7)

i)7J

6

are dependent

(10) (9)

where non-dimensional parameters

on the relative position of CA

6=

L

and defined by the generalized curve (Pig.3) plotted against

the data obtained from [2] and [3]. Irrespective of the CA

position coefficients

c4

an. may be considered as constant, viz., = 0.14 and

=0.954.05.

5. Estimation of hydrodynamic forces

The hydrodynamio fOrce components, the longitudinal one

and the normal , as well as the moment

/7

originating during rotation of the vessel may be estimated

approximately from the for=lae

i

_{± c}

_i)

,

=c(jI+?1) :,

(11

lv1

=

c1

j4i

(I

re coefficients O, and are chosen in c fc.i.

with

recorr.mendationa of Ref.[2], and deter:::.. y

ezpresion

pJL

established under the assumption that the centre of ship's

rotation in the horizontal plane coincides with the athwartshi.p

plane. The second terms of formulaö (11) allow for the presence

of current

(13)

6. Estimation of tension o the anchor chain

At an arbitrary moment of time the longitudinal axis

0,

_T

eorms an angle (fi with the anchor chain sagging plane

(Fig.1). Projections of tension

T

for the latter on the body axis will be expressed by the relationships

The moment of force

T

about the central vertical axis appears to be equal to

=.T..

15

The horizontal component of the anchor chain tension is

re-presented by the sum

T=7+AT

(1G)

Here

7 is taken as

=

_{TCos(+ 9),}

which corresponds to the longitudinal component ''a1 of

the

aerodynamic force for

}3

?= C..

The tension increment â

T

is estimated by the

hawse-hole

shifts

AZ'

in the course of drifting or

yawing

of the vessel The curve of

T

against A

plotted With allowance for the chain line characteristics is

presented in the dimensionless form in Pig.4 for the case when

0

=a

In all other cases the relationship of

0 _ZllW,

is easily determined, using the same figure,

by shifting the origin along the curve to

the

point where the latter is intersected by the straight line

X= X,

7. The final form of differential equations of motion

Taking into account the results given above and converting

the equations (2) to the form where the coefficients for the

second derivatives of Yariables ' , , and. are equal to

unit, the set of differential equations of motion for an ar.i.chored

vessel in the presence of wind and current can be presented in

the following final form which will be convenient for further

analysis:

-10-I

+

I

d0+j)

L'o;(d±j)

-e Co (' +

-

P

T? (B±

)

7"8

1±

±

a

I

-P3

(A)

&(d,±

7'3//7

4;)==_

45T7

+

o)=

-5'A

-t±&,'

+/ftA

A

+

T

(±)±

I

(1.3)

In equations (18) the

values of CG

velocity projections

and

are

determined from

formulae (4) and the following

designations are used:

/77 -f 222

'p77

,0

zr*'1t

I)

/77-t-2,

in

I4

AeJ

/85

2(Y-t255)

2(in-f322) -,

6

2(n+2)

,O

,f.

'J2 2(7-3-266)

10

c0 (4+ i)A Oz

7 + 2'

J4

2 (7+266) .'

J'ooL.. ,0

=

_{'2 (Y+i?g)'}

-

_{.2 (o286)}

hrHn P.4/i

dy

+j66

(19) (20) (21) 8.. Equations of equilibrium

Equations of

equilibrium for anchored vessels subjected. to

wind and current

action can

be derived from differential

equa-tions(].8) providing

=

C;

°='=

0;

0

and

_J

0.

The set of equations thus

obtained

makes it pcib1e

not only to define the

equilibrium position of an

anchored chip,

12

-with the wind and current prescribed, but also to follow the

dependence of this position on the ship's particulars and the

coordinates of the hawee-hole.

In the absence of current

(zç=o) this set of equations

is reduced to a

single

equation which determines the

angle

of equilibrium

A

E

(86

-

4)

-

$'rn/

+

(22)

.CoA=O.

It follows from equation (22) that the angle

ff.

is dependew on the coordinates of the hawse-hole and and the

lengthwise position of the centre of sail area Bg. The

ordinate has no appreciable effect on the equilibrium

position.of the vessel. Setting

h=0 we shall

find that angle

,4

is equal to zero if the hawse-hole. abscissa

satisfies the condition

Otherwise angle R is defined from the formula (Pa

fi=2i (6+-4

c_i

C

___

(24)

Co

Z

9. Stability of equilibrium positions

A vessel nay stay in the positions of equilibrium as

dined above only on condition that these positions *re

13

-Considering the stability of the vessel with respect to yawing

in conformity with A.M. Liapunov's general theory [4], the

following criterion of stability can be obtained:

a°r

Cc4± Ce

Z

? Jo

i$A.

-1-+

/4',

h

₍₂₅₎

where

i2,

is the non-dimensional shifting of the hawse-ho].e in relation to the anchorage depth

,4

as the ship passes from the state of rest in the absence of wind to an equilibrium

position with the wind having the velocity of ZC

It can practically be assumed that = 0.5. In this

case angle

_)%

is equal to zero, which can easily be verified by using formula (23), and the criterion (25) is simplified

taking the form

hi

1Ae\

Co

/

>

o1".S')

'3

_Caç

Taking into account the curves of Pig.3 it can easily be aci

that for the conventional arrangement of the forward hawce-1e

condition (26) is not met, i.e. in the absence of current the

anchored vessel subjected to wind, will not be stable to anxlar

deflections from the course.

Instability of equilibrium of a vessel held in place by

anchor is the iaIn cause of drifting and yawing,which In th::

absence of current and with'constant Wind have the ntur2

14

-auto-oscillations which are syiwnetrical With respect to the

wind, directions.

Pig.5 shows the curves obtained by computer

simulation of the set of equations (18), which characterize the

auto-oscillations of the anchored vessel (

-i--

=

5;

= 0.5;

= 0.07) subjected to constant wind.

(.T

j2 rn/eec) in the

absence of current (

Z

-

0).

Under the simultaneous action of the wind and current the

yawing becomes arizuuetric with respect to the wind provided that

the direction of the wind differs from that of the current.

The

average angle

and average shifting of the háwse-holO

increase with the increase in the flow velocity

Z/'

and angle

The amplitude of steady cyclic yaw is but slightly

dependent on the flow parameters.

On the contrary,the amplitude

of lateral displacement of the hawse-hole is substantially

decreased with the increase of the flow velocity.

The increase in the flow velocity leads, all other things

being equal, to increasing angle

4.

in consequence, as is

seen from expression (25), the position of the vessel's

equilibrium may change from being instable in respect of yawing

to a stable one, which will involve complete ceasing of its

oscillations due to yawing and drifting.

In the example giver

the oscillations of the electronic model of an anchored VC1

ceased at a flow velocity exceeding 0.8 nVsec.

It. is obvious from equations (18) that period

L

of th

oscillations under consideration is mainly dependent on th.

depth

H

at nchore (Pig.6).

At the same tifle there i

a

clearly defined dependence of this period on wind veici-ty.

In consequence of the ship's motions and wave action

the resistance to drift and yaw must increase

much like the resistance of a ship moving in a seaway, which

is not taken into account by the set of equations (18).

Addi-tional resistance to drift and yaw in a seaway brings about an

appreciable reduction in drifting velocity and, consequently,

an increase in the period of auto-oscillations of an anchored

vessel, all other things being equal. Hence, seaways may be

considered as the cause of significant weakening of the

relationship between the period of yawing oscillations and the

velocity of wind. According to full-scale data, the period of

oscillations due to, strong wind slightly differs from that

when the wind force is 3-4 (on Beaufort scale).

10. Ways to eliminate the auto-oscillations of anchored ve.ssei

Solution of equations (18) indicates that the intensity of

auto-oscillations for the given depth at anchorage and wind

velocity may be in direct relation to the extent of instability

of the ship's equilibrium position. This latter is defined by

the difference. between the right-hand and left-hand sides of

inequalities (25) and (26). In similar anchorage conditions

the left-hand side of these inequa].itiesis substantially

dependent upon the position of the hawse-hole along the ship's

length. The right-hand side of the inequalities is eventually

characterized by the initial (for = 0) value of the

positional aerodynamic derivative coefficient (9): 15

16

-i.e. by the lengthwise position of the centre of sail area.

Pig.7 gives an indication of the rel8tionahip between the

intensity of yawing and the extent of instability of the ship's

equilibrium position. The intensity of yawing is characterized

n this figure by the relative amplitude

-

versus the

1,778

derivative L,

Here Am = dimensional amplitude of yaw,

= dimensional amplitude of yaw for

the vessel with

=

5.0,

₄

0.068.

The curve of

_fi,

against the anchorage

depth is pre8ented in Pig.8.

Thus the elimination of the wind-induced auto-oscillations

of an anchored vessel may be brought about if stability of Its

equilibrium position is ensured.

This latter can be ensured,

as evidenced by the analysis of condition (25), by shifting aft

both the centre of sail area and .the hawse-hole.

This 8ame

condition, along with (26), gives the quantitative value of the

required shifting of the above points.

When the hawse-hole is located near the forward

perpendi-cular, the aulo-oscillations of the anchored vessel subjected

to wind may be eliminated at the cost of shifting the centre of

sail area well aft.

As angle

₄₌

0 correspons -in this case

to the ship's equilibrium position, and the.

,ro

ratio is

rather large, so the stability of equilibrium position, as

follows from inequality (26), can practically be ensured if the

right-hand side of this inequality is close to zero or negative

This will be the case if

-I

&25.

17

-So considerable a shifting of the centre of sail area, however,

adversely affects the controllability of the vessel in wind.

The shifting of the hawse-.hole aft of the forward

perpendi-cular must be greater than that where the ship's equilibrium is

possible with the value, of different from zero. As the

angle )

increases, the instability of equilibrium position

decreases, and at a certain value of the position

becomes stable, viz, inequality (25) is satisfied. Thus, with

(T/th:\

L

hJcr

the auto-oscillations of the anchored vessel, are eliminated.

Even so, this conclusion based on the analysis of small

perturba-tion stability quite 8atisfactorily characterizes moperturba-tion in

general.

The test results shown in Pig.9(a) and (b) for an electronic

model of an anchored vessel ( = 0.068,

4-

-

5.0) give an idea of the effect the longitudinal arrangement of the hawse-hole hcs

on the intensity of yaw and drift. The dashed lines in the

region of unstable equilibrium represent the curves of yaw

amplitudes against abscissa During the tests no

au-to-oscillations were observed at L'va].ues beyond the left en

of these curves. The solid lines indicate the aid

parameters for the ship's equilibrium position. These

relation-ships were calculated from static equilibrium equations (see

section 8); thern results obtained, from equations (18) are

illustrated by point3 cn the solid lines.

It is obvious that the results presented are in good

agreement with the boundary of auto-oscillations region

deter-mined by calculation from formula (29). Setting O for

the critical abscissa of the hawee-hole the following formula

can be obtaifled:

In contrast, for eliminating the auto-oscillations of the

anchored vessel by shifting the hawse-holeaft it is desirable

that the centre of aerodynamic pressure (CP) should be shifted

forward.

Really, in the position of ship's equilibrium the line of

aerodynamic action coincides with the anchor chain horizontal

projection and passes through the hawse-hole. Hence, no

auto-oscillations are present if the following inequalities are met

aimultaneou8ly:

where = abscissa of CP, But

Crr7a

Co a'

=

,

LC3/A.

from which we obtain

z

Zjcr

-18--%

(4±4)

CoIJBo

21

It is evident frci relationships (24), (30) aiid (32) t.at

if CP is shifted forward, given the position of the

'.e-ci

(30)

(31)

19

-along the ship's length, this will involve an increase of

A

and a reduction of which will allow satisfaction of the

second inequality (31). So, the farther forward CP is

dis-placed, the less is the necessity of shifting the hawse-hole

aft of the stem so as to eliminate the wind-induCed

auto-oscillations of the anchored vessel.

The shifting of the anchor hole aft of the stem is

equi-valent to springing the vessel as is accepted in. maritime

practice.

Let us consider the scheme (Fig.1O) showing the springing

technique The lengths of the forward

'4

4'

and after

,4, h

portions of the spring mut be chosen so that in the ship's

equilibrium position they will be tensioned. As long as the

spring remains.tensione4 during oscillations, its presence

will be equivalent to the hawse-hole shifting to point *'g, ,

and the tension line of the anchor chain will intersect the

centre line plane at point.k which is Coincident with CP.

It is evident that. the position of equilibrium will not be

disturbed if a single anchor rope is secured to the vessel at

the point

As far as research and. fishing vessels are concerned for

which lying at anchor at various places of the water area is

the basic condition of operation, it may prove to be convenient

that a special anchor gear be de8igned so that the point where

the anchor chain is secured to the vessel is shifted aft of the

stem when at station. This point must satisfy the ccndition3

(31). In the case of a fishing vessel it was ford that

(Fig.1O) covering the centre of sail area in order to eliminate

yawing of the anchored ship. This region is likely to be equal

for ships which do not differ much in respect of the deck-house

architecture. It is expected that such an anchor gear, if

properly designed, will create favourable conditions for the

operation of the above-mentioned ships.

References:

Bassln, AL:

"Khodkost' i upravliayemost' sudov"

(Performance and controllability of ships),

Izd. Transport, Leningrad, 1968 (in

Russian).

Voitkunsky, LI.,; "Spravochnik pa teorii korablia"

Pershitz, R.Y.; (Reference book on ship theory),

Sudprom-Titov, l.A. giz, Leningrad, 1960 (in Russian).

Gofman, A.D.; Zaikov, V.1.; Semionova-Tian-Shanskaya, A.V.

Liapunov, A.M.:

Savelov, A.A.:

-20-.

"K rasohetu upravliayemosti sudna pri

vetre" (Analysis of controllability of

veseels subjected to wind), Trudy LIIVT,

vyp.8l, Réchizdat, Leningrad, 1965 (in

Russian).

"Obshchaya zadacha ob ustoichivosti

dvizheniya" (Stability of ship's motion:

General problem), ONTI, Moskva, 1933 (in

Russian).

"Ploskiye iviye" (Plane curves), GIPML,

/

/

/

/

/

/

I.

_/

/

Pig.1

Syztci of coordinates ±cr th

dczcript..i

a)

7

1

/

/

//

44

C

xi-

_i1,

N

"01

Pig.2.

Sytter of coordiriatea for the doncription of the anchor chain

poaitioning in space.

4'25

2O

-/0

-5

₀

/0

IJi

0/

1?L,f0

3.

Coofficionts

C0

_and

veriiw the 1enthviice poition

A:.

40

40

fO

-24-a -

I

AZ

/

Pj.4. Diniofl1S3 tension o

the

ii cin

i:i.

Ad8,

4/77.

/00... 50

o0-'0

80

J8

20lb/C

-20..-

-"7--

-no--so-

--

50

I

'of

0

/0

20

- 30

10

25

-Pig.50

Development of auto-oscillations of the

anchored vessel (sea depth

H

= 100 m,

wind velocity

12 rn/eec).

4

Fig.6.

Soa depth cffccton the poriccI. of otci11tion8

(iind ve1city f

7

/uec).

to

47/0

=0

:.-I/f

?20

430

Fig.7.

The intermity of yaving vercus the derivative

of aercyrt1ic iaoinnt.

/00

L0

/

/

/

/

IL' 'Sek.

-I

/0

18

a')

4

/08

0

5)

.9:44/77

50

25

29

-I

f,n

ifek

-Pig. 9.

The effect of 1onitidina1 shifting

of the havrse-ho].e on th

intensity

of yaw (a) and drift (b).

-3O