stability floating body

8/9/2019 Stability Floating Body

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5.

THE

STABILITY OF A FLOATING BODY

l roduct on

When designing a vessel such as a ship, which is to float on water,

it

is clearly

necessary to be able to establish beforehand that it will float upright in stable

equilibrium.

Fig 5 a shows such a floating body, which is in equilibrium under the action of

two equal and opposite forces, namely, its weight W acting vertically downwards

through its centre

of

gravity G, and the buoyancy force, of equal magnitude W, acting

vertically upwards at the centre

of

buoyancy

B

This centre

of

buoyancy is located at

the centre of gravity

of

the fluid displaced by the vessel. When

in

equilibrium, the

points

G and B lie in the same vertical line. At first sight, it may appear that the

condition for stable equilibrium would be that G should lie below However, this is

not so.

B

w

I

I

G

J

a

b

Stable

Fig 5 J Forces acting

on

jlo ting body

c Unslable

To establish the true condition for stability, consider a small angular displacement

from the equilibrium position, as shown in Figs 5 b and 5 c . As the vessel tilts, the

centre of buoyancy moves sideways, remaining always at the centre of gravity of the

displaced liquid. If,

as

shown on Fig 5 b , the weight

and

the buoyancy forces

together produce a couple which acts to restore the vessel to its init ial position,

the

equilibrium is

stable.

however, the

couple acts

to move the vessel even

further from its initial position, as in Fig S c , then the equilibrium is unstable.

The special case when the resulting couple is zero represents the condition of neutral


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I

ex

L

i _

x

. f UX

cb

x

' , '/

_

x. .J

Fig

5.2

Derivation

o

conditionsor stability

w

a

W=wV

WX

x

g

W

Fig 5.2(a) shows a body

of

total weight W floating on even keel. The centre of

gravity G may be shifted sideways by moving a jockey

of

weight

\Vj

across the width

of the body. When the jockey is moved a distance xi>

as

shown in Fig 5.2(b), the

centre of gravity of the whole assembly moves to G The distance GO', denoted

by

x

g

, is given from elementary statics

as

(5.1)

Experimental Determination

o

Stability

In thc following text, we shall show how the stability may he investigated

experimentally, and then how a theoretical calculation can be used to predict the

results.

stability. It will be seen from Fig

5 1

(b) that it is perfectly possible

to

obtain stable

equilibrium when the centre

of

gravity G is located above the centrc

of

buoyancy B


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The shift of the centre of gravity causes the body to

tilt

to a new equilibrium position.

al a small angl e

e

the vertical, as shown in Fig 5.2 b), with an associated movement

of the centre of buoyancy from B to B

t

The point B

t

must lie vertically below

since the body is in equilibrium in the tilted position. le t the vertical l ine of the

upthrust through B intersect the original line of upthrust S at tbe poi nt M. c al le d the

metacentre We may now regard the

jockey

movement as having caused the floating

body

to

swing about the point

M.

Accordingly, the

equilibrium

is

stable

if the

mctacentre lies

above

G. P ro vi de d that

e is

small, the distance GM is given

by

X

g

~ -

S

where e is in c irc ula r measure. Sub st it ut ing for x

g

from Equation 5.1) gives the

result

W.

GM = .

W S

5.2)

The

dimension GM

is

called the metacentric height.

In

the experiment described

below. it is mea sured direc tly from the slope of a graph of Xj against obmined by

moving a

jockey

across a pontoon.

Analy i al De ermina ion

A quite separate theoretical calculation

of

the position

of

the met acent re ca n be made

as follows.

The movement

of

the centre of buoyancy to B

t

produces a moment

of

the buoyancy

force a bou t the original ce ntre

of

buoyancy

B.

To establish the magnitude of this

moment. first consider the element

of

moment

exened

by a small cl eme nt of change

in

di sp la ce d volume, as indicated on Fig 5.2 c).

An

element of width 8x, lying

at

distance x from

B.

has a n additional depth xdue to the tilt of the body. Its length.

as shown

in

the plan view on Fig 5.3 c), is

L

So the volume

OV cfthe

element

is

8V = S.x.L.ox = SLx8x

and the element of additional buoyancy force 8F is


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8F

w.8V = we Lx8x

where

\\

is the specific weight of water. The element of momenl about B produced by

the element of force is 8M. where

oM = of.x

w8Lx

2

x

The total moment about 8 is obtained

by

integration over the whole of the plan area

of the body. in the plane

of

the water surface:

M = we fLx

1

dX = weI

5.3)

In this,

1

re pre sen ts the s ec on d mome nt , about the a xis of symmetry,

of

the water

plane area of the body.

: \ow this moment represents the movement

of

the upthrust wV from B to B

t

namely,

wV.BB . Equating this the expression for M in Equation 5.3)

wV.BB weI

From the geometry of the figure,

we

see that

BB

=

e.BM

and eliminating BS between these last two equations gives 8M as

BM

I

V

5.4)

For the particular case ofa body with a rectangular planfonn of width 0 and length L,

the second moment I is readily found as:

0 1 0 1 [

]0/1

I;

fLx

dX = L fx

1

dx

= L =

r

Df2 D 2

42

L

12

5.5)


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Now the distance BG may be found from the computed or measured positions of B

and of G, so the metaccntric height GM follows from

Equation

S.4 and

geometrical relationship

GM

M

5.6

This gives an independent check on the result obtained experimentally by traversing a

jockey weight across the floating body.

xperimental rocedure

T he pon toon shown in Fig 5.3 has a re cta ng ul ar p la tf on n, a nd is prov ide d w it h a rigid

sail. jockey weight

t

ma y be traversed in preset steps nd at various heights across

the pontoon, along slots in the sail. Angles

of

tilt are shown by the movement

of

a

plumbline over an angular scale. as indicated in Fig 5.3 a).

The height of the centre

of

gravity

of

the whole floating assembly is first measured.

for one chosen height of the jockey weight.

he

pon to on is s usp en de d from a hole at

one side

of

the sail, as indicated

in

Fig 5.3 b), and the jockey weight

is

placed at such

a position on the line of symmetry as to cause the pontoon to hang with its base

roughly vertical. A p um bl in e is h un g from the su spe nsi on point.

he

height of the

centre of gravity G of the w ho le s usp end ed asse mb ly t he n lies at the point whe re the

plumbline intersects the line of symmetry of the pontoon. This establishes the

position of G for this particular jockey height. he position

of

G for any o th er j oc ke y

height may then be calculated from elementary statics, as will be seen later.

After measuring the external width and length

of

the pontoon. and noting the weights

of

the various components. the pontoon is floated

in

water.

Wilh the j oc ke y weight on the line of symmetry, small magnetic weights are used

trim the a sse mb ly to e ve n keel. i ndi ca te d by a zero reading on the a ngu la r scale.

jockey

is

then moved in steps across the width of the pontoon. the corresponding

t In some equipmenls. two jockey weights


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angle

of

tilt (over a range which

is

typically 8 being recorded

at

each step. This

procedure is then repeated with the jockey traversed at a number

of

different heights.

}

Jockey

weight

Angular

/

_/ scale

a Floating pontoon tilted

by movement

jockey

weight

, Suspension

l

f

l = = -

U

=

- .

-

Plumb

line

b Determination position

o centre

gravity

Fig

5.3

Sketch

a pontoon

Results

and

Calculations

Weight and imensions Pontoon

Weight

of

pontoon (excluding jockey weight) W

p

Weight

of jockey Wj

Total weight

of

floating assembly W

=

W

p

Wj

P

d

I V W 2.821

ootoon ISP acement

=

w

1000

Breadth of pontoon D

Length

of

pontoon L

Area

of

pontoon in plane

of water

surface

A LO

0.3601

x

0.2018

3

0.360 I x 0.20183

Second Moment

of

Area I

=

= ---,.,-- --- -- -

12 12

V 2.821

x

10-

3

Depth

of

immersion OC

= - =

c: ccc:: c

A

7.267 x

1

2

2.430 kgf

0.391

kgf

2.821

kgf

2.821 x IO 3

m

3

201.8

mm

=

0.2018

m

360.1

mm

0.3601

m

7.267 x 10-

2

m

2

3.88

X

10-

2

m

=

38.8 mm


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Height

of

centre

of

buoyancy B above 0

=

BC

= OC

2

Height o Centre Gravity

19.4

mm

When the pontoon was suspended as shown in Fig

5.3 b and with the jockey weight placed in the uppennost slot

of

the sail, the

following measurements were made:

=

G

c

B

o

i 5

I

Y

I

I

Fig 5.4 shows schematically the positions of the

centre of buoyancy B. centre of gravity G. and

metacentre M 0 is a reference point on the

external surface

of

the pontoon, and C is the point

where the axis

of

symmetry intersects the plane

of

the water surface. The thickness of the material

from which the pontoon is made is assumed to be

2 mm. The height of G above rhe reference point

o

is

OG. The height

of

the jockey weight above

o is Yj

Height

of

jockey weight above 0 Yj 345 mm

Corresponding height

of

G above 0 OG 92 mm

The value

of

may now be detcnnined for any other value

of Yj

If

Yj

changes by

6Yj then this will produce a change

in of

Wj ..6y/W. The vertical separation

of

the slots

in

the sail is 60 mm, so

will change

in

steps

[ 391

x 60/2.821

=

8.32

mm Table 5

J

shows the values

of

calculated in this way for the 5 different

heights Yj

of

the jockey weight.

Table

5 1

Heights

o G

above base

Do pontoon

Yi

OG

mm

mm

105

58.7

165

67.1

225

75.4

285

83.7

345

92.0

45


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BM BG+GM OG-OB+GM OG+GM-19.4mm

Table 5.2 ngles

of

lilt caused byjockey displacement

45.7

.391

x

330.0

2 821

5.76 mm/deg 5.76 x 57.3 3 0.0 mm/rad

GM

dX

l The preset sleps in

Xj

shown in the table are 15 rnm To provide accuracy, this has been reduced

to

7.5 mm in later versions of the equipment.

Inserting this into Equation 5.2 ,

This value, and corresponding values for other jockey heights, are entered

in

the

fourth column

of

Table 5.3. Values

of 8M

are also shown, derived as follows refer

to

Fig 5.4 for notation :

xperimental determination ofmetaeentric height

Jockey Jockey Displacement from Centre, Xj

Height

mm

mm

-45 -30

-15

0 15 30

45

105

-7.8 -5.2 -2.7

0.0

2.6 5.2 7.8

165

-6.2

-3.1 0.0 3.2 6.2

225.

-7.7

-3.8

0.0 3.9 7.8

285

-5.2

0.0 5.2

345 -7.5 -0.1

7.4

Table 5.2 shows the re.sults obtained when the pontoon was tilted by traversing the

jockey weight across its width

l

.

These results are shown graphically on Fig 5.5. For each of the jockey heights, the

angle of tilt is proportional to the jockey displacement. The metacentric height may

now

be found from Equation 5.2 , using the gradients of the lines

in

Fig 5.5. For

example, when 105 mm, the gradient is

330.0 mm/rad


9/10

40

E

2

-

-

E

e

O l ~ ~ : : : . .. = ~ I

~

is.

.

~

20

u

o

..,

-40

-8 -6 -4

-2 0 2

Angle

of

tilt

4 6

8

ig Variation

0

angle

of

tilt

with jockey displacement

~ M

-

-

-

0

-

-

-

-

-

~

E 60

c >

~ .

~

40

+-

6

2 0 - - - - - - - - - - - - - - - - - - - - - - - - -

o

radient ofstability IiDe dx de mml )

ig 6 Variation st bilitywith me acentric height

47


10/10

Jockey

OG

xj 9

Metacentric

BM

height

height GM

mm)

mm)

mm/ )

mm)

mm)

105 58.7 5.76 45.7 85.0

165

67.1 4.82 38.3 86.0

225 75.4 3.88 30.8 86.8

285 83.7 2.88 22.9 87.2

345 92.0

2 01

16.0

88

Table

5.3

Me/acentric height derived experimentally

As 8M depends only on the mensuration and total weight

of

the pontoon its value

should be independent

of

the jockey height and this is seen

to

be reasonably verified

by the experimental results. The value computed from theory is

8M

1

V

2.466 X 10

4

2.821 X 10

8.74 X 10

2

m

87.4

mm

which is in satisfactory agreement with the values obtained experimentally.

Another way of expressing the experimental results is presented in Fig 5.6 where the

height BG

of

the centre

of

gravity above the centre

of

buoyancy is shown as a

function of the slope

x e

The experimental points lie on a straight line which

intersects the BG axis at the value 90 mm. As BG approaches this value x S .

Namely the pontoon may be then tilted by an infinitesimal movement of the jockey

weight;

it

is in the condition of neutral stability. Under this condition the centre of

gravity coincides with the metacentre viz. BM

=

BG. So from Fig 5.6 we see that

BM =

90 mm. This experimentally detennined value again is in satisfactory

agreement with the theoretical value

of

87.4 mm.

Di cussiOIl sults

The experiment demonstrates how the stability

of

a floating body is affected by

changing the height of its centre of gravity and how the metacentric height may be

established experimentally by moving the centre of gravity sideways across the body.

The value established in this way agrees satisfactorily with that given by the

analytical result BM =

JlV

stability floating body

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