5. THE STABILITY OF A FLOATING BODY
11l1roduct;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 5.1 (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 itscentre 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 8. However, this is not so. B
w
II'
!~
GJ
(a) (b)StableFig5.J Forces acting on "jloating 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 initial position, the equilibrium is stable. If however, the couple acts to move the vessel even further from its initial position, as in Fig S(c), then the equilibrium is unstable.
I
r=
ex
L!
' , i-.-._.•/.x '."f UXl: (c) (b) x. '",""""'/ ._-- ~--{ 'x.--.J,
Fig5.2 Derivation ofconditions for stability w (a) W=wV W·X· ~ _ J _ J xg 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 xg , is given from elementary statics as
(5.1) Experimental Determination ofStability
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 buoyancyB.
I' 1
j1
r
1
1
.
l,
,
The shift of the centre of gravity causes the body to tilt to a new equilibrium position. al a small angle
e
[0 the vertical, as shown in Fig 5.2(b), with an associated movementof the centre of buoyancy from B to Bt
• The point B t
must lie vertically below G. since the body is in equilibrium in the tilted position. let the vertical line of the upthrust through B' intersect the original line of upthrust SG at tbe point M. called 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. Provided thate is small, the distance GM is givenby
Xg
GM~
S
where
e
is in circular measure. Substituting for xg from Equation (5.1) gives theresult
W. x· GM = - ' . - '
W S
(5.2)
The dimension GM is called the metacentric height. In the experiment described below. it is measured directly from the slope of a graph ofXjagainste,
obmined bymoving a jockey across a pontoon.
Analy/i£:al De/ermina/ion of 8M
A quite separate theoretical calculation of the position of the metacentre can be made as follows.
The movement of the centre of buoyancy to Bt
produces a moment of the buoyancy force about the original centre of buoyancy B. To establish the magnitude of this moment. first consider the element of moment exened by a small clement of change in displaced volume, as indicated on Fig 5.2(c). An element of width 8x, lying at distance x from B. has an additional depth
e.x
due 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 is8F ~ w.8V
=
we Lx8xwhere \\" is the specific weight of water. The element of momenl about B produced by the element of force is 8M. where
oM
=
of.x :::
w8Lx2
OxThe total moment about 8 is obtained by integration over the whole of the plan area of thebody. in the plane of the water surface:
M = we fLx1dX = weI
(5.3)
In this, '1' represents the second moment, about the axis of symmetry, of the water, plane area of the body.
,
:'\ow this moment represents the movement of the upthrust wV from B to Bt
, namely,
wV.BB'. Equating this[0 the expression for M in Equation (5.3)
wV.BB' ~ weI
From the geometry of the figure, we see that
BB'
=
e.BMand 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 fx1dx = L
~
=-Dr-
-Df2 -D/2 42 LD' 12 (5.5)-I _
I _
l-I .,
I
.,
I.
I.
l-I
I
I
I
--,
I
I
-I ..
I
I
-I
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 th~
geometrical relationship
GM ~ BM· BG
(5.6)
This gives an independent check on the result obtained experimentally by traversing a jockey weight across the floating body.Experimental Procedure
The pontoon shown in Fig 5.3 has a rectangular platfonn, and is provided with a rigid sail. A jockey weightt may be traversed in preset steps and at various heights across the pontoon, along slots in the sail. Angles oftilt 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. The pontoon is suspended 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 pumbline is hung from the suspension point. The height of the centre of gravity G of the whole suspended assembly then lies at the point where the plumbline intersects the line of symmetry of the pontoon. This establishes the position of G for this particular jockey height. The position of G for any other jockey 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 jockey weight on the line of symmetry, small magnetic weights are used [0
trim the assembly to even keel. indicated by a zero reading on the angular scale. Th~
jockey is then moved in steps across the width of the pontoon. the corresponding
t In some equipmenls. two jockey weights <Ire provided. These gi\'e sLope for slightly diifcr.:nl
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.
I'
}
Jockey weight Angular / _/ scale(a) Floating pontoon tilted by movement ofjockey weight
, Suspension
l
f
rl
=
=
-U
Gi - ~=
-- ~---: ~-~ ~-- . --- ~-.~ -- -- -,--- Plumbline (b) Determination ofposition o/centre ofgravityFig5.3 Sketch a/pontoon
Results and Calculations
Weight and Dimensions ofPontoon
Weight of pontoon (excluding jockey weight) Wp Weight of jockey Wj
Total weight of floating assembly W= Wp
+
Wj PootoondISP· Iacement V = -W ~ - -2.821w 1000
Breadth of pontoon D Length of pontoon L
Area of pontoon in plane of water surface A ~ LO ~ 0.3601 x 0.2018
L03 0.360I x 0.20183 Second Moment of Area I = - -
=
---,.,--'---'--"'-12 12 V 2.821 x10-3 Depth of immersion OC
= -
= c:-ccc::--c-, A 7.267 x 10 2 2.430 kgf 0.391 kgf 2.821 kgf 2.821 x IO-3 m3 201.8 mm = 0.2018 m 360.1 mm ~ 0.3601 m 7.267 x 10-2m2 3.88 X 10-2m= 38.8 mmHeight of centre of buoyancy B above 0 08
=
BC = OC 2Height o/Centre
0/
Gravity19.4 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:
-= Gc
Bo
Fig 5.4I
Y, II
•
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 be2 mm. The height ofG above rhe reference point
o
is OG. The height of the jockey weight aboveo
is Yj'-Height of jockey weight above 0 Yj 345 mm Corresponding height of G above 0 OG 92 mm
The value of 00 may now be detcnnined for any other value of Yj. If Yj changes by .6Yj. then this will produce a change in 00 of Wj ..6y/W. The vertical separation of the slots in the sail is 60 mm, so 00 will change in steps 0[0.391 x 60/2.821 = 8.32
mm. Table 5.J shows the values of 00 calculated in this way for the 5 different heights Yj of the jockey weight.
Table 5.1 Heights DC o/G above base Do/pontoon
-YiOG
(mm) (mm) 105 58.7 165 67.1 225 75.4 285 83.7 345 92.0-
45BM ~ BG+GM ~ OG-OB+GM OG+GM-19.4mm Table 5.2 Angles oflilt caused by jockey displacement
= 45.7 mm 0.391 x 330.0 2.821 5.76 mm/deg ~ 5.76 x 57.3 ~ 3330.0 mm/rad GM dXj
dO
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):
Experimental determination ofmetaeentric height GM
Jockey Jockey Displacement from Centre,Xj
Height (mm) y; (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 widthl.
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, whenYi= 105 mm, the gradient is
40 ~ E
,§,
20.--=
~ Ee
Ol--~---~---:::...
-=--~---I ~ is..'"
"
>, ~ -20 u o..,
-40 -8 -6 -4 -2 0 2 Angle oftilteo
4 6 8Fig5.5 Variation
0/
angle oftiltwith jockey displacement,
100•
BG~BM-
--80
--+
~ ~+-E J E 60~
~ c.:>+~
.
'"
,
+~
40+-6 5 4 3 2 1 2 0 + - - - - ' - - - - ' - - - - ' - - - - ' - - - - ' - - - '
o
Gradient of stability IiDe dx/de (mml")
Fig 5.6 Variation a/stability withme/acentric height
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 886Table5.3 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
•
.,
.,
.,
..
u u 8M ~ 1 V 2.466 X 10-4 2.821 X 10-3 8.74 X 10-2m 87.4mm uwhich 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 ----). O. 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 satisfactoryagreement with the theoretical value of 87.4 mm.
Di.,.cussiOIl ofResults
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. u
