Polytechnic University of the Philippines College of Engineering Department of Civil Engineering
STABILITY OF FLOATING BODIES LABORATORY EXPERIMENT NO. 2
GICOM, Angelica F. BSCE IV - 2 FLUID MECHANICS Th 10:30 AM – AM – 1:30 1:30 PM
October 17, 2016
Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
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Exercise A Objective To locate the position of the Metacentre (M) of a floating body, by determining its distance from the centre of gravity (G); this distance (GM) is known as the Metacentric Height. Note: It Note: It is a condition for stable equilibrium that M remains higher than G. Method The Centre of Gravity of the pontoon assembly can be varied by adjusting the position of a sliding weight of the mast. For a fixed position of the Centre of Gravity, the metacentric height (GM) can be determined experimentally by measuring the angle of heel (θ) produced at each position of an inclining weight, as it is traversed across the width of the pontoon. Equipment Required In order to complete the demonstration of the Metacentric Height apparatus we need a number of pieces of equipment.
The F1-10 Hydraulics bench (or a suitable depth of undisturbed free water surface) The F1-14 Metacentric Height Apparatus Weigh Balance – Balance – max. max. 2.0 kg (not supplied) Ruler (not supplied) Knife edge or fine string (not supplied)
Theory The theoretical results given here are derived from first principles in many first-degree level fluid mechanics text-books. For static equilibrium of the pontoon, the total weight (W) which acts through the centre gravity (G) must be equal to the buoyancy force which acts through the Centre of Buoyancy (B) which is located at the centroid of the immersed cross-section. W hen the pontoon heels through a small angle (θ) the Metacentre (M) is identified as the point of intersection between the line of action of the buoyancy force (always vertical) and BG extended. For stable equilibrium, M must be above G. M
M
θ
G
G’
G
P W
W P
B
B
P
P
B’ P
K K
Section through floating pontoon
P
Stability of Floating Bodies
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FLUID MECHANICS October 17, 2016
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When the inclining weight is moved to one side, the centre of gravity G shifts to a new position G’ and the centre of buoyancy B also shifts to position B’. Since the shift in the centre of gravity was caused by moving the weight P through a distance x, we can write: Px = W(G G’) from the figure above, we can see that: G G’ = GM tanθ hence: GM =
P×x W×tan θ
Note that this equation cannot be used when θ = 0 It is also possible to calculate the metacentric height, GM, from first principles, by calculating the shift in the centre of buoyancy of the vessel. M
M
v
G
vtanθ θ
W B
B
B’
z
K
The center of buoyancy of the vessel (centroid of the displaced water) can be found by talking the moments. In the initial upright condition:
∙̅ =∫ where y is the lateral position of the centre of buoyancy and V is the immersed volume. When the vessel heels (rototates about the x axis), the new centre of buoyancy is given by:
∙̅ = ∫ (+tan) +tan) Subtracting the first equation from the second gives:
∙ (̅−̅ ) = ∫ tan =tan( =tan ( )
Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
3 8
where Ak 2 is the second moment of area of the plane of floatation about x-axis (I). For small angular displacements:
̅−̅ =tan hence:
= where:
=×× hence:
3 × = 12××× The immersed volume V can be determined by calculation. Since the buoyancy force (up thrust) is equal to the total weight W of pontoon and its load:
= The depth of immersion (d) can be found from:
= × Finally, the centre of buoyancy B at a distance KG above the base.
= from the base.
The centre of gravity G is at distance KG above the base. Hence:
==+ Note: For Note: For body with a fixed displacement, the position of the Metacentric will remain constant but the Metacentric Height GM will reduce if the Center of Gravity is raised. Equiment Set Up Weigh the inclining weight (= P kg), used for traversing across the pontoon with. Assemble the pontoon, pontoon, mast and both weights and and determine determine the total total weight (= W kg). At the start start of the experiment, experiment, position position the sliding sliding weight on on the mast mast to give give the location of the centre centre of of gravity (G) for the whole assembly at the level of the top of the pontoon i.e y = 75mm. The position of G can be determined by:
Use of a knife-edge.
Stability of Floating Bodies
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FLUID MECHANICS October 17, 2016
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Tying a light string tightly around mast and carefully carefully allowing the whole assembly to be suspended from it, adjusting the position of the suspension point until the mast direction becomes horizontal.
CG Edge of Steel Rule or te tens nsio ione ned d str strin in Measure the distance of G from the base of the pontoon and record this distance y. If using the F-1-10 Hydraulic bench fill the volumetric tank with water, then remove the filling tube from the tank (to prevent back siphoning through to the main tank). Procedure Move inclining weight to the centre of the pontoon, indicated by 0mm on the linear scale then tighten the securing screw. Float the pontoon in the water and measure the immersed depth d for comparison with the calculated value (see Theory). If necessary adjust the tilt of the mast (by slackening securing screws which pass through slotted holes) to ensure that the plumb line aligns with the angular scale without rubbing. Tighten the screws. Traverse the inclining weight to the right in 10mm increments to the end of the scale and note the angular displacement (θ) of the plumb line for each position of the weight. Repeat this procedure traversing the inclining weight to the left of the centre. Angles should be designated +ve to one side and-ve to the other to avoid confusion when analyzing the readings. Change the position of the pontoon centre of gravity by moving the sliding weight up the mast. Suggested positions are at the maximum height and then a location midway between maximum height and the position used in the first test. A lower position with the weight at the bottom of the mast (G inside the pontoon) may also be evaluated. For each new position of G, repeat the above test and determine the metacentric height, GM. Hence, locate the position of the Metacentre M (= y + GM) from the base of the pontoon, using the results from all test. Note: The equation for the calculation of GM cannot be applied when θ = 0, so this must be determined graphically as described in the Results section below.
Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
Results (CG @ full height) t t c e
n h
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ig
p x
M
at h
E
0
8
2
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7
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5 8
Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
Results (CG @ 10.3)
t t c e
n h
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p x
M
at h
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4
0
9
5
6
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3
1
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6 8
Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
Results (CG @ midway)
t t c e
n h
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gi
p x
) M
at
m(
h
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M
5
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g c
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t c w
h nI l
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th o g el
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2
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7
7
7
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7
7
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5
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3.
For each position of the Center of Gravity, plot a graph of Metacentric Height GM against angle heel θ. From this graph, extrapolate the position of GM when θ = 0.
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Stability of Floating Bodies
FLUID MECHANICS October 17, 2016
8 8
Conclusions
When G is above the metacenter, it has stable equilibrium position and when the G is below the metacenter, it has unstable equilibrium. The values of GM at lowest values of θ are likely to be less accurate because the lower the angle, the lower the GM. The lower the GM, the smaller the righting moment which makes it less accurate. The unstable equilibrium might be achieved when the moment is overturning or upsetting and has zero or negative megacentric height.
