Eastern Mediterranean University Department of Mechanical Engineering Laboratory Handout
COURSE: Fluid Mechanics (MENG353) Semester: SPRING (2014-2015) Name of Experiment: Stability of a Floating Body Instructor: Assoc. Prof. Dr. Hasan Hacışevki Assistant: Amir Teimourian Submitted by: Student No: Group No: Date of experiment: Date of submission: ------------------------------------------------------------------------------------------------------------
EVALUATION Activity During Experiment & Procedure
30 %
Data , Results & Graphs
35 %
Discussion, Conclusion & Answer to Questions
30 %
Neat and tidy report writing
5%
Overall Mark
Honor Pledge: By electronically submitting this report I pledge that I have neither given nor received unauthorized assistance on this assignment. __________ Date
______________ Signature
OBJECTIVES To determines the stability of a pontoon with its center of gravity at various heights.
LEARNING OUTCOME At the end of this lab session, students should be able to 1. Understand the parameter affecting the stability of a floating body 2. Determine the stability of a floating body with different CG height
THEORY
(a)
(b)
(c)
Figure 1 Derivation of Stability of Floating Pontoon
Consider the rectangular pontoon shown floating in equilibrium on an even keel, as shown in the cross section of Figure 1(a). The weight of the floating body acts vertically downwards through its centre of gravity G and this is balanced by an equal and opposite buoyancy force acting upwards through the centre of buoyancy B , which lies at the centre of gravity of the liquid displaced by the pontoon. To investigate the stability of the system, consider a small angular displacement δθ from the equilibrium position as shown on Figure 1(b). The centre of gravity of the liquid displace by the pontoon shifts from B to B1 . The vertical line of action of the buoyancy force is shown on figure and intersects the extension of line BG at M , the metacentre. The equal and opposite forces through G and B1 exert a couple on the pontoon, and provided that M lies above G (as shown in Figure 1(b)) this couple acts in the sense
of restoring the pontoon to even keel, i.e. the pontoon is stable. If, however, the metacentre M lies below the centre of gravity G , the sense of the couple is to increase the angular displacement and the pontoon is unstable. The special case of neutral stability occurs when M and G coincides. Figure 1(b) shows clearly how the metacentric height GM may be established experimentally using the adjustable weight (of mass ω ) to displace the centre of gravity sideways from G . Suppose the adjustable weight is moved a distance δx1 from its central position. If the weight of the whole floating assembly is W , then the corresponding movement of the centre of gravity of the whole in a direction parallel to the base of the
ω
δx . If this movement produces a new equilibrium position at an angle of a list W 1 δθ , then in Figure 1(b), G1 is the new position of the centre of gravity of the whole, i.e.
pontoon is
ω
δx1
(1)
GG1 = GMδθ
(2)
GG1 =
W
Now, from the geometry of the figure:
Eliminating GG1 between these equations we derive: ω δx1 GM = W δθ or in the limit: ω dx1 GM = W dθ The metacentric height may thus be determined by measuring (
(3)
(4) dx1 ) knowing dθ
ω and
W . Quite apart from experimental determinations, BM maybe calculated from the measurement of the pontoon and the volume of liquid which it displaces. Referring again to Figure 1(b), it may be noted that the restoring moment about B , due to shift of the centre of buoyancy to B1 , is produced by additional buoyancy represented by triangle AA1C to one side of the centre line, and reduced buoyancy represented by triangle FF1C to the other. The element shaded in Figure 1(b) and (c) has and area δs in plan view and a height xδθ in vertical section, so that its volume is xδsδθ . The weight of liquid displaced by this element is wxδsδθ , where w is the specific weight of the liquid, and this is the additional buoyancy due to the element. The moment of this elementary buoyancy force about B is wx2δsδθ , so that the total restoring moment about B is given by the expression: wδθ ∫ x 2 ds where the integral extends over the whole area of the pontoon at the plane of the water surface. The integral may be referring to as I , where: I = ∫ x 2 ds (5) where I is the second moment of area of s about the axis X-X.
The total restoring moment about B may also be written as the total buoyancy force, wV , in which V is the volume of liquid displaced by the pontoon, multiplied by the lever arm BB1 . Equating this product to the expression for total restoring moment derived above: wVBB1 = wδθ ∫ x 2 ds (6) Substituting from Equation 5 for the integral and using the expression:
(7)
BB1 = BMδθ which follows from the geometry of Figure 1(b), leads to: BM =
I
(8)
V This result, which depends only on the measurement of the pontoon and the volume of liquid which it displaces, will be used to check the accuracy of the experiment. It applies to a floating body of any shape, provided that I is taken about an axis through the centroid of the area of the body at the plane of the water surface, the axis being perpendicular to the place in which angular displacement takes place. For a rectangular pontoon, B lies at a depth below the water surface equal to half the total depth of immersion, and I may readily be evaluated in terms of the dimensions of the pontoon as: D/2 1 I = ∫ x 2 ds = ∫ x 2 Ldx = LD 3 (9) 12 −D / 2 APPARATUS The apparatus consists of an open plastic box (‘barge’) which floats in water and carries a mast (Figure 2). A plumb-bob suspended from the mast provides a means of measuring the angle of inclination of the barge. The vertical position of the center of gravity is controlled by a weight Wy which may be moved to different heights on the mast. The horizontal position of the center of gravity is controlled by a second weight ω which may be moved to different horizontal positions on the barge.
Figure 2: A schematic plot of the barge and experimental apparatus
PROCEDURE The stability criterion for the pontoon may be described as follows. First, with the pontoon in horizontal position, a small tilt of 𝜃 is provided. The weight of the floating body provides a rotating moment and tend to destabilize the system. On the other hand, the buoyancy force provides a counter moment that tends to stabilize the system. For the stability test, the horizontal weight ω can be moved to the mast of the pontoon. Thus xG = 0 in this experiment. By trial and error, find the minimum vertical position of the weights at which the apparatus becomes unstable. This procedure is somewhat time consuming due to the sensitivity of the floating body close to the stability limit. Try to get more than one independent measurement. Start over each time with weights located well below the stable limit, raising them in small incremental steps each time checking the stability. [Note: A stable pontoon has a rolling period. As the weights on the mast are moved upward the rolling period increases until the pontoon becomes unstable at the critical position. Wy
Metacentric height
ω yb yb X1
Figure 3 Dimensions of Pontoon
Labels of dimensions of the pontoon are given in Figure 3, which are to be referred in the calculation section.
STABILITY OF A FLOATING BODY EXPERIMENTAL DATA The total weight of the apparatus (including the two magnetic weights) is stamped on a label affixed to the sail housing. The adjustable weight (ω) has its weight engraved on its side. The addition of these two values will give the total weight W of the pontoon. Total Mass of floating assembly ( W ) Adjustable mass ( Wy ) Jockey mass (ω) Breadth of pontoon ( D ) Length of pontoon ( L ) Second moment of area ( 𝐼 =
𝐿𝐷 3 12
= 2.6 [ kg ] = 0.5 [ kg ] = 0.2 [ kg ] = 206 [ mm ] = 360 [ mm ]
)
=…….……[ m4 ]
Volume of water displaced ( V=W/ρ ) Height of metacenter above center of buoyancy ( BM= I/V )
=…….……[ m3 ] = …………[ m ]
𝑉
Depth of immersion of pontoon (2yb ) ( 𝐿𝐷 )
= …………[ m ]
Depth of center of buoyancy (yb ) (
= …………[ m ]
𝑉
2𝐿𝐷
)
Determination of height of G: When the pontoon was suspended and with the jockey weight placed in the uppermost slot of the sail the following measurements were made: Height of adjustable weight above base: y1 = 480 mm Corresponding measured height of G above base: y=120mm The value of height of G may now be determined for any other value of y1. If y1 changes by Δy1, then this will produce a change in height of Wy Δy1 / W.
Table 1 List angles for height and position of adjustable weight Height of adjustable weight y1 [ mm]
Angles of list ( ) for adjustable weight lateral displacement from sail centre line x1 mm -75
-60
-45
-30
-15
0
15
30
45
60
75
Table 2 Derivation of metacentric height from experimental results Metacentric height GM
Height of G X1 / θ from figure
dx1 W d
y
[mm]
[mm/deg]
[mm/rad]
BG=y-yb [mm]
BM = BG+GM [mm]
[mm]
Plot a graph of lateral position of the adjustable weight, x1 against angle of list, 𝜃 for each height of adjustable weight, y1 (on the same graph sheet). 𝑑𝑥
Plot the value of 𝑑𝜃1 , which is the slope of the graph of x1 against , against the height of centre of gravity above water line, CG. Extrapolation of this plot will indicate the limiting value of CG at which
𝑑𝑥1 𝑑𝜃
= 0.
DISCUSSION 1. Base on the graph of x1 against angle of list, , discuss the stability of pontoon. 2. Base on the graph of
𝑑𝑥 𝑑𝜃
versus CGwhen is the pontoon become unstable? Justify.
3. When is the pontoon is naturally stable ? 4. Two ways to improve the apparatus reading are ? 5. What will happen if we change the density of the fluid, using brine, say, instead of fresh water ?
