ME2134-1

ME2134-1

STABILITY OF FLOATING BODY (WS2-01-47)

SEMESTER 3 2011/2012

NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MECHANICAL ENGINEERING

CONTENTS TABLE OF CONTENTS

i

LIST OF ILLUSTRATIONS

i

LIST OF SYMBOLS

ii

INTRODUCTION

1

THEORY OF OPERATION

1

DESCRIPTION OF EQUIPMENT

5

EXPERIMENTAL PROCEDURE

6

ANALYSIS AND DISCUSSION

8

REFERENCES

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LIST OF ILLUSTRATIONS Figure 1

Static stability of a floating body.

2

Figure 2

Inclined experiment to determine GM and KG.

3

Figure 3

Illustration depicting plane of flotation.

4

Figure 4

Effects of free surface.

5

Figure 5

Ballasting of 20 compartments of barge with water (top view).

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i

LIST OF SYMBOLS B

Center of buoyancy

B’

Displaced center of buoyancy

d

Transverse distance of displacement of added mass

FB

Buoyancy force or upthrust

G

Center of gravity of floating body

G’

Displaced center of gravity of floating body

G1

Displaced center of gravity of floating body with free surface

GZ

Moment arm of restoring couple

g

Acceleration due to gravity

IOy

Second moment of area of plane of floatation about its longitudinal axis

i

Second moment of area for free surface tank

K

Keel

KOy

Mass radius of gyration of floating body about its longitudinal axis

M

Transverse metacenter

m

Mass

T

Period of oscillation

W

Weight of floating body

Greek Symbols:

θ

Angle of heel or inclination

ρ

Density of seawater

ρf

Density of liquid in tank

ii

INTRODUCTION Stability is a measure of the tendency of an ocean vehicle to return to its upright configuration if inclined or perturbed by an external force (Figure 1). For different operating conditions, stability can be classified into the following categories: Intact stability (static stability and dynamic stability) and damage stability. It is imperative to ascertain the overall stability of a floating body during the design phase. Objectives The objectives of this experiment are: (a)

To experimentally determine the center of gravity (C.G.) and metacentric height of a body floating on water.

(b)

To investigate the effects of placing a weight vertically above the C.G. on the stability of a floating body.

(c)

To investigate the effects of free surfaces on the stability of a floating body.

Scope In this experiment, only intact static stability of a vessel at small inclination angles (θ < 10°) will be investigated. The theory of operation and the experimental procedure adopted to evaluate the static stability are provided in this manual. THEORY OF OPERATION (I)

Static Stability of Floating Body

Referring to Figure 1, the weight W of the floating body passes through its center of gravity G. The upthrust or buoyancy force FB acting on the floating body passes through the center of buoyancy B, which corresponds to the centroid of the displaced fluid. When the floating body is subjected to a small angular displacement or perturbation θ about its equilibrium upright configuration, the center of buoyancy shifts from B to B’, while the center of gravity of the floating body remains unchanged at G. A vertical line drawn upward from B’ intersects the line of symmetry at M, known as the metacenter. GM is known as the metacentric height. (a)

If M is above G (GM > 0), a restoring couple acts on the floating body in its displaced position tending to restore it to its original position. Hence, the body is in stable equilibrium.

(b)

If M is below G (GM < 0), an overturning couple acts on the body. Hence, the body is in unstable equilibrium.

(c)

If M coincides with G (GM = 0), the resultant couple is zero, and the body has no tendency to return to, nor move further away from its original position. Hence, the body is in neutral equilibrium.

1

Figure 1

Static stability of a floating body.

If the body floats stably, it may be shown that the period of oscillations for small angles of displacement θ is given by  KOy  T = 2π  (1)  ,  g ⋅ GM  where KOy is the radius of gyration of the floating body about its longitudinal axis. Hence, larger values of GM give rise to more rapid oscillations, thus subjecting passengers to higher levels of discomfort. However, the larger the value of GM, the more stable the floating body is. The above are two conflicting requirements for the choice of GM. A good design should thus entail adequate but not excessive values of GM. (II)

Inclined Experiment to Determine GM and KG

In this experiment, a mass m is moved transversely across the deck through a known distance d, as shown in Figure 2.

2

Figure 2

Inclined experiment to determine GM and KG.

For small inclination angles θ, the metacentric height mgd GM = (2) ( ρ gVsub ) tan θ , where Vsub is the volume of fluid displaced, and ρ is the density of the fluid in which the body floats. 3

From geometry,

KM = KB + BM = KG + GM .

Hence,

KG = KB + BM − GM .

(3)

The metacentric radius BM may be evaluated using the expression I Oy BM = , Vsub where IOy is the second moment of area of the plane of floatation about the longitudinal axis Oy of the floating body (see Figure 3).

Figure 3

Illustration depicting plane of flotation.

KB may be evaluated using hydrostatic calculations. In this experiment, a floating body with a rectangular cross section is used. In this case, 1 KB = ( draught ) , 2 where the draught (draft) corresponds to the depth of submergence. (III)

Effects of Free Surfaces

Consider a tank on board an ocean vehicle which contains a liquid (freshwater, seawater, fuel, engine oil, etc.) of density ρ f . If the tank is not completely filled with the liquid, the liquid will move across the tank in the same direction as the vessel during rolling (see Figure 4). The center of gravity of the vessel is no longer fixed, but will be shifted away from the centerline plane from G to G1. The moment-arm of the restoring couple will thus be reduced from GZ to G1Z1 . Note that G1Z1 = G2 Z 2 .

4

Figure 4

Effects of free surface.

It can be shown that the virtual reduction in metacentric height due to effects of the free surface is ρ f ∑i GG2 = , (4) ρ Vsub where i is the second moment of area of the liquid’s free surface in the tank about the tank’s own centerline. Note that ρ f is the density of the liquid in the tank. If there is more than one tank not completely filled with liquid of density ρ f , the second moments of area of the liquids’ free surfaces due to the individual tanks will have to be summed up. Hence, the effective metacentric height is G2 M = GM − GG2 .

(5)

DESCRIPTION OF EQUIPMENT The equipment consists of a rectangular barge with water-tight sub-division compartments. The inclining moment is provided by means of masses which can be moved transversely on either side of the amidship section of the barge, of which the distance from the centerline can be measured. The angle of heel is taken by means of a precision inclinometer.

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EXPERIMENTAL PROCEDURE (I)

Determination of GM & KG

1.

Ballast the barge evenly around its C.G. with the given 10 x 1 kg masses and place the given 5 kg mass at the C.G of the barge.

2.

Check that the barge is at even keel with the spirit level meter.

3.

Slide the 3 kg mass along the mast onto the deck.

4.

Measure the mean draught dl, and hence determine the displacement of the barge. Note that the length (l) of the barge is 1.0 m, whereas the width (b) of the barge is 0.5 m.

5.

Displace the 5 kg masses from the C.G. transversely through a distance d of approximately 200 mm and note down the heeling angles (angles of inclination) θ of the vessel for a trimming moment of md = 5 kg x 0.2 m = 1.0 kg-m, say.

6.

Find the metacentric height GM of the vessel using equation (2).

7.

Time the period of oscillation T for small heeling angles using a stopwatch and compare the result obtained with equation (1). Note that the radius of gyration of the barge about its longitudinal axis is KOy = 0.29 m (without the raised mass).

8.

Calculate the metacentric radius BM and KB. Note that the second moment of area of the plane of floatation about the 1 3 longitudinal axis Oy of the floating body I Oy = lb , where l = 1.0 m and 12 b = 0.5 m . I Oy 1 Also, note that BM = and KB = ( draught ) . Vsub 2

9.

Hence obtain KG using equation (3).

10.

Slide and fasten the 3 kg mass along the mast vertically at a distance of about 700 mm above the deck.

11.

Repeat steps (4) to (9). Note that with the raised mass, the radius of gyration of the barge about its longitudinal axis is KOy = 0.39 m .

12.

Compare the values of GM and KG obtained in (6) and (9) (with and without the raised mass) and comment on the stability of the vessel when a mass is raised vertically above the C.G.

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(II)

Effects of Free Surfaces

13.

Place the 3 kg mass along the mast back to the deck of the barge and also remove all the ballasting weights.

14.

Place the 5 kg mass at the C.G. of the vessel.

15.

Evenly ballast the following 20 compartments of the barge with 1/2 litres of water each, as shown in Figure 5.

Figure 5

Ballasting of 20 compartments of barge with water (top view).

(Note: 1 litre of water is approximately equal to 1 kg of mass, so the draught at even keel should be approximately the same as in step (4).) 16.

Repeat steps (4) to (9).

17.

What conclusions can be made regarding the effects of free surface on the stability of the vessel with the same displacement and same KG of a similar vessel?

18.

Using equation (4), check the virtual reduction in metacentric height using analytical (theoretical) calculations. Useful information: • Length of each compartment lc = 0.195 m • Width of each compartment bc = 0.097 m • In the presence of the free surfaces, the radius of gyration of the barge about its longitudinal axis is KOy = 0.28 m . • Second moment of area of the free surface in each compartment about 1 3 the compartment’s centerline is given by i = lc bc . 12 • Note that there are a total of 20 compartments filled with water.

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ANALYSIS AND DISCUSSION 1.

At large heeling angles (say θ >> 10°), does the value of the metacentric height GM remain constant? If not, please explain why.

2.

What is the difference between static and dynamic stability?

3.

A pontoon of constant rectangular cross section is 50 m long, 10 m wide and 5 m in depth. It floats in sea water at a draught of 2 m with its C.G. 2.5 m above the base. Without causing the pontoon to become unstable, determine the maximum load which can be added at a height of 4 m above the base.

4.

A mass of 10 tonnes is moved 20 m across the deck of an ocean vehicle of 10,000 tonnes displacement. The ocean vehicle has a rectangular tank 20 m long and 10 m wide containing sea water. The center of gravity of the vessel is 5 m above the keel and the transverse metacenter is 6 m above the keel. Calculate the angle of heel (inclination) if:

(a)

the tank is completely filled with sea water and there is no free surface in the tank.

(b)

the tank is half filled with sea water of density ρ = 1025 kg/m3.

(c)

a longitudinal centerline division is installed in the half-filled tank.

(d)

a transverse division is installed in the half-filled tank.

(e)

the tank is half filled with fresh water of density ρf = 1000 kg/m3.

5.

Allowing for the presence of entrained water and assuming that the angular damping is proportional to the angular velocity of the floating body, the equation of motion for rolling in still water of the floating body can be written as

(

2 ρVsub KOy 1 + σ Oy

2

) ddt θ2 + B ddtθ + ρVsub g ( GM ) θ = 0 ,

where 2 ρVsub KOy σ Oy = augmentation of rolling inertia of the floating body due to the t B

entrained water = time = damping coefficient

Determine the period of oscillation of the above damped rolling motion of the floating body in still water.

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REFERENCES Comstock J. P., “Principles of Naval Architecture”, Society of Naval Architects and Marine Engineers, 1967. Muckle W., “Naval Architecture for Marine Engineers”, Newnes-Butterworths, 1975. Rawson K. J. and Tupper E. C., “Basic Ship Theory”, Longman, 4th Edition, 1994. Teo C. J., Lecture notes for ME2134: Fluid Mechanics I. Stokoe E. A., “Reed's naval architecture for marine engineers”, Thomas Reed, 4th Edition, 1991. Tupper E. C., “Introduction to Naval Architecture”, Butterworth Heinemann, 4th Edition, 2004.

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