Stability & Stability & BuoyancyBuoyancy

ObjectivesObjectives• Principles of StabilityPrinciples of Stability• Archimedes PrincipleArchimedes Principle• Terminology of ship’s Terminology of ship’s

hydrostaticshydrostatics• Stability & moments -> staying Stability & moments -> staying

uprightupright• Metacenter, Center of Gravity, Metacenter, Center of Gravity,

Center of Buoyancy, etc.Center of Buoyancy, etc.• Stability curvesStability curves

Principles of StabilityPrinciples of Stability• Floating object is acted on by forces of gravity and forces of Floating object is acted on by forces of gravity and forces of

buoyancybuoyancy• Static equilibrium Static equilibrium FFii = 0 = 0

• Three conditions of static equilibrium:Three conditions of static equilibrium:• Stable:Stable: return to same position if tipped return to same position if tipped

• Neutral:Neutral: when rotated, will come to rest in any position when rotated, will come to rest in any position

• Unstable:Unstable: will come to rest in new position if force acts on it will come to rest in new position if force acts on it

Archimedes PrincipleArchimedes Principle• Law: a body floating or Law: a body floating or

submerged in a fluid is buoyed submerged in a fluid is buoyed up by a force equal to the up by a force equal to the weight of the water it weight of the water it displacesdisplaces

• Depth to which ship sinks Depth to which ship sinks depends on density of water (depends on density of water ( = 1 ton/35ft= 1 ton/35ft33 seawater) seawater)

Archimedes PrincipleArchimedes Principle• Ship sinks until weight of Ship sinks until weight of

water displaced by the water displaced by the underwater volume is equal to underwater volume is equal to the weight of the shipthe weight of the ship• Forces of gravity: Forces of gravity: G = G =

mmshipshipg =Wg =Wshipship

• Forces of buoyancy: Forces of buoyancy: B = B = waterwaterVVdisplaceddisplaced

WWship ship = = waterwaterVVdisplaceddisplaced

Archimedes PrincipleArchimedes Principle• Forces act everywhere on ship Forces act everywhere on ship

-> too tough to analyze-> too tough to analyze• Center of Gravity (G):Center of Gravity (G): all all

gravity forces as one force gravity forces as one force acting downward through acting downward through ship’s geometric centership’s geometric center

• Center of Buoyancy (B):Center of Buoyancy (B): all all buoyancy forces as one force buoyancy forces as one force acting upward through acting upward through underwater geometric centerunderwater geometric center

Archimedes PrincipleArchimedes Principle• Center of Gravity (G):Center of Gravity (G):

• Changes position only by Changes position only by change/shift in mass of shipchange/shift in mass of ship

• Does not change positionDoes not change position with with movement of shipmovement of ship

• Center of Buoyancy (B):Center of Buoyancy (B):• Changes positionChanges position with movement with movement

of ship -> underwater geometric of ship -> underwater geometric center movescenter moves

• Also affected by displacementAlso affected by displacement

G

Hydrostatics Hydrostatics TerminologyTerminology

• Displacement:Displacement: total weight of ship = total weight of ship = total submerged volume of ship total submerged volume of ship (measured in tons)(measured in tons)

• Draft:Draft: vertical distance from waterline vertical distance from waterline to keel at deepest point (measured in to keel at deepest point (measured in feet)feet)

• Reserve Buoyancy:Reserve Buoyancy: volume of volume of watertight portion of ship above watertight portion of ship above waterline (important factor in ship’s waterline (important factor in ship’s ability to survive flooding)ability to survive flooding)

• Freeboard:Freeboard: vertical distance from vertical distance from waterline to main deck (rough waterline to main deck (rough indication of reserve buoyancy)indication of reserve buoyancy)

Hydrostatics Hydrostatics TerminologyTerminology

• As draft & displacement increase, As draft & displacement increase, freeboard and reserve buoyancy decreasefreeboard and reserve buoyancy decrease

MomentsMoments• Def’n: tendency of a force to Def’n: tendency of a force to

produce rotation or to move an produce rotation or to move an object about an axisobject about an axis• Distance between the force and Distance between the force and

axis of rotation is the moment armaxis of rotation is the moment arm• Couple: two forces of equal Couple: two forces of equal

magnitude in opposite and magnitude in opposite and parallel directions, separated parallel directions, separated by a perpendicular distanceby a perpendicular distance• G and B are a coupleG and B are a couple

MomentsMoments• Depending on location Depending on location

of G and B, two types of G and B, two types of moments:of moments:• Righting moment:Righting moment:

tends to return ship to tends to return ship to upright positionupright position

• Upsetting moment:Upsetting moment: tends to overturn shiptends to overturn ship

• Magnitude of righting Magnitude of righting moment:moment:• RM = W * GZRM = W * GZ (ft-tons) (ft-tons)• GZ: moment armGZ: moment arm (ft) (ft)

MetacenterMetacenter

• Def’n: the Def’n: the intersection of two intersection of two successive lines of successive lines of action of the force action of the force of buoyancy as of buoyancy as ship heels through ship heels through small angles (M)small angles (M)• If angle too large, M If angle too large, M

moves off centerlinemoves off centerline

MetacenterMetacenter• Metacentric Metacentric

Height (GM)Height (GM)• Determines size of Determines size of

righting/upsetting righting/upsetting arm (for angles < arm (for angles < 77oo))

GZ = GM*sinGZ = GM*sin• Large GM -> large Large GM -> large

righting arm (stiff)righting arm (stiff)• Small GM -> small Small GM -> small

righting arm righting arm (tender)(tender)

MetacenterMetacenter• Relationship between G and MRelationship between G and M

• G under M: ship is stableG under M: ship is stable• G = M: ship neutralG = M: ship neutral• G over M: ship unstableG over M: ship unstable

STABLESTABLE UNSTABLEUNSTABLE

Metacenter v. Stability Metacenter v. Stability CurvesCurves

• At this point, we could use At this point, we could use lots of trigonometry to lots of trigonometry to determine exact values of determine exact values of forces, etc for all angles -> forces, etc for all angles -> too much work too much work

• GM used as a measure of GM used as a measure of stability up to 7°, after that stability up to 7°, after that values of GZ are plotted at values of GZ are plotted at successive angles to create successive angles to create the stability curvethe stability curve

Stability CurveStability Curve

Stability CurveStability Curve• Plot GZ (righting arm) vs. angle of heelPlot GZ (righting arm) vs. angle of heel

• Ship’s G does not change as angle changesShip’s G does not change as angle changes• Ship’s B always at center of underwater Ship’s B always at center of underwater

portion of hullportion of hull• Ship’s underwater portion of hull changes Ship’s underwater portion of hull changes

as heel angle changesas heel angle changes• GZ changes as angle changesGZ changes as angle changes

QuestionsQuestions??