Stability & Buoyancy

ObjectivesPrinciples of StabilityArchimedes PrincipleTerminology of ships hydrostaticsStability & moments -> staying uprightMetacenter, Center of Gravity, Center of Buoyancy, etc.Stability curves

Principles of StabilityFloating object is acted on by forces of gravity and forces of buoyancyStatic equilibrium SFi = 0Three conditions of static equilibrium:Stable: return to same position if tippedNeutral: when rotated, will come to rest in any positionUnstable: will come to rest in new position if force acts on it

Archimedes PrincipleLaw: a body floating or submerged in a fluid is buoyed up by a force equal to the weight of the water it displacesDepth to which ship sinks depends on density of water (r = 1 ton/35ft3 seawater)

Archimedes PrincipleShip sinks until weight of water displaced by the underwater volume is equal to the weight of the shipForces of gravity: G = mshipg =WshipForces of buoyancy: B = rwaterVdisplaced

Wship = rwaterVdisplaced

Archimedes PrincipleForces act everywhere on ship -> too tough to analyzeCenter of Gravity (G): all gravity forces as one force acting downward through ships geometric centerCenter of Buoyancy (B): all buoyancy forces as one force acting upward through underwater geometric center

Archimedes PrincipleCenter of Gravity (G):Changes position only by change/shift in mass of shipDoes not change position with movement of shipCenter of Buoyancy (B):Changes position with movement of ship -> underwater geometric center movesAlso affected by displacement

Hydrostatics TerminologyDisplacement: total weight of ship = total submerged volume of ship (measured in tons)Draft: vertical distance from waterline to keel at deepest point (measured in feet)Reserve Buoyancy: volume of watertight portion of ship above waterline (important factor in ships ability to survive flooding)Freeboard: vertical distance from waterline to main deck (rough indication of reserve buoyancy)

Hydrostatics TerminologyAs draft & displacement increase, freeboard and reserve buoyancy decrease

MomentsDefn: tendency of a force to produce rotation or to move an object about an axisDistance between the force and axis of rotation is the moment armCouple: two forces of equal magnitude in opposite and parallel directions, separated by a perpendicular distanceG and B are a couple

MomentsDepending on location of G and B, two types of moments:Righting moment: tends to return ship to upright positionUpsetting moment: tends to overturn shipMagnitude of righting moment:RM = W * GZ (ft-tons)GZ: moment arm (ft)

MetacenterDefn: the intersection of two successive lines of action of the force of buoyancy as ship heels through small angles (M)If angle too large, M moves off centerline

MetacenterMetacentric Height (GM)Determines size of righting/upsetting arm (for angles < 7o)GZ = GM*sinfLarge GM -> large righting arm (stiff)Small GM -> small righting arm (tender)

MetacenterRelationship between G and MG under M: ship is stableG = M: ship neutralG over M: ship unstableSTABLEUNSTABLE

Metacenter v. Stability CurvesAt this point, we could use lots of trigonometry to determine exact values of forces, etc for all angles -> too much work GM used as a measure of stability up to 7, after that values of GZ are plotted at successive angles to create the stability curve

Stability Curve

Stability CurvePlot GZ (righting arm) vs. angle of heelShips G does not change as angle changesShips B always at center of underwater portion of hullShips underwater portion of hull changes as heel angle changesGZ changes as angle changes

Questions?