Stability & Buoyancy

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## Stability & Buoyancy

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**Objectives**• Principles of Stability • Archimedes Principle • Terminology of ship’s hydrostatics • Stability & moments -> staying upright • Metacenter, Center of Gravity, Center of Buoyancy, etc. • Stability curves**Principles of Stability**• Floating object is acted on by forces of gravity and forces of buoyancy • Static equilibrium SFi = 0 • Three conditions of static equilibrium: • Stable: return to same position if tipped • Neutral: when rotated, will come to rest in any position • Unstable: will come to rest in new position if force acts on it**Archimedes Principle**• Law: a body floating or submerged in a fluid is buoyed up by a force equal to the weight of the water it displaces • Depth to which ship sinks depends on density of water (r = 1 ton/35ft3 seawater)**Archimedes Principle**• Ship sinks until weight of water displaced by the underwater volume is equal to the weight of the ship • Forces of gravity: G = mshipg =Wship • Forces of buoyancy: B = rwaterVdisplaced Wship = rwaterVdisplaced**Archimedes Principle**• Forces act everywhere on ship -> too tough to analyze • Center of Gravity (G): all gravity forces as one force acting downward through ship’s geometric center • Center of Buoyancy (B): all buoyancy forces as one force acting upward through underwater geometric center**G**Archimedes Principle • Center of Gravity (G): • Changes position only by change/shift in mass of ship • Does not change position with movement of ship • Center of Buoyancy (B): • Changes position with movement of ship -> underwater geometric center moves • Also affected by displacement**Hydrostatics Terminology**• Displacement: 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 ship’s ability to survive flooding) • Freeboard: vertical distance from waterline to main deck (rough indication of reserve buoyancy)**Hydrostatics Terminology**• As draft & displacement increase, freeboard and reserve buoyancy decrease**Moments**• Def’n: tendency of a force to produce rotation or to move an object about an axis • Distance between the force and axis of rotation is the moment arm • Couple: two forces of equal magnitude in opposite and parallel directions, separated by a perpendicular distance • G and B are a couple**Moments**• Depending on location of G and B, two types of moments: • Righting moment: tends to return ship to upright position • Upsetting moment: tends to overturn ship • Magnitude of righting moment: • RM = W * GZ (ft-tons) • GZ: moment arm (ft)**Metacenter**• Def’n: 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**Metacenter**• Metacentric Height (GM) • Determines size of righting/upsetting arm (for angles < 7o) GZ = GM*sinf • Large GM -> large righting arm (stiff) • Small GM -> small righting arm (tender)**Metacenter**• Relationship between G and M • G under M: ship is stable • G = M: ship neutral • G over M: ship unstable STABLE UNSTABLE**Metacenter v. Stability Curves**• At 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**• Plot GZ (righting arm) vs. angle of heel • Ship’s G does not change as angle changes • Ship’s B always at center of underwater portion of hull • Ship’s underwater portion of hull changes as heel angle changes • GZ changes as angle changes