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Chapter 4: Stability!

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**External Forces Acting on a Vessel (4.1)**

OVERALL STABILITY External Forces Acting on a Vessel (4.1) In Chapter 4 we will study five areas: The concept of a ship’s Righting Moment (RM), the chief measure of stability. KG and TCG changes and their effects on RM. How Stability is effected by Damage to the Hull using the “Added Weight” method. Effects of a “Free Surface”. Effects of Negative GM on ship stability.

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OVERALL STABILITY Why?

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**OVERALL STABILITY Internal Righting Moment (4.2)**

The ”Big Picture” in Understanding the Overall Stability of a Vessel: The horizontal distance between the positions of the ship’s displacement vector and the buoyant force vector help determine stability. The relationship changes when a ship is heeled by an external moment.

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OVERALL STABILITY Internal Righting Moment (4.2)

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**OVERALL STABILITY Internal Righting Moment (4.2)**

EXTERNAL FORCES cause a vessel to heel. Recall Force x Dist = Moment External Moment can be caused by wind pushing on one side of the vessel and water resisting the motion on the other side. Each distributed force can be resolved into a resultant force vector. The wind acts above the waterline and the water resistance acts below the waterline.

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**OVERALL STABILITY Internal Righting Moment (4.2)**

The two forces create a couple because they are equal in magnitude, opposite in direction, and not aligned. The couple causes rotation or heeling. The vessel will continue to rotate until it returns to Static Equilibrium (i.e. an Internal Moment is created which is equal in magnitude and opposite in direction). Giving M=0!

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**OVERALL STABILITY Internal Righting Moment (4.2)**

Internal Forces create a Righting Moment to counter the Upsetting Moment of the External Forces. The two internal forces are the weight of the vessel (s) and the resultant buoyant force (FB). Recall the directions they act.

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OVERALL STABILITY Internal Righting Moment (4.2)

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**OVERALL STABILITY Internal Righting Moment (4.2)**

The perpendicular distance between the Weight and the Buoyancy Force vectors is defined as the RIGHTING ARM (GZ). The moment created by the resultant Weight and the resultant Force of Buoyancy is defined as the RIGHTING MOMENT (RM). It may be calculated by:

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**OVERALL STABILITY Internal Righting Moment (4.2) Where:**

RM is the internal righting moment of the ship in ft-LT. s is displacement of the ship in LT. Fb is the magnitude of the resultant buoyant force in LT. GZ is the righting arm in feet.

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**GZ changes with heel angle**

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**OVERALL STABILITY Curve of Intact Statical Stability (4.3)**

Curve of Intact Statical Stability or the Righting Arm Curve Shows the Heeling Angle () versus the righting arm (GZ). Assumes the vessel is heeled over quasi- statically in calm water (i.e. external moments are applied in infinitely small steps).

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**OVERALL STABILITY Measure of Overall Stability (4.3)**

Curve of Statical Stability

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**Curve of Intact Statical Stability (4.3)**

OVERALL STABILITY Curve of Intact Statical Stability (4.3) Caveats! Predictions made by the Curves of Intact Statical Stability are not accurate for dynamic seaways because additional external forces and momentum are not included in the analysis. (”Added Mass”) However, it is a simple, useful tool for comparison and has been used to develop both intact and damaged stability criterion for the US Navy.

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**OVERALL STABILITY Curve of Intact Statical Stability (4.3)**

Typical Curve of Intact Statical Stability Vessel is upright when no external forces are applied and the Center of Gravity is assumed on the centerline. (Hydrostatics) As an external force is applied, the vessel heels over causing the Center of Buoyancy to move off the centerline. The Righting Arm (GZ) is no longer zero.

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**OVERALL STABILITY Curve of Intact Statical Stability (4.3)**

Typical Curve of Intact Statical Stability (cont.) As the angle of heel increases, the Center of Buoyancy moves farther and farther outboard (increasing the Righting Arm). The max Righting Arm will happen when the Center of Buoyancy is the furthest from the CG. This is max stability. If the vessel continues to heel, the Center of Buoyancy will move back towards the CG and the Righting Arm will decrease.

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**OVERALL STABILITY Curve of Intact Statical Stability (4.3)**

Typical Curve of Intact Statical Stability (cont.) Since stability is a function of displacement, there is a different curve for each displacement and KG! These are called the Cross Curves. Cross Curves of Stability (different presentation, more data) X-Axis: Displacement of the Vessel in LT. Y-Axis: Righting Arm in Feet.

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**Cross Curves Example At 2000 LT, the ship Has a RA of 2.5’ @10o**

Righting Arm (feet) For KG=0 30 degrees heel 5 10 degrees heel 2.5 At 2000 LT, the ship Has a RA of Heel and 1000 2000 3000 Displacement (LT)

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**OVERALL STABILITY Cross Curves (4.3) Cross Curves of Stability (Cont.)**

Each curve is for one angle of heel. Plot assumes KG = 0ft. (Does this make sense?) A Curve of Intact Statical Stability can be determined by extracting the values at a given draft, plotting and then applying the sine correction for the correct KG.

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Lab 5 Full Size!

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

From the Curves of Intact Stability the following Measures of Overall Stability can be made: Range of Stability Dynamical Stability Maximum Righting Moment Angle of Maximum Righting Moment Measure of Tenderness or Stiffness

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Range, Max, Angle of Max, AVS, Initial Slope, Dynamic

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Range of Stability The range of angles for which there exists a positive righting moment. The greater the range of stability, the less likely the ship will capsize. If the ship is heeled to any angle in the range of stability, the ship will exhibit an internal righting moment that will right the ship if the external moment ceases.

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Dynamical Stability: The work done by quasi-statically rolling the ship through its range of stability to the capsizing angle. Can be calculated by the equation: This is equal to the product of the ship’s displacement with the area under the Curve of Intact Statical Stability. Not shown directly by the Curve of Intact Statical Stability. Does not account for the actual dynamics, because it neglects the impact of waves and momentum.

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Maximum Righting Moment The largest Static Moment the ship can produce. Calculated by multiplying the displacement of the vessel times the maximum Righting Arm. The larger the Maximum Righting Moment, the less likely the vessel is to capsize.

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Angle of Maximum Righting Arm The angle of inclination where the maximum Righting Arm occurs. Beyond this angle, the Righting Arm decreases. It is desirable to have a larger angle. Why? Waves!

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**OVERALL STABILITY Measure of Overall Stability (4.4)**

Measure of Tenderness or Stiffness The initial slope of the intact statical stability curve indicates the rate at which a righting arm is developed as the ship is heeled over. This slope is GM! A steep initial slope indicates the rapid development of a righting arm and the vessel is said to be stiff. Stiff vessels have short roll periods and react strongly to external heeling moments. A small initial slope indicates the slower development of a righting arm and the vessel is said to be tender. Tender vessel have longer roll periods and react sluggishly to external heeling moments.

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OVERALL STABILITY Effect of a Vertical Shift in the Center of Gravity on the Righting Arm (4.5) A rise in KG decreases the righting arm (GZ). This change in GZ can be found from: Where: Gv is the final vertical location of the center of gravity. G0 is the initial location of KG. (Could be from Cross Curves or from an earlier calculation)

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OVERALL STABILITY Effect of a Vertical Shift in the Center of Gravity on the Righting Arm (4.5) Sine Correction: So GoGv is the shift in KG!

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OVERALL STABILITY Effect of a Vertical Shift in the Center of Gravity on the Righting Arm (4.5) EXAMPLE DDG-51 From the Cross Curves at 20 degrees heel And disp = 8000 LT GoZo is 10 ft. If KG is 22 ft, then the actual Righting Arm is: So, GZ actually equals 2.5 feet!

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DDG-51 Cross Curves

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OVERALL STABILITY Effect of a Vertical Shift in the Center of Gravity on the Righting Arm (4.5) What are the practical implications of KG=48 ft?

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**Effect of Increased Displacement on the Righting Arm (4.5)**

OVERALL STABILITY Effect of Increased Displacement on the Righting Arm (4.5) A higher displacement should increase the Righting Moment as RM= Displacement * RA But, if the added weight is high, then the KG increase could cause a reduction in GZ Weight added low down usually increases stability

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) So far (in Ch. 4) we have only considered the case where the Center of Gravity is on the centerline (TCG=0). The center of gravity may be moved off the centerline by weight additions, removals, or shifts such as cargo loading, ordnance firing, and movement of crew.

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) Notice the improvement crew weight provides!

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OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) Another way to shift TCG! Notice the keel position!

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) The new righting arm created by a shift in TCG may be computed at each angle from the Cosine Correction: Ex. DDG51, disp = 8000 LT, from Cross Curves at 20 degrees heel and KG=22 ft, then GZ is 2.5 ft. If weight is moved such that TCG is now 1 ft, then

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6)

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) The new righting arm (GtZt) created due to the shift in the transverse center of gravity is either shorter or longer than the righting arm created if TCG=0. It depends on which side... The range of stability has decreased on the side that the transverse center of gravity has shifted to (starboard) but has increased on the side it shifted from (port). Recall P-100! What happened when you hiked out on the leeward side?!

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6) EXAMPLE 4-2. DDG51, Disp=8600 LT, KG=23.84 ft, TCG=0.4 At 10 degrees heel...

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**Stability Change for Transverse Shift in the CG (4.6)**

OVERALL STABILITY Stability Change for Transverse Shift in the CG (4.6)

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**STABILITY Damage Stability (4.7)**

“Flooding” - Water ingression such that the vessel has sinkage and trim but no list. May be intentional. “Damage” - Water ingression such that the vessel has sinkage, trim and list.

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STABILITY Damage Stability (4.7) “Flooding” or “Damage”?

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**STABILITY Damage Stability (4.7) Analyze Stability after Damage using:**

Lost Buoyancy Method OR Added Weight Method (we use this one in EN400)

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**Lost Buoyancy Method (4.7.1) (not the one we use in EN400, but common)**

STABILITY Lost Buoyancy Method (4.7.1) (not the one we use in EN400, but common) Analyzes damage by changes in buoyancy versus changes in the Center of Gravity. Premise is that the ship’s CG does not move. Since weight does not change, total buoyant volume must also be constant. Therefore, the ship makes up any lost buoyant volume from damage by listing, trimming, and draft changes. Need to calculate new volumes, which require special programs to be efficient.

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**STABILITY Added Weight Method (4.7.2) (the one we use in EN400 !)**

Damaged Ship Modeled as Undamaged But with Water-Filled Spaces (Compartments) that add weight. Average Distances of Space from Keel, Midships, and Centerline Known & Water Density Known. Therefore We Can Solve for Shifts in “G” as a Weight Addition Problem (which we know how to do!)

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**STABILITY Added Weight Method (4.7.2)**

Independently Solve for Damaged Condition KGnew TCGnew Draft and Trim (new) Must know compartment contents to find Total of (Water Weight) Added (unless it is a storage tank). This involves a “Permeability” factor.

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**STABILITY Permeability (4.7.2.1)**

Compartments are rarely 100% flooded during damage, due to trapped air, equipment, etc. Ratio of volume occupied by water to the total gross volume is defined as “permeability”. Permeability = Volume Available for Flooding Total Volume Permeability is always < or = to 100%! Usually it is 65-95%.

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**Damage Stability Design Criteria (4.7.3)**

Guiding Rules for vessel design. Note that the criteria used in static analysis will neglect the impact of dynamic forces such as wind and waves, so the criteria are simply relative.

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**Damage Stability Design Criteria (4.7.3)**

Three Main Criteria “MARGIN LINE” “LIST” “EXTENT OF DAMAGE TO HULL”

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**Damage Stability Design Criteria (4.7.3)**

MARGIN LINE LIMIT Highest permissible location of any damaged waterplane. Must be at least 3 inches (0.075 m) below top of the bulkhead deck at the side.

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**Damage Stability Design Criteria (4.7.3)**

LIST LIMIT Heel by damage 20 degrees. Naval machinery to operate indefinitely at a permanent list 15 degrees (most will function up to ~25 degrees for a few hours). Assumes personnel can continue damage control efforts effectively at a permanent list of 20 degrees. Ship must possess adequate stability against weather to be towed when at 20 degree list.

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**STABILITY Damage Stability Design Criteria (4.7.3)**

EXTENT OF DAMAGE TO THE HULL LIMIT 100 ft LOA: must withstand flooding in one space. ft LOA: flooding in two adjacent compartments. Warships, troop transports and hospital ships over 300 ft LOA: hull opening up to 15 % of Lpp. Others 300 ft: hull opening up to 12.5% of Lpp.

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**Foundering and Plunging (4.7.4)**

STABILITY Foundering and Plunging (4.7.4) A vessel as result of “damage” or other events can be lost several ways: Insufficient transverse stability. It rolls over. (Could be static or dynamic.) Insufficient longitudinal stability. “Plunging” If insufficient buoyancy. It sinks. “Foundering”

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**Free Surface Correction (4.8)**

STABILITY Free Surface Correction (4.8) Free Surface - A “fluid” that moves freely. Example: water, fuel and oil Question: sand, wheat, fish? Fluid Shift is a weight and causes the CG to shift in both the vertical and horizontal directions. Vertical shift is small for small angles and is usually ignored. Horizontal shift always causes a reduction is the righting arm (GZ), as water flows downhill!

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**Free Surface Correction (4.8)**

STABILITY Free Surface Correction (4.8) Free Surface Correction (FSC) - The distance the center of gravity would have to rise to cause a reduction in the righting arm equivalent to that caused by the actual transverse shift. "Virtual" center of gravity (Gv) - The effective position of this new VCG. Effective Metacentric Height (GMeff) - The distance from the virtual center of gravity (Gv) to the metacenter. Note: dynamic effects are neglected.

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**Free Surface Correction (4.8)**

STABILITY Free Surface Correction (4.8) The Big Picture

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**Free Surface Correction (4.8)**

STABILITY Free Surface Correction (4.8) The free surface correction to GM for small angle hydrostatics is: where: t is the density of the fluid in the tank in lb s2/ft4 s is the density of the water the ship is floating in lb s2/ft4 it is the transverse moment of area of the tank's free surface area in ft4 . s is the underwater volume of the ship in ft3.

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**Free Surface Correction (4.8)**

STABILITY Free Surface Correction (4.8) It is calculated for a rectangular tank as: breadth fwd length Length is in the ship’s x-axis Breadth is in the y-axis The dimensions are for the free surface, not the tank!

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**STABILITY Free Surface Effect (4.8.5)**

The new, effective VCG is Gv, so a sine correction is applied to get the statical stability curve

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**Minimizing Free Surface Effects (4.8.4)**

STABILITY Minimizing Free Surface Effects (4.8.4) Practical Solutions More, smaller compartments “Pocketing” = Topping off tanks Keeping tanks either full or empty by transfering (“30% Rule”). Avoid damage or flooding!

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**STABILITY Metacentric Height (4.9)**

Recall the Metacenter?!! (Hint, GM and KM) The metacentric height is NEGATIVE if the center of gravity (G) is above the metacenter (M). A negative metacentric height will result in an upsetting moment at small angles, but does not necessarily indicate the overall stability. Although we often say that a boat will capsize if it has negative GM, technically we can only say that it is unstable at that point. It may become stable later!

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**Stability One of the least and most stable boats for its size!**

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**STABILITY Metacentric Height (4.9)**

Recall that Overall Stability is measured by: Range of Stability Dynamical Stability Maximum righting moment The angle at which the maximum righting moment occurs. And the initial slope can tell us GM

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STABILITY Metacentric Height (4.9)

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**Initial Slope of the Curve of Intact Stability (4.9)**

At small angles, a right triangle is formed between G, Z, and M. The righting arm may be computed: As 0, if the angle is given in radians the equation becomes:

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**Initial Slope of the Curve of Intact Stability (4.9)**

Metacentric height can then be found from the initial slope of the Curve of Intact Statical Stability: GM (for angles <10 degrees) = GZ at one radian (57.3 degrees)!

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**Initial Slope of the Curve of Intact Stability (4.9)**

To find the slope either: Find the change in the y-axis over a given change in the x-axis. Draw a straight line with the initial slope and read the value of GZ at an angle of degrees (i.e. one radian).

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**Initial Slope of the Curve of Intact Stability (4.9)**

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**STABILITY Metacentric Height (4.9) LET US EXAMINE EACH GM CONDITION**

GM Positive GM Zero GM Negative

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STABILITY Metacentric Height (4.9)

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STABILITY Metacentric Height (4.9)

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STABILITY Metacentric Height (4.9)

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**STABILITY Metacentric Height (4.9) SUMMARIZING GM CONDITIONS**

GM Positive = Positive Stability GM Zero = Neutral Stability GM Negative = Negative Stability Metacentric Height is only a good indicator of stability over small angles. GM is initial slope of Curve Intact Stability

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**The End of Stability (Ch. 4)!**

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