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Identifies Cross Curves (KN Curves and MS Curves) Derives The Formula G2 M8 + GM Sine Derives The Formula G2 KN KG Sine

The document discusses cross curves (KN and MS curves) that are essential for evaluating a ship's stability under different conditions. It derives two important formulas for calculating the righting lever (GZ) based on the ship's heel angle, incorporating parameters such as the distance from the keel to the buoyant force and the metacentric height. These formulas are crucial in naval architecture for ensuring operational safety and stability of vessels.

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wavewalker (Muhammad Syarif)
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22 views2 pages

Identifies Cross Curves (KN Curves and MS Curves) Derives The Formula G2 M8 + GM Sine Derives The Formula G2 KN KG Sine

The document discusses cross curves (KN and MS curves) that are essential for evaluating a ship's stability under different conditions. It derives two important formulas for calculating the righting lever (GZ) based on the ship's heel angle, incorporating parameters such as the distance from the keel to the buoyant force and the metacentric height. These formulas are crucial in naval architecture for ensuring operational safety and stability of vessels.

Uploaded by

wavewalker (Muhammad Syarif)
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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- identifies cross curves (KN curves and MS curves)

- derives the formula G2 = M8 + GM sine


derives the formula G2 = KN KG sine

Here’s a detailed explanation for each point:

1. Identifies cross curves (KN curves and MS curves)

Cross Curves of Stability (KN Curves)

KN curves represent the righting arm (KN) of a ship for different angles of heel at various displacements.

• K is the keel of the ship, and N is the intersection of the vertical line through the center of buoyancy at a
given angle of heel.

• KN is the distance from the keel to the line of action of the buoyant force, measured perpendicular to the
keel.

KN curves are essential for determining the ship's stability, as they help compute the righting lever (GZ) for
different conditions.

MS Curves

MS curves, or metacentric stability curves, show the metacentric height (GM) for a ship at various displacements
or draughts.

• The GM value indicates the ship's initial stability and resistance to rolling.

• MS curves help assess the ship’s stability margin under different loading conditions.

Both KN and MS curves are critical for evaluating a vessel's stability, loading, and operational safety.

2. Derives the formula GZ=KN−KGsin⁡(θ)GZ = KN - KG \sin(\theta)

Definition of terms:

• GZ: Righting lever or arm, the horizontal distance between the center of gravity (G) and the line of action
of buoyancy (B).

• KN: Distance from the keel to the line of action of buoyancy at a given heel angle (θ\theta).

• KG: Distance from the keel to the center of gravity.

• θ\theta: The angle of heel of the ship.

The derivation proceeds as follows:

1. The righting arm (GZGZ) is given by the horizontal component of the distance between the center of
gravity (G) and the point where the buoyant force acts (N).
2. From the geometry of the situation, GZGZ can be expressed as: GZ=KN−KGsin⁡(θ)GZ = KN - KG
\sin(\theta)

3. Here:

o KNKN is the total distance from the keel to the buoyant force line.

o KGsin⁡(θ)KG \sin(\theta) is the vertical projection of the center of gravity above the keel.

This formula relates the righting arm to the ship's heel angle and stability characteristics.

3. Derives the formula GZ=MB+GMsin⁡(θ)GZ = MB + GM \sin(\theta)

Definition of terms:

• GZ: Righting arm or lever.

• GM: Metacentric height, the distance between the center of gravity (G) and the metacenter (M).

• MB: Distance between the metacenter (M) and the center of buoyancy (B) at the given heel angle
(θ\theta).

• θ\theta: Angle of heel.

The derivation proceeds as follows:

1. The total righting arm (GZGZ) can be divided into components: GZ=MB+GMsin⁡(θ)GZ = MB + GM
\sin(\theta)

2. Here:

o MBMB: The vertical distance between the center of buoyancy (B) and the metacenter (M), which
depends on the ship's geometry.

o GMsin⁡(θ)GM \sin(\theta): The component of the metacentric height contributing to the righting
arm.

The formula shows how the stability of a ship depends on both the metacentric height and the heel angle.

These formulas are fundamental in naval architecture, enabling precise calculations of a vessel’s stability under
varying conditions. Let me know if you’d like any further elaboration!

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