حساب زراع الاستعدال او Gz Arm للسفن

Section 4.2 - 4.3

Righting Arm

&

Statical Stability Curves

A ship in static equilibrium is effected by outside forces that will alter its state of equilibrium.

The forces of wind- and the opposing force of the water below the waterline- will cause an external moment couple about the ship’s center of flotation.

CL

G

MT

Wind

Water Resistance

F

CL

G

MT

B

Wind

Water Resistance

The ship reacts to this external moment couple by pivoting about F, causing ashift in the center of buoyancy.

The center of buoyancy will shift because the submerged volume will change.

Note that there is no change in weight or it’s distribution so there is NO change in the location of G!

F

CL

G

MT

B

Because the location of B changes, the location of where the FB is applied also

changes. Because G does not move, the location of the s force does not change.

s

FB

The displacement force and the buoyant for are no longer aligned. The heeling over causes the creation of an internal moment couple.

F

CL

G

MT

B

s

FB

The external moment couple causes the creation of the internal moment couple to oppose it.

Wind

Water Resistance

As a result, the ship is now back into equilibrium, even as it heels over due to the wind force.

F

CL

We are concerned with the created internal moment caused by the offsetting of the ship’s weight and the buoyant force.

The offset distance of the applied forces, GZ, is called the MOMENT ARM. The length of this moment arm is a function of the heeling angle,

G

MT

B

FB

s

Z

Remember that a moment is created when a force acts at a

distance from a given point.

The created moment is called the internal

RM = GZs = GZFB

In the case of the created internal moment couple, we have the two force, s and FB, acting over the distance GZ.

CL

G

MT

B

Z

FB

s

This illustrates just one potential moment arm based upon one particular angle of . There are an infinite number of angles possible, therefore, an infinite number of moment arms that vary with the degree of heel, .

If we can plot the heeling angle versus the created moment arm GZ, we can create

the Intact Statical Stability Curve.

This is a typical curve. Notice that it plots the angle of heel on the x-axis and therighting arm on the y-axis.

The curve is in both the 1st and 4th quadrants (the 4th shows a heel to port). Typically only the curve showing a heel to starboard is shown as it is symmetrical.

Intact Statical Stability

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 25 50 75 85Heeling Angle

Mo

me

nt

Arm

GZ

The above chart plots the data presented in the text on p. 4-6 an 4-7.


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

With at 0 degrees, the moment arm is also is 0. The buoyant force and the ship’sweight are aligned. No moment is created.


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

As the angle of heel increases, the moment arm also increases. At 25degrees, shown here, GZ is 2.5ft.


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

As the angle increases, the moment arm increases to a maximum… here it is 4ft.As increases beyond this point the moment arm begins to decrease and the ship becomes in danger of capsizing…


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

...Remember, the internal moment couple created here is in response to the external couple created by outside forces. At GZ max the ship is creating its maximum internal moment. If the external moment is greater than the internal moment, then the ship will continue to heel over until capsized.


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

The angle of heel continues to increase, but the moment arm GZ, and thus the internal moment couple, decreases.


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5


Mo

me

nt

Arm

GZ

The angle has now increased to the point that G and B are now aligned again,but not in a good way. GZ is now at 0 and no internal moment couple is present. Beyond this point the ship is officially capsized, unable to right itself.

For any ship there exists the CROSS CURVES OF STABILITY. Like the Curves of Form, they are a series of curves presented on a common axis.

• The x-axis is the ship’s displacement, s, in LT• The y-axis is the righting arm, GZ, in ft• A series of curves are presented, each representing a different angle of heel

By plotting the data from the Cross Curves of Stability for a given displacement, you can create an Intact Statical Stability Curve.

Example: Plot the Intact Statical Stability Curve for an FFG-7 displacing 5000LT

Step #1. From the Cross Curves of Form, find the 5000LT displacement valueon the x-axis.

Step #2. Record the righting arm value for each curve, from = 0 to 80 degrees

Step #3. Draw the curve, using as x-axis, and GZ as y-axis

Intact Statical Stability, FFG-7

0

5

10

15

20

25

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90Heeling Angle f

Mo

me

nt

arm

GZ

GZ0 0.005 2.0010 3.8015 5.8020 7.7525 9.7530 11.7535 13.3040 14.7545 16.1050 17.2055 18.0060 18.6065 19.0070 19.3080 19.50

Intact Statical Stability Curve for FFG-7 @ s = 5000LT

… But a correction must still be made!!

In the Cross Curves of Stability, the data is presented assuming that:

KG = 0 (on the keel)

This is, of course, not realistic. It is done this way so that the curves may begeneralized for all drafts.

Once the curve data is recorded and plotted, a sine correction factor must be applied,shifting the KG to its correct position in order to get the

TRUE MOMENT RIGHTING ARM VALUE.

We will cover how to adjust the values within the table and the curve in the next section...

...To be continued….

حساب زراع الاستعدال او Gz Arm للسفن

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