Initial Transverse Stability(Ổn định ngang ban đầu)

Nguyễn Anh TuấnNaval Architecture and Marine Engineering

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Training Course TSL_Oct2010

Main Sections

Basic knowledge and essential conceptsTransverse stability (Ổn định ngang)Example for calculating initial transverse

stability

Section 1:Basic Knowledge and Essential Concepts

Archimedes’s Principle

According to Archimedes’ Principle, Buoyancy Weight (Displacement)W = =. Note:is mass density of fresh water 1 t/m^3 and of salt water 1.025 ÷ 1.03 t/m^3 is displaced volume [m^3] , W [ton]

Example: A box (a mass of 4000kg, a displaced volume of 8 cu.m) is floating in fresh water. The balance state sets when buoyancy weight equals to displaced volume.

How does ship float on water when sticking needle can not float?

DOF of ShipA floating ship in free fluid has 6 degrees of freedom. Translation: Fore and aft (is termed surge), transverse (is termed sway) and vertical

( is termed heave) Rotation: Fore and aft axis ( is termed heel / roll ), transverse axis ( is termed trim /

pitch) and vertical axis (is termed yaw)

http://upload.wikimedia.org/wikipedia/en/5/59/Translations.PNGhttp://upload.wikimedia.org/wikipedia/en/4/49/Rotations.PNG

heel

Figure 2.2 - The Projection of Lines onto 3 Orthogonal Planes

Projection of ship’s line

Top view (Half-Breath plan ~ Hình chiếu bằng)

(Đường nước)

(Đường sườn)

Front viewBody plan ~ Hình chiếu chính

(Đường cắt dọc)

Side viewSheer plan ~ Hình chiếu cạnh

Baseline ~ Đường cơ sở

Center line ~ Đường tâm

O)(Amidships

(After ~ Phía lái ~ Phía đuôi tàu)

(Forward ~ Phía mũi tàu)

Ship is floating when F = W

Buoyancy force F acts at center of buoyancy B: F = Gravity force W acts at center of gravity G: W =

Where:• Mass density [ton/m^3]• Acceleration of gravity g [m/s^2]• Volume of displacement [ton]

Center of Gravity G• Longitudinal centroid (Hoành độ trọng tâm) from amidships LCG ~ XG [m]• Vertical centroid (Chiều cao trọng tâm) from the keel VCG ~ KG ~ ZG [m] • Transverse centroid TCG is nearly in the centerline plane, thus it is often assumed to be in

that plane TCG = 0

Center of Buoyancy B• Longtitudinal center of buoyancy (Hoành độ tâm nổi) from amidships LCB ~ XB [m]• Height of center of buoyancy (Chiều cao tâm nổi) from the keel KB ~ ZB [m]• Transverse center of buoyancy from centerline TCB = 0

Base line

TCBLCB

KB

Center line

Section 2:Initial Transverse Stability

Transverse Stability Conditions

Stable equilibrium is that position of metacenter M is above the center of gravity G, Righting Moment (Moment hồi phục) acts ship to return its back original position.

Neutral equilibrium is the center of gravity and metacenter that are same location. The ship inclines 7 – 10 degrees (Small list).

Unstable equilibrium is that metacanter locates lower than center of gravity. Ship will be capsized by negative righting moment.

Note: Metacenter M is defined by intersection of the vertical line through centerline plane and the vertical line through buoyancy force line (Tâm định khuynh hay tâm quay M là điểm giao giữa đường lực qua tâm nổi và đường thẳng đứng qua mặt phẳng chứa đường tâm tàu).

Initial Transverse Stability (Ổn định ngang ban đầu)

Note: : area of waterplane at height z above the keel [m^2] : Transverse Second moment ~ Transverse Moment of inertia (Moment quán tính hình học) [m^4]BMT: Transverse metacenter above center of buoyancy ~ Transverse metacentric radius (Bán kính tâm nghiêng ngang) is the distance between the center of buoyancy B and metacenter M [m]GMT : Distance between centroid and metacenter (Chiều cao tâm nghiêng ban đầu) [m]

KG

BMT =

KMT = KB +BM

KB =

GMT = KM - KG

Note: GZ= GMT sin = (KB +BMT – KG) sin = BMT sin – (KG – KB) sin

Initial Transverse Stability (Ổn định ngang ban đầu)

Righting Arm (cánh tay đòn phục hồi ~ cánh tay đòn ổn định) GZ = GMT sin [m]

Righting Moment (moment phục hồi ~ moment ổn định) RM = GZ . [ton.m]

GMT = KMT - KG

GZ = GMT sin

RM = GZ .

Cánh tay đòn ổn định trọng lượng (p.53 [5])Cánh tay đòn ổn định hình dáng (p.53 [5])

GM

RM

GZ

Is Positive Stability GOOD?

Section 3:Example for Calculating Initial Transverse Stability

Example 6 -1 p. 117 [2]

Saltwater1. Displacement (Trọng tải) = 922 ton2. Length overall (Chiều dài toàn bộ tàu) L = 240 ft3. Beam (Bề rộng tàu) B = 28 ft4. Draft (Mớn nước ~ Chiều chìm) d = 8 ft5. Vertical Center of Buoyancy above the keel KB = 4.2 ft6. Vertical Center of Gravity above the keel KG = 8 ftThe offsets from 8 ft waterline following station spacing of 20 ft, see the below tableDetermine the initial transverse stability of ship (BM, KM, GM)

x z x^3 Multiflier (Trapezoidal Rule) Function of f(I)  0.5 8 0.125 0.5 0  2.5 28 15.625 1 16  6.6 48 287.496 1 287  

10.1 68 1030.301 1 1030  12.5 88 1953.125 1 1953  13.7 108 2571.353 1 2571  

14 128 2744 1 2744  13.9 148 2685.619 1 2686  

13 168 2197 1 2197  10.8 188 1259.712 1 1260  

6.6 208 287.496 1 287  2.1 228 9.261 1 9  0.5 248 0.125 0.5 0  

      Sum 15,041.11 ft^3      h 20 ftMoment Inertia of waterplane about the longitudinal centerline I 200,548.17 ft^4

𝐼=∫0

𝐿 23𝑟3𝑑𝑥=¿ 2

3h 𝑓 ( 𝐼 )=2

3h∑ 𝑥3.𝑑𝑥¿ 0 4 8 12 16

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

x is half ordinate [ft] (Tọa độ nữa bề rộng tàu)

Example 6 -1 p. 117 [2]

Volume of Displacement (Thể tích chiếm nước)

Transverse metacenter radius (Bán kính tâm nghiêng ngang)BM =

Height of metacenter from keelKM = KB + BM = 4.2 + 6.22 = 10.4 [ft]

Metacenter heightGM = KM – KG = 10.4 – 8 =2.4 [ft]

Note: = is specific volumeSpecific volume of saltwater 35 [ft^3/ton] [2]

Example of Cross Curves of Stability (Đồ thị ổn định ngang)

Next Sections

Effect of changes in the weightInitial Longitudinal Stability with small trimCurves of StabilityAnd so on

References

[1] Eric Tupper. 2002 . Introduction to Naval Architecture. 3rd ed. Butterworth Heinemann. Great Britain.[2] Thomas C. Gillmer and Bruce Johnson. 1982. Introduction to Architecture. Naval Institute Press. USA[3] The Society of Naval Architects and Marine Engineers. 1988. Principles of Naval Architecture Volume I -Stability and Strength. The Society of Naval Architects and Marine Engineers. USA[4] D.R. Derrett and C.B. Barrass. 2001. Ship Stability for Masters and Mates. Butterworth Heinemann. Great Britain[5] Trần Công Nghị. 2006. Lý thuyết tàu –Tập 1 Tĩnh học và Động lực học Tàu. NXB ĐH Quốc Gia. Việt Nam[6] ireference.ca, ship stability, viewed 2 Oct 2010, < http://www.ireference.ca/search/Ship%20stability/ >[7] Federation of American Scientists, Damage Control Training: Stability and Buoyancy Lessons, viewed 3 Oct 2010 < http://www.fas.org/man/dod-101/navy/docs/swos/dca/stg4-01.html >