SHIP BUOYANCY AND STABILITY. Lecture 02 Ship equilibrium and introduction to ship hydrostatics

Transcription

1 SHIP BUOYANCY AND STABILITY Lecture 02 Ship equilibrium and introduction to ship hydrostatics 1

2 Literature J. Matusiak: Laivan kelluvuus ja vakavuus Biran A. B., Ship Hydrostatics and Stability, 2003 J. Matusiak: Short Introduction to Ship Theory (Part 1) Rawson, K. J J. Basic Ship Theory Volume Buoyancy and Stability of a Ship, Finnish text book A Ship Stability book in Enlish. Shorten version of the Finnish textbook, in Enlish Check the library 2

3 BEFORE THIS LECTURE General notes on ship stability Notes on hydrostatic pressure Floatin bodies: Archimede s law Notes on Ship definitions Definition of Buoyancy and center of buoyancy Equilibrium of a ship Now, you should be able to: Estimate the hydrostatic pressure in any points of a fluid Describe the influence of fluid density on the immersed volume Describe the parameters of an equilibrium floatin condition for a ship Describe how the draft chane by chanin the ship weiht 3

4 Introduction Preparation for the laboratory test Dynamic stability Followin lecturers Ship equilibrium and introduction to ship hydrostatics Center of Buoyancy Center of Flotation Ship initial Stability The stability curve (GZ curve) Second eneration of intact stability criteria Ship Damae Stability (Part I & II) Stability special topics 4

5 A W : water plan area ; F : center of the water plan area; S W :wetted surface; :immersed volume; B: center of buoyancy Floatin plane, coincident with the undisturbed sea surface Waterplane Waterline 5

6 Reference System 1 vesiviivataso Waterplane (pinta-ala A W ) z x B h K y L T z 0 Center of buoyancy K: oriin of the reference system XYZ; B:oriin of the reference system xyz. midship 6

7 Reference System 2 : heel anle : trim anle :yaw anle Note that ship immersed hull i.e. displacement does not vary with the yaw anle 7

8 T F - T A TRIM of the ship T F > T A Bow down, stern up T F < T A Stern down, bow up 8

9 How is the trim in this case? ZERO TRIM, ZERO HEEL i.e. EVEN-KEEL CONDITION 9

10 Reference System --- NOTE Z M K M Y There is no difference in terms of equations if we draw the ship inclined or the floatin plane inclined. What matters is that the reference system, fixed with the ship, remains body-fixed and the ravity remains normal to the sea surface. 10

11 Equilibrium of the ship W 0 ( G B) W 0 W ( G B)parallel tow The weiht W is applied in the center of ravity G. That means that for equilibrium weiht and buoyancy vectors have to stay on the same line, that is normal to the floatin plane. 11

12 How can I define the floatin condition of the ship at the equilibrium? Z W WL O K X 12

13 Floatin plane The weiht and the buoyancy are normal to the equilibrium floatin plane i.e. the sea surface; Usin the eneral equilibrium equation we need to determine the floatin plane, i.e. the draft and the trim and heel anle; Anyway, it is possible to individuate the floatin plane even with equivalent parameters; Let s introduce some eometric features reardin an inclined floatin plane with respect to the reference system OXYZ parallel to the main reference system KXYZ 13

14 The W W 0 ( G B) W 0 G x, y, z B x, y, z The The ( G B) x y z x y z W W W cos tan cos tan cos The 14

15 The Geometric Features - O is the intersection point between WL and Z - s is the intersection line between WL and YZ - t is the intersection line between WL and XY The - r is the intersection line between WL and XZ - is the anle between the line s and the Y axis The - is the anle between the line r and X axis - is the anle between the line n (the normal to the plane WL) and the Z axis The - is the anle between the line t and the X axis 15

16 Gleijeses Equations equilibrium equation of a ship W 0 W W ( G B) W 0 Mx My Mz ( y ( x ( x y ) cos ( z x) cos ( z x) cos tan ( y z) cos tan 0 z) cos tan 0 y) cos tan 0 tan tan tan tan ( y ( z ( x ( z ( y ( x y x 16 ) z) ) z) y ) x)

17 02/11/ Maria Acanfora Equilibrium equation-floatin condition The tern T 0,, determine the equilibrium floatin position of the ship (midship draft and floatin plan) The mathematical problem is characterized by three equations for three variables. Ship immersed volume and its center of buoyancy vary with the floatin condition. ) ( ) ( tan ) ( ) ( tan z z x x z z y y ),, ( );,, ( );,, ( ),, ( T z T y T x T T 0 = OK

18 Archimedes Principle Center of Gravity (G): Chanes position only by chane/shift in mass of ship Does not chane position with movement of ship Center of Buoyancy (B): G Chanes position with movement of ship -> underwater eometric center moves Also affected by displacement 18

19 Ship equilibrium with no trim and heel In order to have and equal to zero, from the equilibrium equations comes out that: y G = y B = 0, x G = x B. Even-keel condition The ship bein symmetrical respect to the diametral plan will have the transversal coordinates of the center of buoyancy as zero for a null. This condition ensures that Weiht and Buoyancy forces act on the same line. The ship assumed to have no trim and heel need to comply only the first equation to find W equilibrium. W Ship will submere and emere till it doesn t find the draft at which the displaced volume balances the ship weiht G 19

20 How to find numerically the equilibrium equation? Analytically volume and centre of buoyancy approximated expressions - Cylindric bodies method, small anles - Metacentric method i.e. wede method Grafically volume and centre of buoyancy curves evaluated for different draft, trim and heel anles - Interpolation of hydrostatics calculation Numerically volume and centre of buoyancy numerically evaluated step by step - Iterative computer proram ( T ( y 0,, ) tan x( T,, ); (,, ); (,, ) ( z 0 y T0 z T0 ( x tan ( z 20 y ) z) x) z)

21 z Floatin bodies Aw x B y Due to the external actions, the body is forced to move: waterplane of the floatin body is free to chane as its displaced volume. The center of buoyancy for a eneric body, is linked to the eometry of the immersed volume, accordin to the body position. The ship eometry is not a box, not a cylinder, not described by an analytical equation. The centers of buoyancy do not belon to a known surface Some assumption could be done in analyzin the center of buoyancy variation: Qualitatively analysis Approximation of the center of buoyancy curve with a circumference for small anle 21

22 ISOHULL CONCEPT A ship, in normal operational conditions has a constant displacement. This means that under external moments, the buoyancy vector has always the same manitude, but applied in different points. The immersed volume of the ship, i.e. the amount of the displaced water is constant. A family of hull with different shapes, characterized by the same volume, is called ISOHULL 22

23 HOW IS THE CURVE OF THE CENTER OF BUOYANCY FOR A SHIP? HOW DOES THE CENTER OF BUOYANCY VARY? HOW WE CAN EXIMATE IT? 23

24 Example of a center of buoyancy curve Z Let s assume fixed volume and fixed trim: = 0 (for small anles the draft is the same) θ=0 Let s incline the ship from 0 to 20 4 B=B(, φ, θ)=b( 0, φ, 0) φ ϵ [0, 20 ] T B 0 K For a hull with a constant volume, inclinin the ship only transversally, the center of the immersed volume (center of buoyancy) belon to a curve in the space. 24

25 Projection of the curve Z B=B( 0, φ, 0) φ ϵ [0, 20 ] B 0 K Z Y X Y XY Projection on the horizontal plane XY The curve of the center of buoyancy for finite anles comes out from the YZ plane Measured from the aft perpendicular 25

26 Z Projection of the curve B 0 T Projection on the vertical plane XZ Z B=B( 0, φ, 0) φ ϵ [0, 20 ] K XZ X Measured from the aft perpendicular 26

27 Z Projection of the curve B 0 K T Y M Projection on the lateral plane YZ B=B( 0, φ, 0) φ ϵ [0, 20 ] YZ BM is called Metacentric radius. B 0 27

28 HOW CAN WE EXIMATE HOW THE CENTRE OF BUOYANCY CHANGES WITH THE FLOATING PLANE? approximated methods: 1. vertical variation 2. variation with heel and trim: wede method 28

29 T Vertical center of buoyancy variation. How does the center of buoyancy chane only with ship draft? z z, z' z, z' W G T W + W W y' W T= = = A W T G G y' B y K B' T y B' T y K K a) b) c) W KB = KB' KB = 1 ' ' z d 1 z d KB = 1 + z d + z d 1 z d z d T + T 2 = KB - + T + T 2 KB 29

30 Variation with heel: Wede method VERTICAL SIDE ASSUMPTION + v 2 = + v 1 = is the ship volume Equivalent hull or Isohull assumption v 1 = v 2 = v Emered wede=immersed wede is the common volume (white part in the picture) y' z' z' v 1 1 F N B v 2 2 ' B 0 B B 0 B φ (δy B, δz B ) 1 2 (y, z ) Vector oin from B 0 to B B 0 B' z' B T y' B 1 2 Vector connectin the wede centers K 30 y'

31 z' v 1 1 Wede method- transversal y' B 0 F K z' N y' B B B' z' B v 2 2 ' T y' WEDGE METHOD Pure transversal inclination. Small anle assumptions, metacentric assumption. The volume between the new waterplane and the initial waterplane are two wedes. The volume v 2 is now emered while the volume v 1 is now immersed with respect to the initial position For the equivalent hull hypothesis v 1 = v 2! The variation of the center of buoyancy (observed in the transversal plan) is proportional to the variation of the centers of the wedes by means of v/ y' B = 0 + v y' v y', josta y' B = v y' y' B /z' B = y' /z' z' B + z' B = z' B + v 2 T + z' 2 v 1 T z' 2, josta z' B = v z' B 0 B 1 = v

32 Demonstration of the metacenter definition The metacentric radius for small heelin anle could be written as: 1 v y B 0 B 1 r M 0 y v 2 pinta-ala s y B 0 M 0 = B 0B 1 = v 1 2 Assumin y as half breadth s = 0.5 y (y ) The distance between centers of the wedes is the two r = 2 (2/3) y. The moment of inertia of the waterplan, I T is iven by: B 0 M 0 = B 0B 1 = v 1 2 B 0 M 0 = I T I T = 2 3 L v 1 2 = 1 2 y y dx = 2 3 L y 3 dx L y 3 dx Y y(x ) y(x ) X

33 What is the center of Floatation of a ship? How it is defined? Let s introduce the topic startin from the SMALL ANGLE assumption. 33

34 Emered wede Isohull properties: Euler Theorem Immersed wede α and α floatin planes a axis of inclination, intersection of the two planes ϕ anle of inclination Assumin ϕ infinitesimal anle If the anle of inclination ϕ is small, meanin that the floatin plane are very close, then it is possible to demonstrate the Euler theorem: Euler Theorem The intersection of two very close floatin planes is also very close to their centers. That means that the axis of inclination is central for both the waterplanes. 34

35 Euler Theorem The intersection of two very close floatin planes is also very close to their centers. That means that the axis of inclination is central for both the waterplanes. F is the eometrical center of the initial waterplane F belons to the axis of inclination x: this axis is a central axis of inertia or central. After a small inclination, the center the new waterplane is very close to F They are considered to be the same The x axis is baricentral also for the new waterplane 35

36 Euler Theorem applied to ship A ship that inclines of small anles, rotates around a constant central axis in F and the immersed volume remains constant!!! Transversal inclination F Lonitudinal inclination 36

37 PLEASE NOTE The center of floatation F is the centroid of the waterplane area. It is a POINT, which means it has 3 coordinates (x F, y F, z F ). For small anles F is assumed to be constant and equal to F calculated in the even-keel condition! In eneral the most important coordinate is the lonitudinal one: x F (or LCF); For symmetrical hull y F =0, F belons to the lonitudinal axis! F belons to the waterplane area i.e. to the floatin plane, then z F =draft! 37

38 Center of floatation In order to have an isohull lonitudinal inclination, the ship will rotate around a transversal axis y, passin thorouh the point F. The lonitudinal axes x x The point F is called center of floatation. Assumin this point as oriin of the x axis then: F y(x') x' dx' L F = y(x') x' dx' = 0 L A x F is the lonitudinal coordinate of the center of flotation and depends on the reference frame. If the reference frame has its oriin in F then x F =0 A w L/ 2 L/ 2 y( x) dx L F L A y( x') dx' x F LCF / 2 1 L A w L/ 2 y( x) xdx The waterplane area does not chane with the reference frame 38

39 WL WL 0 a) Cental Moments of Inertia of the waterplane: I T and I L v 2 L A F K z dv = x' da W = 2 y(x) x' dx' L F v 1 x =0 x=x is of symmetry for the waterplane; Ixx is a central moment of inertia. b) dixx = y(x) 3 dx di yy = x 2 da W = x 2 2 y(x) dx I T = Ixx di L = x x F 2 daw = x 2 da W 2 x x F da W + x F 2 da W WL 0 A W y' F x F y dx y(x) The moments of inertia of the waterplane represent important characteristics in assessin the ship initial stability. In the main reference frame Kxyz, let s define Ixx and Iyy x, x' I L = x 2 da W 2 x F x da W + x F 2 da W = Iyy 2 x F 2 A W + x F 2 A W = Iyy x F 2 A W. Ixx = dixx = 2 3 y 3 (x) dx ja Iyy = diyy = 2 y(x) x 2 dx L L 39

40 To Keep in Mind: For a iven draft (even-keel) A w = area of the waterplane (m 2 ) F= center of flotation, i.e. center of the waterplane (m) LCF= lonitudinal coordinate of the center of flotation (m) I T = moment of inertia of the waterplane about the lonitudinal centroidal axis(m 2 ) I L = moment of inertia of the waterplane about the transversal centroidal axis(m 2 ) BM T = BM metacentric radius (sometimes transversal metacentric radius) (m) BM L lonitudinal metacentric radius (m) These values, after small inclinations, remain constant!!! These values, for lare inclinations, CHANGE!!! 40

41 Central axes of inertia Let s assume I T and I L respectively as the transversal and the lonitudinal central moments of inertia of the waterplane. EVEN-KEEL floatin contidion with =0 and =0 I I L T I xx I y' y' I yy x 2 F Aw Still valid for small anle inclinations from the E-K. Basically the waterplane for small inclination anles remains the same. Z Generic floatin contidion with 0 The waterplane for finite inclination anles does not remain the same T K Product of inertia I xy = A W x y da W = 1 2 arctan 2 I xy I yy I xx. I xyα =0 y * y α y 41 A W F x x x * α

42 Center of Buoyancy curve Center of Flotation Wede method Small anle assumption SUMMARY Metacenter and metacentric radius You should be able to: Describe qualitatively how the center of buoyancy varies, accordin to draft chane or heelin anle Describe the ship inclination axes, defined by the center of flotation Use the wede method results Describe the main outcomes of the small anle assumption Define the metacenter and its relation with the curve of the center of buoyancy 42