2018-08-07 Lecture Note of Naval Architectural Calculation Ship Stability Ch. 7 Inclining Test Spring 2018 Myung-Il Roh Department of Naval Architecture and Ocean Engineering Seoul National University 1 Naval Architectural Calculation, Spring 2018, Myung-Il Roh Contents þ Ch. 1 Introduction to Ship Stability þ Ch. 2 Review of Fluid Mechanics þ Ch. 3 Transverse Stability Due to Cargo Movement þ Ch. 4 Initial Transverse Stability þ Ch. 5 Initial Longitudinal Stability þ Ch. 6 Free Surface Effect þ Ch. 7 Inclining Test þ Ch. 8 Curves of Stability and Stability Criteria þ Ch. 9 Numerical Integration Method in Naval Architecture þ Ch. 10 Hydrostatic Values and Curves þ Ch. 11 Static Equilibrium State after Flooding Due to Damage þ Ch. 12 Deterministic Damage Stability þ Ch. 13 Probabilistic Damage Stability 2 Naval Architectural Calculation, Spring 2018, Myung-Il Roh 1 2018-08-07 How can you get the value of the KG? K: Keel G: Center of gravity Ch. 7 Inclining Test 3 Naval Architectural Calculation, Spring 2018, Myung-Il Roh The Problem of Finding an Accurate Vertical Center of Gravity (KG) The problem of finding an accurate KG for a ship is a serious one for the ship’s designer. FG G ü Any difference in the weight of structural parts, equipment, or welds in different ship will produce a different KG. K There is an accurate method of finding KG for any particular ship and that is the inclining test. 4 Naval Architectural Calculation, Spring 2018, Myung-Il Roh 2 2018-08-07 Required Values to Find the KG (2/3) Heeling moment produced by total weight Righting moment produced by buoyant force Static equilibrium of moment t =F × GZ Inclining test formula r B 6 Naval Architectural Calculation, Spring 2018, Myung-Il Roh Required Values to Find the KG (1/3) GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG The purpose of the inclining test is to determine the position of the center of mass of the ship in an accurately known condition. z z¢ t e G: Center of total mass (gravity) B: Center of buoyancy at initial position * Angle of inclination should be within the range of inclinations G1: New position of center of total mass (gravity) after the weight has been moved where the metacenter (M) is not changed (about 7°~10°). B1: New position of center of buoyancy after the ship has been inclined Z: The intersection of vertical line through the center of buoyancy with the transversally parallel line to a waterline through center of total mass (gravity) M: The intersection of a vertical line through the center of buoyancy at previous position(B) with a vertical line through the present position of the center of buoyancy(B1) after the τe : Heeling moment ship has been inclined transversally through a small angle Required values to find the KG τr : Righting moment M f d • Draft FG Z • Total weight (FG) G 1 y • Hydrostatic values (KB, BM) • Weight (w) y¢ • Distance (d) B 1 • Angle of inclination (f )* FB t r 5 Naval Architectural Calculation, Spring 2018, Myung-Il Roh t r=F B × GZ GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG z ¢ z w× d t e GG1 = FG M f d th=F G × GG1 cos f Shift of center of total mass (gravity) FG Z G 1 y tr=F B × GZ » F B × GM sin f y¢ FGB× GG1 cosf = F × GM sin f B1 (QStatic equilibrium: FGB= F ) FB t r 3 Inclining experiment formula 2018-08-07 The angle of inclination can be measured when we perform the inclining test. Derivation of Inclining Test Formula (1/5) t r=F B × GZ GZ» GM ×sinf (at small angle f ) 8 Naval Architectural Calculation, Spring 2018, Myung-Il Roh GM= KB +BM - KGKG z w× d KG z¢ GM = t e FG × tanf M f d KG= KB + BM - GM FG Z w× d G Required Values to Find the KG (3/3) 1 y =KB + BM - FG × tanf y¢ B1 FB t r 7 Naval Architectural Calculation, Spring 2018, Myung-Il Roh t r=F B × GZ GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG d y ¢ d z ¢ z B B B z n B n z¢ 1 M ¢ ¢ KBsinf dyB cos f dzB sin f f KG, N0 NZ, Suppose that the center of gravity is located at K. Then the y y KN represents the righting arm. f d y ¢ B B B y¢ B1 FG F d z ¢ G F B B FB FB w KG, N NZ, KG, 0 GZ = KN =KN0 + N 0 N ¢ ¢ = KB ×sinf +d yBB ×cosf + d z × sin f These terms have positive effect to the restoring moment arm. 4 2018-08-07 t r=F B × GZ GZ» GM ×sinf (at small angle f ) 10 Naval Architectural Calculation, Spring 2018, Myung-Il Roh GM= KB +BM - KGKG ¢ KG= d zG z z z¢ z¢ M M f f FG G DerivationZ of Inclining Test Formula (2/5) y y d z ¢ G f d y ¢ d y ¢ B B B B y¢ y¢ B1 B1 FG ¢ FG ¢ d zB d zB FB FB FGB ' FB KG, NZ, N N0 K N ¢ d0zG sin f GZ = KN GZ =KN - KG ' =KN + N N ¢ ¢ 0 0 = KB ×sinf +d yBB ×cosf + d z × sin f ¢ ¢ = KB ×sinf +d yBB ×cosf + d z × sin f 9 Naval Architectural Calculation, Spring 2018, Myung-Il Roh This term has negative effect to the restoring moment arm. t r=F B × GZ GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG ¢ ¢ KG= d zG G d yG z zn GG= d y ¢ z¢ Z z¢ 1 G M M G1 ¢ Z1 dyG cos f f f d FG G ' G1 ' N FG FG G Z G Z ¢G Z1 y d yG 1 y d z ¢ G f f d y ¢ d yB B B B y¢ y¢ B1 B1 FG d z ¢ d zB B ¢ G ' dyG cos f FGB ' FB FGB ' 1 FB Derivation of Inclining Test Formula (3/5) K N¢ N K N¢ N dzG0sin f dzG 0sin f GZ =KN - KG ' GZ1 1 =KN - KG''' - G G1 ¢ ¢ ¢ ¢ = KB ×sinf +d yBB ×cosf + d z × sin f = KB ×sinf +d yBB ×cosf + d z × sin f ¢ ¢ -dzG ×sin f -dzG ×sin f This term has negative effect to the restoring moment arm. 5 2018-08-07 Derivation of Inclining Test Formula (5/5) Calculation of moment using “GZ” 12 Naval Architectural Calculation, Spring 2018, Myung-Il Roh t r=F B × GZ GZ» GM ×sinf (at small angle f ) GM= KB +BM - KGKG ¢ KG= d zG z z GG= d y ¢ z¢ z n n n b ¢ 1 G rPz=b R b r P z M M écosf- sin f ù n ê ú = Rb ()f ësinf cos f û f y¢ ü Rotational transformation matrix (b to n frame) f f y d FG FG FG z z¢ G DerivationZ of Inclining Test Formula (4/5) G Z Z1 cos(---f ) sin( f ) ¢G1 é ù n d yG y ê ú =Rb () -f yn ësin(--f ) cos( f ) û d zG f ü Rotational transformation matrix (b to n frame) -f y d y ¢ B B y¢ y y¢ b B1 d z ¢ ¢ B dyG cos f G ' FG ' G1 ' F é¢ ù B B édyBB ù Kécosf sinN f ù d y K N N = ê ú 0 ¢ ê ú ê ú -dzG sin f ëd zB û ë-sinf cos fû ê¢ ú ëd zB û GZ1 1 =KN - KG''' - G G1 ¢ ¢ dyBBB= d ycos f + d z sin f ¢ ¢ = KB ×sinf +d yBB ×cosf + d z × sin f ¢ ¢ ¢ ¢ d y dyGGG= d ycos f + d z sin f -dzG ×sin f -dyG ×cos f B 11 Naval Architectural Calculation, Spring 2018, Myung-Il Roh -d yG ü In n-frame, because the forces are acting on vertical direction, moment arm is yn-component. w× d KG= d z ¢ GG = t r=F B × GZ G 1 F GG= d y ¢ G GZ» GM ×sinf (at small angle f ) 1 G GM= KB +BM - KGKG z z¢ GZ =KN - KG''' - G G t e 1 1 1 ¢ ¢ =KB ×sinf + d yBB × cos f + d z × sin f M -dz ¢ ×sin f -dy ¢ ×cos f f d G G = KN -KG sinf -GG cosf F 1 G = GZ Z G f <<1 1 y = GM sinf : Inclining test formula =GMsinf - GG1 cos f y¢ d MGZ= D × B1 1 1 = D ×(GM sinf - GG1 cos f ) FB GMsinf= GG1 cos f Moment produced by total weight & buoyant force t GG w× d r \GM =1 = tanfFG × tan f : Assume that 6 Static equilibrium of moment 2018-08-07 Deflection Mark ! Mark ! Method of Measuring the Angle of Inclination (2/2) * Reference: http://www.mchl.fr 14 Naval Architectural Calculation, Spring 2018, Myung-Il Roh w× d Method of Measuring GM = F × tanf the Angle of Inclination (1/2) G How can you measure the angle of inclination when you perform the inclining test? z z¢ Inclining test formula f y L ¢ Deflection y L: Length of plumb line Plumb: The line which is exactly vertical or perpendicular to a level horizontal line Batten: Long and thin strip of wood Deflection \tanf = L 13 Naval Architectural Calculation, Spring 2018, Myung-Il Roh 7 2018-08-07 Precautions to be Taken During the Inclining Test Certain precautions must be taken to ensure that accuracy is obtained.