23/06/2017 Chapter IV (Ship Hydro-Statics & Dynamics) Floatation & Stability 4.1 Important Hydro-Static Curves or Relations (see Fig. 4.11 at p44 & handout) • Displacement Curves (displacement [molded, total] vs. draft, weight [SW, FW] vs. draft ( T )) • Coefficients Curves ( C B , C M , C P , C WL , vs. T ) • VCB (KB, Z B ): Vertical distance of Center of Buoyancy (C.B) to the baseline vs. T • LCB (LCF, X B ): Longitudinal Distance of C.B or floatation center (C.F) to the midship vs. T 1
23/06/2017 4.1 Important Hydro-Static Curves or Relations (Continue) • TPI: Tons per inch vs. T (increase in buoyancy due to per inch increase in draft) • Bonbjean Curves (p63-66) a) Outline profile of a hull b) Curves of areas of transverse sections (stations) c) Drafts scales d) Purpose: compute disp. & C.B., when the vessel has 1) a large trim, or 2)is poised on a big wave crest or trough. 2
23/06/2017 How to use Bonjean Curves • Draw the given W.L. • Find the intersection of the W.L. & each station • Find the immersed area of each station • Use numerical integration to find the disp. and C.B. 4.2 How to Compute these curves • Formulas for Area, Moments & Moments of Inertia L a ) Area d ydx A ydx , A 0 L b ) Moments d xydx M xydx , M 0 Center of Floatation x M A / 2 2 c ) Moments of Inertia d x d x ydx I A L 2 2 I x ydx , I I x A 0 C F . 0 0 3
23/06/2017 Examples of Hand Computation of Displacement Sheet (Foundation for Numerical Programming) • Area, floatation, etc of 24’ WL (Waterplane) •Displacement (molded) up to 8’ WL •Displacement (molded) up to 24’ and 40 ‘WL (vertical summation of waterplanes) •Displacement (molded) up to 24’ and 40 ‘WL (Longitudinal summation of stations) •Wetted surface •Summary of results of Calculations Red sheet will be studied in 24’wl 8’wl detail 16’wl 4’wl area area area area •1-6 Areas & properties (F.C., Up to Up to Ic, etc) of W.L 8’wl 32’wl 4’wl 40’wl area •7-11 Displacement, Z B , and area Up to X B up W.L., vertical 24’ & 40 integration. wl •12-15 Transverse station area, MTI longitudinal integration for Disp. Up Disp. Up displacement, Z B , and X B to 16’wl MTI to 24’wl Wetted •16-18 Specific Feature (wetted surface surface, MTI, etc. Disp. Up Disp. Up Summary •19 Summary to 40’wl to 32’wl 4
23/06/2017 2 m S 2 1 3 S The distance between the two stations 2 Simpson's 1st 3 2 1 y 2 y 0 1 3 2 y 2 y 2 3 2 1 y y 0 12 3 4 3 y 2 y .. 1 2 4 2 Symmetric Formulas for the remaining coefficents m i 2 2 m S 2 2 3 S the distance between the two stations 2 Simpson's 1st rule coeff.; 2 - Symmetric 3 2 3 m S 2 3 3 3 1 2 1 y m S 2 ( from ) 4 3 3 3 3 2 h 2 h 2 m S 2, m S 2 5 6 3 3 3 3 5
23/06/2017 Illustration of Table 4 : C1 Station FP-0 AP-10 (half station) C2 Half Ordinate copy from line drawing table ( 24’ WL). (notice at FP. Modification of half ordinate) C3 Simpson coefficient (Simpson rule 1) (1/2 because of half station) C4 = C3 x C2 (area function) displacement C5 = Arm (The distance between a station and station of 5 (Midship) C6 = C5 x Function of Longitudinal Moment with respect to Midship (or station 5) C7 = Arm (same as C5) C8 = C6 x C7 Function of Longitudinal moment of inertia with respect to Midship. C9.= [C2] 3 C10. Same as C3. (Simpson Coeff.) C11. = C9 x C10. Transverse moment of inertia of WL about its centerline Table 5 is similar to Table 4, except the additional computation of appendage. 6
23/06/2017 Illustration of Table 8 For low WLs, their change is large. Therefore, it is first to use planimeter or other means to compute the half-areas of each stations up to No. 1 WL (8’ WL). C1. Station C2. Half area (ft 2 ) of the given station C3.C3/(h/3) ( divided by h/3 is not meaningful, because it later multiplying by h/3) (h = 8’ the distance between the two neighboring WLs) C4.½ Simpson’s Coeff. C5. C4 x C3 C6.Arm distance between this station and station 5 (midship) C7 C5 x C6 f(M) 7
23/06/2017 Illustration of Table 9 C1. WL No. C2. f(V) Notice first row up to 8’. f(v) C3. Simpson’s coeff. C4. C2 x C3 C5. Vertical Arm above the base C6. C4 x C5. f(m) vertical moment w.r.t. the Baseline. * Notice up the data in the first row is related to displacement up to 8’ WL. The Table just adding V) 8
23/06/2017 Illustration of Table 12 C1. Station No. C2. under 8’ WL. (From Table 8) C3. 8’ WL x 1 C4. 16’ WL x ¼ (SM 1 + 4 + 1) C5. 24’ WL x 1 C6. (C2 + C3 + C4 + C5) Function of Area of Stations C7. Arm (Distance between this station to midship) C8. C7 x C6 (Simpson rule) C9. C6*h/3 9
23/06/2017 4.3 Stability A floating body reaches to an equilibrium state, if 1) its weight = the buoyancy 2) the line of action of these two forces become collinear . The equilibrium: stable, or unstable or neutrally stable . • Stable equilibrium : if it is slightly displaced from its equilibrium position and will return to that position. • Unstable equilibrium : if it is slightly displaced form its equilibrium position and tends to move farther away from this position. • Neutral equilibrium : if it is displaced slightly from this position and will remain in the new position. 10
23/06/2017 • Motion of a Ship: 6 degrees of freedom - Surge - Sway - Heave - Roll - Pitch - Yaw Translation Rotation Axis x Longitudinal Surge Neutral S. Roll S. NS. US y Transverse Sway Neutral S. Pitch S. z Vertical Heave S. (for sub, N.S.) Yaw NS Righting & Heeling Moments A ship or a submarine is designed to float in the upright position . • Righting Moment : exists at any angle of inclination where the forces of weight and buoyancy act to move the ship toward the upright position. • Heeling Moment : exists at any angle of inclination where the forces of weight and buoyancy act to move the ship away from the upright position. 11
23/06/2017 For a displacement ship, W.L G---Center of Gravity, B---Center of Buoyancy M--- Transverse Metacenter, to be defined later. If M is above G, we will have a righting moment, and if M is below G, then we have a heeling moment. For submarines (immersed in water) B G G If B is above G, we have righting moment If B is below G, we have heeling moment 12
23/06/2017 Upsetting Forces (overturning moments) • Beam wind , wave & current pressure • Lifting a weight (when the ship is loading or unloading in the harbor.) • Offside weight (C.G is no longer at the center line) • The loss of part of buoyancy due to damage (partially flooded, C.B. is no longer at the center line) • Turning • Grounding Longitudinal Equilibrium For an undamaged (intact) ship, we are usually only interested in determining the ship’s draft and trim regarding the longitudinal equilibrium because the ship capsizing in the longitudinal direction is almost impossible . We only study the initial stability for the longitudinal equilibrium . 13
23/06/2017 Static Stability & Dynamical Stability Static Stability: Studying the magnitude of the righting moment given the inclination (angle) of the ship*. Dynamic Stability: Calculating the amount of work done by the righting moment given the inclination of the ship. The study of dynamic Stability is based on the study of static stability. • Static Stability 1) The initial stability (aka stability at small inclination) and, 2) the stability at large inclinations . • The initial (or small angle) stability: studies the right moments or right arm at small inclination angles. • The stability at large inclination (angle): computes the right moments (or right arms) as function of the inclination angle, up to a limit angle at which the ship may lose its stability (capsizes). Hence, the initial stability can be viewed as a special case of the latter. 14
23/06/2017 Initial stability • Righting Arm: A symmetric ship is inclined at a small angle dΦ. C.B has moved off the ship’s centerline as the result of the inclination. The distance between the action of buoyancy and weight, GZ , is called righting arm . • Transverse Metacenter: A vertical line through the C.B intersects the original vertical centerline at point, M . GZ GM sin d GMd if d 1 Location of the Transverse Metacenter Transverse metacentric height : the distance between the C.G. and M ( GM ). It is important as an index of transverse stability at small angles of inclination. GZ is positive, if the moment is righting moment. M should be above C.G, if GZ >0. If we know the location of M , we may find GM , and thus the righting arm GZ or righting moment can be determined given a small angle dΦ. How to determine the location of M ? 15
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