On the Ship s Trimming using Moments of the Gravity and Buoyancy Forces of High Order

Transcription

1 ABS TECHNICAL PAPERS 8 On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order Luben D Ivanov ), John E. Kokarakis ) ) American Bureau of Shipping, USA, Livanov@eagle.org ) Bureau Veritas, Greece, john.kokarakis@gr.bureauveritas.com Originall presented at the nd Int. Smposium on Ship Operations, Management and Economics held in Athens, Greece, September 7-8, 8 Abstract The present stud introduces a technique b which the hull girder shear forces and bending moments can be minimized. It is based on the equalit of the moments of higher order of the gravit and buoanc forces. Cargo hold weights determined on the basis of these equalities can be shown to result to minimum bending moments and shear forces. The proposal is depicted in the simplified case of a barge and a 4 K DWT bulk carrier.. Introduction Traditionall (Watson D G M, 998), when trimming a ship, two equations are used: zero moment (i.e., equalit of the total ship s weight and displacement) and first moment (i.e., equalit of the static moment relative to selected point of the total ship s weight and displacement). Thus, two unknown values can be determined the cargo in two cargo holds/tanks. Naturall, the cargo holds/tanks are more than two. Hence, to trim the ship, one should assume given cargo in the remaining cargo/holds/tanks in order to have onl two cargo holds/tanks with unknown cargo. This practice works relativel well but it could be improved in the era of fast computers used onshore and onboard ships. There will be no trim, hull girder bending and shear if the distribution of the gravit forces (the total ship s weight) and buoanc forces (ship s displacement) are eactl the same. This is not possible in real ships. However, it is clear that the more high order moments of the two forces are equal, the smaller the hull girder bending and shear will be. It means that one can increase the number of equations representing the equalit of moments of the ship s gravitational and buoanc forces relative to an selected point up to the number of cargo holds/tanks to be loaded. Logicall, one can epect that the hull girder bending and shear will be smaller when more than two moments of the gravit and buoanc forces are equal.. Calculation of the Moments of An Order The mathematical base of the calculations is the integration b parts (Murra R Spiegel, 968), i.e.: b u( ) v '( ) d = u( ) v( ) u' a ( ) v( ) d () a Eq. () can be applied to an moment (zero moment, first moment, second moment, etc.). For eample: Elementar area (see Fig. as an eample): da( ) = ( ) d () Elementar static moment relative to ais X: ds ( ) = da( ) = ( ) d (3) Elementar moment of inertia relative to ais X: di ( ) = da( ) = da( ) = ds ( ) (4) Area: ( ) = ( ) = ( ) ( ) (5) A d ' d ( ) ( ) ( ) u = v' = d (6) ( ) ( ) ( ) u' = ' v = d= (7) Static Moment relative to ais X: ( ) = ( ) = ( ) ( ) (8) S da A A d ( ) ( ) ( ) u = v' = da (9) ( ) = ( ) = ( ) = ( ) u' v d A A () Moment of inertia relative to ais X: ( ) = ( ) = ( ) () I di ds ( ) ( ) ( ) u = v' = ds () ( ) = ( ) = ( ) = ( ) (3) ( ) = ( ) = ( ) ( ) (4) u' v ds d S I ds S S d Geometric interpretation: I () = area OAD, S () = area OADE S ( ) d = EC = OB = area ODE I () = area OAD = area OADE area ODE (5) For the eample: b b a On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 85

2 ABS TECHNICAL PAPERS 8 c ( ) = c ' ( ) = h h 3 (6) ch ch ch A = SX( = h) = IX( = h) = (7) 6 The error of the simplified method (when trapezoidal rule for numerical integration is used) relative to the accurate analtical calculations of the rectangular triangle is.%. d Y ( ) c Fig. : Eample for rectangular triangle The same principle can be applied to real ships. An eample is given in Figs. 3~5 which refer to a 47K DWT product/oil tanker. The distance from point O to an transverse section is presented in dimensionless form for convenience, i.e., ξ = /L BP (8) The first dimensionless integral function of the lightweight, η(ξ), is: Fig. : Illustration of the simplified procedure where OA = ED = S () and OB = EC = I () = area ODE h X tons / m C O ξ p(ξ) B A ξ = / LBP [ - ] light weight η(ξ) Fig. 3: Lightweight distribution and its first dimensionless integral function η(ξ) of a 47K DWT product/oil tanker ξ ηξ ( ) = P( ξ )/PLW = p( ) d P ξ ξ (9) LW where P(ξ) = lightweight aft of ξ; P LW = total lightweight p(ξ) in Fig. 3 is the lightweight distribution, whereas P(ξ) is the integral function of p(ξ). The second integral function relative to point O at an section ξ will be: ξ ( ) m( ξ) = ξp( ξ) ηξdξ= area OABC area OAB () An eample for application of the procedure (e.g., m()) is given in Fig [t] ξ = / LBP [-] buoanc forces at even keel η(ξ) Fig. 4: Buoanc forces distribution and its first dimensionless integral function η(ξ) of a 47K DWT product/oil tanker On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 86

3 ABS TECHNICAL PAPERS [ - ] ξ = / LBP η(ξ) available cargo volume Fig. 5: Distribution of the available cargo volume and its first dimensionless integral function η(ξ) of a 47K DWT product/oil tanker () [t] [m] 5 5 η m () [t.m] m Fig. 6: Integral functions η() and m() for the lightweight of a 47K DWT product/oil tanker When working with absolute values, the difference between the curves is ver big. If drawn with the same scale, one of them is barel seen. Therefore, two different scales are used. The ship s equilibrium is represented b the following equations ( n = number of cargo holds to be loaded): MP, = MB, MP, = MB, () MP,n = MB,n where the subscript P stands for gravit forces, the subscript B for buoanc forces. The second subscript denotes the order of the corresponding moment (e.g., stands for zero moment, stands for the first moment, and n for the n th moment.). The zero moment is equal to the displacement or ship s weight. The first moment is equal to the static moment of the displacement or of the ship s weight, etc. If one assumes that the ship will be loaded up to its design draught, one can use the curve of the immersed cross-sectional areas to calculate the required moments of the buoanc forces b following the outlined procedure. The unknown parameters are in the ship s weight. These are the unknown cargo weights in the cargo holds/tanks envisaged for loading (e.g., two, three, four, etc. depending on the ship s size and tpe). For convenience in the calculations, the moments of the ship s weight are presented in the following form: M P,i = M Pi,k + M P,i,u () where the subscript k stands for all known moments of the ship s weight components (e.g., lightweight, weight of the fuel oil, lubricants, potable water and water for the machiner, weight related to the crew, etc.). The subscript u stands for the unknown moments of the cargo in the cargo holds/tanks envisaged for loading. Thus, one can present M P,i,k in the following wa MP,i,k = MLW, i + Mf, i + Mw, i + ML,i + Mc,i (3) where M LW,i = i th order of the moment of the lightweight, M f,i = i th order of the moment of the fuel oil, M w,i = i th order of the moment of the water, M L,i = i th order of the moment of the lubricants, M c,i = i th order of the moment of the weight related to the crew. It is convenient to represent the unknown moments of the cargo weights in the cargo holds/tanks envisaged for loading b the moments of the corresponding cargo holds/tanks volumes. In principle, these moments can be accuratel calculated when the capacit plan is known. Another wa to perform the calculations is to simplif the curve representing the cargo holds/tanks volume distribution within the length of each cargo hold/tank envisaged for loading. The geometric figure that could be used in the simplified approach is the trapezoidal one. If one assumes that the unknown cargo is to be loaded in three cargo holds/tanks, one can write for M P,i,u the following equation: MP,i,u =λ M V,i +λ M V,i +λ 3M V3,i (4) where λ, λ and λ 3 are the unknown coefficients (the have the same dimension as the specific gravit of the cargo, i.e. [t/m 3 ]) to be determined b Eq. (); M V,i, M V,i and M V3,i are the corresponding i th moments of the total volume in the first, second and third cargo hold/tank envisaged for loading. At this junction, it is worth noticing that moments of order higher than one cannot be calculated as product of the weight and the corresponding lever (powered to, 3, etc.). The correct calculation requires integration (the above given procedure is applicable for their calculation). Some simplification can be done if the distribution of the volume available for cargo, fuel oil, lubricants, etc. is represented in the form of trapezoid (see Fig. 7). Then, the equation for an n th moment, M n, relative to On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 87

4 ABS TECHNICAL PAPERS 8 the ais for comparison is: Ais for comparison () Fig. 7: Simplification of the volume distribution for cargo, fuel oil, lubricants, etc. n+ n+ Mn = ( ) + n+ (5) n+ n+ + ( ) n+ ( ) = + ( ) (6) When n =, M = trapezoid s area; when n =, M = static moment of the trapezoid to the ais for comparison; when n =, M = moment of inertia of the trapezoid to the ais for comparison; when n = 3, M 3 = trapezoid s moment of third order to the ais for comparison, etc. The definitions above can be easil envisioned if we consider the case = = of a rectangular distribution. 3. Simplistic Eample Fig. 8: Eample for a floating homogeneous parallelepiped To facilitate comparison between different cases, it is convenient to present the gradius r n in dimensionless format in the following wa: rn = ρ n / l (8) where l = length of the corresponding bod, figure, etc. The shear force and bending moment distribution for the simplistic case are shown in Fig. 9 The equations for q(), Q() and M() are: When l / q( ) ( c a) = 4 l (9) Before appling the proposed procedure for ship s trimming, simple eample for a homogeneous floating parallelepiped is given hereafter. The floating bod is balanced, i.e. its weight is equal to the buoanc forces and its centroid is equal to the center of buoanc. A parametric stud has been performed b changing graduall the value of b (see Fig. 8) starting from b = till b = a while keeping the same weight (gravit forces) and its centroid (in the eample, it is at midlength). For each combination of b and c, the maimum bending moment acting on the parallelepiped has been calculated together with the moments up to sith order. To avoid working with ver large numbers, one can present the n th moment in the following wa: Fig. 9: Load intensit q(), Shear Force Q() and Bending Moment M() for the simplistic eample n n n P n n M n M =ρ ρ = (7) P where P = gravit force, buoanc force, volume, etc. Q( ) ( c a) = l M( ) = ( c a) 3 l (3) (3) On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 88

5 ABS TECHNICAL PAPERS 8 When l / q( ) = ( c a) ( l) l l l Q( ) = ( c a) + 3 l ( ) ( ) l 5 7 (3) (33) M = C+ c a l + (34) 6 3 l C= ( c a) l (35) 4 The effect of the difference between radii of grations of gravit and buoanc forces with different order on the maimum of the bending moments is summarized in Fig.. One can observe in Fig. the linear dependenc between the bending moment and the difference between the second order radii of gration of gravit and buoanc forces, which coincides with the findings in (Ivanov L D, 7). The dependenc between the bending moment and the difference between higher order radii of grations of gravit and buoanc deviates from linear. For orders greater than three, the effect of the difference between the radii of gration of gravit and buoanc forces on the change of bending moments almost does not change. One can also observe in Fig. that the smaller the difference between moments of higher order, the smaller the bending moment relative reduction of B.M. [ - ] absolute difference between rg and rb [ - ] for r(g) - r(b) for r4(g) - r4(b) for r6(g) - r6(b) for r3(g) - r3(b) for r5(g) - r5(b) Fig. : Change of the bending moment vs. the difference between radii of gration of different order. G stands for gravit and B for buoanc forces For eample, if the difference between the dimensionless second order radii of gration of the gravit and buoanc forces is.3, the bending moment reduces b 5%; if the difference between the dimensionless third order radii of gration of the gravit and buoanc forces is also.3, the bending moment reduces b 4%; if the difference between the dimensionless fourth order radii of gration of the gravit and buoanc forces is again.3, the bending moment reduces b 46%. Further reduction of the difference between higher order radii of gration does not produce significant reduction of the bending moment (the reduction of the bending moment is calculated relative to the basic case for which b =, (see Fig. 8). 4. Eample for a 4K DWT Bulk Carrier The eample refers to 4K DWT bulk carrier designed as a mathematical model developed b the authors to test the application of the proposed methodolog. Its main dimensions are: length L BP = 7. m, L ma = 78.4 m, L WL = m, width B =.8 m, depth D = 4. m. However, the methodolog is general and can be applied to an real ship. The ship has five cargo holds. 4. Calculations for Lightweight (Gravit Forces) The lightweight of the ship is P LW = 79 t, its abscissa relative to midship is -.745L BP. The dimensionless integral curves used to calculate the values of the zero moment (lightweight), first moment (static moment relative to ship s enc) and second moments (moment of inertia relative to ship s end) m, m, and m are given in Fig. (the are calculated following the procedure above). The dimensionless format of the integral curves is obtained in the following wa (in this case, the ship s maimum length L ma is used): m m ( ) M ( ξ) P ( ξ) LW ξ = = (36) PLW PLW ( ) ( ξ) ( ξ) M M ξ = ( ξ ) = (37) P L P L m LW ma LW ma The function μ(ξ) depicted in Fig. is the moment of weight (area, force, etc.) aft of section ξ relative to the same section ξ. The non-dimensional parameter, α, shown in Fig. is the farthest point of the lightweight distribution curve from the Aft Perpendicular. The absolute values of the moments for the gravit forces (lightweight) are given in Table together with data for the buoanc forces and the consumables (fuel oil, diesel oil, water, lubricants, weight related to crew). It is worth noticing here that the dimensionless function m (ξ) is the same as the function η(ξ) used for analtical calculation of shear forces (Ivanov, 6), i.e.: m ξ = η ξ (38) ( ) ( ) The relationship between the dimensionless integral function μ(ξ) used for analtical calculation of bending moments (Ivanov, 6) and the newl introduced dimensionless integral function m (ξ) is: μ( ξ) m ( ξ) = η( ξ) ξ α (39) ηξ ( ) On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 89

6 ABS TECHNICAL PAPERS 8 consumables are shown in Fig. 3. The absolute values of the moments for this part of the deadweight are given in Table. Fig. : Dimensionless integral functions μ(ξ), m (ξ), m (ξ), and m (ξ) for the ship s lightweight These equations allows for calculating the shear forces and bending moments simultaneousl with the calculations related to ship s trimming. 4. Calculations for Displacement (Buoanc Forces) The ship s displacement (i.e., ship s buoanc forces) is Δ = 393 t, its abscissa relative to midship is.4l BP. The dimensionless integral curves used to calculate the values of the zero moment (lightweight), first moment (static moment relative to ship s enc) and second moments (moment of inertia relative to ship s end) m, m, and m are given in Fig. (the are calculated following the same procedure as for the gravit forces). The absolute values of the moments for the buoanc forces are also given in Table. The moments for gravit and buoanc forces are calculated relative to different aes for comparison. Therefore, the have to be recalculated using the formulae in Appendi B. In this case, the n th moments of the buoanc forces are recalculated relative to the aft end of L ma (the n th moments of the gravit forces were calculated relative to the same point). The difference a (see Appendi B) between the aft ends of L ma and L WL is a =.7 m. Then, using the equations in Appendi B, the magnitudes of n th moments were recalculated as given in Table. 4.3 Calculations for Deadweight (Gravit Forces) The deadweight is split in two parts: weight of consumables (fuel oil, diesel oil, water, lubricants, weight related to crew) and cargo weight. For the sample ship, the weight of consumables is: heav fuel oil in Main Engine Room = 4 t, Heav fuel oil outside Main Engine Room = 968 t, diesel oil = 7 t, lubricants 4 t, water for machiner = 4.6 t, weight related to crew = 45 t. The cargo weight is 37 t. The dimensionless integral functions of the Fig. : Dimensionless integral functions m (ξ), m (ξ), and m (ξ) for the buoanc forces The second part of the deadweight refers to the weight of the cargo (in the eample, 37 t). The sample bulk carrier has five cargo holds. Fig. 3: Dimensionless integral functions m (ξ), m (ξ), m (ξ) and μ(ξ) for the consumables and weight related to crew The eample in the paper refers to case when the ship is loaded in alternate cargo holds, i.e. in cargo holds, 3 and 5. These cargo holds should be so loaded that the ship will be on even keel and the still water shear forces and bending moments will be minimal. The amount of cargo in each cargo hold is unknown et. To determine it, one should solve a sstem of three algebraic equations using moments of zero, first and second order. To solve this problem, it is convenient to work with the volume of each cargo hold and formulate as unknown parameter the specific gravit of the cargo loaded in the corresponding cargo hold. When the three unknown virtual specific gravities (quotation marks are used because these specific gravities are mathematicall derived and ma be different for each cargo hold which is not the case in the real ship On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 9

7 ABS TECHNICAL PAPERS 8 operation). However, once obtained, the product of the cargo hold s volume and the corresponding virtual specific gravit will provide the absolute value of the cargo in the cargo hold. Then, using the stowage factor of the cargo, one can determine the required portion of the cargo hold s volume needed for carring the calculated cargo. In other words, the mathematicall derived virtual specific gravit consists of two components specific gravit of the real cargo and coefficient of cargo holds volume usage. Following Eq. () and using the numerical data obtained for the ship s lightweight, displacement and first part of the deadweight, one can write the following equation: m Vn,i i = Cn (4) i= C = M M M (4) n n, Δ n,dw n,lw If onl three cargo holds are used for carring the cargo in alternate loading pattern (hold No. 5, 3 and ), Eq. (4) takes the following form: V + V + V = C V + V + V = C V + V + V = C,5 5,3 3,, 5 5, 3 3,,5 5,3 3, (4) where the letter V is used to distinguish the moments for cargo volumes from the moments for gravit and buoanc forces. The first subscript denotes the order of the moment and the second one the number of the cargo hold. For convenience in the calculations, Eq. (4) can be presented in the following wa: +α +α = N 5 +α, 3 3 +α, = N +α +α = N 5,3 3, 5,3 3, (43) where: V,3 V, C α,3 = α, = N = V,5 V,5 V,5 V, 3 V, C α, 3 = α, = N = (44) V, 5 V, 5 V, 5 V,3 V, C α,3 = α, = N = V,5 V,5 V,5 The roots 5, 3 and can be found b an specialized computer program. If not available, one can use the following formulae: ( N N)( α,3 α,3) -( N N)( α,3 α,3) = (45) α α α α - α α α α 3 (,, )(,3,3 ) (,3,3 )(,, ) ( ) N N α α = α α,,, 3, 3 5, 3 3, (46) = N α α (47) where i is the mathematicall derived specific gravit of the cargo in the corresponding i th cargo hold (the input data and the results are shown in Tables ~ 5). The calculation of the n th moments for the cargo holds volumes requires availabilit of the capacit curve, which provides information for the volumes in each cargo hold and its longitudinal center of gravit. For the sample ship, the volumes available for cargo are shown in Fig. 4 together with the volumes of the forepeak, after peak and the total volume below ship s deck. In the eample, alternate loading pattern is considered (hold Nos., 3 and 5). For this case, the cargo holds volumes are shown in Fig. 5. The nondimensionalizing parameter is the total volume available for cargo. The total volume of holds,3 and 5 is 589 m 3. Table : Data for M n Μ,Δ Μ,Δ Μ,Δ Μ,DW Μ,DW Μ,DW Μ,LW Μ,LW Μ,LW Table : Data for the n th moments of the volumes of cargo holds in alternating loading V,5 V,3 V, V,5 V,3 V, V,5 V,3 V, Table 3: Data for the dimensionless coefficients in the sstem of equations in Eq. (43) α,3 α, α,3 α, α,3 α, Table 4: Data for the parameters C i and N i C C C N N N Table 5 Roots of Eq. (43) On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 9

8 ABS TECHNICAL PAPERS The calculations for the n th moments of the volumes of each cargo hold are performed b the trapezoidal rule following the formulae in Eq. (5) and (6). The numerical eample in the paper refers to stowage factor of the iron ore.4 m 3 /t. For this case, the weight of the cargo is distributed as it follows: hold No t, hold No t, hold No t. Thus, the coefficients for usage of the cargo volumes will be: hold No..5, hold No. 3.9 and hold No Coefficient of usage is the ratio between the used volume and the total volume. The load intensit, still water shear forces (SWSF) and still water bending moments (SWBM) are shown in Figs. 6~8. It is worth noticing that the method is efficient when the stowage factor of the cargo is such that none of the cargo holds volume is full used. Fig. 6: Distribution of the gravit and buoanc forces Fig. 7: Distribution of the load intensit Fig. 4: Distribution of the total cross-sectional area and the area available for cargo loading Fig. 8: Still water shear forces and bending moments Fig. 5: Distribution of the volume available for cargo (the shaded areas for holds No., 3 and 5 are used for alternate loading) and the dimensionless integral functions m (ξ), m (ξ), m (ξ) and μ(ξ) If, in the eample, the stowage factor is.435 m 3 /t, the volume of cargo hold No. 3 will be full utilized. If the stowage factor is >.4 m 3 /t (in the eample), onl two unknowns eist the cargo in cargo hold No. 5 and No.. In this case, onl two equations are needed to balance the ship and the traditional practice for trimming the ship is to be followed. Nevertheless, the proposed methodolog still has some usefulness because it indicates the best location of the cargo although it ma On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 9

9 ABS TECHNICAL PAPERS 8 not be possible to follow it due to limitations b the available cargo holds volumes. It is also noted that it might be difficult to implement the optimum solution in eisting ships because of the limitation imposed b the maimum cargo per hold (in the design stage, the strength of the inner bottom and inner hull are alread adapted to withstand the cargo as computed from the variet of loading patterns). Furthermore, in case of negative or ver high densities, the calculated with the proposed procedure mathematical optimum cannot be achieved. 5. Conclusions A method for ship s trimming and loading is proposed that uses moments of high order for the gravit and buoanc forces. It is shown that the more high order moments of the gravit and buoanc forces are equal, the smaller the hull girder bending and shear will be. The eistence of onshore and onboard ships computers facilitates the application of the method. References Ivanov L D, (6), On the relationship between maimum still water shear forces, bending moments, and radii of gration of the total ship s weight and buoanc forces, Ships and Offshore Structures, vol., No., pp Ivanov L D, Wang Ge, (7), An approimate analtical method for calculation of the still water bending moments, shear forces and the ship s trim in the earl design stages, RINA Transactions, International Journal of Maritime Engineering, vol. 49, Part A, pp. -4 Moor D I, Parker M N, Pattulo R N M, (96), The BSRA Methodical Series an Overall Presentation. Geometr of Forms and Variations of Resistance with Block Coefficient and Longitudinal Centre of Buoanc, Quarterl Transactions of the Roal Institution of Naval Architects, vol. 3, No. 4, pp Spiegel M R, (968), Mathematical Handbook of Formulas and Tables, Schaum s Outline Series, McGraw-Hill, New York. Watson D G M, (998), Practical Ship Design, Elsevier. On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 93

10 ABS TECHNICAL PAPERS 8 Appendi A: Distribution of Buoanc Forces If insufficient data for the buoanc forces are available for the calculations, one can use the following formulae as a first approach (see Fig. A.). Fig. A.: Approimation of the buoanc forces distribution r ( r) = A +δ r ( r r ) r e ( e) = A n + e( e e ) e 4 δ= V-A 4 + ( e r) e n r E n = 4 e ( e+ r) + ( ) (A.) (A.) (A.3) (A.4) L L E = S + A r e mid 3 e r ( )( + L) + + (A.5) 7 L + r V A ( e r) V = buoanc forces; S mid = static moment of the buoanc forces relative to midship (positive when the longitudinal center of buoanc (LCB) is in front of the midship and negative when the LCB is aft of midship. S mid is the product of the buoanc forces and the LCB relative to midship, i.e. S = V.ζ L (A.6) mid B, i BP where ζ B,i is the dimensionless LCB relative to midship as a portion of LBP. If the LCB relative to the midship section for an draft T i is not given, one can calculate it in dimensionless format (as a portion of L BP ) b the following formula based on data of Moor et al.(96): T i ζ B, i = C B C B T Ti Ti T i C B 6.4C B (A.7) T T T Ti T i 3.3C B C B T T Note that Eq. (A.7) is valid for a ship on even keel, which is ver convenient for the purpose of this stud. On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 94

11 ABS TECHNICAL PAPERS 8 Appendi B: Transfer of n th Another Ais for Comparison Moment to Assume that n th moment of given figure is calculated relative to given ais for comparison (i.e., ais passing though point A in Fig. B.). If the n th moment is to be calculated for another ais for comparison (i.e., ais passing through point B in Fig. B.), one can use the binomial formulae (Spiegel, 968): B New ais for comparison a Old ais for comparison A () Fig. B.: Transfer of n th moment relative to new ais for comparison n n n n n n ( a+ ) = a + a + a + (B.) n n 3 3 n n + a n where the binomial coefficients are (Spiegel, 968): ( )( ) ( ) n n n n... n k+ = = k k! (B.) n! n = where!= = k! ( n k )! n k The binomial coefficients up to n = 5 are shown in Table B. following the Pascal s triangle (Spiegel, 968). The calculation of the binomial coefficients for moments of higher order can be done using an standard mathematical handbook (e.g., Spiegel, 968). With the binomial coefficients given in Table B., the following equations for n th moment (for up to n = 5) can be used: Table B.: Binomial coefficients up to n = 5 k n ( ) ( ) ( ) M = am + M (B.3),B, A ( ) ( ) ( ) ( ) M,B = a M + am,a + M,A (B.4) 3 M3,B( ) = a M ( ) + 3a M,A ( ) + + 3aM, A ( ) + M3, A ( ) (B.5) 4 3 M4,B( ) = a M ( ) + 4a M,A ( ) + + 6a M, A ( ) + 4aM3, A ( ) + M4, A ( ) (B.6) 5 4 M5,B( ) = a M ( ) + 5a M, A ( ) a M, A ( ) ++ a M3, A ( ) + + 5aM4, A ( ) + M5, A ( ) (B.7) where the subscript A stands for n th moment relative to the ais passing through point A; the subscript B for the n th moment relative to the ais passing through point B. The figures,,.5 stand for the order of n th moments. On the Ship s Trimming using Moments of the Gravit and Buoanc Forces of High Order 95

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