Intro to Naval Architecture 3E Episode 2 docx

20 SHIP FORM CALCULATIONS Figure 3. 1 situation is illustrated in Figure 3.1. The area of the shaded trapezium is: Any area can be divided into two, each with part of its boundary a straight line. Such a line can be chosen as the axis about which moments are taken. This simplifies the representation of the problem as in Figure 3.2 which also uses equally spaced lines, called ordinates. The device is very apt for ships, since they are symmetrical about their Figure 3.2 SHIP FORM CALCULATIONS 21 middle line planes, and areas such as waterplanes can be treated as two halves, Referring to Figure 3.2, the curve ABC has been replaced by two straight lines, AB and BC with ordinates y 0 , y 1 and y 2 distance h apart. The area is the sum of the two trapezia so formed: The accuracy with which the area under the actual curve is calculated will depend upon how closely the straight lines mimic the curve. The accuracy of representation can be increased by using a smaller interval h. Generalizing for n+1 ordinates the area will be given by: In many cases of ships' waterplanes it is sufficiently accurate to use ten divisions with eleven ordinates but it is worth checking by eye whether the straight lines follow the actual curves reasonably accurately. Because warship hulls tend to have greater curvature they are usually represented by twenty divisions with twenty-one ordinates. To calculate the volume of a three dimensional shape the areas of its cross sectional areas at equally spaced intervals can be calculated as above. These areas can then be used as the new ordinates in a curve of areas to obtain the volume. SIMPSON'S RULES The trapezoidal rule, using straight lines to replace the actual ship curves, has limitations as to the accuracy achieved. Many naval architectural calculations are carried out using what are known as Simpson's rules. In Simpson's rules the actual curve is represented by a mathematical equation of the form: The curve, shown in Figure 3.3, is represented by three equally spaced ordinates y 0 , y 1 and y 2 . It is convenient to choose the origin to be at the base of y 1 to simplify the algebra but the results would be the same 22 SHIP FORM CALCULATIONS Figure 33 wherever the origin is taken. The curve extends from x = -h to x. = +h and the area under it is: It would be convenient to be able to express the area of the figure as a simple sum of the ordinates each multiplied by some factor to be determined. Assuming that A can be represented by: SHIP FORM CALCULATIONS 23 These equations give: This is Simpson's First Rule or 3 Ordinate Rule. This rule can be generalized to any figure defined by an odd number of evenly spaced ordinates, by applying the First Rule to ordinates 0 to 2, 2 to 4, 4 to 6 and so on, and then summing the resulting answers. This provides the rule for n + 1 ordinates: For many ship forms it is adequate to divide the length into ten equal parts using eleven ordinates. When the ends have significant curvature greater accuracy can be obtained by introducing intermediate ordi- nates in those areas, as shown in Figure 3.4. The figure gives the Figure 3,4 Simpson multipliers to be used for each consecutive area defined by three ordinates. The total area is given by: where y 1 , y 3 , y 11 and y 13 are the extra ordinates. The method outlined above for calculating areas can be applied to evaluating any integral. Thus it can be applied to the first and second 24 SHIP FORM CALCULATIONS Figure 3,5 29.8 = 9.93 m 2 moments of area. Referring to Figure 3.5, these moments about the y-axis, that is the axis through O, are given by: First moment = xy dx about the y-axis Second moment = x 2 y dx about the y-axis = I y The calculations, if done manually, are best set out in tabular form. Example 3.1 Calculate the area between the curve, defined by the ordinates below, and the x-axis. Calculate the first and second moments of area about the x- and y-axes and the position of the centroid of area. Solution There are 9 ordinates spaced one unit apart. The results can be calculated in tabular fashion as in Table 3.1. Hence: SHIP FORM CALCULATIONS 25 Table 3.1 X 0 1 2 3 4 5 6 7 8 y 1.0 1.2 1.5 1.6 1.5 1.3 1.1 0.9 0.6 SM 1 4 2 4 2 4 2 4 ] Totals F(A) 1.0 4.8 3.0 6.4 3.0 5.2 2.2 3.6 0.6 29.8 xy 0 1.2 3.0 4.8 6.0 6.5 6.6 6.3 4.8 f(My) 0 4.8 6.0 19.2 12.0 26.0 13.2 25.2 4.8 111.2 x*y 0 1.2 6.0 14.4 24.0 32.5 39.6 44.1 38.4 Fry 0 4.8 12.0 57.6 48.0 130.0 79.2 176.4 38.4 546.4 f 1.0 1.44 2.25 2.56 2.25 1.69 1.21 0.81 0.36 f(M x ) 1.0 5.76 4.50 10.24 4.50 6.76 2.42 3.24 0.36 38.78 / 1.0 1.728 3.375 4.096 3.375 2.197 1.331 0.729 0.216 F(4) 1.0 6.912 6.750 16.384 6.750 8.788 2.662 2.916 0.216 52.378 First moment about ^axis Centroid from y-axis First moment about x-axis Centroid from x-axis Second moment about y-axis Second moment about as-axis The second moment of an area is always least about an axis through its centroid. If the second moment of an area, A, about an axis x from its centroid is 7 X and / xx is that about a parallel axis through the centroid: In the above example the second moments about axes through the centroid and parallel to the ^-axis and y-axis, are respectively: 26 SHIP FORM CALCULATIONS Where there are large numbers of ordinates the arithmetic in the table can be simplified by halving each Simpson multiplier and then doubling the final summations so that: Other rules can be deduced for figures defined by unevenly spaced ordinates or by different numbers of evenly spaced ordinates. The rule for four evenly spaced ordinates becomes: This is known as Simpson's Second Rule. It can be extended to cover 7, 10, 13, etc., ordinates, becoming: A special case is where the area between two ordinates is required when three are known. If, for instance, the area between ordinates y 0 and ji of Figure 3.3 is needed: This is called Simpson's 5, 8 minus 1 Rule and it will be noted that if it is applied to both halves of the curve then the total area becomes: as would be expected. Unlike others of Simpson's rules the 5, 8, -1 cannot be applied to moments. A corresponding rule for moments, derived in the same way as those for areas, is known as Simpson's 3, 10 minus 1 /Iwfeand gives the moment of the area bounded by $> and yi about yo, as: SHIP FORM CALCULATIONS 27 If in doubt about the multiplier to be used, a simple check can be applied by considering the area or moment of a simple rectangle. TCHEBYCHEFFS RULES In arriving at Simpson's rules, equally spaced ordinates were used and varying multipliers for the ordinates deduced. The equations con- cerned can equally well be solved to find the spacing needed for ordinates if the multipliers are to be unity. For simplicity the curve is assumed to be centred upon the origin, x - 0, with the ordinates arranged symmetrically about the origin. Thus for an odd number of ordinates the middle one will be at the origin. Rules so derived are known as Tchebycheff rules and they can be represented by the equation: Span of curve on #-axis X Sum of ordinates A = — Number of ordinates Thus for a curve spanning two units, 2/i, and defined by three ordinates: The spacings required of the ordinates are given in Table 3.2. Table 3,2 Number of ordinates 2 3 4 5 6 7 8 9 10 Spacing each side of origin -r 0.5773 0 0.1876 0 0.2666 0 0.1026 0 0.0838 0.7071 0.7947 0.3745 0.4225 0.3239 0.4062 0.1679 0.3127 the half length 0.8325 0.8662 0.5297 0.5938 0.5288 0.5000 0.8839 0.8974 0.6010 0.9116 0.6873 0.9162 28 SHIP FORM CALCULATIONS GENERAL It has been shown 1 that: (1) Odd ordinate Simpson's rules are preferred as they are only marginally less accurate than the next higher even number rule, (2) Even ordinate Tchebycheff rules are preferred as they are as accurate as the next highest odd ordinate rule. (3) A Tchebycheff rule with an even number of ordinates is rather more accurate than the next highest odd number Simpson rule. POLAR CO-ORDINATES The rules discussed above have been illustrated by figures defined by a set of parallel ordinates and this is most convenient for waterplanes. For transverse sections a problem can arise at the turn of bilge unless closely spaced ordinates are used in that area. An alternative is to adopt polar co-ordinates radiating from some convenient pole, O, on the centreline. Figure 3.6. If the section shape is defined by a number of radial ordinates at equal angular intervals the area can be determined using one of the Figure 3.6 Polar co-ordinates SHIP FORM CALCULATIONS 29 approximate integration methods. Since the deck edge is a point of discontinuity one of the radii should pass through it. This can be arranged by careful selection of O for each transverse section. SUMMARY It has been shown how areas and volumes enclosed by typical ship curves and surfaces, toether with their moments, can be calculated by approximate methods. These methods can be applied quite widely in engineering applications other than naval architecture. They provide the means of evaluating the various integrals called up by the theory outlined in the following chapters. Reference 1. Miller, N, S. (1963-4) The accuracy of numerical integration in ship calculations, rrnss. [...]... differentiating the equation for KM with respect to T and equating to zero That is, KM is a Table 4.1 d 0.5 d 18.75d I 0.5 1.0 1.5 2. 0 2. 5 3.0 18.75 9.37 6 .25 4.69 3.75 3. 12 2 3 4 5 6 Figure 4.9 Metacentric diagram KM 19 .25 10.37 7.75 6.69 6 .25 6. 12 40 FLOTATION AND STABILITY minimum at T given by: In the example KM is a minimum when the draught is 6.12m Vessel of constant triangular section Consider... is the moment that causes a trim t, so the moment to cause unit change of trim is: WGML/L This moment to change trim, MCT, one unit is a convenient figure to quote to show how easy a ship is to trim The value in SI units would be 'moment to change trim one metre' This can be quite a large quantity and it might be preferred to work with the 'moment to change trim one centimetre' FLOTATION AND STABILITY... 1 025 kg/m3 and g= 9.81 m/s2 increase in displacement per metre increase in draught is: 1 025 X 9.81 X 1 X A = 10055Anewtons, In imperial units the value quoted was usually the added tons per inch immersion, TPI As it was assumed that 35 ft3 of sea water weighed 1 ton, for A in ft2: The increase in displacement per unit increase in draught is useful in approximate calculations when weights are added to. .. displacement to Wj L} It has been seen that because ships are not symmetrical fore and aft they trim about F As shown in Figure 4. 12, the displacement to W0Lo is less than that to W} Lj, the difference Figure 4, 12 being the layer Wj L! L^ W2 where W2L2 is the waterline parallel to Wj LI through F on W0Lo- If A is the distance of Fforward of amidships then the thickness of layer=A X t/L where t= T a - Tf... curve can be drawn as in Figure 4 .2 The underwater volume is: If immersed cross-sectional areas are calculated to a number of waterlines parallel to the design waterline, then the volume up to each can be determined and plotted against draught as in Figure 4.3 The volume corresponding to any given draught T can be picked off, provided the waterline at T is parallel to those used in deriving the curve... displacement can be determined Heel due to moving weight In Figure 4.14 a ship is shown upright and at rest in still water If a small weight wis shifted transversely through a distance h, the centre of gravity of the ship, originally at G, moves to Gj such that GGj = wh/W The ship will heel through an angle . 3.1 X 0 1 2 3 4 5 6 7 8 y 1.0 1 .2 1.5 1.6 1.5 1.3 1.1 0.9 0.6 SM 1 4 2 4 2 4 2 4 ] Totals F(A) 1.0 4.8 3.0 6.4 3.0 5 .2 2 .2 3.6 0.6 29 .8 xy 0 1 .2 3.0 4.8 6.0 6.5 6.6 6.3 4.8 f(My) 0 4.8 6.0 19 .2 12. 0 26 .0 13 .2 25 .2 4.8 111 .2 x*y 0 1 .2 6.0 14.4 24 .0 32. 5 39.6 44.1 38.4 Fry 0 4.8 12. 0 57.6 48.0 130.0 79 .2 176.4 38.4 546.4 f 1.0 1.44 2. 25 2. 56 2. 25 1.69 1 .21 0.81 0.36 f(M x ) 1.0 5.76 4.50 10 .24 4.50 6.76 2. 42 3 .24 0.36 38.78 / 1.0 1. 728 3.375 4.096 3.375 2. 197 1.331 0. 729 0 .21 6 F(4) 1.0 6.9 12 6.750 16.384 6.750 8.788 2. 6 62 2.916 0 .21 6 52. 378 First . 3.1 X 0 1 2 3 4 5 6 7 8 y 1.0 1 .2 1.5 1.6 1.5 1.3 1.1 0.9 0.6 SM 1 4 2 4 2 4 2 4 ] Totals F(A) 1.0 4.8 3.0 6.4 3.0 5 .2 2 .2 3.6 0.6 29 .8 xy 0 1 .2 3.0 4.8 6.0 6.5 6.6 6.3 4.8 f(My) 0 4.8 6.0 19 .2 12. 0 26 .0 13 .2 25 .2 4.8 111 .2 x*y 0 1 .2 6.0 14.4 24 .0 32. 5 39.6 44.1 38.4 Fry 0 4.8 12. 0 57.6 48.0 130.0 79 .2 176.4 38.4 546.4 f 1.0 1.44 2. 25 2. 56 2. 25 1.69 1 .21 0.81 0.36 f(M x ) 1.0 5.76 4.50 10 .24 4.50 6.76 2. 42 3 .24 0.36 38.78 / 1.0 1. 728 3.375 4.096 3.375 2. 197 1.331 0. 729 0 .21 6 F(4) 1.0 6.9 12 6.750 16.384 6.750 8.788 2. 6 62 2.916 0 .21 6 52. 378 First moment about . -r 0.5773 0 0.1876 0 0 .26 66 0 0.1 026 0 0.0838 0.7071 0.7947 0.3745 0. 422 5 0. 323 9 0.40 62 0.1679 0.3 127 the half length 0.8 325 0.86 62 0. 529 7 0.5938 0. 528 8 0.5000 0.8839 0.8974 0.6010 0.9116 0.6873 0.91 62 28 SHIP