Hydrostatics And Stability

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Iooking back at hull development in the history of- yachting, it is obvious that opinion about the optimum shape of a yacht has -/changed many times. This is due in part to the changing rules, but more recently the changes in design trends reflect the increasing knowledge about the physical laws governing the behaviour of sailing yachts. The aim of this book is to present the state of the art in yacht design. While current knowledge does not provide explanations for all phenomena, there is one area where the basic laws have been known for a long time, and where the methods have been in use by designers for centuries. This is the area of hydrostatics and stability.

Hydrostatics and stability represent perhaps the most important aspects of a design since the properties of a yacht in these respects reflect its ability to carry the required weight and to withstand the heeling moment from the sails. It should be stressed that the exact knowledge of stability is restricted to the static case, with no waves on the water surface. We have, however, chosen to include also dynamic stability in this chapter, although the laws are quite different.

We begin this chapter by introducing some simple ways of computing areas. This knowledge is required in subsequent paragraphs dealing with calculations of the wetted surface, displacement and its centre of gravity, the prismatic coefficient, the water plane area and the related mass per mm of immersion as well as the moment per degree of heel and trim. The discussion of dynamic stability includes stability in waves, methods for reducing roll, requirements for offshore yachts and some statistical information on the righting moment of existing yachts.

Calculation of areas For the amateur designer, one way to obtain the area of a closed curve might be to draw it 011 a square grid and just count the number of squares. In most cases this method is accurate enough, but it is tedious and would hardly be used by professionals.

Another convenient way is to make use of the planimeter, as explained in the previous chapter. This method is fast and accurate but few amateur designers have access to this handy instrument.

The best choice for many designers is to compute the area using a simple numerical method, based on the ordinates (y-values) of the curve at certain intervals. Such methods are often included in the subroutine package of electronic calculators, but if this option is not available it is simple to apply the method from first principles.

Fig 4.1 introduces the most common numerical method for computing

Fig 4.1 Simpson's rule

Function Y

Function Y

Simpson's rule :

A = ^ • (Yq+ + 2Y2+ 4Y3+ 2Y4+ 4YS+ 2Y6+ 4Y?+ 2Yb+4Yg + rj£}J

 Ordinate No. Ordinate valua S . M . Product 0 Yo it- Yo 1 4 4Y1 2 Y2 2 2Y2 3 Y3 4Ys 4 Y4 2 2Y4 6 7 a 4Y7 2 Y yio J Sum of products j

areas. It is called Simpson's rule, and is quite popular in naval architecture. Since the sequence of operations is always the same when applying Simpson's rule a special scheme, shown in Fig 4.1, may be employed. The distance between the end points of the interval, in this case X0 and X10, is divided into an even number of equidistant steps, in this case 10. The step size is denoted S. Values of the function Y arc computed for all X-values and may be inserted into the table in the column 'ordinate value'. By multiplying each value by its Simpson multiplier, I for the end values and 4 and 2 alternating for the others, and adding all the products the 'sum of products' is obtained. The area A under the curve Y is then simply obtained as this sum multiplied by the step size divided by 3.

Of course, the number of steps may be other than 10, but the number has to be even in Simpson's rule. In many applications within yacht design the number of steps is indeed 10, due to the standard division of the waterline from station 0 to station 10, but sometimes a

higher accuracy is needed near the ends, where half stations may be introduced. The principle of Simpson's rule may still be used, by-considering end intervals as pairs of halves, but the number of full intervals must always be even, so normally two or four intervals have to be divided. Fig 4.3 shows the change caused by dividing an interval into two halves. In the following discussion, we will always refer to Simpson's rule for area calculations. However, the other methods mentioned above may be used as alternatives.

Wetted surface Due to the three-dimensional nature of the hull an exact calculation of the wetted surface is complicated, but a good approximation may be obtained as explained in Fig 4.2. If the girth length g along the surface from the keel to the waterline is measured at each station, and plotted against the longitudinal position on the hull from bow to stern, the area A under the curve is a reasonably good representation of the wetted surface of one half of the hull. The computation of this area is also shown in Fig 4.2. The values for the YD-40 are given in brackets.

The problem with the computed area is that the longitudinal slope of the hull, as seen in the watcrlines or the diagonals, is not considered. The effect of this is small, but a more accurate result is obtained by adding 2-4%, ie by multiplying by a 'bilge factor' c, which is in the range 1.02-1.04. The bilge factor can be estimated by comparing the length of a typical diagonal with the straight line distance between the end points of the waterline.

To simplify the presentation as much as possible, we have chosen to use full-scale entries for all formulae. Measures obtained from the drawings therefore have to be converted to full scale before being used in the calculations. In this way the somewhat confusing exercise with scale factors of various powers can be avoided in the different formulae. Note also that many calculations, like the present one, are made for onlv one half of the hull. Where this is the case the final value is therefore obtained only after multiplying by 2.

A very fast, but somewhat more approximate method to find the wetted surface is to make use of an empirical formula based on the length, beam, draft, displacement and prismatic coefficient of the canoe body (as shown in Fig 4.2). For smooth hulls this formula is surprisingly accurate, but if a drawing of the hull is available the method above is recommended.

Displacement According to Archimedes" principle the mass of a floating body is equal to the mass of the displaced volume of water. Thus the volume displacement of the yacht, V, multiplied by the density of water, p (ie the weight

Fig 4.2 Calculation of the wetted surface

Ordinate value g7 [ 0.530 ]

Product

3.020 J

4.744 J

length = g

Alternatively ( entirely empirical ) : Sw = [ 1.97 + 0.171 x Lwl lc cu = ---—zr—r [ 0.752 ]

M LWL BWL rc'Cp

Ordinate value

3.020 J

4.744 J

length = g

Alternatively ( entirely empirical ) : Sw = [ 1.97 + 0.171 x Lwl lc cu = ---—zr—r [ 0.752 ]

M LWL BWL rc'Cp

displacement m), has to be equal to the total mass of the yacht.

In this chapter we will deal with the calculation of the volume displacement, while the mass of the yacht will be discussed in Appendix 2. It should be noted, that p is equal to 1000 kg/m3 for fresh water, but varies for salt water, depending on the salinity. As an average value for salt water 1025 kg/m3 may be used.

To obtain the volume, the curve of sectional areas has to be

determined first. This is obtained by plotting the area of each section (the submerged part) at a suitable scale in the half breadth plan, as explained in Chapter 3. A difficulty encountered when applying Simpson's rule to compute the area As of a section is that the ordinates are not known at suitable intervals, so each section has to be properly divided (see Fig 4.3).

The ordinates in Fig 4.3 are the half breadths arranged in such a way that the depth at that section is divided into five parts. Half spacing is used in the lowest interval, since the ordinates vary rapidly in that region. The Simpson multipliers are thus changed in this interval, but otherwise the normal scheme may be used.

Having obtained all the areas of the sections and had them plotted to obtain the curve of sectional areas (as in the lines plan of Fig 3.5), the displacement is obtained as the area under the curve. Fig 4.4 shows how this is computed using Simpson's rule. Note again that full-scale values are used throughout and that the values for the YD-40 are given in brackets.

Centre of buoyancy The moment created by a force with respect to a perpendicular axis is

Fig 4.3 Calculation of the sectional area s/2-l s/2 r

DV/L

DV/L

Fig 4.4 Calculation of the volume displacement

As : Areas calculated as In Fig 4.3

As : Areas calculated as In Fig 4.3

 Ordinate No. Ordfn a ie value 5 • A/• Product 0 A so [ O.OOO ] i Aso [ O.OOO ] 1 As, [ 0.160 ] 4 4AS1 [ 0.640 ] 2 AS2 [ 0.470 J 2 2AS2 f. 0.940 ] 3 ASJ [ °'83Z J 4 4ASJ [ 3.328 ] 4 [ I** J 2 ¿As4 [ 2.288 ] 5 Ass [ ] 4 *As5 [ 5.328 ] 6 AS6 [ 7.341 J 2 2AS6 [ 2.682 ] 7 As7 [ 1. 156 J 4 4AS7 [ 4.624 J 8 A58 [ 0.805 ] 2 2 A sg [ 1.610 ] 9 AS9 [ 0.364 ] 4 4Asg [ 1.456 ] 10 Asw [ 0.000 ] 1 A\$ic f O.OOO J products [ 22.896 ]

the product of the force and the distance to the axis (the lever arm). This concept can be used for finding the centre of gravity of a body. By definition, the centre of gravity is the point where the mass of the body may be assumed concentrated. The gravitational force may be assumed acting at this point.

One way to calculate the distance to the centre of gravity from an arbitrary axis, is to add the moments of the different parts of the body with respect to this axis. This gives a resulting moment, which must be cqual_to that of the concentrated mass at the centre of gravity. This method is explained in Fig 4.5, where the axis chosen is located athwartships at the FP.

A corresponding computation can be performed for the centre of gravity of the displaced volume of water, ie the centre of buoyancy. Let us first compute the longitudinal position, LCB, using the same axis as

Fig 4.5 Methods of finding the centre of gravity before. Each section of the hull may now be considered as contributing to the moment by an amount proportional to its area multiplied by its distance from the FP. Thus a "curve of sectional moments' can be constructed in a similar way to the curve of sectional areas. The area under the new curve represents the total moment, from which the position of the ccntrc of buoyancy can be obtained as explained in Fig 4.6.

There is a simple alternative method, which is used frequently for determining the LCB. If carefully employed, this method is probably as accurate as the numerical one. The sectional area curve is simply cut out in a piece of cardboard and the cut out part is balanced on the edge

Centre of gravity measured from DV/L along z—axis:

Transverse axis at FP

mtotXG ~ m1X1 m2X2'i" m3XJ+ ••• + ml Xj + m. , — m. + n^ -h m^ + ... + m- -f- ...

Centre of gravity measured from FP along x—axls. mto'tg-xG = m1gx1 + m2gx2-h m3gx5 + ... + mfgx. + ...

mtotXG ~ m1X1 m2X2'i" m3XJ+ ••• + ml Xj + m. , — m. + n^ -h m^ + ... + m- -f- ...

Centre of gravity measured from DV/L along z—axis:

x—Values forward of FP and z—values below DWL are negative.

Transverse axis at FP

Fig 4.6 Calculation of the longitudinal centre of buoyancy of the canoe body of a knife at right angles to the longitudinal axis. When the cardboard is balanced, its centre of gravity is on the edge of the knife. This is also the position of the LCB. If the piece is hung on a needle and allowed to rotate, the vertical line through the needle crosses the centre of gravity. By hanging the piece at two positions and using a plumb bob to mark the vertical lines, the centre of gravity is found at their intersection.

For the determination of the vertical position of the centre of buoyancy (VCB), the vertical distribution of sectional moments must be considered. If the areas of several waterlines areTnoWn, the vertical distribution of the volume can be plotted in the form of a curve. This curve can then be

A : Areas calculated as In Fig 4.3

Ordinate

Ordinate value

A : Areas calculated as In Fig 4.3

Ordinate

Ordinate value

Fig 4.7 Calculation of the longitudinal moment of inertia treated in the same way as the sectional area curve and the location of the VCB can be found. However, the areas of the waterlines might not be known, since they are not normally required for other purposes. Another possibility is to cut out all sections of the hull from a piece of paper and glue them together just as in the body plan. The vertical position of the centre of gravity for this paper body is the desired VCB.

Curve of sectional moments of inertia

= 'lfp- \ '

Moment of Inertia around centre of floatation

XF = distance from FP to centre of flotation [ 5.674 m ]

Curve of sectional moments of inertia

Water plane area

Fig 4.8 Calculation of the transverse moment of inertia certain distance; secondly, its centre of gravity is located on the axis around which the hull is trimmed, when moving a weight longitudinally on board; thirdly, the so-called moment of inertia (sometimes called the second moment of area) around a longitudinal axis determines the stability at small angles of heel; and fourthly, the moment of inertia around a transverse axis through the centre of gravity (of the area) yields the longitudinal stability, ie the moment required to trim the hull a certain angle.

The calculation of the area is straightforward, using Simpson's rule exactly as shown in Fig 4.1. If the area is denoted ADWL (full-scale value),

Curve of cubic half beam

Ordinate No bf f o.ooo J

Curve of cubic half beam

Ordinate value

Product

the additional displacement when sinking the hull 1 mm is 0.001 • ADWL m3. The mass of this volume, corresponding to the applied mass on the hull, is p • 0.001 • Adwl, where p is the water density. The mass per mm immersion is thus calculated from this simple formula.

As appears from the previous paragraphs the centre of buoyancy is determined from the geometrical centre of gravity of the sectional area curve. Either a numerical method, like Simpson's, or the simple 'cardboard" method can be used for the calculation. To obtain the geometrical centre of gravity of the water plane area, usually called the 'centre of flotation', the same techniques can be employed.

No simple method is available for finding the moment of inertia, but the numerical calculation is similar to that of the centre of gravity. Let us first calculate the longitudinal moment of inertia ILH, about a transverse axis at the FP. A "curve of sectional moments of inertia* can now be constructed, where each ordinate is the product of the waterline half-width and the square of the distance from the FP. The area of this curve can be used for finding the full-scale moment of inertia (both sides) in the usual way (see Fig 4.7).

In the formula for longitudinal stability, to be presented in the next section, the moment of inertia IL is taken about an axis, not through the FP. but through the centre of flotation. The calculated value ILFP may, however, be converted to Ij_ quite simply, as shown in Fig 4.7.

In principle, the transverse moment of inertia IT around the longitudinal axis, needed for the transverse stability, could be computed in a similar way, but then the water plane area would have to be divided into a set of longitudinal strips, which could be treated like the transverse ones above. This division is impractical, however, since it is not used in any other calculation. An alternative method is therefore shown in Fig 4.8. Note that, for reasons of symmetry the longitudinal axis has to pass through the centre of flotation, so no correction need be applied.

Transverse and The transverse stability of a yacht may be explained with reference to longitudinal stability Fig 4.9. When the yacht is heeled the centre of buoyancy moves to at small angles leeward from B to B'. The buoyancy force, upwards, then creates a couple with the equally large gravity force acting downwards at G. The lever arm is usually called GZ and the righting moment is m • g • GZ. since the gravity force is the mass, m, times the acceleration of gravity, g (9.81 m/s2).

There is another important point marked in the figure: the transverse metacentre, M. This is the intersection between the vertical line through

B' and the symmetry plane of the yacht. For small angles of heel this point may be assumed fixed, which simplifies_the calculations considerably. The distance between G and M, GM. is called the metacentric height and BM is the metacentric radius. A fundamental stability formula (which will not be proven here) says that the metacentric radius is equal to the ratio of the transverse moment of

Fig 4.9 Transverse stability

Transverse stability relations:

GM = BM - BG GZ = GM • sin \$ [ BG - 0.27 m ] [ v = 7.9 m3 J

( Fundamental stability formula ) ( G above B ) ( \$ = heel angle )

Transverse righting moment: [Nm]

Centre of gravity \

Upright centre of buoyancy

Centre of gravity \

Upright centre of buoyancy

M = transverse metacentre

Heeled centre of buoyancy

m-g inertia IT and the volume displacement V. Using this formula and some simple geometric relations the righting moment may be obtained as explained in Fig 4.9.

Since the stability of the yacht is proportional to GM there are two principal ways of increasing it. Either G may be lowered or M may be raised. A low G is found on narrow, heavy yachts with a large ballast ratio, like the 12 m and other R yachts. They have weight stability. Modern racing yachts, on the other hand, are wide and shallow, which raises M. They have form stability.

The method of calculating the longitudinal stability corresponds exactly to that of the transverse stability. Thus, the restoring moment when the hull gets a trim angle, may be computed from the formulae of

Fig 4.10 Longitudinal stability

Longitudinal stability relation:

bml = ( Fundamental stability formula ) [ 11.785 m j

Longitudinal righting moment:

Longitudinal righting moment:

Trim angle in degrees when moving a weight with the mass (W)

a distance (x) longitudinally:

Trim angle in degrees when moving a weight with the mass (W)

a distance (x) longitudinally:

Fig 4.10, which correspond to those of the previous figure. There is also a formula for computing the trim angle obtained when moving a weight longitudinally on board the yacht.

Transverse stability at The calculation of the righting moment at large heel angles is large angles of heel considerably more complicated than that for small angles. One difficulty arises from the fact that the positioning of the heeled hull with respect to the water surface is not known. If the hull is just rotated about the centreline (at the level of the DWL), the displacement will generally become too large and a trimming moment will develop. The only way to overcome this difficulty is by trial and error, ie by trying several attitudes, varying the sinkage and trim systematically, in order to find a position where the displacement and LCB correspond to the original ones.

After finding the right attitude a considerable amount of calculation is needed to find the righting moment, since no simple formulae, like those for small heel angles, are available. In practice, these calculations have seldom been carried out manually even for ships, because before the computer era naval architects made use of a special instrument, called an integrator, a development .of the planimeter. Such an instrument is, however, rarely available to the yacht designer, so wc will propose a slightly more approximate method, which is often accurate enough. The method is illustrated in Fig 4.11. Special care must be taken, however, with very beamy yachts with large fore and aft asymmetry. Such hulls will develop a considerable trim when heeling, and this effect is not considered here.

To find the attitude of the hull, rotate it first around the centreline at DWL to the desired angle. Then calculate the displacement VA up to this waterline located at ZA. This cannot be done, however, without knowing the shape of the sections on both sides of the symmetry plane, so the body plan has first to be completed to include both sides of the hull.

Fig 4.11 Procedure to find the heeled waterline

The displacement VA is bound to be too large, so a new waterline at ZB has to be found. A first estimate of this line can be made by dividing the excess displacement by the area of the original DWL. This gives the approximate distance to the new waterline at ZB, for which the displacement VB is also computed. Not even this is likely to be very accurate, but the final position Z of the waterline can be found by interpolation or extrapolation to the right V, as explained in the figure. In this way the displacement will be quite accurate, although all effects of trim are neglected.

Z ( Location of waterline )

Having found the waterline, the 'cardboard method* is used to find the transverse position of the centre of buoyancy. B' in Fig 4.12. All heeled sections below the waterline arc cut out in cardboard and slued together in their correct positions. The centre of gravity can then be found from the intersection of two lines, obtained using a plumb bob, as explained above.

Fig 4.12 Stability at large angles of heel

Movement of metocentre / M = metocentre at upright position l-*

+1 0

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Responses

• aatifa
How to calculate wetted surface area of a ship?
4 years ago
• Ghenet
How to find transverse moment of inertia of catamaran?
2 years ago
• lee
Where the center of flotation and longitudinal moment of inertia can be located?
2 years ago
• JULIA
Are hydrostatics used in ship making?
5 months ago