Ship Hydrostatics and Stability 2010 Part 13 pot

Computer methods 305 Figure 13.10 shows that a surface can be described by a net of isoparametric curves. One procedure for generating a surface can begin by defining a fam- ily of plane curves, for example ship stations, with the help of Bezier curves, non-rational or rational B -splines, or NURBS, with the parameter u. Taking then the points u = 0 on all curves, we can fit them a spline of the same kind as that used for the first curves. Proceeding in the same manner for the points u = 0.1, . . . , u = 1, we obtain a net of curves. Plane curves can be prop- erly described by breaking them into spline segments and imposing continuity conditions at the junction points. Similarly, surfaces can be broken into patches with continuity conditions at their borders. The expressions that define the patches can be direct extensions of plane curves equations such as those described in the preceding sections. For example, a tensor product Bezier patch is defined by ij -J i|m (u)J J> H, u=[0, 1], ™ = [0, 1] i=0 j=0 where the control points, B^ define a control polyhedron, and Ji^ m (u) and Jj,n ( w ) are tne basis functions we met in the section on Bezier curves. There are more possibilities and they are described in detail in the literature on geometric modelling. 13.2.7 Ruled surfaces A particular case is that in which corresponding points on two space curves are joined by straight-line segments. For example, in Figure 13.11 we consider three of the constant-it; curves shown in Figure 13.10. Then, we draw a straight line from a u = i point on the curve w = 0.6 to the u = i point on the curve w = 0.7, for i — 0, 0.1, , 1. The surface patch bounded by the w = 0.6 and the w = 0.7 curves is a ruled surface. A second ruled-surface patch is shown between the curves w — 0.7 and w = 0.8. Ruled surfaces are characterized by the fact that it is possible to lay on them straight-line segments. 13.2.8 Surface curvatures In Figure 13.12, let N be the normal vector to the surface at the point P, and V, one of the tangent vectors of the surface at the same point P. The two vectors, N and V, define a plane, TTI, normal to the surface. The intersection of the plane TTI with the given surface is a planar curve, say C. The curvature of C at the point P is the normal curvature of C at the point P in the direction of V. We note it by k n . A theorem due to Euler states that there is a direction, defined by the tangent vector V m i n , for which the normal curvature, k m - m , is minimal, and 306 Ship Hydrostatics and Stability u=0 "W=0.8 w=0.7 w=0.6 Figure 13.11 Two ruled surfaces another direction, defined by the tangent vector V max , for which the normal curvature, /c max , is maximal. Moreover, the directions V m i n and V max are perpendicular. The curvatures k m i n and /c max are called principal curvatures. For example, in Figure 13.12 the planes TTI and 7T 2 are perpendicular one to the other and their intersections with the ellipsoidal surface yields curves that have the principal curvatures at the point from which starts the normal vector N. The two curves are shown in Figure 13.13. Figure 13.12 Normal curvatures Computer methods 307 Figure 13.13 Principal curvatures The product of the principal curvatures is known as Gaussian curvature: •**• ~ "'min ' "'max \ij.l and the mean of the principal curvatures is known as mean curvature: ~r (13.20) In Naval Architecture, curvatures are used for checking the fairness of surfaces. A surface with zero Gaussian curvature is developable. By this term we under- stand a surface that can be unrolled on a plane surface without stretching. In practical terms, if a patch of the hull surface is developable, that patch can be manufactured by rolling a plate without stretching it. Thus, a developable surface is produced by a simpler and cheaper process than a non-developable surface that requires pressing or forging. A necessary condition for a surface to be developable is for it to be a ruled surface. Cylindrical surfaces are devel- opable and so are cone surfaces. The sphere is not developable and this causes problems in mapping the earth surface. Readers interested in a rigorous theory of surface curvatures can refer to Davies and Samuels (1996) and Marsh (1999). The literature on splines and surface modelling is very rich. To the books already cited we would like to add Rogers and Adams (1990), Piegl (1991), Hoschek and Lasser (1993), Farm (1999), Mortenson (1997) and Piegl and Tiller (1997). 308 Ship Hydrostatics and Stability 13.3 Hull modelling 13.3.1 Mathematical ship lines De Heere and Bakker ( 1970) cite Chapman (FredrikHenrikaf Chapman, Swedish Vice- Admiral and Naval Architect, 1721-1808) as having described ship lines as early as 1760 by parabolae of the form y = 1 - x n and sections by In 1915, David Watson Taylor (American Rear Admiral, 1864-1940) published a work in which he used 5-th degree polynomials to describe ship forms. Names of later pioneers are Weinblum, Benson and Kerwin. More details on the history of mathematical ship lines can be found in De Heere and Bakker (1970), Saunders (1972, Chapter 49) and Nowacki et al (1995). Kuo (1971) describes the state of the art at the beginning of the 70s. Present-day Naval Architectural computer programmes use mainly B-splines and NURBS. 13.3.2 Fairing In Subsection 1.4.3, we defined the problem of fairing. A major object of the developers of mathematical ship lines was to obtain fair curves. Digital comput- ers enabled a practical approach. Some early methods are briefly described in Kuo (1971), Section 9.3. A programme used for many years by the Danish Ship Research Institute is due to Kantorowitz (1967a,b). Calkins et al (1989) use one of the first techniques proposed for fairing, namely differences. Their idea is to plot the 1st and the 2nd differences of offsets. In addition, their software allows for the rotation of views and thus greatly facilitates the detection of unfair segments. As mentioned in Subsections 13.2.2 and 13.2.8, plots of the curvature of ship lines can help fairing. Surface-modelling programmes like MultiSurf and Sur- faceWorks (see next section) allow to do this in an interactive way. More about cur- vature and fairing can be read in Wagner, Luo and Stelson ( 1 995), Tuohy, Latorre and Munchmeyer (1996), Pigounakis, Sapidis and Kaklis (1996) and Farouki (1998). Rabien (1996) gives some features of the Euklid fairing programme. 13.3.3 Modelling with MultiSurf and SurfaceWorks In this section, we are going to describe a few steps of the hull-modelling process performed with the help of MultiSurf and SurfaceWorks, two products of Aero- Hydro. We like these surface modellers for their excellent visual interface, the Computer methods 309 possibilities of defining and capturing many relationships between the various elements of a design, and the wide range of useful point, curve and surface types. A recent possibility is that of connecting SurfaceWorks to SolidWorks. The programmes described in this section are based on a concept developed by John Letcher; he called it relational geometry (see Letcher, Shook and Shep- herd, 1995 and Mortenson, 1997, Chapter 12). The idea is to establish a hierarchy of dependencies between the elements that are successively created when defin- ing a surface or a hull surface composed of several surfaces. To model a surface one has to define a set of control, or supporting curves. To define a supporting curve, the user has to enter a number of supporting points; they are the con- trol points of the various kinds of curves. Points can be entered giving their absolute coordinates, or the coordinate-differences from given, absolute points. Moreover, it is possible to define points constrained to stay on given curves or surfaces. When the position of a supporting point or curve is changed, any depen- dent points, curves or surfaces are automatically updated. Relational geometry considerably simplifies the problems of intersections between surfaces and the modification of lines. Both MultiSurf and SurfaceWorks use a system of coordinates with the origin in the forward perpendicular, the x-axis positive towards aft, the y-axis positive towards starboard, and the z-axis positive upwards. When opening a new model file, a dialogue box allows the user to define an axis or plane of symmetry, and the units. For a ship the plane of symmetry is y = 0. We begin by 'creating' a set of points that define a desired curve, for example a station. Thus, in MultiSurf, a first point, pOl, is created with the help of the dialogue box shown in Figure 13.14. The last line is highlighted; it contains locked N<ime = pQ1 User data = Layer = 0 Weight = 0.000 Color = 14 Visibility = 1 Figure 13.14 MultiSurf, the dialogue box for defining an absolute three-dimensional point 310 Ship Hydrostatics and Stability Figure 13.15 MultiSurf, points that define a control curve, in this case a transverse section the coordinates of the point, x = 17.250, y = 0.000, z = 3.000. There is a quick way of defining a set of points, such as shown in Figure 13.15. In this example all the points are situated along a station; they have in common the value x = 17.250 m. To 'create' the curve defined by the points in Figure 13.15 the user has to select the points and specify the curve kind. A Bcurve (this is the MultiSurf terminology for B-splines) uses the support points as a control polygon (see Subsection 13.2.4), while a Ccurve (MultiSurf terminology for cubic splines) passes through all support points. Figure 13.16 shows the Bcurve defined by YZ X Figure 13.16 MultiSurf, a curve that defines a transverse section Computer methods 311 \\\ Figure 13.17 MultiSurf, a surface defined by control curves such as those in Figure 13.16 the points in Figure 13.15. The display also shows the point in which the curve parameter has the value 0, and the positive direction of this parameter. Several curves, such as the one shown in Figure 13.16, can be used as support of a surface. To 'create' a surface the user selects a set of curves and then, through pull-down menus, the user choses the surface kind. An example of surface is shown in Figure 13.17. Any point on this surface is defined by the two parameters u and v. The display shows the origin of the parameters, the direction in which the parameter values increase, and a normal vector. To exemplify a few additional features, we use this time screens of the Sur- faceWorks package. In Figure 13.18 we see a set of four points along a station. The window in the lower, left corner of Figure 13.18 contains a list of these points. Figure 13.19 shows the B-spline that uses the points in Figure 13.18 as control points. At full scale it is possible to see that the curve passes only through the first and the last point, but very close to the others. The display shows again the origin and the positive sense of the curve parameter. Figure 13.19 is an axonometric view of the curve. Figure 13.20 is an ortho- graphic view normal to the x-axis. In Figure 13.21, we see the same station and below it a plot of its curvature. In this case we have a simple third-degree B-spline; the plot of its curvature is smooth. In other cases the curve we are interested in can be a polyline composed of several curves. Then, the curva- ture plot can help in fairing the composed curve. Usually, it is not possible to define a single surface that fits the whole hull of a ship. Then, it is neces- sary to define several surfaces that can be joined together along common edges. A surface is defined by a set of supporting curves, for example, the bow profile, some transverse curves, etc. [...]... Mathieu, Lame and Allied Functions Oxford: Pergamon Press ASTM (2001) Guide F1321-92 Standard Guide for Conducting a Stability Test (Lightweight Survey and Inclining Experiment) to Determine the light Ship Displacement and Centers of Gravity of a Vessel http://www astm.org/DATABASE.CART/PAGES/ F1321.htm Attwood, E.L and Pengelly, H.S (1960) Theoretical Naval Architecture, new edition expanded by Sims,...316 Ship Hydrostatics and Stability ® Ship Lines: powerboat-3:3 Figure 13. 26 The lines of a powerboat 13. 4 Calculations without and with the computer Before the era of computers, the Naval Architect prepared a documentation that was later used for calculating the data of possible loading cases The documentation included: • hydrostatic curves; • cross-curves of stability; • capacity... Gawthrop, P.J., Kountzeris, A and Roberts, J.B (1988) Parametric excitation of nonlinear ship roll motion from forced roll data Journal of Ship Research, 32, No 2, 101-11 Gilbert, R.R and Card, J.C (1990) The new international standard for subdivision and damage stability of dry cargo ships Marine Technology, 27, No 2, 117-27 Gray, A (1993) Modern Differential Geometry of Curves and Surfaces Boca Raton,... universities and on the market The software calculates the added masses and damping coefficients, for a series of frequencies, by using potential theory and certain simplifying assumptions Next, the software calculates the response amplitude operators, RAOs, of various motions or events For a wave frequency component, and given ship heading and speed, the programme calculates the frequency of encounter and. .. given by Kim, Chou and Tien (1980) 322 Ship Hydrostatics and Stability 13. 5.1 A simple example of roll simulation Subsection 9.3.2 shows how to implement in MATLAB a Mathieu equation and simulate the roll motion produced by parametric excitation More complicated models can be simulated in a similar manner by writing the governing equations as systems of first-order differential equations and calling an... arm, and MH, a heeling moment We rewrite Eq (13. 21) as rr (13. 22) In this example we neglect added mass and damping, but use a non-linear function for GZ and can accept a variety of heeling moments To represent this equation in SIMULINK we draw the block diagram shown in Figure 13. 29 by putting in blocks taken from the libraries of the software and connecting them by lines that define the relationships... relational data bases Johnson, Glinos, Anderson et al (1990), Carnduf and Gray (1992) and Reich (1994) discuss more types of data bases Many modern ships are provided with board computers that contain the data of the ship and a dedicated computer programme Moreover, the computer can be connected to sensors that supply on line the tank and hold filling heights 13. 4.1 Hydrostatic calculations Some hydrostatic... SNAME Transactions, 97, 85- 113 Cardo, A., Ceschia, M., Francescutto, A and Nabergoj, R (1978) Stabilita della nave e movimento di rollio: caso di momento sbandante non variabile, Tecnica Italiana, No 1, 1-9 Bibliography 329 Carnduff, T.W and Gray, W.A (1992) Object oriented computing techniques in ship design In Computer Applications in the Automation of Shipyard Operation and Ship Design IV, pp 301-14... Helland-Hansen The investigation and recommendations for preventing similar accidents Norwegian Maritime Research, 8, No 3, 2 -13 Dahle, A and Kjaerland, O (1980) The capsizing of M/S Helland-Hansen The investigation and recommendations for preventing similar accidents The Naval Architect, No 3, March, 51-70 Dahle, A.E and Myrhaug, D (1996) Capsize risk of fishing vessels Schiffstechnik /Ship Technology Research,... (compartment and tank volumes, centres of gravity, and free surfaces), cross-curves of stability, damage stability, and longitudinal bending Many examples in this book were obtained with the ARCHIMEDES programme A newer version of the software, ARCHIMEDES II, is described by Soding nd Tongue (1989) Recent programmes have a graphic interface that enables the user to build and change interactively the ship . (1991), Hoschek and Lasser (1993), Farm (1999), Mortenson (1997) and Piegl and Tiller (1997). 308 Ship Hydrostatics and Stability 13. 3 Hull modelling 13. 3.1 Mathematical ship lines De Heere and . Kountzeris and Roberts (1988) and Kat and Paulling (1989). An example of simulation in frequency domain is given by Kim, Chou and Tien (1980). 322 Ship Hydrostatics and Stability 13. 5.1 A. class="bi x0 y0 w3 h10" alt="" 316 Ship Hydrostatics and Stability ® Ship Lines: powerboat-3:3 Figure 13. 26 The lines of a powerboat 13. 4 Calculations without and with the computer Before the