TO DUY ANH SHIP CALCULATION

Transcription

1 TO DUY ANH SHIP CALCULATION

2 Ship Calculattion (1)-Space Cuvers 3D-curves play an important role in the engineering, design and manufature in Shipbuilding. Prior of the development of mathematical and computer models to support engineering, design and manufacturing descriptive geometry was used. Today, many of these geometric design techniques have been carried over into Computer Aid Geometric Design.

3 Ship Calculattion (1) Surfaces are freguently described by a net of curves lying in orthogonal cutting planes plus three dimensional (3D) feature or detail lines. This curve are obtained by digitizing on a physical model or a drawing and then fitting a mathematical curve to the digitized data.

4 Ship Calculattion (1) Interpolation curve are characterized by the fact that the devived mathematical curve passes through each and every data point. Ex: Cubic spline, Parabolic blended curve, B-spline..ect.. Fitting curve are generated with out any prior knowledge of curve shape or form. These are characterized by the fact that few if any point on the curves pass through the control points used to define the curve.

5 Ship Calculattion (1) Curves : Plane curve - Space curve Intersecting two planes Intersecting a plane with a curves surface Intersecting two curved surfaces Straight line Plane curve Space curve Form of Representation Non parametric Parametric Implicit Explicit x= f(t) f(x.y)=0 y= f(x) y= f(t) Plane curve f(x.y.z)=0 y=f(x) x= f(t) g(x.y.z)=0 z= g(x) y=g(x) Space curve z= h(t)

6 Ship Calculattion (1) Non-parametric implicit : F(x.y) = 0 Problem in finding the correct root of an algebraic equation. Non-parametric explicit : y = f(x) Axis dependent no multiple valued curves problems with vertical tangent. Parametric : Each point on the curve is defined by parameter (t) Multiple valued curves Axis independent Problem in case find value of y(x) with a given (x)

7 Ship Calculattion (1) When the composite curve are combined. The curves are determined by the properties according to the continuity condition at the given joint point position. To the n th derivative are continuous at that point. The following table shows the major continuity conditions. C none positional discontinuity, discontinuous (non continuity) C o positional comtimuity,continous y, slope discontinuity (zero order continuity) C slope comtimuity,continous y, curve discountinuity (first order continuity). C 2 curvature,continous y,curve countinuity y (second order continuity) C 3 curvature,continous y,curve countinuity y (third order continuity) C n all derivative are continous.

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9 Ship Calculattion (1) Lagrange Polynomial Curve Given (x o,y o ),(x 1,y 1 ).(x n,y n ) x i < x j and i < j n order interpolational polynomial equation Ex: L( x) L 0 n y f ( x ) y. L i ( x ) i n 0 ( x x1)( x x2)...( x x )( x x )...( x x ) i1 i1 n ( x x1)( x x2)...( x x )( x x )...( x x ) i i i1 i1 n j0 i j ( x) x x i j x x x x1 ( x x )...( x x ) 0 1 j n i

10 Ship Calculattion (1) Property If the number of points are increased to enhance the procision of curve.the degree of polynomical is increased but make increase a posibily of oscillation. The curve is too perfect. Below these oscillations in an interpolating curve of n = 7 The variation of curve affects a whole curve

11 Ship Calculattion (1) This particular value of the parameter t is associated with each interpolating point P i and is called knot As normal: we will give the values of t such as t i = t i or t i-1 - t i = 1 1 named: grid uniform which can be 2 or 3...const... The Curve will be in same. Another way: If we give various grid space We will take different Curve such below:

12 Ship Calculattion (2) Parabolas Blending Basic idea : For 4 given pointsp,,,, 0 P1 P2 P3 Find the two over lapping parabolas and through P P,,, 1 2 P 3 through P, P P, 0 1 2

13 Ship Calculattion (2) Then find a smooth between the two interior points P 1 and P 2 by blending the two over lapping parapolic segments This method was first suggested by over houser.(1968) Parabolic blending is a measure to draw a smooth curve between 2 interior points from 4 given points.

14 Definition Ship Calculattion (2) Parabolic blended curve is given by C (1 t) t n s, (linear form) Where pn g s are parametric parabolas though, and,,, respectively P, P P The parametric representation of p n p P P t 1 2 P 3 g [ n 2 n1][ B ] g s [ s 2 s1][ D] p, g n s is

15 Ship Calculattion (2) Where [B] & [D] are metrics involving the position vectors P, P P, and To determind [B] & [D] that it necessary to establish the relationship between r, t and s P P,,, 1 2 P 3 Assumed: n = k 1 t+k 2 s = k 3 t +k 4 Voting that data is freguently evenly or nearly evenly spaced and assuming that the parameter rauge are normalized:

16 Ship Calculattion (2) To calculate K i we will write the condition at the end of two parabolas and the curve segment. Boundary condition: P(0) = P o P(0,5) = P 1 P(1) = P 2 Q(0) = P 1 Q(0,5) = P 2 Q(1) = P 3 C(0) = P 1 C(1) = C 2 This result in: n(t) = ½(t+1) ; s(t) = ½t Then:

17 Ship Calculattion (2) Generalized Parabolic Blending Normal parabolic blending is assumed a specific value of ½ for parameter r and s at P 1 and P 2 If the postion vector (data) to be fit are not nearly evenly space,more resonable assumption is to use a normalized chord length approximation.

18 Ship Calculattion (2) Generalized Parabolic Blending Using the Boundary consition as concerned we will take the relationship between n,s with t parameter. n t (1 ) t s. t t

19 Ship Calculattion (2) Generalized Parabolic Blending ASSUMSION: The Blending Curve ( Cat- Mull Curve ) starts at the point P 1 and ends at P i-1. To make it pass through over all initial given points, we must add more two point Called : Imagination Points Example:

20 Ship Calculattion (3) Spline Curve Mathematical spline: piecewise polynomical with continuous derivatives at knots. Knots ( given points or offsets) From: Plastic strip drafting spline. (Deforme to minimize energy) Use: Duck lead weight. Theory of spline is a certain generalization of the behavior of the elastic spline.

21 Ship Calculattion (3) Spline Curve *Linearized beam theory: E: young s modulus (deterimed by meterid properties of the beam) I: Moment of Inertia (deterimed by the cross-sectional shape of the beam) Assuming that ducks acts as simple supports, the bending moment M(x) is know to way linearly between supports. Substituting: M(x) = Az + B. Euler s equation becomes: y = Ax 3 +Bx 2 +Cx = D. for the deffection of the beam. The results show that the shape of the physical spline between ducks is mathematically discribled by cubic polynominal.

22 Ship Calculattion (3) Spline Curve A piecewise 3 th degree polynomial with the following characteristics: -Between Ducks are knots. -We need Ducks to decribe the curve. Between Duck Between Knots y constant discontinuous y linear continuous y guardtic continuous y cubic continuous C 2 (second order) continuing at knots. General mathematical spline: a piecewise polynomical of degree m with contiunity of derivatives of order m-1 at knots

23 Ship Calculattion (3) Spline Curve Equation of Cubic Spline The equation for a single parametric cubic spline segment is given by: Where t i t i+1 are the parameter values at the begining and the end of segment:

24 Ship Calculattion (3) Spline Curve is the position vector of anypoint on the cubic segment = [x(t) y(t) z(t)] : Vector values function or cantisian coordinate of position vector. The constant coefficient B i are determined by specified Four Boundary Conditions for Spline Segment. Or = [x (t) y (t) z (t)]

25 Ship Calculattion (3) Spline Curve Assuming without loss of generality, that t i = 0 and applying the Four Boundary Conditions.

26 Ship Calculattion (3) Spline Curve Cubic Spline with Internal points Generaling for n data points to give n-1 piecewise cubic spline segments with position, slop and curvature i.e C 2 -continuity at the internal jonts. The generalized equations for any two adjacent cubic spline segments:

27 Ship Calculattion (3) Spline Curve For any two adjcont spline segment equating the second derivatives at the common interal joints, i.e, letting Applying this equation recusively over all the spline segment yields n-2 equations for the tangent vectors

28 Ship Calculattion (3) Spline Curve

29 Ship Calculattion (3) Spline Curve

30 Ship Calculattion (3) Spline Curve Alternate cubic Spline end conditions.

31 Ship Calculattion (4) Bezier Curve A Bezier curve is determined by a defining A parametric Bezier Curve is defined by Where the Bezier or Bernstein basic or blending.

32 Ship Calculattion (4) Bezier Curve The properties of Bezier Curve -The basic functions are real. -The degree of the polymomial defining the curve segment is one less than the number of defining polygon points. -The curve generally follows the shape of the defining polygon. -The first and last points on the curve are concident with the first and last points of the defining polygon. -The tangent vectors at the ends of the curve have the same direction as the first and last polygon segments respectively. -The curve is contained within the convex hull of the polygon obtainable with the defining polygon ventices. -The curve exhibits the variation diminishing propenty.

33 Ship Calculattion (5) B-Spline The properties of B-Spline Curve B- Spline curve of order K: Peicewise continuos polynomial of degree K-1. Continuously differentiable K-2 times. C 2 continuity As with Bezier, B-spline is uniquely defined by the ventical of a defining polygon The order K of the B-spline and the number of ventical m may be chosen independently of each other a long as m >= K. Bezier curve : m = K B-spline curve : m > k m > k allows local modification

34 Ship Calculattion (5) B-Spline

35 Ship Calculattion (5) B-Spline

36 Ship Calculattion (5) B-Spline