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[4] num integration

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  • 1. Adi Wirawan Husodo
  • 2.  Why numerical integration? ◦ Ship is complex and its shape cannot usually be presented by mathematical equation. ◦ Numerical scheme, therefore, should be used to calculate the ship’s geometrical properties. Which numerical method ? ◦ Trapezoidal rule ◦ Simpson’s 1st rule ◦ Simpson’s 2nd rule
  • 3. - uses 2 data points- assume linear curve : y=ax+b y4 y2 y3 A1=s/2 (y1+y2) y1 A2=s/2 (y2+y3) A1 A3=s/2 (y3+y4) A2 A3 s s x3 s x4 x1 x2Total Area = A1+A2+A3 = s/2 (y1+2y2+2y3+y4)
  • 4. - uses 3 data points - assume 2nd order polynomial curve y(x)=ax²+bx+c Mathematical Integration Numerical Integration yy dx y(x)=ax²+bx+c y1 y2 y3dA A A x x x1 s x2 s x3 x1 x2 x3 x3 s Area : A dA y dx ( y1 4 y2 y3 ) x1 3
  • 5. y y6 y7 y8 y2 y5 y9 y1 y3 y4 s x x1 x2 x3 x4 x5 x6 x7 x8 x9 s s A ( y1 4 y2 y3 ) ( y3 4 y4 y5 ) 3 3 Odd number s s ( y5 4 y6 y7 ) ( y7 4 y8 y9 ) 3 3 s ( y1 4 y2 2 y3 4 y4 2 y5 4 y6 2 y7 4 y8 y9 ) 3 sGen. Eqn. A (y1 4y2 2y3 ... 2yn 2 4yn 1 yn ) 3
  • 6. Simpson’s 2nd Rule (skip) - uses 4 data points - assume 3rd order polynomial curve y y4 y2 y3 y1 y(x)=ax³+bx²+cx+d A x s s x3 x4 x1 x2 3s Area : A ( y1 3 y2 3 y3 y4 ) 8
  • 7. Application of Numerical Integration • Application - Waterplane Area - Sectional Area - Submerged Volume - LCF - VCB - LCB • Scheme - Simpson’s 1st Rule
  • 8. Numerical Calculation• Calculation Steps 1. Start with a picture of what you are about to integrate. 2. Show the differential element you are using. 3. Properly label your axis and drawing. 4. Write out the generalized calculus equation written in the same symbols you used to label your picture . 5. Write out Simpson’s equation in generalized form. 6. Substitute each number into the generalized Simpson’s equation. 7. Calculate final answer. Not optional ! Always follow the above steps!
  • 9. Waterplane Area y y(x) x FP dx AP Lpp AWP 2 dA 2 y( x ) dx 0 area AWP water plane area( ft2 )Factor for Symmetric W.A. dA differential area( ft2 ) y ( x ) y offset(hal - breadth) at x( ft) f dx differential width( ft)
  • 10. Waterplane Area(cont.)• Generalized Simpson’s Equation y x x FP 0 1 2 3 4 5 6 AP 1 AWP 2 x y0 4 y1 2 y2 .. 3 x distance between stations
  • 11. Sectional Area• Sectional Area : Numerical integration of half-breadth as a function of draftz T WL Asec t 2 dA 2 y( z ) dz 0 y(z) areaT dz Asec t sectional area up to z ( ft2 ) dA differential area( ft2 ) y y ( z ) y offset(hal - breadth) at z( ft) f dz differential width( ft)
  • 12. Sectional Area(cont.)• Generalized Simpson’s equation z WL 8 6 z T 4 z distance between waterlines 2 y 0 T Asect 2 dA 2 y ( z ) dz 0 area 1 2 z y0 4 y1 2 y2 .. 4 yn 1 yn 3
  • 13. Submerged Volume : Longitudinal Integration • Submerged Volume : Integration of sectional area over the length of ship • Scheme z x As (x ) y
  • 14. Submerged Volume • Sectional Area Curve As Asec t ( x ) dx x FP AP• Calculus equation Lpp Vsubmerged s dV Asec t ( x )dx volume 0• Generalized equation 1 s x y0 4 y1 2 y2 .. 4 yn 1 yn 3 x distance between stations
  • 15. Longitudinal Center of Floatation (LCF) • LCF - Centroid of waterplane area - Distance from reference point to center of floatation - Referenced to amidships or FP - Sign convention of LCF + WL - + FP
  • 16.  Merupakan titik berat dari luas bidang garis air (water plane area). Suatu titik dimana kapal mengalami heel atau trim. Titik ini terletak pada centre line (dalam arah memanjang), disekitar midship (bisa di depan atau dibelakang midship).
  • 17.  Disebut juga dengan KB (Keel to Buoyancy) Merupakan titik berat dari volume displacement kapal total moment about the keel KB atau VCB = vol. displaceme nt

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