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# Num Integration

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### Num Integration

2. 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. 3. - uses 2 data points - assume linear curve x1 x2 x3 x4 s s s y1 y2 y3 y4 A1 A2 A3 : y=ax+b Total Area = A1+A2+A3 = s/2 (y1+2y2+2y3+y4) A1=s/2 (y1+y2) A2=s/2 (y2+y3) A3=s/2 (y3+y4)
4. 4. - uses 3 data points - assume 2nd order polynomial curve Area : )4( 3 321 3 1 yyy s dxydAA x x x1 x3 y(x)=ax²+bx+c x y A dx x1 x2 x3s y1 y2 y3 x y AdA Mathematical Integration Numerical Integration x2 s y(x)=ax²+bx+c
5. 5. x1 x2 x3 s y1 y2 y3 x y x4 x5 x6 x7 x8 x9 y4 y5 y6 y7 y8 y9 Gen. Eqn. Odd number )y4y2y...2y4y(y 3 s A n1n2n321 )4242424( 3 )4( 3 )4( 3 )4( 3 )4( 3 987654321 987765 543321 yyyyyyyyy s yyy s yyy s yyy s yyy s A
6. 6. - uses 4 data points - assume 3rd order polynomial curve x1 x2 x3 s s y1 y2 y3 y(x)=ax³+bx²+cx+d x y Area : )33( 8 3 4321 yyyy s A A x4 y4 Simpson’s 2nd Rule (skip)
7. 7. Application of Numerical Integration • Application - Waterplane Area - Sectional Area - Submerged Volume - LCF - VCB - LCB • Scheme - Simpson’s 1st Rule
8. 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. 9. Waterplane Area y x dxFP AP y(x) area Lpp WP dxxydAA 0 )(22 )width(aldifferenti )(atbreadth)-foffset(hal)( )area(aldifferenti )area(planewater 2 2 ftdx ftxyxy ftdA ftAWP Factor for Symmetric W.A.
10. 10. Waterplane Area(cont.) • Generalized Simpson’s Equation ..24y 3 1 2 210 yyxAWP stationsbetweendistancex y x FP AP 0 1 2 3 4 5 6 x
11. 11. Sectional Area • Sectional Area : Numerical integration of half-breadth as a function of draft WL z y dz y(z) T area T t dzzydAA 0 sec )(22 )width(aldifferenti )z(atbreadth)-foffset(hal)( )area(aldifferenti )(toupareasectional 2 2 sec ftdz ftyzy ftdA ftzA t
12. 12. Sectional Area(cont.) • Generalized Simpson’s equation swaterlinebetweendistancez nn area T t yyyyz dzzydAA 1210 0 sec 4..24y 3 1 2 )(22 z y WL T 0 2 4 6 8 z
13. 13. Submerged Volume : Longitudinal Integration • Submerged Volume : Integration of sectional area over the length of ship • Scheme z x y )( xAs
14. 14. Submerged Volume • Sectional Area Curve • Calculus equation volume L tssubmerged pp dxxAdVV 0 sec )( x As FP AP dx )(sec xA t • Generalized equation nns yyyyx 1210 4..24y 3 1 stationsbetweendistancex
15. 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 + + - FP WL
16. 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. 17.  Disebut juga dengan KB (Keel to Buoyancy)  Merupakan titik berat dari volume displacement kapal  KB atau VCB = ntdisplacemevol. keelabout themomenttotal