Sii : Simulation of a Ship Ramming Ice

Transcription

1 sii_d_h.mcd Sii : Simulation of a Ship Ramming Ice Claude Daley, Nov. '93 Modified: July, 95, Jan., Mar. 97 a Mathcad 6.+ File File : Sii_D_H - Ship_ice_interaction model for D (head-on) ramming. The model is udated to calculate the cases for the Harmonized Rules. This file calculates the forces on a ship, when ramming a floe of finte thickness and extent. The surge, heave, pitch and bennding of the ship are modelled. The crushing, tipping, and bending of the floe are included. Disclaimer: The model contains idealizations and assumptions. The author believes that the model fairly represents the mechanics that occurs when a ship strikes an ice edge, based on comparison with available data. However, the model contains numerous parameters, many of which have a significant effect on the particular results. Caution should always be used when applying the results.

2 sii_d_h.mcd INPUT: Ship Parameters The INP vector is copied from the excel sheet or can be supplied by the user Main Particulars: length (m) LBP := INP LBP = 3 beam (m) B := INP B = 4.7 draft (m) T := INP T = 5.46 waterline entrance angle alf := INP deg alf =.6 7 initial penetration initpen. Po (Pa) Po INP exponent ex INP deg stem angle (from vert) gam 9 INP 8 := gam =.47 Block coefficient CB := INP 3 CB =.737 Waterplane coefficient CWP := INP 4 CWP =.8 ship speed (m/s) vship := INP 6 vship = 5.3 INPUT: Ice Parameters := Po 3 6 := ex =. = for p_ice = Po A^ex water density (kg/m^3) ro 5 gravity (m/s^) g 9.8 ice density d:= ro g roice := ro.88 ice thickness (m) floe length (m) floe width (m) ice flex strength (Pa) hice := INP hice = 9 Lfloe := INP Lfloe = 4 Wfloe := INP Wfloe = 4 sigf := INP 6 3 sigf = 8 5

3 sii_d_h.mcd 3 Calculated Quantities displacement (m^3) mass of ship (kg) area of waterplane (m^) surge stiffness: Longitudinal Moment of Inertia of Waterplane Disp := LBP B T CB M := kx := Disp ro AWP := B LBP CWP Disp = 5 M =.4 8 AWP = n := n =.54.9 CWP.49 I L n B ( LBP )3 := I L = Heave Stiffness MN/m K H := AWP d Heave Only K H = F bow δ δ Pitch Stiffness MN-m/rad K P := I L d F bow Pitch Stiffness for Bow Force MN/m Pitch Only K B = F bow δ δ K B := K P LBP F bow Vertical Stiffness at Bow ky := + K B K H Heave + Pitch K BOW = F bow δ δ ky = F bow

4 sii_d_h.mcd 4 Vertical Rigid Body Mass at Bow <= 37% of ship's mass**** (includes added mass). + B 3T My :=. AM Disp ro AM := My = AM = Horizontal Rigid Body Mass at Bow Mx := Disp ro.45 <= 5% of ship's mass**** Mx = (includes added mass) Generalized bending stress of Pitch/Heave Mode Generalized Mass of st Mode Mf :=.45 LBP 3 Mf = 85 (kg) 45 σh := LBP σh 5 = Pa/m Generalized Stiffness of st Mode kf := LBP kf =.95 8 (N/m) Natural frequency of st mode om := kf Mf om.55 π Midbody moment of Inertia: = hz Im :=.54 LBP 4 Generalized bending stress of st Mode 97 σf := LBP σf = 3. 8 Pa/m Midbody total Height: Hm :=.5 T Midbody section modulus: Sm := Im Hm

5 sii_d_h.mcd 5 Ice Force Parameters The following functions determine the ship-ice interaction geometry and forces, based on the ship's bow moving horizontally and vertically, and the ice edge moving vertically (it is assumed that the floe is laterally restrained by other ice). Small pitch angles are assumed. penetration function : vertical projection of area pressure in contact Pen( x, y, z) := [ x ( y + z) tan( gam) ] Area v ( xy,, z) := Pen( x, y, z) tan( alf) Pres( x, y, z) := Po Area v ( xy,, z) ( ex) Pen γ α Pen le γ F v elastic layer thickless for ice and ship (m) ( - a numerical requirement) : ice elastic force (during elastic contact): ice crushing force (during crushing contact) resultant vertical ice force: le F el ( xy,, z, p) := ( Pen( x, y, z) p) F cr ( xy,, z) := Area v ( xy,, z) Pres( x, y, z) Area v ( xy,, z) Pres( x, y, z) () F v ( xy,, z, p) := max min F el ( xy,, z, p) F cr ( xy,, z) le F h resultant horizontal ice force: F h ( xy,, z, p) := F v ( xy,, z, p) tan( gam) crushing velocity: (needed to track crushed depth) Crumv( x, y, z, vx, vy, vz, p) := if[ p < Pen( x, y, z) kl le, max( ( Pen( vx, vy, vz) )), ] kl.999

6 sii_d_h.mcd 6 Ice Force Limit Calculations z-mass of floe (kg) Mz :=.45 ro hice Lfloe Wfloe Mz = 4.63 floe tipping stiffness (N/m) kz := roice g 3 kz =.947 Lfloe Wfloe max tipping force (N) Fzmax := kz hice 9 Fzmax = 3.74 floe breaking force (N) Fvbend :=. sigf hice Fvbend = breaking force limited to long floes F vlim := ( Lfloe > hice) Fvbend + ( Lfloe < hice) F vlim = z-mass of cusp (kg) Mcz :=.45 roice hice 3 5 Mcz =.5 cusp tipping stiffness kcz := ro g 3 5 hice max force on broken cusp Fzcmax := kcz hice 9 Fzcmax = function to determine if the floe is broken in bending Flex( x, y, z, p, e) := ( e ) F vlim < F v ( xy,, z, p) + ( e > ).64

7 sii_d_h.mcd 7 Simulation Parameters Damping: surge damping (drag) cx := bow heave damping: critical damping cy cr := ky My cy cr = % of critical damping (N/ m/s) ship flexure damping ζy :=. cy := ζy ky My cf :=. cy = ice heave damping cusp heave damping cz :=. kz Mz ccz :=.5 kcz Mcz Initial conditions: y := initpen vship le = surge position - x = surge velocity = pitch position 3= pitch velocity 4= flexure position -y 5= flexure velocity 6= ice position - z 7= ice velocity 8= crushing penetration 9= crushing velocity =breaking condition = change in breaking cond

8 sii_d_h.mcd 8 Time step: Ty := π My ky <= bow heave period Ty =.798 Tf := om <= st mode period Tf =.943 dt := Tf <= time step (sec.) Time of simulation imax := 3 Tmax := dt imax sec Tmax = <= number of time steps <=total simulation time i :=.. imax ii :=.. imax + t := i idt <= time vector

9 sii_d_h.mcd 9 Simulation Mechanics Response functions for Masses Mx and My These are the accelerations of the masses. Fx( t, Fy( t, Ff( t, := := kx cx y Mx Mx y ky y + My Fz( t, y >. kf My F h y, y, y, y Mx y y 4 cy My y 3,, y, y 6 8 F kf v y y 4 := y y Mf 4 + Mf min( ( kz y Fzcmax 6 )) cz F v ( y, y, y, y Mz Mz y ) := +... <-----intact floe + 7 Mz min( ( kz y Fzcmax 6 )) cz F v ( y, y y ( y >. ) Mcz Mz y,, y 4 6 ) <---broken cusp 7 Mcz Cr( t, := Crumv y, y, y, y, y, y, y Brak( t, Flex y, y, y, y, y := y and Z vector components = surge position - x y = surge velocity Fx( t, = pitch position 3= pitch velocity y 3 4= flexure position -y Fy( t, 5= flexure velocity y 6= ice position - z 5 7= ice velocity Ff( t, Dty (, ) := 8= crushing penetration y 9= crushing velocity 7 =breaking condition Fz( t, = change in breaking cond Cr( t, y Brak( t,

10 sii_d_h.mcd Sketch of system cy Ky - Bow Heave Spring My - Bow Heave Mass Mx - Surge Mass cx y x Kx - Surge Spring Kx= pen γ φ kel Ice Force Fice (elastic + crushing) kcr Coupled R-K Difference Equations: Z rkfixed( y,, Tmax, imax, D) <= The simulation takes place by iteratively solving the simultaneous equations in a stepwise manner, using the built-in function rkfixed. := <= SOLVE

11 sii_d_h.mcd Simulation Results vship = 5.3 Po = 3 6 m/s pa LBP = 3 m Fig. - Bow coordinates vs. time 4 Fig. - Bow coordinates motions (m) Y (m) F v Z Fv := i 3 6 Pen X Y Vertical Force (MN): i, ( Z 5 ) i, ( Z 7 ) i, ( Z 9 ) i Time (s) Horizontal Force (MN): F h Z Fh := i i, ( Z 5 ) i, ( Z 7 ) i, ( Z 9 ) i 3 4 X (m) Total Force (MN): Ft := i ( Fv i ) + Fh i.5 Fig. 3 - Vertical and Total Force vs. Time Force (MN) time (s) Total Force Vertical Force Stress (MPa) stress in st mode: ( ) i Z 3 i sig := Z 5 i Fig 4. Bending Stress vs Time MPa Time(s) σf

12 sii_d_h.mcd Acceleration Functions Ax( x, vx, f, z, cr) F h ( xf,, z, cr) kx Mx x cx Mx vx := Bow Surge Mx Ay( y, vy, f) := ky y + My kf ( f My cy My vy Bow Heave Af( x, y, f, z, cr) := kf ( f Mf + F v ( xf,, z, cr) Mf Bending Calculate Accelerations at Bow AX := Ax Z i i, ( Z ) i, ( Z 5 ) i, ( Z 7 ) i, ( Z 9 ) i ( ) i, ( Z 4 ) i, ( Z 5 ) i ( ) i, ( Z 3 ) i, ( Z 5 ) i, ( Z 7 ) i, ( Z 9 ) i AY := Ay Z 3 i AF := Af Z i Acceleration (m/s) time (s) X -Accel Acceleration (m/s) time (s) Y -Accel Acceleration (m/s) time (s) F -Accel BM i FVmax = 4.95 FTmax = Xmax = Ymax.778 Xmax := max Z Ymax := max Z 5 Zmax := max Z 7 Penmax := max Z 9 = Zmax = Penmax =.46 + le AXmax := max( ( max( AX) min( AX) )) AYmax := max( ( max( AY) min( AY) )) AFmax := max( ( max( AF) min( AF) )) AXmax =.83 AYmax =.747 AFmax =.94 BMmax := := sig Sm imax = 3 i FVmax := max( Fv) FTmax := max( Ft) max( BM) BMmax =. 4

13 sii_d_h.mcd 3 assemble( FV, FT, X, Y, Z, AY, AF, pen, BM, dt, Ft) := U FV U FT U X U Y 3 U Z U pen 5 Ft = U AY 6 U AF U BM 8 U dt for i.. rows( Ft) + 9 U Ft i i U res := assemble( FVmax, FTmax, Xmax, Ymax, Zmax, AYmax, AFmax, Penmax, BMmax, dt, Ft) res = FVmax = 4.95 Fvmax Ftmax Xmax Ymax Zmax Penmax Aymax Afmax Bmmax dt Ft INP LBP B T CB CWP Disp vship alf gam Hice Dice Po ex sigf

14 sii_d_h.mcd 4 pen := Z 9 i i le + +. Ar := i tan alf pen i Confirmation that Force/Area follows P/A relationship pr := i Fv i Ar i 8 pressure [MPa] pr i Ar i Area [m^]