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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

FINITE ELEMENT ANALYSIS OF RESPONSE OF A FLOATING STRUCTURE TO AN UNDERWATER EXPLOSION (UNDEX)

M.Sc. Thesis by

Fatih ARUK

Supervisors: Prof. Dr. Tuncer TOPRAK

Dr. Ergün BOZDAĞ

Examination Date: 09.06.2008

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

What is the importance of UNDEX resistance?

• Warships should last destructive effects of any near underwater explosion.

• Also shipboard systems must be shock hardened to a certain level to ensure combat survivability of both personnel and equipment.

So, shock resistance is a major issue that should be considered during early design phase of...

warships, radars, weapons, torpedos or any other shipboard equipment

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Shock Trials (Physical Tests);

• Extremely Expensive• Dangerous

• Harmful to the surrounding Environment

• Require years of planning and preparation

• Due to safety risk, shock trials do not test up to the ship’s design limits or even the true wartime shock environment.

• These tests are performed after the first ship is already built.

Shock trials of USS WINSTON S. CHURCHILL (DDG 81) in 2001

• United States Navy spent tens of millions of dollars • Years of planning and preparation

Shipboard eqipment testing; MIL-S 901D (Military Specification for shock testing of ship board equipment)

Shock Test Platform

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Impact Tests;

expensive experimental tests on simple cylindrical shells and plate structures.

Computational modeling and response, if perfected, can effectively and accurately replace the experimental procedures used to obtain the UNDEX response.

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX Simulation;The shock response of an immersed or floating structure is obtained when subjected to a near UNDEX loading

• Kwon and Cunningham; dynamic responses of stiffened cylinder and beam elementsDYNA3D [Explicit FE Code]+ Underwater Shock Analysis (USA) [BE Code based on DAA]

• 90s Kwon and Fox;the nonlinear dynamic response of a cylinder subjected to side-on underwater explosion

• Sun and McCoy UNDEX analysis of a composite cylinder ABAQUS + a fluid-structure interaction code• Cichocki, Adamczyk, and Ruchwa implemented fluid-

structure interaction phenomenon, pressure wave distribution, and the radiation boundary conditions into ABAQUS.

Three dimensional ship shock trial simulation of a warship was performed by Shin in 2004

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX PHENOMENA;

0 1.5 12 13.5 15 149 150.5 152Radius

(m)

234.4 Mpa

23.4 Mpa

15.2 Mpa

2.3 Mpa1.1 Mpa

Similitude Relations (Pressure versus Time)

1

( )( , )A

cc

aP R t P

Rf

B

c c

c

a v t

R a

( )f e 1

1.338 1.805( ) 0.8251 0.1749f e e 7

;P;R;ca;t

Pressure

, , , ;c cP v A B

Distance to charge (stand-off distance)

Charge radiusTime

Charge constants

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

0 1 2 3 4 5 6

x 10-3

0

2

4

6

8

10

12

14

P(M

pa)

t(s)

Pressure vs. time history for 25kg of HBX-1 charge, standoff distance of 10m

according to Swisdak

according to Price

UNDEX PHENOMENA;Similitude Relations (Pressure versus Time)

Material Source

TNT (1.52 g/cc) Coles (1946) 1.42 992 0.13 0.18

TNT (1.60 g/cc) Farley and Snay (1978) 1.45 1240 0.13 0.23

TNT (1.60 g/cc) Price (1979) 1.67 1010 0.18 0.18

HBX-1 (1.72 g/cc) Swisdak (1978) 1.71 1470 0.15 0.29

HBX-1 (1.72 g/cc) Price (1979) 1.58 1170 0.144 0.24

Pentolite (1.71 g/cc) Thiel (1961) 1.65 1220 0.14 0.23

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Explosive Gas Bubble

UNDEX PHENOMENA;

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

1

3

5 5

6( 10)

cmT K

D

1

3

max 6 1

3( 10)

cma K

D

Explosive Gas Bubble

UNDEX PHENOMENA;

Gas Bubble Period Max. Gas Bubble Radius

For our case ( MIL-S-901 D; 27.3 kg HBX-1 at 7.3 m depth) T=0.64 s

amax =4.4 m

• Bubble pulses are a strong source of excitation for ships whose bending vibration mode is near to the bubble pulse frequency

• It is especially important for the late time response of the ship

• However, for our case, the first mode of vibration is well above the bubble pulse freq. calcualted above.

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Explosive Gas Bubble, Geers-Hunter Bubble Model (2002)

UNDEX PHENOMENA;

UNDEX= SHOCK WAVE PHASE + BUBBLE OSCILLATION PHASE

SHOCK WAVE PHASE provides initial conditions for BUBBLE OSCILLATION PHASE

ABAQUS includes Geers-Hunter (2002) model for UNDEX loading.(a fourth-order Runge-Kutta integrator to prescribe the pressure variation)

MIL-S-901 D; 27.3 kg HBX-1 at 7.3 m depth 27.3 kg HBX-1 at 65 m depthPressure variation for two cases (7.3 m and 65 m cases) at stand-off distance (8.7 m)

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX PHENOMENA;Cavitation

Cavitation takes place in water when there is area of near-zero absolute pressure (about 206.8 Pa)

Two types of cavitation occur in an UNDEX event;

‘bulk’ cavitation;a large volume of low pressure due to reflections from sea surface

‘local’ cavitation;a small zone of low pressure at fluid-structure interaction surface.

The effect of cavitation on the response of the floating structures is important and must be properly modeled to obtain physically meaningful results.

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

1234

6315.2

6096

7620

9144

12192

2,3,41

UNDEX PHENOMENA;Cavitation

‘bulk’ cavitation

0i atm stc RP P P P Cavitation condition:

2 1j j

cf

R Rt

c

( , ) 0F x y

( , ) 0G x y

Equations of lower and upper cavitation boundaries:

EXPLOSION VARIATIONS ACCORDING TO MIL-S-901 D

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX PHENOMENA;Cavitation

‘local’ cavitation

Taylor plate theory

Assumptions;• The plate is rigid• The shock wave is planar

p i rv v v

i f f i

r f f r

P c v

P c v

pp t i r

dvm P P P

dt

max2 2p tp f f p i

dvm c v P P e

dt

1

max

A

cc

aP P

R

max2( )

(1 )t t

pp

Pv t e e

m

max2( )

1t t

p

PP t e e

f f

p

c

m

As becomes large (a lightweight plate), cavitation occurs faster.

ln

1cavt

1 1maxmax

2p

f f

Pv

c

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX PHENOMENA;Cavitation

‘local’ cavitation

pp atm

dvm P

dt cavt t

max( ) ( )atmp cav p

p

Pv t t t v

m

cavt t

max2( )

(1 )t t

pp

Pv t e e

m

max2( )

1t t

p

PP t e e

cavt t

cavt t

Incident and total pressures behind, and velocity of shock platform subjected to through-centerline UNDEX of 50 kg of HBX-1 charge at 30 m depth.

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;

Acoustic EquationsEquilibrum equation for small motions of a compressible, adiabatic fluid with velocity-dependent momentum loses;

0f ff

p

u u

x The slow flow assumption is accurate for

steady fluid velocities up to Mach 0.1

Acoustic Constitutive Equation;

f fp K

ux

Acoustic Constitutive Equation for cavitating fluid;

v f fp K

ux

max ,v c op p p p linear

nonlinear

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

0f ff

p

u u

x f fp K

u

x

1 10

f f f f

pp p

K K

x x

Introducing a variational field δp, integrating over entire body and applying Green’s theorem yields;

1 10

f fpf f f fV S S

p pp p p dV p T dSK K

xx x

1f

f

pT

x n u nx

Boundary traction term

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

fpSthe value of the acoustic pressure isprescribed;

ftSprescribes motion of the fluidparticles, modeling pressure wave

0f

ftT T x n u

0p

fiSthe radiating acoustic boundary,waves passing exclusively outward

1 1

1 1fiT p p

c a

x

fsS acoustic-structural interactionf m n u n u

mfsT

x n u

Acoustic Boundary Conditions

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

0

1 1

1 1

1 10

f ft

fi fs

f f f fV S

m

S S

p pp p p dV pT dSK K

p p p dS p dSc a

x x

n u

:

0fs t

m m m mc

V V V

m m

S S

dV dV dV

p dS dS

σ u u u u

u n u t

the final variational statement for the acoustic medium

the virtual work statement for a structure.

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

The Discretized Finite Element Equations

interpolation functions

P Pp H p

m N Nuu N 1,2,...N up to the number of displacement degrees of freedom.

1,2,...P up to the number of pressure nodes

[ [ ]] ff f ff sM p C p K p u PS

[ ] [ ]Ts f ss ssM u C u K u p PS

I Sp p p

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Surface-Based Acoustic-Structural Interaction Procedure

( ) ( ( ))fs

m iN N N

iS

p dS A N m

in u n x P x u ( ) ( ( ))fs

m iN N N N

iS

p dS p A N u n n x P x

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

( , ) j oI j x j

f

R Rp t p t p

c

x x

j s jR x x

s ox j

s j

p

x xx

x x

Note that the source point sx should be located out of the fluid domain.

Incident Wave Loading

Elements of UNDEX Simulation;Formulation of Direct Integration, Coupled Acoustic-Structural Analysis

o s oR x x

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Elements of UNDEX Simulation;Mesh Refinement

For reasonable accuracy, at least six representative internodal intervals of the acoustic mesh should fit into the shortest acoustic wavelength present in the analysis. Eight or more will be better.

maxL ; maximum linear element length

f f fc K ; the speed of sound

;the number of linear elements per acoustic wavelength

maxf ; max. frequency of excitation which can be simulated accurately

maxmin max

fcfn L

minn

max

max

1500

8 0.053750

m sf

mf Hz

We used an element size of about 50 mm around the acoustic-structural interface. The element size increases up to 150 mm at outer fluid regions.

Meshing whole fluid medium with 50 mm elements would result in about 16 million elements!

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Explicit Time Integration

Elements of UNDEX Simulation;

( 1) ( )1 1

( ) ( )2 2

( )2i iN N

ii

i

Nt tu uu

( 1) ( ) ( 1) 1( )

2

N N Ni i i

iu u t u

• An explicit central-difference time integration rule is used;

• Each increment is relatively inexpensive because there is no solution for a set of simultaneous equations.

• The time increments must be quite small so that the accelerations are nearly constant during an increment.

• A lumped mass matrix is used because its inverse is simple to compute.

1

( ) ( ) ( )N NJ J Ji i iu M P I

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Advantages of the explicit time integration method;

Explicit Time Integration

Elements of UNDEX Simulation;

• Well-suited to solving high-speed dynamic events

• No global tangent stiffness matrix. Iterations and tolerances are not required.

Stability of Explicit Integration;max

2t

2max max

max

21t

The Stable Time Increment Estimation;

If damping included;

minLt

c

2sc

(1 )(1 2 )

Ev

v v

2(1 )

E

v

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

Structural Damping

Explicit Time Integration

Elements of UNDEX Simulation;

[ ] [ ] [ ]s R s R sC M K

Mass Proportional Damping

Stiffness Proportional Damping

2 2R iR

ii

In names of natural freq.;

damps lower frequencies

damps higher frequencies

Effects of damping on the stable time increment in Explicit Analysis

2max max

max

21t

maxmax

max2 2RR

R has greater effect on stable time increment

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS METHODOLOGY

UNDEX TEST PARAMETERS FROM MIL-S-901D

MIL-S-901Dspecification which covers shock testing requirements for ship board machinery, equipment, system and structures.

heavyweight shock testing platform27.2 kg HBX-1

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX TEST PARAMETERS FROM

MIL-S-901D

UNDEX FE MODEL GENERATION

FLUID-STRUCTURE INTERACTION FE

ANALYSIS

CORRELATION OF UNDEX RESPONSES AND VALIDATION OF

NUMERICAL CODE

CONDUCT UNDEX TEST

SHORT DURATION DYNAMIC RESPONSE

Equivalent?

SHORT DURATION DYNAMIC RESPONSE

NO

MODIFY UNDEX MODEL/ANALYSES

PARAMETERS

YES

UNDEX ANALYSIS METHODOLOGY

UNDEX CORRELATION METHODOLOGY

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS METHODOLOGY

SUBMODELING ANALYSIS

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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MODELING AND ANALYSIS

Weight; about 39 tones

CAD MODELING; CATIA V15

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MODELING AND ANALYSISMESHING; ABAQUS CAE

Number of nodes: 140316

Number of elements:142327 Linear quadrilateral elements of type S4R

Connections were imposed by means of kinematic couplings

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

MODELING AND ANALYSISGENERATING A REDUCED (COARSE) MODEL for tryouts and acoustic mesh convergence studies; HYPERMESH

Number of nodes: 7858

Number of elements: 8229 Linear quadrilateral elements of type S4R538 Linear triangular elements of type S3R

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

MODAL ANALYSIS

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Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH REDUCED (COARSE) MODELFLUID MESH CONVERGENCE ANALYSIS

Boundary Conditions

1

1

f f

f

c K

1

1

2f f f f

fa K

Plane type radiating surfaces;

1f

0

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH REDUCED (COARSE) MODELFLUID MESH CONVERGENCE ANALYSIS

Acoustic-Structural InteractionInitial Static PressureIncident Wave (UNDEX) Loading

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH REDUCED (COARSE) MODELFLUID MESH CONVERGENCE ANALYSIS

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH REDUCED (COARSE) MODELFLUID MESH CONVERGENCE ANALYSIS

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH REDUCED (COARSE) MODELANALYSIS WITH DEFORMABLE PLATFORM

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH REDUCED (COARSE) MODELANALYSIS WITH DEFORMABLE PLATFORM; EFFECT OF DAMPING

1.5R

0.5 6R E

For first two modes; % 0.4

For the first torsional and bending modes ; 0.25 % and 0.15 %

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH REDUCED (COARSE) MODELANALYSIS WITH DEFORMABLE PLATFORM; EFFECT OF DAMPING

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH MAIN (FINE) MODEL

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH MAIN (FINE) MODELEFFECT OF MESH REFINEMENT AROUND ACOUSTIC-STRUCTURAL INTERACTION REGION

Mesh Convergence Analysis;

Element size; 150 mm

Number of nodes; 1059260

DFT Analysis of Incident Pressure Waves;

Element size; 50 mm

Number of nodes; 4198257

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH MAIN (FINE) MODELEFFECT OF MESH REFINEMENT AROUND ACOUSTIC-STRUCTURAL INTERACTION REGION

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH MAIN (FINE) MODELEFFECT OF CAVITATION

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH MAIN (FINE) MODELEFFECT OF STRUCTURAL DAMPING 1.5R 0.5 6R E

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

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UNDEX ANALYSIS WITH MAIN (FINE) MODEL

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

UNDEX ANALYSIS WITH MAIN (FINE) MODEL

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

SUBMODELING ANALYSIS

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Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

CONCLUSION AND OUTLOOK

• Fluid mesh size has important effect on the structural response and should be selected carefully for accurate results; Mesh Convergence Study DFT analysis of loadings

• Though it requires a nonlinear fluid behavior which adds to the cost

of the analyses, including cavitation is a must to obtain physically meaningful results.

• The effect of damping was also shown to be important for peak acceleration estimation in the late time response of the platform.

• Submodeling analysis can bu run to obtain converged stress-strain results at some sub-region of the structure.

• Experimental work is a must for the validation of the numerical code used and of the analysis procedure followed in this study. The work done in this work will provide the basis for the future experimental work.

İTÜ

Faculty of Mechanical Engineering

Mech. Eng. Fatih ARUK

aruk@itu.edu.tr

END OF THE PRESENTATION

QUESTIONS?