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Analysis of added mass effect on surface ships subjected to underwater explosions

Trivedi Ruturaj Radhakrishna

Master Thesis

presented in partial fulfillment of the requirements for the double degree:

“Advanced Master in Naval Architecture” conferred by University of Liege "Master of Sciences in Applied Mechanics, specialization in Hydrodynamics,

Energetics and Propulsion” conferred by Ecole Centrale de Nantes

developed at ICAM in the framework of the

“EMSHIP” Erasmus Mundus Master Course

in “Integrated Advanced Ship Design”

Ref. 159652-1-2009-1-BE-ERA MUNDUS-EMMC

Supervisor: Prof. Hervé Le SOURNE, ICAM

Reviewer: Prof. Lionel Gentaz, Ecole Centrale de Nantes

Nantes, February 2019

ii Ruturaj Radhakrishna Trivedi

Master Thesis developed at ICAM, France

This page is intentionally left blank.

Analysis of added mass effect on surface ships subjected to underwater explosions iii

“EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019

DECLARATION OF AUTHORSHIP

I declare that this thesis and the work presented in it are my own and has been generated by

me as the result of my own original research.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With the exception

of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made clear

exactly what was done by others and what I have contributed myself.

This thesis contains no material that has been submitted previously, in whole or in part, for the

award of any other academic degree or diploma.

I cede copyright of the thesis in favour of ICAM.

Date: Signature

iv Ruturaj Radhakrishna Trivedi



Analysis of added mass effect on surface ships subjected to underwater explosions v


ABSTRACT

Analysis of added mass effect on surface ships subjected to underwater

explosions

By Trivedi Ruturaj Radhakrishna

This study presents a method to determine analytically the added mass used for analysis of

surface ships subjected to underwater explosions. Regarding the action of the first shock wave

generated by the explosion, Taylor’s theory is used to calculate the acoustic pressure which is

applied on the ship hull. The main purpose of the present study is to investigate the effects of

the water inertial forces on the whipping response of the surface ships subjected to underwater

explosions.

A macro is first developed in ANSYS APDL to calculate the added masses to be attached to

the nodes of the wet hull, using strip theory and ellipsoid methods. The Lewis transformation

mapping is used for simplifying the cross section of the ship. The results obtained from the

developed macros are then compared to results from previous researches extracted from the

literature. An implicit ANSYS model including added masses attached on wet hull nodes is

built for modal analysis and then converted to be used for LS-DYNA explicit simulations.

Third, the two models-a semi-cylindrical stiffened like-ship structure and real surface ship, the

material and geometrical characteristics for which are provided by Chantiers de l’Atlantique,

are considered and simulations with added masses calculated analytically are confronted to

simulations based on a fully coupled finite element model where the water is represented by

acoustic elements.

Thus, in this research, a method was developed to calculate added mass components using strip

theory and ellipsoid methods and implement them on a real ship ANSYS model. It was confirmed

that a combination of strip theory and ellipsoid theory is better for obtaining added mass

components. It was confirmed that added mass depends on geometry of the body and fluid

density and plays an important role in underwater explosion analysis. It was also found that the

calculated natural frequency is reduced when added mass is considered. The simulation time

required for nodal mass method proposed in this research is much less than that of the fully

coupled fluid mesh which reduces simulation time and can be used in the initial design stage.

Keywords: Underwater Explosion, Finite Element Method, Fluid Acoustic Elements, Added

Mass, Strip Theory Method, Ellipsoid Method, Lewis transformation

vi Ruturaj Radhakrishna Trivedi



Analysis of added mass effect on surface ships subjected to underwater explosions vii


CONTENTS

DECLARATION OF AUTHORSHIP iii

ABSTRACT v

CONTENTS vii

LIST OF FIGURES ix

LIST OF TABLES xi

1. INTRODUCTION 1

1.1 Motivation 1

1.2 Objective 2

2. LITERATURE REVIEW 3

2.1 Underwater Explosion 3

2.2 Water Added Mass 5

2.3 Methods to Assess Added Masses 9

2.3.1 Elimination of Added Mass Components Due to Symmetry of Ship Hull and Added

Mass Matrix 9

2.3.2 Method of Equivalent Ellipsoid 9

2.3.3 Method of Plane Sections 11

2.3.4 Strip Theory Method 12

2.3.5 Determination of Additional Added mass components 16

3. Added Mass- Calculation, Implementation and Verification 17

3.1 Added Mass Calculation 17

3.2 Implementation of Added Mass Components 19

3.3 Verification of Results 20

4. Model Preparation and Verification 23

4.1 Model Preparation 23

4.1.1 Semi-Cylinder Ship-Like Model 23

4.1.2 Real Surface Ship Model 26

4.2 Verification of Added Mass Calculation 29

4.3 Modal Analysis 31

4.3.1 Semi-Cylinder like-ship model 31

4.3.2 Real Surface Ship Model 35

5. Simulations: Underwater Explosions 40

5.1 Model Preparation 40

5.2 Explosion Parameters And Model Processing 41

5.3 Analysis Process And Results 45

5.3.1 Energies 45

viii Ruturaj Radhakrishna Trivedi


5.3.2 Nodal Vertical Displacements 46

5.3.3 Frequency Analysis from Displacement Plot 47

5.3.4 Accelerations 48

6. CONCLUSIONS 50

7. FUTURE SCOPE 51

8. ACKNOWLEDGEMENTS 52

9. REFERENCES 53

APPENDIX A1 54

Analysis of added mass effect on surface ships subjected to underwater explosions ix


LIST OF FIGURES

Figure 1 Explosion under a surface ship [1] 1

Figure 2 Shockwave pulsation with respect to time [2] 3

Figure 3 Bubble migration [3] 4

Figure 4 Relation between bubble oscillation and pressure variation [3] 4

Figure 5 Surface effect of UNDEX [3] 5

Figure 6 Ship motion in 6 degrees of freedom [8] 8

Figure 7 Ship assumed as an Ellipsoid [8] 10

Figure 8 Strip theory representation of the ship [8] 13

Figure 9 Algorithm-added mass calculation macro 17

Figure 10 Areas on the selected frames 18

Figure 11 Array of geometric parameters [8] 18

Figure 12 Array of added mass components in ANSYS 19

Figure 13 Added mass implementation on wet hull 20

Figure 14 Input data for the macro : extracted from [10] 20

Figure 15 Finite element model of like-ship semi cylinder-dry model [3] 23

Figure 16 Semi-cylinder-nodal mass model 24

Figure 17 Semi-cylinder-fluid mesh model 25

Figure 18 Fluid structure interface mesh 25

Figure 19 Finite element model of surface ship -dry model 26

Figure 20 Surface ship - nodal added masses 27

Figure 21 Surface ship - fluid mesh model 28

Figure 22 Fluid structure interface mesh 28

Figure 23 1st vertical bending mode- Dry model 32

Figure 24 Vertical bending modes 33

Figure 25 Torsional bending 34

Figure 26 Local modes 34

x Ruturaj Radhakrishna Trivedi


Figure 27 1st Vertical bending mode- Dry model 36

Figure 28 Vertical bending modes 37

Figure 29 Transverse bending modes 38

Figure 30 Torsional modes 39

Figure 31 Surface ship in LSDYNA-dry model 40

Figure 32 Added lumped masses on directional springs 41

Figure 33 Location of charge 42

Figure 34 Positions of selected nodes for analysis 43

Figure 35 Positions of selected nodes 43

Figure 36 Positions of selected elements 44

Figure 37 Pressure distribution on selected elements 44

Figure 38 Local co-ordinate system at node 1314 45

Figure 39 Comparison of internal energies 46

Figure 40 Comparison of z displacements 46

Figure 41 Time period measurement-dry model 47

Figure 42 Time period measurement-wet model 48

Figure 43 Acceleration at the bow of the ship 49

Analysis of added mass effect on surface ships subjected to underwater explosions xi


LIST OF TABLES

Table 1 Degrees of freedom 7

Table 2 Comparison of results 21

Table 3 Added mass values 22

Table 4 Parameters of semi-cylinder like-ship 24

Table 5 Parameters of surface ship 27

Table 6 Added mass components of surface ship 29

Table 7 Added mass components of semi-cylinder like-ship 30

Table 8 Modal frequency comparison of models 31

Table 9 Model frequency comparison of models 32

Table 10 Modal frequency comparison of models 35

Table 11 Model frequency comparison of models 36

Table 12 Initial conditions of explosion scenario [3] 41

Table 13 Time period and frequency for dry model 47

Table 14 Time period and frequency for wet model 48

xii Ruturaj Radhakrishna Trivedi



Analysis of added mass effect on surface ships subjected to underwater explosions 1


1. INTRODUCTION

1.1 Motivation

UNDEX-Underwater Explosion is a very common war scenario. All the surface ships in war

zone are at the risk of facing explosions under water in their lifetimes. Thus, it becomes an

important subject of research for the navies of the world. A typical underwater explosion is

shown in figure 1.

Figure 1 Explosion under a surface ship [1]

Work on UNDEX was performed in the second half of the 19th century long before the First

World War. Later, intensive research was carried out during and after the Second World War.

These researches focused on replicating the phenomena using models and testing to improve

the designs. The phenomenon of first shock wave was studied until the bubble oscillation

effect was discovered. Later, mathematical models were used to analyze gas bubble

oscillations along with first shock wave during UNDEX. In recent times advanced methods of

numerical analysis such as Boundary Element Method and Finite Element Method are used

for analyzing UNDEX.

Former EMship student, Navarro in 2015, [2] worked on the effect of first shock wave

generated due to UNDEX on elasto-plastic behavior of flat plate. He found that the effect of

2 Ruturaj Radhakrishna Trivedi


first shock wave is much greater than bubble oscillation and did not take it into account for

analysis. But, it was found that the bubble oscillations may cause resonance of the ship which

may damage on board equipment and the structure of the ship. This effect was considered by

Ssu-Chieh Tsai in 2016 [3]. In 2017, Enes Tasdelen studied the response of ship equipment

during UNDEX [4] using the so-called dynamic design analysis method (DDAM).

The current work is a continuation of these previous works and aims to include the water

inertial effects in the numerical analysis of a surface ship subjected to underwater explosion.

Added masses are calculated analytically using strip theory and ellipsoid methods and Lewis

transformation is applied to simplify the ship cross sections.

1.2 Objective

In order to analyze the effect of explosion on a surface ship due to water inertial effects, the

work has two main objectives. The first is to study an existing analytical method used to

calculate the added masses and write a macro to develop an ANSYS finite element model

including added masses attached on the wet hull nodes. The second objective is to verify the

calculated added masses by confronting ship modal characteristics extracted from Lewis and

fully coupled models, and to convert ANSYS data file to LS-DYNA one in order to carry out

underwater explosion simulations. The corresponding work is divided into following steps:

Study existing analytical methods that allow calculating added masses of bodies floating

on water surface.

Prepare a model of semi-cylinder stiffened like-ship structure to be used for analysis.

Program using ANSYS-APDL (Ansys Parametric Design language) macro to calculate

and implement added masses on wet hull nodes.

Compare the added masses obtained from program with results of previous researches

from literature.

Implement the added mass on the semi cylinder like-ship and real surface ship provided

by Chantiers de l’Atlantique.

Perform modal analyses to verify the models.

Convert the implicit ANSYS model to explicit LSDYNA model for underwater explosion

simulations.

Perform underwater explosion simulations using LS-DYNA explicit solver.



2. LITERATURE REVIEW

2.1 Underwater Explosion

Highly energetic thermochemical reaction inside water is called underwater explosion. This

produces more damage to the vessel as compared to air explosion. It is because water has low

compressibility and it transfers pressure more efficiently than air. Figure 2 shows effect of

bubble pulses and exponential decay pressure under UNDEX.

First, the shock wave is generated due to the charge explosion. It is propagated as a spherical

wave at a speed of around 1500 m/s. The pressure generated is too high and behaves like an

impact on the ship surface. Then, the gas bubble is formed which expands till the gas pressure

is equal to fluid pressure and then contracts again. This expansion and contraction produces

waves of low pressure but the frequency of these bubble oscillations can match with the

natural frequency of the ship and cause hull girder’s resonance or whipping phenomena.

Figure 2 Shockwave pulsation with respect to time [2]

During bubble oscillation process, the changing of bubble size can be modeled as spring mass

system. The bubble oscillations and its migration toward the sea surface are shown in figure 3.

From this figure, it is visible that evolution of pressure level is related to bubble oscillation

phenomena. As the bubble pulsation level decreases, only the first and the second bubble

oscillations are considered for study of UNDEX [2].

Whenever an object is accelerated in a viscous fluid, the water surrounding the object interacts

with the body and exerts additional force opposing the motion of the object. This additional

force is the product of mass of water and the acceleration of the object. This leads to added



inertia to the object due to fluid structure interaction. This inertia is represented by added

mass which depends on the density of the fluid and the geometry of the immersed object. The

same phenomenon is seen with ships and water. This added mass and its interaction with the

ship affects the ship response to underwater explosions. Thus, this research focuses on

calculating this added mass for incorporating it into UNDEX analysis.

Figure 3 Bubble migration [3]

Figure 4 shows the variation of incident pressure according to oscillations and migration of

the gas bubble. The very high pressure observed at the beginning corresponds to the first

shock wave generated by the charge detonation. The pressure variation in the graph after this

high pressure is due to bubble pulses. When bubble radius reaches minimum, pressure peak

occurs.

Figure 4 Relation between bubble oscillation and pressure variation [3]



The surface effect of underwater explosion is shown in figure 5.

Figure 5 Surface effect of UNDEX [3]

2.2 Water Added Mass

Dubua in 1776 [5] first gave the concept of water added mass and studied small oscillations of

a spherical pendulum experimentally. Green in 1833 and Stokes in 1843 [5] obtained an exact

expression for water added mass of a sphere. Stokes studied the motion of a sphere in a finite

volume of fluid. The inertial and viscous properties of the fluid are used to determine

hydrodynamic forces and torques of a body in motion. The added masses of the body can be

used to express inertial forces and torques of a body in motion inside a fluid. Sometimes the

added masses are comparable to the mass of the body and thus have to be considered in the

dynamic analysis of immersed structure. For example, in the study of ships and submarines,

the added masses may be huge and become important [5].

An object moving or vibrating in a fluid displaces the surrounding fluid to accommodate for

its motion, which generates pressure inside the fluid and this pressure acts on the object. The

fluid moving around the object affects natural frequency and damping characteristics of the

object. An object moving with constant velocity in an ideal fluid experiences no resistance.

But, when the velocity changes, i.e. when the object accelerates, it experiences a resistance.

The body behaves like it has some fluid mass which adds to its proper mass. The total force

required to accelerate the body is given by equation (1). [6]

2

2( )a v

u uF m m c

t t

(1)

Where,

m = mass of object



am = added mass (hydrodynamic mass)

vc = viscous damping coefficient

The added mass is proportional to the density of fluid and the volume of object and is given

by [5]

a mm V c (2)

Where,

= fluid density

V= volume of object

mc = added mass coefficient

(m+ am )= virtual/apparent mass of the object

For an inviscid fluid, the viscous damping coefficient vc =0.

The above equation is valid for an object moving in one direction. For the motion of body

with three degrees of freedom, a 6 6 matrix is required. For a body with N degrees of

freedom, number of terms needed to describe the added mass is N(N+1)/2 [6]. The matrix of

N degrees of freedom of an object is represented by [ijm ], where i,j=1,2,...,N. In case of single

object, [ijm ] is symmetric and the eigenvalues of this matrix are called effective

hydrodynamic masses [6].

Floating structures with small motions and linear behavior can be modelled like a spring-

mass-damper system in which forced motion can be described by the following equation: [7]

( )mx bx kx f t (3)

Where,

m = system mass

b = linear damping coefficient

k = spring coefficient

f(t) =force acting on the mass

x = displacement of the mass

= natural frequency of the system

k

m

Physically, this added mass corresponds to the weight added to the system because the body’s

acceleration or deceleration moves some volume of fluid around it. The added mass opposes

the motion and increases the body’s inertia. It can be modeled with the following equation:



( ) mamx bx kx f t x (4)

Rearranging above equation,

( ) ( )am m x bx kx f t (5)

Now, the simple spring mass damping system can be used with a new mass ' am m m and

the natural frequency writes

''

'a

k

m

k

m m

(6)

But, ship is in motion in many directions in 6 degree of freedom and added mass effects can

be seen in one direction due to motion in other direction. Thus, a 6×6 matrix of mass

coefficients is obtained. The force matrix considering 6 degrees of freedom is given as,

11 12 13 14 15 16 1

21 22 23 24 25 26 2

31 32 33 34 35 36 3

41 42 43 44 45 46 4

51 52 53 54 55 56 5

61 62 63 64 65 66 6

m m m m m m u

m m m m m m u

m m m m m m uF

m m m m m m u

m m m m m m u

m m m m m m u

(7)

The motion of a ship in six degrees of freedom is shown in figure 6. The degrees of freedom

of the ship are defined as in Table 1.

Table 1 Degrees of freedom

Degrees of freedom Description Velocities

1 Surge- motion in x direction 1u - Linear

2 Sway-motion in y direction 2u - Linear

3 Heave- motion in z direction 3u - Linear

4 Roll- rotation about x axis 4u - Angular

5 Pitch- rotation about y axis 5u - Angular

6 Yaw- rotation about z axis 6u - Angular



Figure 6 Ship motion in 6 degrees of freedom [8]

The inertial hydrodynamic force is

1 2 3 4 5 6M [u u u u u u ]T

HF (8)

Where, M includes the inertia matrix of the ship and the water added mass.

M=M MS A (9)

MS is the mass and inertia moment matrix of the ship and is given as [8].

0 0 0

0 0 0

0 0 0

0

0

0

g g

g g

g g

s

g g x xy xz

g g yx y yz

g g zx zy z

m mz my

m mz mx

m my mxM

mz my I I I

mz mx I I I

my mx I I I

(10)

M A is the water added mass and added inertia moment matrix. ijm is a component of

hydrodynamic force in the thi direction due to unit acceleration in direction j. M A has 36

components [8].

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

41 42 43 44 45 46

51 52 53 54 55 56

61 62 63 64 65 66

A

m m m m m m

m m m m m m

m m m m m mM

m m m m m m

m m m m m m

m m m m m m

(11)



2.3 Methods to Assess Added Masses

For simple contours many different formulations are available [5]. Explicit formulations for

3D ellipsoid and thin cylindrical aerofoils are available. But, for most real ships it is not

possible to calculate the water added mass explicitly and hence approximate methods have to

be used. A combination of following methods was used to determine the added mass matrix

components analytically in this research.

2.3.1 Elimination of Added Mass Components Due to Symmetry of Ship Hull and Added

Mass Matrix

As a ship is symmetric on port-starboard (xz plane) vertical motion due to heave and pitch do

not induce force in transversal direction [8].

32 34 36 52 54 56 0m m m m m m

The added mass matrix being symmetric, ij jim m ,

23 43 63 25 45 65 0m m m m m m

Considering the same for motions in longitudinal directions due to acceleration in the

direction j=2,4,6:

12 14 16

21 41 61

0

0

m m m

m m m

Thus, for a ship moving with 6 degrees of freedom the added mass is reduced to 18

components [8]

11 13 15

22 24 26

31 33 35

42 44 46

51 53 55

62 64 66

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

A

m m m

m m m

m m mM

m m m

m m m

m m m

(12)

2.3.2 Method of Equivalent Ellipsoid

In this method, the ship is modelled by a solid ellipsoid like a rugby ball. The water added

mass components for this ellipsoid are calculated using the theory of kinetic energy of fluid

[8].

According to theory of kinetic energy of fluid,

j

ij is

m dSn

(13)

Where,



S = Wetted ship area

= density of water

i = flow potential when ship is moving in thi direction with unit speed

To calculate ijm , the ship is assumed to be a 3D body like a sphere, ellipsoid, cylinder, etc.

[8]. To represent a real ship, the most representative of the hull is elongated ellipsoid with

c/b=1 and r = a/b. Here, a and b are semi axis lengths of the ellipsoid as shown in figure 7.

Figure 7 Ship assumed as an Ellipsoid [8]

The added mass components can be defined as [8]

11 11

22 22

m mk

m mk

33 33

44 44

55 55

66 66

xx

yy

zz

m mk

m k I

m k I

m k I

(14)

ijk are called the hydrodynamic coefficients and are given by

011

02

Ak

A

(15)

022

0

033

0

44

2

2

0

Bk

B

Ck

C

k

(16)

2 2 2

0 055 4 4 2 2 2

0 0

( 4 ) ( )

2(4 ) ( )(4 )

L T A Ck

T L C A T L

(17)



2 2 2

0 066 4 4 2 2 2

0 0

(L ) ( )

2( ) ( )(L )

B B Ak

L B A B B

(18)

Where,

2

0 3

2

0 0 2 3

2 2

2 2

2(1 ) 1 1ln

2 1

1 1 1ln

2 1

1 1

e eA e

e e

e eB C

e e e

b de

a L

(19)

Where,

d = maximum diameter (m)

L= overall length (m)

Moment of inertia of displaced water is approximately the moment of inertia of the ellipsoid

2 2

2 2

2 2

1(4 )

120

1(4 )

120

1(B )

120

xx

yy

zz

I LBT T B

I LBT T L

I LBT L

(20)

The accuracy of this method depends on the shape of the body under consideration, in this

case, the ship. The more it is equivalent to the ellipsoid, the better are the results. This method

cannot determine some components of the added mass matrix like 24m , 26m , 35m , 44m , 15m , 51m

[8].

2.3.3 Method of Plane Sections

If a body is elongated along one of its axes (for example the x-axis) the added masses in

orthogonal directions (i.e., along y and z axes) can be computed by the method of plane

sections. In this method, the added masses of all plane sections orthogonal to the x-axis are

computed and then integrated along x. It is assumed that the motion of the fluid in the x-

direction is negligible if the body moves in any direction orthogonal to the x axis. This

assumption is well-satisfied for prolate bodies, when the ratio of the length of the body (L) to

its diameter (B or 2T) is large enough (λ = L/B ≥ 9). When λ gets smaller, the fluid motion

along the x-axis becomes essential, and the added masses computed by the method of plane

sections have to be corrected [5].



The formulas for added masses computed via the method of plane sections can be written as

follows:

2

22 220

1

2

33 330

1

2

24 240

1

( ) ( ) 212

( ) ( )

22

( ) ( ) 2

L

L

L

L

L

L

Lx dx

T

Lx dx

B

Lx dx

T

2

34 340

1

2

44 440

1

23

( ) ( ) 2

4

( ) ( ) 252

L

L

L

L

Lx dx

B

Lx dx

T

2

26 1 220

1

2

35 1 330

1

2

2

55 1 330

1

6

( ) ( ) 262

( ) ( ) 2

7

( ) ( ) 2 8

L

L

L

L

L

L

Lx xdx

T

Lx xdx

B

Lx x dx

B

2

2

6 1 220

1

( ) ( ) 292

L

L

Lx x dx

T

In the above equations, integration is performed between the extremities of the considered

section whose co-ordinates are L1 and L2. μ(λ) and 1 (λ) are correction factors related to fluid

motion along the x-axis. The sign of the added mass in the formula depends on the co-ordinate

system. [5]

This method was further simplified and applied for ships using the conformal mapping

initially proposed by Lewis [9] to simplify the ship cross section. The resulting method, which

was further used by Do [8] for ship water added mass calculation, is called the strip theory

method.

2.3.4 Strip Theory Method

The ''Ordinary Strip Theory Method” was introduced by Korvin-Kroukovsky and Jacobs in

1957[8]. Then it was developed by Tasai in 1969 with a “Modified Strip Theory Method” [8].

According to this method, the ship is supposed to be made of a finite number of transversal

3D slices (see Figure 8). The slice is representing the cross section of the ship at that point

along a given length and the added mass of this component can be calculated. Then, the 2D

added mass value is integrated along the length of the wet hull.



Figure 8 Strip theory representation of the ship [8]

2

22 22

1

2

33 33

1

2

24 24

1

( ) 30

( )

31

( )

L

L

L

L

L

L

m m x dx

m m x dx

m m x dx

32

2

44 44

1

2

26 22

1

35 33

( ) 33

( )

34

( )

3

L

L

L

L

m m x dx

m m x dx

m m x dx

2

1

2

46 24

1

2

66 22

1

5

( ) 36

( ) 37

L

L

L

L

L

L

m m x dx

m m x dx

Where, ( )ijm x is added mass of 2D cross section at location x.



For a real ship, the cross section may be complex. Lewis transformation mapping method is

thus used to map the complex cross section to a unit semi-circle. This method is detailed in

Appendix A1. The ratios H(x) and ( )x are defined as,

2 2

2 2

( ) 1( ) 38

2 ( ) 1

( ) 1 3(

) 39 ( ) ( ) 4(

4 1 )

B x p qH x

T x p q

A x p qx

B x T x q p

Where,

B(x) = breadth of the cross section

T(x) = draft of the cross section

A(x) = area of cross section

p and q are parameters defined with the ratio of H(x) and ( )x .

2

2

2

3(1 )

4 4 21

(1 )q

(40)

( 1)p q q (41)

4

1

1

H

H

(42)

The added mass components for each section may be written as

2 2 2 2

22 222

2 2 2 2

33 332

( ) (1 ) 3 ( )( ) ( )

2 (1 ) 2

( ) (1 ) 3 ( )m

43

( ) ( )8 (1

4)

48

T x p q T xm x k x

p q

B x p q T xx k x

p q

32 2

24 2

( ) 1 8 16 4 4m p(1 ) (20 7 ) q (1 ) (1 )(7 5 ) (45)

2 (1 ) 3 35 3 5

T xp q p p p p

p q

3

24 24

4 2 2 2 4

44 444

( )

m ( )2

( ) 16[ (1

46

) 2 ( )

( )256 (

1 ) 256

47

T xk x

B x p q q B xm k x

p q



2

26 22

2

35 33

3

46 24

48

49

( )( ) ( ) xdx

2

( )m ( ) ( )

8

( )( ) ( )

2

T xm x k x

T xx k x xdx

T xm x k x

2

2

66 22

50

( )( ) ( ) x d

x 51 2

T xm x k x

The total ijm for a section the extremities of which are located at L1 and L2 becomes:

2

2

22 1 22

1

2

2

33 1 33

1

3

24 1 24

( ) ( ) ( ) 522 2

( ) B( ) ( ) 53

8

( ) ( ) ( )2 2

L

L

L

L

Lm T x k x dx

T

Lm x k x dx

B

Lm T x k x dx

T

2

1

2

4

44 1 44

1

2

2

26 2 22

1

2

2

35 2 33

1

54

( ) B( ) ( ) 552 256

( ) ( ) ( ) 562 2

( ) B(

) ( )8

L

L

L

L

L

L

L

L

Lm x k x dx

T

Lm T x k x xdx

T

Lm x k x xdx

B

57

2

3

46 2 24

1

( ) ( ) ( ) 582 2

L

L

Lm T x k x xdx

T

2

2 2

66 2 22

1

( ) ( ) ( ) 592 2

L

L

Lm T x k x x dx

T

Where, 1( ) and 2 ( ) are the correction factors related to the fluid motion along x-axis

due to elongation of the body. 1( ) is the correction factor for the added mass known as

the Pabst correction, the most well-known experimental correction [8].

1 22 ( ) 1 0.425 6011

2 ( ) is the correction factor for added moment of inertia using theoretical formula

2 66 2

1( ) ( , ) 1k q q

(61)



Where,

2c Tq

b B (62)

And c, b, T, B are shown in figure 7. 66k is obtained from the ellipsoid method explained in

section 2.3.2.

Re-entrant and asymmetric forms are not acceptable for applying Lewis coefficient and these

equations are not applicable for ships with such forms.

The coefficient is bounded by a lower limit to omit re-entrant Lewis forms and by upper

limit to omit asymmetric Lewis forms.

( ) 1.0

3 1 1(2 ) (10 ( ) )

32 ( ) 32 ( )

( ) 1.0

3 1(2 ( )) (10 ( ) )

32 32 ( )

H x

H xH x H x

H x

H x H xH x

(63)

2.3.5 Determination of Additional Added mass components

As the component 13m is small compared to the total added mass, it is neglected, i.e.

13m = 31 0m

The components 15m and 24m cannot be determined by above methods, hence they are

determined approximately considering that they are caused by hydrodynamic force due to 11m

and 22m with the force centre at the centre of buoyancy of the hull BZ [8].

15 51 11 Bm m m Z (64)

24 42 22 Bm m m Z (65)

4215 51 11

22

mm m m

m (66)

Where, 24m and 42m are determined by using strip theory method.



3. ADDED MASS- CALCULATION, IMPLEMENTATION AND

VERIFICATION

3.1 Added Mass Calculation

The formulae in [8] as described in section 2.3 were used to calculate the added masses

analytically. They were implemented using a computer program. Initially, they were tested

using a program in C. Later, due to better applicability of the ANSYS-APDL software for

reading data from model, implementing data on the model, editing the model and programing,

a new macro was created using ANSYS Parametric Design Language.

The algorithm developed in the macro is as shown in figure 9.

Figure 9 Algorithm-added mass calculation macro

The program opens the ANSYS model from the database. Then, it asks the user for ship parameters,

Length-L (m), Beam-B (m), Draft-T(m) and the displacement of the ship (Metric Tons). These

parameters can also be set in the macro if the ship under analysis is the same and if they do not change

throughout the analysis. For example, a constant value of fluid density of 1025 3/kg m was used

throughout this research. The macro depends on a component of elements on the wet hull named

“carene”. Before importing the model and running the macro, another code is run to create this

component.

The macro runs according to selected frames on the ship. These frames divide the ship into slices or

strips as in the strip theory method. The x location of these frames is stored in an array. Then some

blank arrays are initialized to store the geometric parameters like length, draft, beam and area. A local

co-ordinate system is defined at each frame on the water line and area is created at each of the selected

frames. The more the number of frames, the more accurate the results. These areas calculated at



selected frames are stored in an array. They are shown in figure 10. Thus, the input table is obtained

which is shown in figure 11.

Figure 10 Areas on the selected frames

Here, X is the distance of each slice from ship center of gravity and dx is the thickness of each

slice/strip. H and are ratios as defined in section 2.3. These values are further used for

calculation of water added mass matrix components.

Figure 11 Array of geometric parameters [8]



The added mass components are calculated using the strip theory and ellipsoid method and are

taken into consideration as recommended by Do and Tran [8]. The obtained added masses are

in tons as per the formulations and are converted into SI unit (kilogram) before implementing

on the model. The added mass components 11m , 22m and 33m can be directly applied in three

directions as ANSYS supports directional masses in x,y and z directions. An example output

array of the added mass components on frames is shown in figure 12.

Figure 12 Array of added mass components in ANSYS

The rows represent the frames selected on the ship and columns represent the added mass

components. The first column is deliberately kept blank and does not represent anything. The

columns are named from left to right starting with column 2 as 22m ,

33m , 24m , 44m , 26m , 35m , 46m and 66m .

3.2 Implementation of Added Mass Components

In this part of the program, the nodes on each frame selected for calculations, as shown in

figure 10, are re-selected and the value of added mass distributed on the nodes. For this,

lumped mass elements are created and attached to each of the wet hull nodes and the values of

added mass in three directions are implemented. Thus, we have a distribution of added masses

on the wet hull as shown in figure 13. The blue points in the figure represent the created mass

elements.



Figure 13 Added mass implementation on wet hull

3.3 Verification of Results

For further underwater explosion analysis, it was necessary to validate the calculated added

masses and their effect as lumped masses to the wet hull nodes. This was done using previous

literature. The ship model by Do and Tran [10] was used for validation.

Figure 14 Input data for the macro : extracted from [10]



The table of ship parameters as shown in figure 14 was used as input for the macro and the

added mass components were calculated. In the research, strip theory method is suggested as

it can determine most of the added mass components with high accuracy [10]. Hence, the

table of comparison is as per the methods suggested in the research for different added mass

components.

Table 2 Comparison of results

Added Mass Coefficients mij

Sr No Added Mass Coefficients Obtained In paper [10] Method Used Error (%)

1 m11 0.035 0.035 Ellipsoid 0

2 m22 1.113 1.113 Strip theory 0

3 m33 1.440 1.44 Strip theory 0

4 m24 0.814 0.814 Strip theory 0

5 m44 0.013 0.014 Strip theory 0.09

6 m55 0.034 0.034 Ellipsoid 0

7 m26 0.028 0.028 Strip theory 0

8 m35 0.002 0.002 Strip theory 0

9 m46 0.092 0.092 Strip theory 0

10 m66 0.065 0.065 Strip theory 0

11 m15 -0.026 -0.026 Strip theory 0

It can be seen from table 2 that the values of added mass coefficients obtained using the

developed macro match exactly with the values obtained in previous research for the same

ship model [10]. Thus, the developed macro gave good results and could be used for further

analysis.

Table 3 shows the added mass values obtained for the reference ship in metric tons. By

comparing the added mass values to the displacement of the ship, it can be seen that the

components 22m , 33m and 24m are considerably large and may affect the underwater explosion

simulations.



Table 3 Added mass values

Added Mass

Sr No Added Mass Component Added Mass (Tons)

1 m11 261.92

2 m22 8213.47

3 m33 10629.62

4 m24 6010.57

5 m44 93.78

6 m55 251.19

7 m26 205.18

8 m35 12.91

9 m46 675.93

10 m66 479.13

11 m15 -191.67

Displacement of Ship 9178

The negative sign for some added mass components is due to the coordinate system used and

has no physical significance.



4. MODEL PREPARATION AND VERIFICATION

Underwater explosion analysis including the added mass effect was to be carried out with the

surface ship finite element model provided by Chantiers de l’Atlantique. But, before

implementing the new method developed for including added mass effect on the surface ship,

a semi-cylinder like-ship model was to be used which was re-used from research of Tsai [3].

Both the models were prepared and tested to check their suitability for underwater explosion

analysis.

4.1 Model Preparation

4.1.1 Semi-Cylinder Ship-Like Model

The model of semi-cylinder like-ship structure is as shown in figure 15. It represents a 150-

meter-long surface ship, the beam of which is 20 meter and the draft 8 meter. The internal

stiffening system has transverse bulkheads every 10 meters along the ship length and

longitudinal bulkheads every 3 meters along the beam.

Figure 15 Finite element model of like-ship semi cylinder-dry model [3]

The deck plate is 10 mm thick, transverse and longitudinal bulkheads are 20 mm and hull

thickness is 80 mm to account the presence of stiffeners. The density of the hull is increased

to 20000 3/kg m to include the effect of on board equipment and make it similar to a real

ship. The shell element mesh size is 1m. The principal parameters of the ship are shown in

table 4. This model will be hereafter referred as “dry model” in this chapter.

The developed macro to read the model, calculate and implement added mass was run in

ANSYS on this model and a new model with added masses distributed on wet hull nodes was

obtained as shown in figure 16.



Table 4 Parameters of semi-cylinder like-ship

Sr No Item Description

1 Length 150 m

2 Breadth 20 m

3 Draft 8 m

4 Displacement 4276.56 MT

The blue points in the picture are the nodal mass elements distributed on the wet hull. There

are 16 frames as seen in the figure and the cross section at each frame is the same as the

model is semi-cylindrical. This model now considers the effect of added mass calculated

using the strip theory and ellipsoid methods. This model will hereafter be referred to as “nodal

mass model” in this chapter.

Figure 16 Semi-cylinder-nodal mass model

To have a comparison of the results obtained using this model and to validate them, another

model with fluid acoustic element was prepared. A macro was developed in APDL to develop

this fluid mesh around the model. The fluid mesh has two types of acoustic fluid elements.

The ones near the fluid also include the effect of fluid structure interaction. The element

selected for this mesh is Fluid30. It is selected as it is an acoustic fluid element and includes

the effect of fluid structure interactions which can be incorporated by defining appropriate

KEYOPT values. The semi-cylinder model with acoustic fluid elements is shown in figure 17.



The interaction between the fluid and the structure is modelled using a special acoustic

element mesh shown in figure 18. The pink elements represent the acoustic fluid elements

which consider the fluid structure interaction. This model will be referred as “fluid mesh

model” in this chapter.

Figure 17 Semi-cylinder-fluid mesh model

Figure 18 Fluid structure interface mesh



Thus the semi-cylinder like-ship structure was ready for further validation before being used

for underwater explosion analysis.

4.1.2 Real Surface Ship Model

The model of real surface ship as shown in figure 19 represents a 100.9-meter-long surface

ship provided by Chantiers de l’Atlantique, the beam of which is 15.53 meters and draft is

4.75 meters. The material used and the internal stiffening system have not been disclosed by

Chantiers de l’Atlantique as they are considered confidential.

Figure 19 Finite element model of surface ship -dry model

The density of different shell elements of the model has been modified to take into account

the extra weight of equipment. At some nodes on the ship, some nodal masses are defined

which also represent the position and weight of on-board equipment like engine, motor, etc. A

coarse mesh is used as it is considered sufficient for the current analysis. The mesh size used

varies along the length and is around 2.1 meters. The model consists of beam44, shell63 and

mass21 elements. The principal characteristics of the ship are shown in table 5. This model

will be hereafter referred as “dry model” in this chapter.



Table 5 Parameters of surface ship

Sr No Item Description

1 Length 100.9 m

2 Breadth 15.53 m

3 Draft 4.75 m

4 Displacement 3512.478 MT

The macro developed to read the ship model data, calculate and implement the water added

mass was run in ANSYS and a new model including lumped masses distributed on the wet

hull nodes was obtained as shown in figure 20. This model will hereafter be referred to as

“nodal mass model” in this chapter.

Figure 20 Surface ship - nodal added masses

The red points in the picture are the nodal added mass elements on the wet hull. The cross

section varies along the length of the ship. This model now considers the effect of added mass

calculated using the strip theory and ellipsoid methods. To validate the results obtained using

this model, another model with fluid acoustic element was prepared. A macro was developed

in APDL to build the fluid mesh around the ship. As for the semi cylinder, the fluid mesh

includes two types of acoustic fluid elements. The ones near the structure allow to account for

the fluid structure interaction. The element type selected for this mesh is Fluid30. The ship

model with surrounding acoustic fluid elements is shown in figure 21. The interaction



between the fluid and the structure is taken into account thanks to the pink acoustic elements

as shown in figure 22. This model will be referred as “fluid mesh model” in this chapter.

Figure 21 Surface ship - fluid mesh model

Figure 22 Fluid structure interface mesh



Thus the real surface ship was ready for further validation before being used for underwater

explosion analysis.

The three models each for semi-cylinder like-ship and real ship will be referred hereafter in

this chapter as follows:

a. Dry model – no added mass

b. Nodal mass model – includes lumped added mass calculated by strip theory and

ellipsoid method and attached to wet hull nodes

c. Fluid mesh model - includes acoustic fluid element mesh

4.2 Verification of Added Mass Calculation

The values of added mass components obtained for the semi-cylinder and the ship were

checked before proceeding for further simulations.

Table 6 Added mass components of surface ship

Added Mass Component Added Mass (Tons)

m11 140.89

m22 2723.57

m33 5537.33

m24 257.59

m44 159.08

m55 74.49

m26 3499.8

m35 28.83

m46 1891.26

m66 284.76

m15 181.05

Displacement 3512.48

As seen from table 6, the added mass components 22m , 33m , 26m and 46m are significant and

should affect the underwater explosion analysis. Comparing tables 3 and 6, we see different

added mass components to be significant. This can be because of the fact that added mass is a

function of geometry of the body and the density of the surrounding fluid. As the fluid density



in this research work is kept constant, it can be concluded that added mass depends on

geometry of the body.

The added mass components for the semi-cylinder are as shown in table 7. The added mass

components 22m , 33m , 24m and 66m are significant and should also affect the underwater

explosion analysis. Some added mass component values are 0 and these can be due to the

exact symmetry of the semi-cylinder along x direction. The added mass components 22m , 33m

are significant for underwater explosion analysis and they can be applied to the nodes.

Table 7 Added mass components of semi-cylinder like-ship

Added Mass

Sr No Added Mass Component Added Mass (Tons)

1 m11 135.00

2 m22 15589.27

3 m33 24094.78

4 m24 7497.31

5 m44 28.22

6 m55 537.45

7 m26 0.00

8 m35 0.00

9 m46 0.00

10 m66 941.23

11 m15 -125.97

Displacement of Ship 4181.69

The value of some added mass components is much higher than the displacement of the like-

ship structure. This again proves that added mass is a function of geometry and density of

fluid and not related to the mass of the body.



4.3 Modal Analysis

Modal analysis of the real ship and the semi-cylinder like-ship structure was performed up to

100 modes. The hull girder modes (vertical and horizontal bending as well as torsion) were

obtained for dry model, nodal mass model and fluid mesh model for the semi-cylinder and the

ship.

4.3.1 Semi-Cylinder like-ship model

The modal analysis of the three models as discussed in section 4.1.1 was carried out.

Table 8 Modal frequency comparison of models

Frequency Difference

Sr

No Mode shape

Dry

Model

(Hz)

Nodal Mass

model (Hz)

Fluid Mesh

model (Hz)

Dry and Nodal

mass models

(%)

Dry and Fluid

mesh models

(%)

1 1st Vertical

Bending 1.46 0.74 0.84 49 42

Table 8 shows the comparison of first vertical bending mode of the three models. The natural

frequency for this mode is less for the nodal mass model and fluid mesh model as compared

to the dry model. This shows that added mass has significant effect on the vertical bending of

the body. The difference here in this case is as high as 49 % between the dry and wet models.

In order to verify the obtained results, it is usual to consider as a very first approximation that

the vertical added mass of the ship is the same than its mass: m33 = m. Then, the difference

between the natural frequencies of dry and wet models for first vertical bending may be

approximately relied by

2

dry

wet

Where, = natural frequency

In this case, substituting the appropriate values the wet natural frequency is 1.03 Hz, which is

27% more than the obtained values with the fluid mesh model and nodal masses model. The

discrepancy in the three models will be discussed in the further part of this chapter.

The first vertical bending mode for the dry model is shown in figure 23.



Figure 23 1st vertical bending mode- Dry model

The natural frequencies of the nodal mass model and the fluid mesh model were compared as

shown in table 9.

Table 9 Model frequency comparison of models

Frequency (Hz)

Sr No Mode shape Nodal mass model Fluid mesh

model Error (%)

1 1st Vertical Bending 0.74 0.84 12

2 2nd Vertical Bending 1.97 2.31 14

3 1st Torsional 1.50 2.53 41

Table 9 shows the natural frequencies for the first two vertical bending modes and the first

torsional modes for the two models under study- nodal masses and fluid mesh. The other

global mode shapes could not be obtained for the semi-cylinder as mostly the local mode

shapes were excited. According to Chantiers de l’Atlantique, this is due to absence of local

stiffening in the scantling of the semi-cylinder. The difference between the fluid mesh is

acceptable for the vertical bending modes and quite high for the torsional modes.

The reason maybe that the added masses were calculated for rigid body motion. Deformation

related to each hull girder mode should be taken into account while calculating the added

masses as proposed by Basic [12].



The discrepancies in vertical bending natural frequencies are due to the following reasons:

a. Strip theory and ellipsoid methods are approximate [5].

b. Number of strips selected on the model for calculation is limited to 16.

c. The distance between nodes on each frame is not equal. Hence, the mass on some

nodes must be greater and some must be smaller on the same frame. But, the current

macro does not take this into account.

The mode shapes at the frequencies shown in table are compared in figures 24-25.

Wet model- Nodal added mass Wet model- fluid mesh

1st Vertical Bending (2 nodes)

(7th mode, f=0.74 Hz) (1st mode, f=0.84 Hz)

2nd Vertical Bending (3 nodes)

(18th mode, f=1.97 Hz) (48th mode, f=2.31 Hz)

Figure 24 Vertical bending modes




Figure 25 Torsional bending

After modal analysis of the semi-cylinder like-ship structure, it was found that most of the global

mode shapes are absent within first 100 mode shapes. According to the experience of Chantiers de

l’Atlantique, real ships of this length have natural frequencies for global mode shapes around 5-6 Hz.

But, with the semi-cylinder model, very few global mode shapes were obtained and the other mode

shapes were mostly local as shown in figure 26.

Figure 26 Local modes

This seems to indicate that the semi-cylinder like-ship model does not represent a real ship in a

dynamic point of view. The local modes can also be due to the absence of longitudinals and girders



which support the deck between bulkheads and are an essential part of a real ship scantling. Thus, this

model was not considered for underwater explosion simulations.

4.3.2 Real Surface Ship Model

Table 10 shows the comparison of first vertical bending mode of the three models.

Table 10 Modal frequency comparison of models

Frequency Difference

Sr

No

Mode

shape

Dry

Model

(Hz)

Nodal mass model

(Hz)

Fluid

Mesh

model

(Hz)

Dry and nodal

mass models (%)

Dry and Fluid

mesh models

(%)

1

1st

Vertical

Bending

3.66 2.46 2.7 33 26

The natural frequency for this mode is less for the nodal mass and fluid mesh models as

compared to the dry model. This shows that added mass has significant effect on the vertical

bending of the body. The difference here in this case is 33 % between the dry and wet models.

According to the experience of Chantiers de l’Atlantique it is around 35%. Hence the results

are acceptable.

In order to verify the obtained results, it is usual to consider as a very first approximation that

the vertical added mass of the ship is the same than its mass: m33 = m. Then, the difference

between the natural frequencies of dry and wet models for first vertical bending may be

approximately relied by

2

dry

wet

Where =natural frequency.

In this case, substituting the appropriate values, the wet natural frequency is 2.59 Hz which is

4.98% more than the obtained values with the fluid mesh and nodal masses. This is quite less

and acceptable. The first vertical bending mode for the dry model is shown in figure 27.



Figure 27 1st Vertical bending mode- Dry model

The natural frequencies obtained from the nodal added mass model were compared to the

ones obtained from the fluid mesh model.

Table 11 Model frequency comparison of models

Frequency (Hz)

Sr No Mode shape Nodal masses model Fluid mesh model Error (%)

1 1st Vertical Bending 2.46 2.7 9

2 2nd Vertical Bending 3.98 4.29 7

3 1st Transverse Bending 2.89 3.7 22

4 2nd Transverse Bending 5.19 6.66 22

5 1st Torsional mode 4.9 6.99 28

6 2nd Torsional mode 6.92 9.19 35

Table 11 shows the natural frequencies for the first two vertical, transverse and torsional

modes for the two models under study- nodal masses and fluid mesh. The difference between



the fluid mesh is acceptable for the vertical bending modes and is high for transverse and

torsional modes. The discrepancies in vertical and transverse bending natural frequencies are

due to the same reasons as explained in section 4.3.1.


1st Vertical Bending (2 nodes)

(7th mode, f=2.46 Hz) (1st mode, f=2.7 Hz)

2nd Vertical Bending (3 nodes)


Figure 28 Vertical bending modes



The mode shapes at the frequencies shown in table are compared in figures 30-30.


1st Transverse bending (2 nodes)

(8th mode, f=2.89 Hz) (2nd mode, f=3.7 Hz)

2nd Transverse bending (3 nodes)


Figure 29 Transverse bending modes

It was concluded from the global mode shapes that they were almost the same for the nodal mass and

the fluid model. It was also concluded that the nodal mass was realistic as compared to the fluid mesh

model and gave results for modal analysis with acceptable difference due to known reasons. Moreover,

this analysis also confirmed the presence of added masses on the nodal mass model and thus this

model was used for underwater explosion simulations.




1st Torsion


2nd Torsion


Figure 30 Torsional modes



5. SIMULATIONS: UNDERWATER EXPLOSIONS

An underwater explosions phenomenon is transient and dynamic. ANSYS solver cannot be

used for these simulations as it has an implicit solver. Tsai [3] tried using modal superposition

method to obtain the results but the results were not acceptable. Hence, LS-DYNA explicit

solver was used for analysis in this research.

5.1 Model Preparation

In this part of the research, the model of the real ship provided was converted to LS-DYNA

format “. k” file using a macro developed in Chantiers de l’Atlantique. Figure 31 shows the

converted dry model of the surface ship.

Figure 31 Surface ship in LSDYNA-dry model

As LS-DYNA does not support directional masses, the added mass components 11m , 22m and

33m were applied in x, y and z directions using discrete spring elements of very high stiffness

of 101 10 N/m. The springs are directional that is they act only in one direction, x, y or z. This

was the second model used in this analysis i.e. the wet model. It is shown in figure 32. Thus in

this analysis we have 2 models:

a. Dry model (no added masses considered)

b. Wet model (added masses considered)

This terminology is used throughout this chapter.



Figure 32 Added lumped masses on directional springs

5.2 Explosion Parameters And Model Processing

The details of the initial conditions of the explosion scenario are listed in table 12. In this

case, a charge of 500 kg is exploded at a depth of 50 m below the free surface. The

corresponding shock factor is 0.49 obtained from Tsai [3] as explained in Appendix A1.

Table 12 Initial conditions of explosion scenario [3]

Description

mc TNT charge mass, mc= 500 kg

di Distance from charge to free surface, di= 50 m

r Distance from charge to standoff point, r = 45.25 m

c Density of charge, c = 1600 kg/m3

SF Shock factor = 0.49



The charge is assumed to be located at the origin of the co-ordinate. Figure 33 shows the

location of the charge below the ship. A macro developed by Tsai [3] was used to calculate

the pressure distribution on the wet hull. This macro runs in ANSYS-APDL and generates a

pressure.k file to be used for LS-DYNA simulations.

Figure 33 Location of charge

To post-process the results of the simulations three nodes and 3 shell elements are selected.

The three selected nodes are: one at the center of the second deck, one at the stern and one at

the bow of the ship are as shown in figure 34-35. Three elements are selected at 3 locations at

the bottom of the ship as shown in figure 36. They are used to study the pressure distribution

along the wet hull when subjected to an underwater explosion. The elements are selected at



the center aft and fore of the ship. The pressure distribution at each element is different and is

shown in figure 37.

Figure 34 Positions of selected nodes for analysis

Figure 35 Positions of selected nodes



Figure 36 Positions of selected elements

Figure 37 Pressure distribution on selected elements

The pressure at element 114 is the highest at 13.3610 Pa, the second highest at the stern,

element 105055, 5.78610 Pa and the lowest at the bow on element 927, 2.23

610 Pa. This

variation is due to the location and orientation of element with respect to the charge. If the

element is at an angle and is not parallel to the xy plane, the incident pressure is less. Hence,

the element at bow (927) has less pressure compared to element at stern.

The pressure, which is the highest at time step 0, is the pressure due to first shock wave. It

occurs at the beginning of the explosion. The simulation starts just as this shock wave reaches

the ship hull surface. Hence, high pressure is seen at time 0. The first bubble pulse is seen at

time 0.5 seconds and the second pulse at time 1.05 seconds. The main focus of this research

was the study of bubble pulsation as it may excite the hull girder natural mode of the ship.

Though the pressure is quite less in bubble pulse phase as compared to first shock wave, the



excitation of a natural mode leads to global bending of the ship which can damage the on

board equipment and sometimes break the ship into two parts.

Figure 38 Local co-ordinate system at node 1314

The ship model was not subjected to gravity in the current LSDYNA simulations as it should

either have a fluid mesh or the equilibrium force between mass of ship, gravity and buoyancy

forces applied on the model, which was complicated and not in the scope of this research.

Hence, when explosion pressure is applied, non-physical overall displacement of the ship

occurs along with (physical) whipping. This leads to large displacement plot where the ship

displacement is also plotted. To avoid this and for better post-processing of the results, a local

co-ordinate system is defined at node 1314 using two more nodes, 1339 and 1318, in the x-y

plane. Black lines in figure 38 show this local co-ordinate system. Now, the displacement plot

of selected nodes gives the local displacements of the nodes during the whipping phenomena.

5.3 Analysis Process And Results

This section presents the results of the LSDYNA explicit underwater explosion simulations.

5.3.1 Energies

The plot of internal energy time evolutions obtained from the dry and wet models when

subjected to underwater explosion is shown in figure 39. As see in the plot, the internal energy

increases at time 0.5 second and 1.05 second, which represents the phenomena of first and

second bubble pulsations. The energy of dry model is more than that of the wet model.



Figure 39 Comparison of internal energies

This is because, energy is directly proportional to force and displacement. As the same

pressure is applied on both the models, the force on nodes is same. Hence, energy depends on

displacement. The displacement of wet model is less than that of dry model as added mass

increases the inertia of the model. Hence, the internal energy of the wet model is also less than

the dry model as seen in the plot.

5.3.2 Nodal Vertical Displacements

The average displacement at the extreme ends of the ship, obtained by taking the average of

the vertical displacements of nodes (with respect to the local coordinate system) at bow and

stern extremities (109 and 1171), is plotted for the two models in figure 40.

Figure 40 Comparison of z displacements



The time period of oscillation is less for the wet model as compared to dry model which represents the

effect of added mass. Thus, added mass acts as added inertia and decreases the natural frequency and

displacement and increases the time period during the whipping response of a ship subjected to

underwater explosion.

5.3.3 Frequency Analysis from Displacement Plot

The average displacement of the extreme nodes of the dry model was plotted with respect to

time. This plot is sinusoidal and the time period between two oscillations was studied. The

inverse of this time period gives the frequency of vibration of the dry model of the ship. The

time period was measured as shown in figure 41.

Figure 41 Time period measurement-dry model

The time period and frequencies are shown in table 13.

Table 13 Time period and frequency for dry model

Sr No Time period (s) Frequency (Hz)

1 0.27 3.70

2 0.27 3.70

3 0.26 3.85

4 0.26 3.85

5 0.27 3.70

Average 3.76

The average of natural frequencies obtained from the displacement plot is 3.76 Hz. But, as

seen from table 10, the first natural frequency for dry model is 3.66 Hz. This frequency is near

but not the same. This indicates that the ship is not vibrating at it natural frequency, but near

the first natural frequency. If the ship vibrates at the first natural frequency, the first mode

ship will be excited. This will be the worst case scenario of underwater explosion and can be



obtained by varying the location of explosive charge to match the bubble pulsation frequency

with the first natural frequency.

Figure 42 Time period measurement-wet model

Similar analysis was carried out with the wet model and the measurements are shown in

figure 42. Table 14 lists the natural frequencies from the displacement plot.

Table 14 Time period and frequency for wet model

Sr No Time period (s) Frequency (Hz)

1 0.445 2.25

2 0.576 1.74

3 0.44 2.27

4 0.37 2.70

5 0.37 2.70

Average 2.33

The natural frequency obtained here is 2.33 Hz, whereas from section 4.3.2, the natural

frequency of the wet model is between 2.46 Hz and 2.69 Hz. Hence, again the natural

frequency of the ship was not excited and it can be excited by varying the depth of charge to

study the worst case scenario.

5.3.4 Accelerations

The acceleration plot at specific nodes on the ship model can be used to obtain the shock

spectrum. This shock spectrum data can be used to analyze the embarked equipment when the

ship is subjected to underwater explosions. A research on this was carried out by Tasdelen [4]

in 2018. An example acceleration plot for the wet model is shown in figure 43.



Figure 43 Acceleration at the bow of the ship

The shock spectrum data can be used to manufacture and install on board equipment to resist

the effect of shock from underwater explosions.



6. CONCLUSIONS

In this research, a method was developed to calculate added mass components using strip

theory and ellipsoid methods and implement them on a real ship ANSYS model. The process

of converting ANSYS model to LS-DYNA and performing explicit underwater explosion

simulations was further explored. The main conclusions from this research are summarized

below.

1. A detailed study of the available methods to calculate added mass of a real ship was

conducted. It was found that the strip theory gives more accurate results as compared

to Ellipsoid method and it was confirmed that a combination of strip theory and

ellipsoid theory is better for obtaining added mass components.

2. With the help of previous literature, the developed APDL macro was validated and

found to give acceptable results.

3. The modal analysis proved the proper application of added masses on nodes of the wet

hull. It was confirmed that the calculated natural frequencies of the body under

analysis are reduced when effect of added mass is considered.

4. Post-processing the underwater explosion results, it was found that an explosion below

the ship leads to ship hull whipping phenomena, as expected. If the frequency of

bubble oscillation matches with that of the ship, one or several natural bending modes

may be excited and it is the worst case scenario for a ship subjected to underwater

explosion. The added mass plays an important role increasing the inertia of the ship.

5. The displacements during oscillations are less when added mass is considered. The

energy level and its fluctuation also reduces. The time period of oscillations is reduced

due to the added mass effect. Thus, there is probability of resonance of the ship at

frequencies lower than previously calculated using dry model. Hence, it is important to

consider the added mass effect during underwater explosion analysis.

6. The simulation time required for nodal mass method proposed in this research is much

less than that of the fully coupled fluid mesh. This saves time and gives good results.

This method can be used in the early design stage, the data from which can be verified

during the detailed analysis in later stages of design.



7. FUTURE SCOPE

In the underwater explosion analysis, nodal displacements, energy and acceleration plots were

studied. The stress and strain analysis can also be studied as part of future analysis. The

velocities and accelerations can be used to calculate the shock response spectrum of the on

board equipment and can be supplied to the manufacturer to manufacture equipment that can

withstand explosion shocks. The position of charge can be changed to simulate different

conditions to find the response of the ship in the worst case scenario.

The attempt to test the model with fluid mesh for underwater explosions was not successful in

this research as the behavior fluid elements was not normal at the fluid structure interface.

There is scope for further research to find the reason for this behavior of elements and find a

way to model the fluid elements right taking into account the fluid structure interaction. Then

the results obtained by strip theory and ellipsoid methods i.e. nodal masses can be compared

with the fluid mesh. It would also be very interesting to modify the strip and ellipsoid theory

formulations to match the results with the numerical simulations using fluid mesh.



8. ACKNOWLEDGEMENTS

Firstly, I would like to express my sincere gratitude to my supervisor Prof. Hervé Le Sourne

for his guidance during the research and the time spent in ICAM.

Besides, I would also like to thank all members in the department of Acoustic and Vibrations

in Chantiers de l’Atlantique, for giving me an opportunity to work on their project: the head

of Acoustic and Vibrations department Mr. Sylvain Branchereau, Mr. Simon Paroissien and

Mr. Clement Lucas. All of them have improved my research from various perspectives by

contributing with insightful comments and encouragement.

Moreover, my sincere thanks to Prof. Philippe Rigo, Ms. Christine Reynders and all members

who gave me an opportunity to be a part of EMSHIP program. This research would not have

been possible without their precious support.

This thesis was developed in the frame of the European Master Course in “Integrated

Advanced Ship Design” named “EMSHIP” for “European Education in Advanced Ship

Design”, Ref.:159652-1-2009-1-BE-ERA MUNDUS-EMMC.



9. REFERENCES

1. Hendershot john and Kaczmarek Robert, 2014. Defense AT & L [online]. Source. Available

from: https://apps.dtic.mil/dtic/tr/fulltext/u2/1015915.pdf [20th December 2018].

2. Navarro M.G., 2015. Rules and Methods for Dimensioning Embarked Materials for Surface

Ships When Subjected to UNDEX. Thesis (MSc). EMship.

3. Tsai S.C., 2017. Numerical simulation of surface ship hull beam whipping response due to

submitted to underwater explosion. Thesis (MSc). EMship.

4. Tasdelen Enes., 2018. Shock Analysis of On-board Equipment Submitted to Underwater

Explosion. Thesis (MSc). EMship.

5. Korotkin A.I., 2009. Added Masses of Ship Structures. St. Petersburg, Russia: Springer

Science + Business Media B.V.

6. Chen S.S. and Chung H., 1976. Design Guide for Calculating Hydrodynamic Mass Part I:

Circular Cylindrical. Illinois, Thesis (MSc). Argonne National Laboratory.

7. Techet, A.H., 2005. Hydrodynamics [online]. Source. Available from:

http://web.mit.edu/2.016/www/handouts/2005CourseInfo.pdf [20th December 2018].

8. Do Thanh Sen and Tran Canh Vinh, 2016, Determination of Added Mass and Inertia

Moment of Marine Ships Moving in 6 Degrees of Freedom, International Journal of

Transportation Engineering and Technology, 2(1), 8-14.

9. Lewis, F.M.,1929, The inertia of the water surrounding in a vibrating ship, SNAME 37, 1–

20.

10. Do Thanh Sen and Tran Canh Vinh, 2016, Method to Calculate Components of Added

Mass of Surface Crafts, Journal of Transportation Engineering and Technology, Vol 20.

11. Ciobanu C., Caţă M. Anghel A.R., 2006, Conformal Mapping in Hydrodynamic, Bulletin

of The Transilvania University of Braşov. Vol. 13(48).

12. Bašić Josip and Parunov Joško, 2016, Analytical and Numerical Computation of Added

Mass in Ship Vibration Analysis, UDC 519.6:629.5.015.

http://web.mit.edu/2.016/www/handouts/2005CourseInfo.pdf



APPENDIX A1

Conformal Mapping

The cross section of ships may be complex and must be simplified to be used in the strip

theory method. The advantage of conformal mapping is that the velocity potential of the fluid,

flowing around an arbitrary shape of a cross section in a complex plane, can be derived from

the more convenient circular section in another complex plane [11]. Thus, the coefficients of

conformal mapping are used to solve hydrodynamic problems like calculating the

hydrodynamic coefficients. The general transformation formula may be written as:

12

0

12 )(

k

n

k

ks ZaZf (67)

Where

( )f Z z and z x iy is the plane of the ship's cross section

iZ ie e - is the plane of the unit circle

s = scale factor

1 1a

2 1ka are the conformal mapping coefficients ( 1, ... ,k n )

n = number of parameters

(2 1)

2 1

0

( ) ,n

i k

s k

k

x iy a ie e

(68)

(2 1) (2 1) 2 12 1

0

( ) [cos( ) sin( )]n

k k x

s k

k

x iy a i e i

(69)

(2 1)2 1

0

( 1) [ cos(2 1) sin(2 1) ]n

k k

s k

k

x iy a e i k k

(70)

From the relation between the coordinates in the z - plane (the ship's cross section) and the

variables in the Z - plane (the circular

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