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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 This page is intentionally left blank. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions iii 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 iv Ruturaj Radhakrishna Trivedi This page is intentionally left blank. Master Thesis developed at ICAM, France 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 vi Ruturaj Radhakrishna Trivedi This page is intentionally left blank. Master Thesis developed at ICAM, France 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 2.31 Elimination of Added Mass Components Due to Symmetry of Ship Hull and Added Mass Matrix 9 2.32 Method of Equivalent Ellipsoid 9 2.33 Method of Plane Sections 11 2.34 Strip Theory Method 12 2.35 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 4.1 Model Preparation 23 23 4.11 Semi-Cylinder Ship-Like Model 23 4.12 Real Surface Ship Model 26 4.2 Verification of Added Mass Calculation 29 4.3 Modal Analysis 31 4.31 Semi-Cylinder like-ship model 31 4.32 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 5.31

Energies “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 45 viii Ruturaj Radhakrishna Trivedi 5.32 Nodal Vertical Displacements 46 5.33 Frequency Analysis from Displacement Plot 47 5.34 Accelerations 48 6. CONCLUSIONS 50 7. FUTURE SCOPE 51 8. ACKNOWLEDGEMENTS 52 9. REFERENCES 53 APPENDIX A1 54 Master Thesis developed at ICAM, France 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 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 Master Thesis developed at ICAM, France 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 xii Ruturaj Radhakrishna Trivedi This page is intentionally left blank. Master Thesis developed at ICAM, France 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 3 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 4 Ruturaj Radhakrishna Trivedi 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] Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 5 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] F  (m  ma )   2u u  cv 2 t

t Where, m = mass of object “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 (1) 6 Ruturaj Radhakrishna Trivedi ma = added mass (hydrodynamic mass) cv = viscous damping coefficient The added mass is proportional to the density of fluid and the volume of object and is given by [5] ma    V  cm (2) Where,  = fluid density V= volume of object cm = added mass coefficient (m+ ma )= virtual/apparent mass of the object For an inviscid fluid, the viscous damping coefficient cv =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 [ mij ], where i,j=1,2,.,N In case of single object, [ mij ] 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 springmass-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: Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 7 mx  bx  kx  f (t )  m a x (4) (m  ma ) x  bx  kx  f (t ) (5) Rearranging above equation, Now, the simple spring mass damping

system can be used with a new mass m '  m  ma and the natural frequency writes ' k m' (6) k ' m  ma 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,  m11   m21 m F   31  m41  m51   m61 m12 m22 m32 m42 m52 m62 m13 m23 m33 m43 m53 m63 m14 m24 m34 m44 m54 m64 m15 m25 m35 m45 m55 m65 m16   u1    m26   u2  m36   u3    m46   u4  m56   u5    m66   u6  (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 u1 - Linear 2

Sway-motion in y direction u2 - Linear 3 Heave- motion in z direction u3 - Linear 4 Roll- rotation about x axis u4 - Angular 5 Pitch- rotation about y axis u5 - Angular 6 Yaw- rotation about z axis u6 - Angular “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 8 Ruturaj Radhakrishna Trivedi Figure 6 Ship motion in 6 degrees of freedom [8] The inertial hydrodynamic force is FH  M [u1 u 2 u3 u 4 u5 u 6 ]T (8) Where, M includes the inertia matrix of the ship and the water added mass. M=M S  M A (9) M S is the mass and inertia moment matrix of the ship and is given as [8].  m   0  0 Ms    0  mz g   my g  0 m 0 mz g 0 mxg 0 0 m myg mxg 0 0 mz g myg Ix  I yx  I zx mz g 0 mxg  I xy Iy  I zy myg   mxg  0    I xz   I yz   I z  (10) M A is the water added mass and added inertia moment matrix. mij is a component of

hydrodynamic force in the i th direction due to unit acceleration in direction j. M A has 36 components [8].  m11   m21 m M A   31  m41  m51   m61 m12 m22 m32 m42 m52 m62 m13 m23 m33 m43 m53 m63 m14 m24 m34 m44 m54 m64 m15 m25 m35 m45 m55 m65 m16   m26  m36   m46  m56   m66  Master Thesis developed at ICAM, France (11) Analysis of added mass effect on surface ships subjected to underwater explosions 9 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. 2.31 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]. m32  m34  m36  m52  m54  m56  0 The added mass matrix being symmetric, mij  m ji , m23  m43  m63  m25  m45  m65  0 Considering the same for motions in longitudinal directions due to acceleration in the direction j=2,4,6: m12  m14  m16  0 m21  m41  m61  0 Thus, for a ship moving with 6 degrees of freedom the added mass is reduced to 18 components [8]  m11   0 m M A   31  0  m51   0 0 m22 0 m42 0 m62 m13 0 m33 0 m53 0 0 m24 0 m44 0 m64 m15 0 m35 0 m55 0 0   m26  0   m46  0   m66  (12) 2.32 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, mij      i s  j n dS Where, “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 (13) 10 Ruturaj Radhakrishna Trivedi S = Wetted ship area  = density of water  i = flow potential when ship is moving in i th direction with unit speed To calculate mij , 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] m11  mk11 m22  mk22 m33  mk33 m44  k44 I xx m55  k55 I yy (14) m66  k66 I zz kij are called the hydrodynamic coefficients and are given by k11  A0 2  A0 k22  B0 2  B0 k33  C0 2  C0 (15) (16) k44  0 k55 

( L2  4T 2 )2 ( A0  C0 ) 2(4T 4  L4 )  (C0  A0 )(4T 2  L2 )2 Master Thesis developed at ICAM, France (17) Analysis of added mass effect on surface ships subjected to underwater explosions k66  (L2  B 2 )2 ( B0  A0 ) 2( L4  B 4 )  ( A0  B0 )(L2  B 2 )2 11 (18) Where, A0  2(1  e2 )  1  1  e    2 ln  1  e   e  e3     1 1  e2  1  e  B0  C0  2  ln   e 2e3  1 e  e  1 (19) b2 d2  1  a2 L2 Where, d = maximum diameter (m) L= overall length (m) Moment of inertia of displaced water is approximately the moment of inertia of the ellipsoid 1  LBT (4T 2  B 2 ) 120 1 I yy   LBT (4T 2  L2 ) 120 1 I zz   LBT (B2  L2 ) 120 I xx  (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 m24 , m26 , m35 , m44 , m15 , m51 [8]. 2.33 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 xdirection 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]. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019

12 Ruturaj Radhakrishna Trivedi The formulas for added masses computed via the method of plane sections can be written as follows: 22   (  L2 L ) 220 ( x)dx 2T L1 L2 L B L1 33   (  )  330 ( x)dx L2 L 24   (  )  240 ( x)dx 2T L1 L2 L B L1 34   (  )  340 ( x)dx L2 L 44   (  )  440 ( x)dx 2T L1 L2 L 26  1 (  )  220 ( x) xdx 2T L1 L2 L B L1 35   1 (  )  330 ( x) xdx L2 L 55  1 (  )  330 ( x) x 2 dx B L1 66  1 (  L2 L )  220 ( x) x 2 dx 2T L1  21  22   23  24   25  26   27   28   29  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. 2.34 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to

underwater explosions 13 Figure 8 Strip theory representation of the ship [8] L2 m22   m22 ( x)dx  30  L1 L2 m33   m33 ( x)dx  31 L1 L2 m24   m24 ( x)dx  32  L1 L2 m44   m44 ( x)dx  33 L1 L2 m26   m22 ( x)dx  34  L1 L2 m35    m33 ( x)dx  35 L1 L2 m46   m24 ( x)dx  36  L1 L2 m66   m22 ( x)dx L1 Where, mij ( x) is added mass of 2D cross section at location x. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019  37  14 Ruturaj Radhakrishna Trivedi 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, H ( x)  B( x) 1  p  q  2T ( x) 1  p  q  38   ( x)  A( x)  1  p 2  3q 2  B( x)T ( x)

4 4(1  q) 2  p 2  39  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 ) . 3          (1   2 ) 4 4 2 q 1    (1   2 ) 2 (40) p  ( q  1) q   (41)  4 H 1  H 1 (42) The added mass components for each section may be written as m22 ( x)  m33 ( x)  m24  T ( x)3 2  T ( x)2 (1  p) 2  3q 2 2 (1  p  q) 2  B( x)2 (1  p) 2  3q 2 8 (1  p  q) 2    T ( x) 2 2  T ( x) 2 8 k22 ( x)  43 k33 ( x)  44  1  8 16 4 4   p(1  p)  q 2 (20  7 p)  q  (1  p)2  (1  p)(7  5 p)   (45) 2  (1  p  q)  3 35 5 3  m 24  T ( x )3 k24 ( x) 2  B( x) 4 16[ p 2 (1  q ) 2  2q 2  B ( x) 4 m44    k44 (

x) 256 (1  p  q) 4 256 Master Thesis developed at ICAM, France  46   47  Analysis of added mass effect on surface ships subjected to underwater explosions m26 ( x)   T ( x) 2 k22 ( x) xdx 2  T ( x) 2 m35 ( x)   k33 ( x) xdx 8 T ( x ) 3 m46 ( x)  k24 ( x) 2  T ( x) 2 m66 ( x)  k22 ( x) x 2dx 2 15  48  49   50   51 The total mij for a section the extremities of which are located at L1 and L2 becomes: m22  1 (  L  ) T ( x) 2 k22 ( x)dx 2T 2 L1  52  m33  1 (  L  ) B( x) 2 k33 ( x)dx  B 8 L1  53 m24  1 (  L  ) T ( x)3 k24 ( x)dx 2T 2 L1  54  m44  1 (  L  ) B( x) 4 k44 ( x)dx  2T 256 L1  55 L  m26  2 (  ) T ( x) 2 k22 ( x) xdx 2T 2 L1  56  L  ) B( x) 2 k33 ( x) xdx  B 8 L1  57  L  m46  2 (  ) T ( x)3 k24 ( x) xdx  2T 2 L1

58 L  m66  2 (  ) T ( x)2 k22 ( x) x 2 dx 2T 2 L1  59 L2 L2 L2 L2 L2 m35   2 (  L2 L2 L2 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 ( )      1  0.425   1 2  1 2   60   2 ( ) is the correction factor for added moment of inertia using theoretical formula   2 ( )  k66 ( , q)q 1  1   2  “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 (61) 16 Ruturaj Radhakrishna Trivedi Where, q c 2T  b B (62) And c, b, T, B are shown in figure 7. k66 is obtained from the ellipsoid method explained in section 2.32 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. H ( x)  1.0 3 1  1 (2  )    (10  H ( x)  ) 32 H ( x) 32 H ( x) H ( x)  1.0 3  1 (2  H ( x))    (10  H ( x)  ) 32 32 H ( x) (63) 2.35 Determination of Additional Added mass components As the component m13 is small compared to the total added mass, it is neglected, i.e m13 = m31  0 The components m15 and m24 cannot be determined by above methods, hence they are determined approximately considering that they are caused by hydrodynamic force due to m11 and m22 with the force centre at the centre of buoyancy of the hull Z B [8]. m15  m51  m11Z B (64) m24  m42  m22 Z B (65) m42 m22 (66) m15  m51  m11 Where, m24 and m42 are determined by using strip theory method.

Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 17 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 kg / m3 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 18 Ruturaj Radhakrishna Trivedi 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 23 These values are further used for calculation of water added mass matrix components. Figure 11 Array of geometric parameters [8] Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 19 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 m11 , m22 and m33 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 m22 , m33 , m24 , m44 , m26 , m35 , m46 and m66 . 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. “EMSHIP”

Erasmus Mundus Master Course, period of study September 2017 – February 2019 20 Ruturaj Radhakrishna Trivedi 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] Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 21 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 m22 , m33 and m24 are considerably large and may affect the underwater explosion simulations. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 22 Ruturaj Radhakrishna Trivedi 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 23 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 4.11 Semi-Cylinder Ship-Like Model The model of semi-cylinder like-ship structure is as shown in figure 15. It represents a 150meter-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 kg / m3 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. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 24 Ruturaj Radhakrishna Trivedi Table 4 Parameters of semi-cylinder like-ship Sr No Item Description 1 Length 150 m 2 Breadth 20 m 3 Draft 8m 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 25 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 26 Ruturaj Radhakrishna Trivedi Thus the semi-cylinder like-ship structure was ready for further validation before being used for underwater explosion analysis. 4.12 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 27 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 28 Ruturaj Radhakrishna Trivedi 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 29 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 m22 , m33 , m26 and m46 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 30 Ruturaj Radhakrishna Trivedi 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 m22 , m33 , m24 and m66 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 m22 , m33 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 likeship 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 31 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. 4.31 Semi-Cylinder like-ship model The modal analysis of the three models as discussed in section 4.11 was carried out Table 8 Modal frequency comparison of models Frequency Sr No 1 Dry Mode shape Model (Hz) 1st Vertical Bending 1.46 Difference Nodal Mass Fluid Mesh model (Hz) model (Hz) 0.74 0.84 Dry and Nodal Dry and Fluid mass models mesh models (%) (%) 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 wet  dry 2 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. “EMSHIP” Erasmus Mundus Master Course,

period of study September 2017 – February 2019 32 Ruturaj Radhakrishna Trivedi 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) Fluid mesh Sr No Mode shape Nodal mass model 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 model Error (%) 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]. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 33 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 34 Ruturaj Radhakrishna Trivedi (8th mode, f=1.50 Hz) (78th mode, f=2.53 Hz) 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 Master

Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 35 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. 4.32 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 Sr Mode No shape Dry Model Difference Fluid Nodal mass model Mesh Dry and nodal (Hz) model mass models (%) (Hz) Dry and Fluid mesh models (Hz) (%) 1st 1 Vertical 3.66 2.46 2.7 33 26 Bending 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 wet  dry 2 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 36 Ruturaj Radhakrishna Trivedi 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 37 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 4.31 Wet model- Nodal added mass Wet model- fluid mesh 1st Vertical

Bending (2 nodes) (7th mode, f=2.46 Hz) (1st mode, f=2.7 Hz) 2nd Vertical Bending (3 nodes) (9th mode, f=3.98 Hz) (4th mode, f=4.29 Hz) Figure 28 Vertical bending modes “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 38 Ruturaj Radhakrishna Trivedi The mode shapes at the frequencies shown in table are compared in figures 30-30. Wet model- Nodal added mass Wet model- fluid mesh 1st Transverse bending (2 nodes) (8th mode, f=2.89 Hz) (2nd mode, f=3.7 Hz) 2nd Transverse bending (3 nodes) (18th mode, f=5.19 Hz) (16th mode, f=6.66 Hz) 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions Wet model- Nodal added mass Wet model- fluid mesh 1st Torsion (17th mode, f=4.9 Hz) (17th mode, f=6.99 Hz) 2nd Torsion (24th mode, f=6.92 Hz) (28th mode, f=9.19 Hz) Figure 30 Torsional modes “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 39 40 Ruturaj Radhakrishna Trivedi 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 m11 , m22 and m33 were applied in x, y and z directions using discrete spring elements of very high stiffness of 1  1010 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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 41 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 42 Ruturaj Radhakrishna Trivedi 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 43 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 44 Ruturaj Radhakrishna Trivedi Figure 36 Positions of selected elements Figure

37 Pressure distribution on selected elements The pressure at element 114 is the highest at 13.3 106 Pa, the second highest at the stern, element 105055, 5.78 106 Pa and the lowest at the bow on element 927, 223 106 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 105 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 45 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. 5.31 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 105 second, which represents the phenomena of first and second bubble pulsations. The energy of dry model is more than that of the wet model “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 46 Ruturaj Radhakrishna Trivedi 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. 5.32 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 47 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. 5.33 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 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 48 Ruturaj Radhakrishna Trivedi 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 432, the natural frequency of the wet model is

between 2.46 Hz and 269 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. 5.34 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 49 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. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 50 Ruturaj Radhakrishna Trivedi

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. Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 51 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. “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 52 Ruturaj Radhakrishna Trivedi 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 Master Thesis developed at ICAM, France Analysis of added mass effect on surface ships subjected to underwater explosions 53 9. REFERENCES 1. Hendershot john and Kaczmarek Robert, 2014 Defense AT & L [online] Source Available from: https://apps.dticmil/dtic/tr/fulltext/u2/1015915pdf

[20th December 2018] 2. Navarro MG, 2015 Rules and Methods for Dimensioning Embarked Materials for Surface Ships When Subjected to UNDEX. Thesis (MSc) EMship 3. Tsai SC, 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 AI, 2009 Added Masses of Ship Structures St Petersburg, Russia: Springer Science + Business Media B.V 6. Chen SS and Chung H, 1976 Design Guide for Calculating Hydrodynamic Mass Part I: Circular Cylindrical. Illinois, Thesis (MSc) Argonne National Laboratory 7. Techet, AH, 2005 Hydrodynamics [online] Source Available from: http://web.mitedu/2016/www/handouts/2005CourseInfopdf [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, FM,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 AR, 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 519.6:6295015 “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 54 Ruturaj Radhakrishna Trivedi 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: n f ( Z )   s  a 2 k 1 Z 2 k 1 (67) k 0 Where f ( Z )  z and z  x  iy is the plane of the ship's cross section Z  ie e  i - is the plane of the unit circle  s = scale factor a 1  1 a2 k 1 are the conformal mapping coefficients ( k  1, . , n ) n = number of parameters n x  iy  s  a2 k 1 (ie ei ) (2 k 1) , (68) k 0 n x  iy  s  a2 k 1 (i) (2 k 1)e (2 k 1) [cos( )  i sin( )]2 x1 (69) k 0 n x  iy  s  (1)k a2 k 1e (2 k 1) [i cos(2k  1)  sin(2k  1) ] (70) k 0 From the relation between the coordinates

in the z - plane (the ship's cross section) and the variables in the Z - plane (the circular cross section): n x  s  (1)k a2 k 1e (2 k 1) sin(2k  1) (71) k 0 n y  s  (1)k a2 k 1e (2 k 1) cos(2k  1) . k 0 Now by using conformal mapping approximations, the contour of the ship's cross section, follows from setting   0 in previous sections. We get: n x0   s  (1)k a2 k 1 sin(2k  1) , k 0 Master Thesis developed at ICAM, France (72) Analysis of added mass effect on surface ships subjected to underwater explosions 55 n y0  s  (1)k a2 k 1 cos(2k  1) (73) k 0 The breadth on the waterline of the approximate ship's cross section is defined by n B0  2s  , with    a2 k 1 (74) k 0 and the draft is defined by n D0  2 , with    (1)k a2 k 1 (75) k 0 The breadth on the waterline is

obtained for    2 , that means: n n k 0 k 0 x 2  s  (1)k a2 k 1 sin(2k  1)  2, hencex 2  s  a2 k 1 and B0  2 x /2 The scale factor is  s  B0 and the draft is obtained for   0 : 2 n n k 0 k 0 y0  s  (1)k a2 k 1 cos(2k  1)  0, hencey0  s  (1)k a2 k 1 and D0  y0 with  s  (76) d0  (77) . Lewis Conformal Mapping With n=2, a simple transformation can be obtained of the hull form. The two-parameter Lewis transformation of a cross section is defined by: z  f (Z )  s a1Z  s a1Z 1  s a3Z 3 (78) Where, a1  1,  s = scale factor a1 and a3 = conformal mapping coefficients, also called Lewis coefficients. Then, for z  x  iy and Z  ie e  i , that is Z  ie cos     i sin    , we have: x  s [e sin( )  a1e sin  

a3e3 sin 3 ] (79) x  s (e sin   a1e sin   a3e3 sin 3 ) (80) y  s [e cos( )  a1e cos   a3e3 cos3 ] (81) y  s (e cos   a1e cos   a3e3 cos3 ) (82) For   0 we obtain the contour of the so- called Lewis form expressed as: x0  s (sin   a1 sin   a3 sin 3 ) , “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 (83) 56 Ruturaj Radhakrishna Trivedi y 0   s (cos  a1 cos  a3 cos 3 ) (84) or as: x0  s [(1  a1 )sin   a3 sin 3 ] (85) y0   s ((1  a1 ) cos   a3 cos 3 ) (86) Ds Bs or s  2(1  a1  a3 ) 2(1  a1  a3 ) (87) Where, the scale factor  s is: s  Where, Bs = sectional breadth on the load waterline Ds = sectional draft. The half breadth to draft ratio H 0 is: H0  Bs 1  a1  a3  2 Ds 1  a1

 a3 (88) An integration of the Lewis form delivers the sectional area coefficient  s : s  As  1  a12  3a32   Bs Ds 4 (1  a3 )2  a12 (89) Where, As = area of the cross section, d  0 2 As  2  xdy   s2 (1  a12  3a32 ) (90) Bs  Ds  2[(1  a3 ) 2  a12 ] (91) The coefficients a1 , a3 and the scale factor  s are determined in such that the breadth, the draft and the area of the approximate cross section and of the actual cross section are identical. From expression for H 0 , we obtain, a1  ( H 0  1)(1  a3 ) 1  H0 (92) Substituting it into the expression of  s , we get a quadratic equation in a3 2 2  H 1     H 1  3  s (1  a3 )   0  (1  a3 )2  s    0  (1  a3 )2  a32 4 4  H0 1  4  H0 1  2 Therefore Master Thesis developed at ICAM, France (93) Analysis of added mass effect on surface ships subjected to

underwater explosions  57 2 2 2 8 s   4 s  H 0  1   2    H 0  1  8  H 0  1  3     a3   1  a  2     3 s 4     H 0  1   4   H 0  1    H 0  1        4 s  H  1   4 s    4  1     s   0   1  0 4      H 0  1  2  (94)  Now,      H 1  m  3  4  s  1  4 s   0       H0 1  2 (95) 2      H 1  n  8  s   2  8 s   0   2m  6      H0 1  (96) 2      H 1  p  4  s  1  4 s   0   1  m  4      H0 1  (97) The equation obtained is, ma32  na3  p  0 (98) a3   m

 3  9  2m m (99) The Lewis form with the other solution of a3 is: a3   m  3  9  2m m (100) Calculation of the Incident Pressure due to Explosion As per the EMSHIP master thesis work of Ssu Chieh Tsai [3], the DAA (Doubly Asymptotic Approximation) method was found to be the best to calculate the incident pressure due to both the shock wave and the gas bubble pulses generated by an underwater explosion. Hence, in this research, the macro developed in the framework of Tsai’s master thesis was used for ship pressure loading calculation. All the related formulations and theory can be found in Tsai [3] Shock Factor According to Ried (1996) [3], for large shock wave angle, keel shock factor [KSF] can be equivalent to hull shock factor [HSF]. According to experiments and theory, KSF can be assumed proportional to the vertical velocity of the ship considered as a rigid body. The shock factors can be described as follows: [3] Wn D W n 1  sin  HSF  D 2

KSF  “EMSHIP” Erasmus Mundus Master Course, period of study September 2017 – February 2019 58 Ruturaj Radhakrishna Trivedi Where, W = the mass of explosive in TNT equivalence (kg) D = the standoff distance from target (m) n = varies slight different value from underwater experiments θ= shock wave angle between a horizontal line and distance from the measured point In ISSC (2006) [3] it was observed that the maximum shock factor (SF) can be evaluated by following equation: W D W 1/2 1  sin  HSF   D 2 SF  The above equation was used in this research work to calculate shock factor as mentioned in section 5.2 Master Thesis developed at ICAM, France