-1- On the Dynamics of Low Tension Marine Cables by Waldir Terra Pinto A thesis submitted for the degree of Doctor of Philosophy in The Faculty of Engineering, University of London. Department of Mechanical Engineering University College London January 1995 ( eilL) -2- -3- Abstract This thesis is concerned with the dynamics of low tension marine cables. These cables are widely used in the ocean environment for signal and power transmission applications. There are two main issues in the dynamic analysis of such cables. When the tension is zero, which is often the situation encountered at the seabed during cable laying, the cable geometric stiffness matrix becomes singular. The other issue is that the transformation from local co-ordinates to global co-ordinates made through Euler angles leads to a greater number of unknowns than the number of differential equations. The former problem can be overcome by taking into account the flexural rigidity of the cable. The latter problem can be overcome by assuming that one of the Euler angles is known. However, this procedure can introduce singularities on the formulation of the problem. A new three dimensional model for the dynamics of marine cables is presented in this thesis. The model takes into account the bending stiffness of the cable in order to overcome singularities in the geometric stiffness matrix. In order to overcome the problem owing to the use of Euler angles, a new displacement approach is introduced. This new displacement approach uses the differential geometry definition of curvature and torsion in order to establish the transformation from the local co-ordinates to the global co-ordinates. The general formulation of the dynamics of marine cables presented in this thesis is applicable to a wide range of cases such as towed cables, cable installation and cable recovery. In order to illustrate this new formulation the cases of towed cables and cable installation are investigated in the some detail. Solutions for the differential equations of motion are presented for two and three dimensions. The two dimensional solution is obtained through a finite -4- element based technique which uses a weak Galerkin formulation for integration in space and the Newmark method for integration in time. The model's results are compared with full scale measurements. Simulations of the dynamic response of marine cables to vessel wave induced motions and vessel changes in speed are also presented. The three dimensional solution is obtained by expressing the equations of motion as functions of the Euler angles. The space integration is also performed by a finite element model but it uses a finite difference scheme for the time integration. This solution is then used to study the influence of sheared cross-currents in the cable's configuration. Finally, conclusions and suggestions for further research are presented. -5- TO MY FAMILY -6- Acknowledgements I would like to express my gratitude to my supervisor, Dr J . A. Witz, for his encouragement and valuable suggestions throughout this work. I very grateful to my sponsors, The British Council, Brazilian Post-Graduate Federal Agency (CAPES) and University of Rio Grande for their financial support throughout this research. I am very indebted to my wife, Vera, and my son, Humberto, for their love, support and encouragement. 1 am also very indebted to my parents, Francisco and Irene, and my sister Rosa for their love. I would like to specially thank my friend, Janito, for his valuable help. Finally I would like to thank colleagues and staff from the Department of Mechanical Engineering, University College London, for their support. -7- Table of Contents Abstract .3 Acknowledgements .6 Tableof Contents..........................................................................................................7 Listof Figures................................................................................................................10 Listof Tables .................................................................................................................13 Nomenclature................................................................................................................14 1.1 - A Brief Review of the History of Submarine TelecommunicationCables ...................................................23 1.2 - Engineering Operations Related to Submarine Cable Systems.....................................................................................25 1.2.1- Cable Installation........................................................................26 1.2.2- Cable Protection .........................................................................27 1.2.3 - Cable Recovery..........................................................................28 1.3 - Scope of the Work ....................................................................................30 2- Previous Work.........................................................................................................32 2.1 - Stationary Solutions for Cables..............................................................35 2.1.1 - Solution for the Catenary Cable..............................................35 2.1.2 - Stationary Solution for Cable Laying with Zero BottomTension.....................................................................................36 2.1.3 - Kinematics of Stationary Cable Laying..................................39 2.2 - Literature Review of Numerical Solutions for Marine Cables ..........40 3- A Three Dimensional Model for the Dynamic Analysis of Marine Cables.............................................................................................................................47 3.1 - Introduction ..............................................................................................47 3.2 - Assumptions.............................................................................................49 3.3 - Kinematics of Marine Cables..................................................................50 3.3.1 - Reference System.......................................................................51 3.3.2 - Position Vector...........................................................................54 3.3.3 - Absolute and Relative Velocity Vectors.................................55 AbsoluteVelocity Vector.........................................................55 RelativeVelocity Vector ..........................................................56 3.3.4 - Absolute and Relative Acceleration Vectors.........................58 AbsoluteAcceleration Vector.................................................58 RelativeAcceleration Vector...................................................58 3.3.5 - Equation of Geometric Compatibility....................................59 3.4 - Kinetics of Marine Cables .......................................................................59 3.4.1 - Dynamic Equilibrium for an Element of Marine Cable.......................................................................................................60 -8- 3.4.2 - Vector Equation of Motion .63 Self-Weight and Buoyancy Forces .........................................64 DragForce .................................................................................64 AddedMass Force....................................................................65 D'Alembert Force......................................................................66 3.4.3 - Scalar Equations of Motion......................................................66 3.4.4 - Boundary Conditions ...............................................................68 3.5 - Consistency and Generality of the Model ............................................69 4- Two Dimensional Analysis of Low Tension Marine Cables.............................72 4.1 - Introduction ..............................................................................................72 4.2 - Two Dimensional Equations of Motion................................................73 4.2.1 - Two Dimensional Assumptions..............................................73 4.2.2 - Two Dimensional Scalar Equations of Motion .....................75 4.2.3 - Two Dimensional Reference System and Curvature...........76 Relationship between the Reference Frames........................77 Determinationof the Curvature.............................................79 4.2.4 - Determination of the Velocity and Acceleration Vectors....................................................................................................81 4.2.5 - Boundary Conditions and Final Expression for the Equationsof Motion.............................................................................83 4.3 - Solution of the Differential Equation of Motion..................................84 4.3.1 - Space Integration of the Equation of Motion ........................85 4.3.2 - Time Integration of the Equation of Motion .........................88 4.3.3 - Solution of the Non-Linear System of Algebraic Equations...............................................................................................91 Determination of the Global Force Vector............................93 ShapeFunctions........................................................................94 Vertical Co-ordinate and its Derivatives...............................95 Horizontal Co-ordinate and its Derivatives .........................96 Tangent and Normal Components for the Velocity andAcceleration.......................................................................98