Nonlinear and Stochastic Dynamics of Compliant Offshore Structures

by Han, Seon Mi; Benaroya, Haym

• ISBN13: 9781402005732
• ISBN10: 1402005733
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Kluwer Academic Pub

Summary

The purpose of this monograph is to show how to model a compliant offshore structure in an ocean environment. Such structures are of increasing importance because of the desire to access the deep ocean. A brief history and an extensive review of the four existing transverse beam models are presented. The four beam models are Euler-Bernoulli, Rayleigh, shear and Timoshenko models. These are the fundamental models for the preliminary studies of offshore structures. The complete analytical solutions are given for each model. This review is followed by a chapter on the stochastic modelling of environmental forces. The waves are assumed to be random, and the fluid force is modelled using the Morison Equation. In subsequent chapters, the models are expanded to include the axial motion as well as the transverse motion in two and three dimensions.Numerical results are obtained using both deterministic and random forces. A detailed description of the numerical methods is provided so that the reader can adapt these methods to his or her particular problem. Researchers in the fields of structural mechanics and ocean engineering, especially those with an interest in nonlinear hybrid beam/cable models, will find this a useful book.

Table of Contents

Preface	ix
Acknowledgments	xiii
Introduction	1	(12)
Tension Leg Platforms	2	(1)
Mathematical Models for Dynamic Responses	3	(6)
Single Degree of Freedom Models	4	(2)
Six Degree of Freedom Model	6	(2)
Rigid and Elastic Models for Tendons	8	(1)
Outline	9	(4)
Principle of Virtual Work, Lagrange's Equation And Hamilton's Principle	13	(16)
Introduction	13	(2)
Virtual Work	15	(5)
The Principle of Virtual Work	17	(2)
D'Alembert's Principle	19	(1)
Lagrange's Equation	20	(3)
Lagrange's Equation for Small Oscillations	22	(1)
Hamilton's Principle	23	(1)
Lagrange's Equation with Damping	24	(1)
Application to Longitudinally Vibrating Beams	25	(3)
Chapter Summary	28	(1)
Overview of Transverse Beam Models	29	(66)
Literature Review and Underlying Assumptions	29	(4)
Nomenclature	33	(1)
Equation of Motion and Boundary Conditions Via Hamilton's Principle	34	(17)
Euler-Bernoulli Beam Model	34	(4)
Rayleigh Beam Model	38	(3)
Shear Beam Model	41	(6)
Timoshenko Beam Model	47	(4)
Natural Frequencies and Mode Shapes	51	(24)
Symmetric and Antisymmetric Modes	51	(3)
Euler-Bernoulli Beam Model	54	(1)
Rayleigh Beam Model	55	(6)
Shear Beam Model	61	(5)
Timoshenko Beam Model	66	(9)
Comparisons of Four Models	75	(3)
Free and Forced Response	78	(15)
Orthogonality Conditions for the Euler-Bernoulli, Shear, and Timoshenko Models	79	(3)
Orthogonality Conditions for the Rayleigh Model	82	(2)
Free and Forced Response via Method of Eigenfunction Expansion of the Euler-Bernoulli, Shear, and Timoshenko Models	84	(2)
Free and Forced Response via Method of Eigenfunction Expansion of the Rayleigh Model	86	(2)
Sample Responses	88	(2)
Discussion of the Second Frequency Spectrum of the Timoshenko Beam	90	(3)
Chapter Summary	93	(2)
Environmental Loading-Waves and Currents	95	(16)
Nomenclature	95	(1)
Fluid Forces - General	96	(2)
Fluid Forces I: The Morison Equation	98	(9)
Wave Velocities	101	(5)
Current Velocity in the Ocean, Uc	106	(1)
Wind Velocity, Uw	107	(1)
Fluid Force II: Vortex Induced Oscillations	107	(3)
Chapter Summary	110	(1)
Coupled Axial and Transverse Vibration In Two Dimensions	111	(76)
Nomenclature	112	(1)
Mathematical Formulation	113	(13)
Displacements, Strains, and Stresses	113	(3)
Lagrangian	116	(3)
Equations of Motion and Boundary Conditions via Hamilton's Principle	119	(1)
Non-dimensionalization	120	(2)
Linear Transverse Vibration with Tension	122	(1)
Linear Response without Tension	123	(1)
Longitudinal Motion	123	(1)
Transverse Motion	124	(2)
Free and Damped-Free Response using the Two-Dimensional Coupled Model	126	(32)
Free Response - Displacements, Phase Plots, and Spectral Density Plots	129	(7)
Free Response - Potential and Kinetic Energies	136	(10)
Damped-Free Response - Displacements, Phase Plots, and Spectral Density Plots	146	(7)
Damped-Free Response - Potential and Kinetic Energies	153	(4)
Effect of Varying Fluid Coefficients	157	(1)
Forced Response using the Two-Dimensional Coupled Model	158	(25)
Harmonic Forcing	160	(1)
Subharmonics	161	(6)
Effects of Current	167	(3)
Effect of Random Waves	170	(13)
Chapter Summary	183	(4)
Three-Dimensional Vibration	187	(42)
Nomenclature	187	(1)
Mathematical Formulation	188	(12)
Rigid Model	188	(5)
Elastic Model	193	(1)
Displacements, Strains and Stress	193	(2)
Potential and Kinetic Energies	195	(1)
Equations of Motion and Boundary Conditions Using Variational Principles	196	(3)
Linearization of Equations of Motion	199	(1)
Results on the Free Vibration	200	(12)
Three-Dimensional Rigid Model	202	(8)
Three-Dimensional Elastic Model	210	(2)
Sample Results for the Forced Response of the Elastic Model	212	(12)
Case I: Harmonic Loading in the y Direction	213	(5)
Case II: Harmonic and Non-harmonic Loadings in the Perpendicular Directions	218	(6)
Chapter Summary	224	(5)
Summary	229	(3)
Appendices	232	(29)
Fourier Representation of a Gaussian Random Process	233	(4)
Physically Plausible Initial Displacements	237	(4)
Finite Difference Method	241	(16)
1. Two Dimensional Equations of Motion and Boundary Conditions	241	(1)
1.1 Discretized Equations of Motion	242	(6)
2. Three-Dimensional Equations of Motion and Boundary Conditions	248	(1)
2.1 Discretized Equations of Motion	249	(3)
2.2 Sample MATLAB Codes for 3D System	252	(1)
2.2.1 Main Program	252	(2)
2.2.2 Function Used in the Main Program	254	(3)
Energy Loss Over One Cycle In Damped Case	257	(2)
Steady-State Response Due to Ocean Current	259	(2)
References	261	(6)
Index	267