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Summary
Summary
The purpose of this monograph is to show how a compliant offshore structure in an ocean environment can be modeled in two and three di mensions. The monograph is divided into five parts. Chapter 1 provides the engineering motivation for this work, that is, offshore structures. These are very complex structures used for a variety of applications. It is possible to use beam models to initially study their dynamics. Chapter 2 is a review of variational methods, and thus includes the topics: princi ple of virtual work, D'Alembert's principle, Lagrange's equation, Hamil ton's principle, and the extended Hamilton's principle. These methods are used to derive the equations of motion throughout this monograph. Chapter 3 is a review of existing transverse beam models. They are the Euler-Bernoulli, Rayleigh, shear and Timoshenko models. The equa tions of motion are derived and solved analytically using the extended Hamilton's principle, as outlined in Chapter 2. For engineering purposes, the natural frequencies of the beam models are presented graphically as functions of normalized wave number and geometrical and physical pa rameters. Beam models are useful as representations of complex struc tures. In Chapter 4, a fluid force that is representative of those that act on offshore structures is formulated. The environmental load due to ocean current and random waves is obtained using Morison's equa tion. The random waves are formulated using the Pierson-Moskowitz spectrum with the Airy linear wave theory.
Table of Contents
Preface | p. ix |
Acknowledgments | p. xiii |
1. Introduction | p. 1 |
1. Tension Leg Platforms | p. 2 |
2. Mathematical Models for Dynamic Responses | p. 3 |
2.1 Single Degree of Freedom Models | p. 4 |
2.2 Six Degree of Freedom Model | p. 6 |
2.3 Rigid and Elastic Models for Tendons | p. 8 |
3. Outline | p. 9 |
2. Principle of Virtual Work, Lagrange's Equation and Hamilton's Principle | p. 13 |
1. Introduction | p. 13 |
2. Virtual Work | p. 15 |
2.1 The Principle of Virtual Work | p. 17 |
2.2 D'Alembert's Principle | p. 19 |
3. Lagrange's Equation | p. 20 |
3.1 Lagrange's Equation for Small Oscillations | p. 22 |
4. Hamilton's Principle | p. 23 |
5. Lagrange's Equation with Damping | p. 24 |
6. Application to Longitudinally Vibrating Beams | p. 25 |
7. Chapter Summary | p. 28 |
3. Overview of Transverse Beam Models | p. 29 |
1. Literature Review and Underlying Assumptions | p. 29 |
2. Nomenclature | p. 33 |
3. Equation of Motion and Boundary Conditions Via Hamilton's Principle | p. 34 |
3.1 Euler-Bernoulli Beam Model | p. 34 |
3.2 Rayleigh Beam Model | p. 38 |
3.3 Shear Beam Model | p. 41 |
3.4 Timoshenko Beam Model | p. 47 |
4. Natural Frequencies and Mode Shapes | p. 51 |
4.1 Symmetric and Antisymmetric Modes | p. 51 |
4.2 Euler-Bernoulli Beam Model | p. 54 |
4.3 Rayleigh Beam Model | p. 55 |
4.4 Shear Beam Model | p. 61 |
4.5 Timoshenko Beam Model | p. 66 |
5. Comparisons of Four Models | p. 75 |
6. Free and Forced Response | p. 78 |
6.1 Orthogonality Conditions for the Euler-Bernoulli, Shear, and Timoshenko Models | p. 79 |
6.2 Orthogonality Conditions for the Rayleigh Model | p. 82 |
6.3 Free and Forced Response via Method of Eigenfunction Expansion of the Euler-Bernoulli, Shear, and Timoshenko Models | p. 84 |
6.4 Free and Forced Response via Method of Eigenfunction Expansion of the Rayleigh Model | p. 86 |
6.5 Sample Responses | p. 88 |
6.6 Discussion of the Second Frequency Spectrum of the Timoshenko Beam | p. 90 |
7. Chapter Summary | p. 93 |
4. Environmental Loading -Waves and Currents | p. 95 |
1. Nomenclature | p. 95 |
2. Fluid Forces - General | p. 96 |
3. Fluid Forces I: The Morison Equation | p. 98 |
3.1 Wave Velocities | p. 101 |
3.2 Current Velocity in the Ocean, U[subscript c] | p. 106 |
3.3 Wind Velocity, U[subscript w] | p. 107 |
4. Fluid Force II: Vortex Induced Oscillations | p. 107 |
5. Chapter Summary | p. 110 |
5. Coupled Axial and Transverse Vibration in Two Dimensions | p. 111 |
1. Nomenclature | p. 112 |
2. Mathematical Formulation | p. 113 |
2.1 Displacements, Strains, and Stresses | p. 113 |
2.2 Lagrangian | p. 116 |
2.3 Equations of Motion and Boundary Conditions via Hamilton's Principle | p. 119 |
2.4 Non-dimensionalization | p. 120 |
2.5 Linear Transverse Vibration with Tension | p. 122 |
2.6 Linear Response without Tension | p. 123 |
2.6.1 Longitudinal Motion | p. 123 |
2.6.2 Transverse Motion | p. 124 |
3. Free and Damped-Free Response using the Two-Dimensional Coupled Model | p. 126 |
3.1 Free Response - Displacements, Phase Plots, and Spectral Density Plots | p. 129 |
3.2 Free Response - Potential and Kinetic Energies | p. 136 |
3.3 Damped-Free Response - Displacements, Phase Plots, and Spectral Density Plots | p. 146 |
3.4 Damped-Free Response - Potential and Kinetic Energies | p. 153 |
3.5 Effect of Varying Fluid Coefficients | p. 157 |
4. Forced Response using the Two-Dimensional Coupled Model | p. 158 |
4.1 Harmonic Forcing | p. 160 |
4.1.1 Subharmonics | p. 161 |
4.2 Effects of Current | p. 167 |
4.3 Effect of Random Waves | p. 170 |
5. Chapter Summary | p. 183 |
6. Three-Dimensional Vibration | p. 187 |
1. Nomenclature | p. 187 |
2. Mathematical Formulation | p. 188 |
2.1 Rigid Model | p. 188 |
2.2 Elastic Model | p. 193 |
2.2.1 Displacements, Strains and Stress | p. 193 |
2.2.2 Potential and Kinetic Energies | p. 195 |
2.2.3 Equations of Motion and Boundary Conditions Using Variational Principles | p. 196 |
2.3 Linearization of Equations of Motion | p. 199 |
3. Results on the Free Vibration | p. 200 |
3.1 Three-Dimensional Rigid Model | p. 202 |
3.2 Three-Dimensional Elastic Model | p. 210 |
4. Sample Results for the Forced Response of the Elastic Model | p. 212 |
4.1 Case I: Harmonic Loading in the y Direction | p. 213 |
4.2 Case II: Harmonic and Non-harmonic Loadings in the Perpendicular Directions | p. 218 |
5. Chapter Summary | p. 224 |
7. Summary | p. 229 |
Appendices | p. 232 |
Fourier Representation of a Gaussian Random | p. 233 |
Process Physically Plausible Initial Displacements | p. 237 |
Finite Difference Method | p. 241 |
1. Two Dimensional Equations of Motion and Boundary Conditions | p. 241 |
1.1 Discretized Equations of Motion | p. 242 |
2. Three-Dimensional Equations of Motion and Boundary Conditions | p. 248 |
2.1 Discretized Equations of Motion | p. 249 |
2.2 Sample MATLAB Codes for 3D System | p. 252 |
2.2.1 Main Program | p. 252 |
2.2.2 Function Used in the Main Program | p. 254 |
Energy Loss Over One Cycle In Damped Case | p. 257 |
Steady-State Response Due to Ocean Current | p. 259 |
References | p. 261 |
Index | p. 267 |