Stochastic Dynamic Response and Stability of Ships and Offshore Platforms [1 ed.] 9819958520, 9789819958528

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Ocean Engineering & Oceanography 27

Yingguang Wang

Stochastic Dynamic Response and Stability of Ships and Offshore Platforms

Ocean Engineering & Oceanography Volume 27

Series Editors Manhar R. Dhanak, Florida Atlantic University SeaTech, Dania Beach, USA Nikolas I. Xiros, University of New Orleans, New Orleans, LA, USA

The Ocean engineering & Oceanography (OEO) series publishes state-of-art research and applications in the two related and interdependent areas of ocean engineering and oceanography. The series contains monographs, lecture notes, edited volumes and selected PhD theses focusing on all different theoretical as well as applied aspects and subfields of ocean engineering and oceanography research. Topics in the series include marine electronic engineering, concerned with the design of electronic devices for use in the marine environment, offshore engineering (or offshore construction), concerned with the design of fixed and floating marine structures, pipelines and other underwater equipment, and oceanography, concerned with physical, chemical, geological and biological processes throughout the marine environment. All books published in the series are submitted for consideration in Web of Science.

Yingguang Wang

Stochastic Dynamic Response and Stability of Ships and Offshore Platforms

Yingguang Wang School of Naval Architecture, Ocean and Civil Engineering Shanghai Jiao Tong University Shanghai, China

ISSN 2194-6396 ISSN 2194-640X (electronic) Ocean Engineering & Oceanography ISBN 978-981-99-5852-8 ISBN 978-981-99-5853-5 (eBook) https://doi.org/10.1007/978-981-99-5853-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

The contents of this book have been continuously developed and improved since the author began teaching several courses in naval architecture and ocean engineering at Shanghai Jiao Tong University (SJTU) in 2003. The subject of stochastic dynamic response and stability of ships and offshore platforms is so broad that it is not possible to cover every aspect of the subject in a single textbook. However, most of the key knowledge of this subject has been included in this book, and the topics covered in this textbook have been made as complete as possible so that they can be studied in their entirety without having to refer to other works. This book is intended to have at least three applications. Firstly, it is expected that the book may be used as a textbook in both the undergraduate and graduate studies for students majoring in the fields of civil, mechanical, and ocean engineering. Secondly, researchers interested in interdisciplinary studies that combine the challenging theories of nonlinear stochastic dynamics with investigations of applied ocean engineering systems will immediately find the book to be a “perfect fit” for them. Finally, some of the book’s content was written based on this author’s decadelong ship design practices in industry before joining the faculty at SJTU in 2003. Therefore, the book should also be useful to the marine engineers and naval architects, as well as civil engineers and mechanical engineers who work on the design and operation of ships and ocean engineering systems. In brief, the organization of this book is as follows: Although Chap. 1 is only titled “Introduction”, the detailed theories regarding the statistical linearization method and the perturbation method for stochastic dynamic response analysis of nonlinear ocean engineering systems are thoroughly elucidated. Chapter 2 mainly introduces spectral analysis and numerical simulation of random ocean waves, fluid forces on slender offshore structures as well as other types of random wave loads on marine structures. Chapter 3 elucidates the Monte Carlo Simulation method for stochastic dynamic response analysis of marine structures. The theoretical background of the fourth-order Runge-Kutta method for numerical integration during the Monte Carlo simulation is also included. In the meantime, the theory for the level up-crossing rate and the concept of the slow drift extreme responses of marine structures are also presented in a concise and clear manner. Chapter 4 explains the path integration v

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method for analyzing response problems of marine structures. A detailed explanation of the theory of Markov diffusion processes is given in this chapter. A new compound path integration method based on the Gauss-Legendre integration scheme is also proposed in this chapter. Furthermore, this chapter also provides the detailed derivation processes for obtaining the equations to calculate the exceedance probability of the extreme response of an offshore structure. In Chap. 5, the traditional methods and a modern geometric method for ship stability analysis are presented. The explanation of the unexpected capsizing mechanism of a ship in the Melnikov method is given in this chapter, followed by deriving the Melnikov criteria for ship capsizing under harmonic excitations and the validation of the Melnikov criteria by the Lyapunov exponents. In addition, a stochastic Melnikov method is developed and a mean-square criterion is obtained in this chapter for analyzing a ship’s stability in random waves. Chapter 6 explains the first passage theory for ship stability analysis and provides the derivation processes for obtaining the generalized Pontryagin equation. The emphasis of this chapter is on elucidating a stochastic averaging method based on generalized harmonic functions. Chapter 7 is a summarization of the entire contents in this book. Finally, Appendix A elucidates the derivation processes for obtaining the Kolmogorov equation and Appendix D provides the detailed procedures for the derivation of the Melnikov function in a clear manner. A knowledge of dynamics, ship hydrostatics and stability, probability, and stochastic processes is the prerequisite for the complete use of this book. It is a pleasure to acknowledge the help I received from some of my colleagues and graduate students during my teaching career here at the School of Naval Architecture, Ocean and Civil Engineering of SJTU. I would also like to take this opportunity to express my gratitude to the State Key Laboratory of Ocean Engineering of SJTU for its generous support of my research over the years. Teaching and research are two sides of the same coin, and it is only through an active and fruitful research program that teaching can be made vibrant and relevant. Shanghai, China

Yingguang Wang [email protected]

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction to Analysis Methods for Stochastic Dynamic Response of Ships and Offshore Platforms . . . . . . . . . . . . . . . . . . . . . 1.1.1 Response Spectrum of a Linear Stochastic Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Statistical Linearization Method . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Monte Carlo Simulation (MCS) Method . . . . . . . . . . . . . . . . . 1.1.5 Numerical Path Integration Method . . . . . . . . . . . . . . . . . . . . . 1.1.6 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Introduction to Analysis Methods for Stochastic Dynamic Stability of Ships and Offshore Platforms . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Geometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 First Passage Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Contents of This Book and the Key Issues to Be Addressed . . . 1.3.1 Monte Carlo Simulation Method for Analyzing Response Problems of Marine Structures . . . . . . . . . . . . . . . . 1.3.2 Numerical Path Integration Method for Analyzing Response Problems of Marine Structures . . . . . . . . . . . . . . . . 1.3.3 Global Geometric Method for Analyzing Stability Problems of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 The First Passage Theory for Analyzing Stability Problems of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Random Loads Acting on Marine Structures . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Random Waves in the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Randomness of Sea Waves and Their Probabilistic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 10 17 24 26 30 30 32 33 34 35 38 40 42 44 46 51 51 52 52

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2.2.2 Spectral Analysis of Random Ocean Waves . . . . . . . . . . . . . . 2.2.3 Wave Spectrum Mathematical Formulations . . . . . . . . . . . . . 2.2.4 Numerical Simulation of Random Ocean Waves . . . . . . . . . . 2.3 Fluid Forces on Slender Offshore Structures . . . . . . . . . . . . . . . . . . . . 2.3.1 Morison Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Morison Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Morison Inertia Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Combined Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Generalized Form of the Morison Formula . . . . . . . . . . . . . . . 2.4 Other Types of Random Wave Loads on Marine Structures . . . . . . . 2.4.1 Second-Order Slow-Drift Stochastic Excitations on Compliant Offshore Structures . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Spectral Density Function of Random Wave Loads on Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Simplified Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 60 64 67 67 68 68 69 69 70 70 72 74 74 75 76

3 Monte Carlo Simulation Method for Dynamic Response of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.1 Nonlinearities Handled by the Monte Carlo Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.2 Differential Equation of Motion for an Offshore Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.3 The Fourth-Order Runge–Kutta Method for Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.4 Motion Response Analysis of the Platform by Monte Carlo Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.5 Response Statistics Analysis of the Offshore Structure by the Monte Carlo Simulation Method . . . . . . . . . . . . . . . . . 88 3.2.6 Level Up-Crossing Rates of the Structural Motion Responses by the Monte Carlo Simulation Method . . . . . . . . 97 3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift Extreme Responses of Marine Structures Under Random Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.1 Concept of the Slow Drift Extreme Responses of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.2 Method for Treating External Random Excitations During Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . 105

Contents

3.3.3 Analysis of the Slow Drift Extreme Response of a Moored Floating Cylinder by Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Highlighting the Low Accuracies of the MCS Forecasted Extreme Responses of Marine Structures . . . . . . 3.3.5 Highlighting the Low Efficiency of the MCS Method to Forecast the Extreme Responses of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Shortcomings of Monte Carlo Simulation to Analyze the Stability of Marine Structures Under Random Excitations . . . . . 3.4.1 Quantitative Description of the Inefficiencies When Analyzing Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Path Integration Method for Analyzing Response Problems of Marine Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Development History of Path Integration Method . . . . . . . . . . . . . . . 4.3 Theoretical Background of the Path Integration Method . . . . . . . . . . 4.3.1 Markov Diffusion Process Theory . . . . . . . . . . . . . . . . . . . . . . 4.3.2 A New Path Integration Method Based on Gauss–Legendre Integration Scheme . . . . . . . . . . . . . . . . . 4.3.3 Effectiveness of the Path Integration Method for Strongly Nonlinear Stochastic Problems . . . . . . . . . . . . . . 4.4 Slow Drift Oscillation Responses of a Moored Ship by the Path Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Slowly Varying Wave Drift Force . . . . . . . . . . . . . . . . . . . 4.4.2 The System Equation for a Moored Ship . . . . . . . . . . . . . . . . 4.4.3 Path Integration Solution of the System Equation . . . . . . . . . 4.5 Analysis of Slow Drift Extreme Response of Offshore Structures by the Path Integration Method . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Analyzing the Same Offshore Structure Ever Used in the Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Comparing the Accuracy of the Path Integration Solutions with the MCS Solutions When the System Equation Has Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Obtaining Accurate Forecasts of the Slow Drift Extreme Response of an Offshore Structure by the Path Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Concurrently Obtaining the Exceedance Probabilities of the Slow Drift Extreme Responses of Offshore Structures by the Path Integration Method . . . . . . . . . . . . . . .

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118 129 129 131 133 136 137 137 139 144 144 148 153 161 161 163 170 173 173

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4.5.5 A Proposed Compound Path Integration Method to Find the Slow Drift Extreme Responses of Offshore Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.6 Compound Path Integration Method for the Extreme Response of a Compliant Platform in the General Case . . . . 4.6 Evaluation of the Path Integration Method for Analyzing the Slow Drift Extreme Responses of Offshore Structures Subjected to Random Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Superiority Over the MCS Method in Terms of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Superiority Over the MCS Method in Terms of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Points to Note When Applying the Path Integration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Ship Stability Analysis: Traditional Methods Versus Geometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ship Intact Stability Analysis—The Traditional Methods . . . . . . . . . 5.2.1 Ship Flotation and Related Calculations . . . . . . . . . . . . . . . . . 5.2.2 The Initial Stability of an Intact Ship and Related Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Large Angle Stability of an Intact Ship and Related Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Melnikov Criteria for Ship Capsizing Under Harmonic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Derivation of a Ship’s Differential Equation of Rolling Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Explanation of the Unexpected Capsizing Mechanism of a Ship in the Melnikov Method . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Melnikov Criteria for Ship Capsizing Under Harmonic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Application of the Melnikov Method for Further Case Studies . . . . 5.4.1 Analysis of the Damping Required for a Barge to Prevent Capsizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Validation of the Melnikov Criteria by the Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Stochastic Melnikov Analysis and the Mean Square Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.5 Evaluation of the Melnikov Method for Analyzing the Stability of Marine Structures Under Random Excitations . . . . . 5.5.1 Superiority over the MCS Method in Terms of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Problems to Be Noted When Applying the Melnikov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 First Passage Theory for Ship Stability Analysis . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stochastic Averaging Method Based on Generalized Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Generalized Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Stochastic Averaging Procedure . . . . . . . . . . . . . . . . . . . . . . . . 6.3 First Passage Forecasting of Ship Capsizing . . . . . . . . . . . . . . . . . . . . 6.3.1 First Fassage Failure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Stochastic Averaging of the Ship’s Rolling Equation . . . . . . 6.3.3 First Passage Time of Ship Capsizing . . . . . . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 293

7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Main Contents of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Novel Aspects in This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Directions for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 315 322 325 327

295 295 296 299 299 302 308 312 314 314

Appendix A: Derivation of the Kolmogorov Equation . . . . . . . . . . . . . . . . . 329 Appendix B: Derivation of Eq. (5.44) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Appendix C: Gravity Center Movement Principle . . . . . . . . . . . . . . . . . . . . 339 Appendix D: Derivation of the Melnikov Function . . . . . . . . . . . . . . . . . . . . 341

Chapter 1

Introduction

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response of Ships and Offshore Platforms The jacket platform, gravity-based platform and jack-up platform are some types of marine structures built in the early days of the offshore oil development process. These types of platforms were designed to operate in shallow waters. The philosophy of designing these types of platforms is to ensure that their structures have sufficient strength and stiffness to resist the action of environmental loads (wind, wave and current loads). Therefore, the natural frequencies of these types of platforms are much higher than the corresponding frequencies of the dominant waves [1] and these platforms do not resonate under the action of environmental loads (mainly wave loads), i. e., their amplitude of motion under the action of environmental loads is very small. Thus, when doing mechanics analysis on these types of platforms, it is almost not necessary to perform dynamic analysis, but only to do the quasi-static analysis. With the offshore oil development moving to the deep sea, the design of platforms according to the above philosophy will inevitably lead to oversized structures, which is not economically viable. Therefore, people’s design philosophy has been changed to the opposite extreme: that is, designing some kinds of platforms to “comply” with the action of environmental loads, the representative types are semi-submersible platforms with catenary mooring and floating production storage and offloading systems and Tension Leg Platforms with tension leg mooring. The natural frequencies of these deepwater platforms are much lower than the corresponding frequencies of the first-order wave forces, but the natural frequencies of these platforms are difficult to avoid coinciding with the corresponding frequencies of the second-order slowdrift wave forces [1], so that they often produce harmonic resonance under the action of the second-order slow-drift wave forces and result in large motion amplitudes. Because the space that can be reserved for the helicopter landing pad (i.e., the size of the helicopter landing pad) is limited when designing the layout of these platforms, providing accurate predictions of the motion response of these platforms is critical to improving the success of helicopter landing. This is an important reason for the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5_1

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2

1 Introduction

need of dynamic analysis of these platforms. Of course, dynamic analysis is also important to ensure the strength and reliability of these platforms. The dynamic analysis of a deep-water platform requires two tasks: on the one hand, the added mass, structural damping and hydrodynamic external loads of the platform are required, which is not the focus of this book (there are many good commercial software available to deal with this aspect); on the other hand, the rigid body equations of motion of the platform are established, and then the equations of motion are solved under the given initial and boundary conditions. This is the focus of this book. This aspect of the work is the focus of this book because so far industry has only been using some approximations to deal with this aspect of the problem. Even in the academic world, the research in this area is not very mature [2]. Since no two wave trains are recorded precisely similar in the real-world sea conditions, mathematical statistics is the only valid way to describe waves. The wellknown concept of wave spectrum is to express the irregular waves on the sea surface in the language of stochastic processes and mathematical statistics. Since the waves on the sea surface are random, the wave loads acting on a deep-water platform are also random. This makes the established rigid body equation of motion of the platform a stochastic differential equation. In dealing with the response of marine structures under random loads, a method widely used in industry is to perform spectral analysis in the frequency domain. The frequency domain analysis method uses the spectral equation to derive a closed-form solution to the stochastic differential equation of motion, the output of which is the response spectrum, and then the response spectrum is used to calculate the time history of motion responses, response statistics and motion extrema. The principle of this method is simple to understand and easy to implement in the engineering practice, and the mathematical background of this method is briefly described below.

1.1.1 Response Spectrum of a Linear Stochastic Dynamic System If the equations of motion for a dynamic system take the form of linear differential equations with constant coefficients, the system is said to be linear and time-invariant. A single second-order ordinary differential equation can describe a system with a single degree of freedom. Suppose the dynamics of an offshore platform can be ideally expressed using the following differential equation: } { M x¨ + C x˙ + K x = Re f ei ω t

(1.1)

where M, C and K respectively represent the mass coefficient, damping coefficient and stiffness coefficient of the system, which are assumed to be constants. x, x˙ and x¨ are the offshore platform response }displacement, response velocity and response { acceleration, respectively. Re f ei ω t is the real part of f ei ω t . Equation (1.1) can

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

3

also be written in terms of the natural frequency of the undamped oscillation of the offshore system, denoted by ωn , as } { x¨ + 2 ζ ωn x˙ + ωn2 x = Re f 0 ei ω t

(1.2)

where √ ωn = K /M = natural frequency of the undamped oscillation. ζ = C/(2Mωn ) = damping factor. f 0 = f /M = complex amplitude of the excitation per unit mass (or moment of inertia). Given certain initial conditions, integrating the differential equation of motion (1.2) can provide a specific solution for x(t) if f 0 (t) is a given deterministic time history. The solutions to linear differential equations with constant coefficients can be found using a variety of methods. The use of a complex frequency–response function in conjunction with the Fourier integral and the use of an impulse-response function in conjunction with the superposition or convolution integral will both be covered in this book. Since these two approaches are essentially Fourier transforms of one another, they are closely related. When the excitation is steady state simple harmonic motion, a characteristic of linear time-invariant systems is that the response is also steady state simple harmonic motion at the same frequency. In general, the frequency affects the response’s amplitude and phase. Giving the complex frequency–response function G ( ω ) is a succinct way to explain how the amplitude and phase depend on frequency. This has the characteristic that the response is the real part of G( ω )e i ω t when the excitation is the real part of e i ω t . For deriving the general solution of the Eq. (1.2) we need to consider (a) the general solution of the homogenous equation (i.e., free oscillation without external excitations in Eq. 1.2), and (b) the particular solution of the non-homogenous equation (i.e., with external excitations in Eq. 1.2). The former yields a damped free oscillation expressed as follows: x(t) = J1 e−ζ ωn t e

(√ ) ζ 2 −1 ωn t

+ J2 e−ζ ωn t e

(√ ) − ζ 2 −1 ωn t

(1.3)

In the above equation J 1 and J 2 are determined from the initial conditions. By carefully studying Eq. (1.3) we can find that the oscillation damps out with time under each of the three possible conditions of damping (underdamping (ζ < 1), critical damping (ζ = 1), and overdamping (ζ > 1)) and under all possible initial conditions. Therefore, it is not a steady state oscillation. Consequently, this is not the solution we are looking for. Thus, the general solution of the Eq. (1.2) eventually reduces to the particular solution of the non-homogenous equation (i.e., Eq. 1.2), which represents the steady-state oscillation. In order to derive the particular solution of the non-homogenous equation, let us write } { x(t) = Re x0 e i ω t

(1.4)

4

1 Introduction

Then, by combining Eqs. (1.2) and (1.4) we can derive the following expression f0 ) x0 = ( 2 ωn − ω2 + 2iζ ωn ω

(1.5)

By setting f 0 = 1 in Eq. (1.5) we can obtain the complex frequency–response function G(ω) as follows: 1 ) G(ω) = ( 2 2 ωn − ω + 2iζ ωn ω

(1.6)

The magnitude of G(ω) becomes: |G(ω)| = /(

ωn2 − ω2

1 )2

(1.7) + (2ζ ωn ω)2

All the information required to determine the response x(t) to an arbitrary known excitation f (t) is contained in the knowledge of the complex frequency–response function G(ω) for all frequencies. This statement is based on the principle of superposition that is applicable to linear systems. Here, Fourier’s method is used to perform the superposition in the frequency domain. When f (t) has a period, a Fourier series can be formed by splitting f (t) up into sinusoids. Each sinusoid’s response is provided separately by G(ω), and these responses combine to create a new Fourier series that represents the response x(t). When f (t) has a Fourier transform but is not periodic, an analogous superposition is valid. The total response of the system can be obtained by superposition since the actual excitation f (t) is provided by the superposition of components with various frequencies: 1 x(t) = G(ω) f (t) = G(ω) 2π =

1 2π

{∞ −∞

G(ω) ei ω t dω

{∞ −∞ {∞

F(ω) ei ω t dω

f (τ ) e−i ω τ dτ

(1.8)

−∞

In theory, using Eq. (1.8) to evaluate the explicit response x(t) for an arbitrary excitation f (t) is possible given the knowledge of G(ω). For the purpose of considering stochastic oscillations, a general input–output relation like Eq. (1.8) is highly preferred. By superposing unit solutions in the time domain, a general solution for the response of a linear time-invariant system can also be found. In this book we choose to use the unit impulse, so we assume that the forcing function f (t) is made up of a series of impulses of varying magnitude, as shown in Fig. 1.1a. The unit impulse

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

5

f(t) f(τ)

t (a)

τ

f(t)

τ+dτ

to ∞

δ(t-τ) Unit area

t (b)

τ

x(t)

g(t-τ)

t (c)

τ

Fig. 1.1 When the excitation is unit impulse (b), the response is the impulse response function (c), and the forcing function f (t) is expressed in the form of a series of impulses (a)

δ(t − τ ) is the Dirac delta function, which, as shown in Fig. 1.1b, is zero everywhere except at t = τ , where it encloses a unit area. In Fig. 1.1b the ordinate is infinite at time t = τ . In Fig. 1.1c, g(t − τ ) is called the impulse-response function, denoting the system response x(t) to the unit-impulse excitation δ(t − τ ). As shown in Fig. 1.1c, it is assumed that x(t) is zero before t = τ , allowing the differential equations of motion to be solved with δ(t − τ ) as the excitation and with zero initial conditions for t < τ to yield g(t − τ ).

6

1 Introduction

Once obtained, the impulse-response function g(t − τ ) contains all the details required to determine the response x(t) to an arbitrary known excitation f (t). This comment is based on the principle of superposition once more. This time the superposition is carried out in the time domain. This is elucidated in in Fig. 1.1a, which depicts the division of an arbitrary excitation f (t) into differential elements. A typical area element at t = τ possesses a height f (τ ) and a width dτ . An impulse with a magnitude of f (τ )dτ applied at time t = τ can be utilized to approximate this typical area. At time t = t, the system response to this impulse would be [ f (τ )dτ ]g(t − τ ). Then, because of the linearity, the response x(t) due to all such elements in the past of f (t) is given by the following superposition or convolution integral (also called the Duhamel integral in the existing literature): {t x(t) =

f (τ )g(t − τ ) dτ

(1.9)

−∞

Since g(t − τ ) = 0 when t < τ (i.e., the response prior to the application of the impulse is zero), the upper limit of integration in Eq. (1.9) can be changed to ∞ so that {∞ x(t) =

f (τ )g(t − τ ) dτ

(1.10)

−∞

Furthermore, by changing the integration variable from τ to θ = t − τ , Eq. (1.10) can be rewritten as {∞ x(t) =

f (t − θ )g(θ ) dθ

(1.11)

−∞

Once the impulse-response function of the system g(t) is known, Eqs. (1.9), (1.10), or (1.11) can be used to find the response of the system x(t) for any arbitrary excitation f (t). Once the system’s complex frequency–response function, G(ω ), is known, the Fourier integral, Eq. (1.8), can be used to determine the response of the system. The afore-mentioned two approaches are closely related to one another despite appearing to be different. Let us consider the system’s excitation to be a unit impulse δ(τ ) in Eq. (1.8) in order to see how they are related. By definition, the response is g(t), and Eq. (1.8) gives 1 x(t) = g(t) = 2π

{∞ G(ω) e −∞

i ωt

{∞ dω −∞

δ(τ ) e−i ω τ dτ

(1.12)

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

7

It should be noted that we have the following result: {∞

δ(τ ) e−i ω τ dτ = 1

(1.13)

−∞

because δ(τ ) is zero everywhere except at τ = 0, where it has a unit area and e−i ωτ is also unity at τ = 0. Combining Eqs. (1.12) and (1.13) we obtain: 1 g(t) = 2π

{∞

G(ω) e i ω t dω

(1.14)

−∞

It should be noticed that the mathematical Eq. (1.14) is just the Fourier integral representation of g(t) in which G(ω ) is the Fourier transform of g(t): {∞ G(ω) =

g(t) e−i ω t dt

(1.15)

−∞

For arbitrary known excitations f (t), the relationships between excitation and response were derived in the previous paragraphs. When the excitation is a stationary stochastic process, we consider similar relationships in the following paragraphs. The response in this situation will also be a stationary stochastic process. Both the impulse-response (time domain) and frequency–response (frequency-domain) approaches are used to analyze the relationship between the excitation and the response. The relationship between the autocorrelation functions of the excitation and the response can be determined using the following procedures. To start with, for a particular pair of input and output functions we use Eq. (1.11) twice to write {∞ x(t)x(t + τ ) = −∞ {∞

⎡ f (t − θ1 )g(θ1 ) dθ1 × ⎣

{∞

⎤ f (t + τ − θ2 )g(θ2 ) dθ2 ⎦

−∞

{∞

=

f (t − θ1 ) f (t + τ − θ2 )g(θ1 ) g(θ2 )dθ1 dθ2

(1.16)

−∞ −∞

Notice that θ is substituted with θ1 and θ2 instead in order to avoid confusion in the above equation. The autocorrelation function of x(t) is given by

8

1 Introduction

{∞ {∞ E[ f (t − θ1 ) f (t + τ − θ2 )] g(θ1 )g(θ2 )dθ1 dθ2

R x (τ ) =E[x(t)x(t + τ )] = −∞ −∞

{∞ {∞ =

R f (τ + θ1 − θ2 ) g(θ1 ) g(θ2 )dθ1 dθ2

(1.17)

−∞ −∞

Equation (1.17) is the desired relationship between the input autocorrelation function R f (τ ) and the output autocorrelation function R x (τ ). Starting from the relation (1.17) between autocorrelation functions and using the Wiener-Khintchine theorem to transform to spectral densities, we can also obtain an input–output relation for spectral densities. Let us denote the spectral density function of the output process by Sx (ω). Then, according to the Wiener-Khintchine theorem, Sx (ω) is given by 1 Sx (ω) = 2π

{∞

Rx (τ ) e−i ω τ dτ

(1.18)

−∞

Substitution of Eq. (1.17) into Eq. (1.18) results in: 1 Sx (ω) = 2π

{∞ e

−i ω τ

−∞

{∞ {∞ R f (τ + θ1 − θ2 ) g(θ1 ) g(θ2 )dθ1 dθ2

dτ

(1.19)

−∞ −∞

Introducing ei ω θ1 e−i ω θ2 e−i ω(θ1 −θ2 ) = 1

(1.20)

into Eq. (1.19) leads to {∞ Sx (ω) =

g(θ1 )e

−∞

⎡

1 ×⎣ 2π

i ω θ1

{∞ dθ1

g(θ2 )e−i ω θ2 dθ2

−∞

{∞

⎤

R f (τ + θ1 − θ2 ) e−i ω (τ +θ1 −θ2 ) dτ ⎦

(1.21)

−∞

θ1 and θ2 are constants in the third integral on the right side of Eq. (1.21), and introducing a new integration variable μ as μ = τ + θ1 − θ2 results in

(1.22)

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

1 2π

{∞ R f (τ + θ1 − θ2 ) e −∞

−i ω (τ +θ1 −θ2 )

1 dτ = 2π

{∞

9

R f (μ) e−i ω μ dμ = S f (ω)

−∞

(1.23) It can be noticed that the second integral on the right-hand side of Eq. (1.21) is the Fourier transform G(ω ) of the impulse response g(θ2 ) as given in Eq. (1.15). Similarly, the first integral on the right-hand side of Eq. (1.21) can be recognized as the complex frequency–response function G(−ω ). Equation (1.21) can thus be expressed as: Sx (ω) = G(−ω)G(ω)S f (ω)

(1.24)

Because G(−ω ) is the complex conjugate of G(ω ) and noting that the product of G(ω ) and its complex conjugate may be written as the square of the magnitude of G(ω ), a slightly more compact form for Eq. (1.24) can be achieved as follows: Sx (ω) = |G(ω)|2 S f (ω)

(1.25)

The relationship between the power spectral densities of the excitation and the response is given by this Eq. (1.25). It can be noted that after the magnitude of |G(ω)| is calculated by utilizing Eq. (1.7), the response spectrum Sx (ω) of the offshore platform can then be obtained by utilizing Eq. (1.25) if the excitation spectrum S f (ω) is known beforehand. From the response spectrum Sx (ω), the time history of the platform motion responses, response statistics, and motion extrema can subsequently be computed. The aforementioned spectral analysis procedure is based on the ideal assumption that the offshore system is linear and time invariant. However, in the real world, the rigid body equations of motion of all the deepwater platforms are essentially nonlinear. The nonlinearities of these deepwater systems come from two sources: on the one hand from the nonlinear material properties of the structure, the nonlinear interactions between the structure and the sea floor rock and soil, the nonlinear behaviors of catenary moorings and tension legs, etc. These nonlinearities can be collectively called as structural nonlinearities (or system nonlinearities). On the other hand, the nonlinearities of these deepwater systems also come from the nonlinear external force excitations. For example, wind force is a nonlinear function of wind speed, and the well-known Morison drag force is a nonlinear function of water particle velocity. These nonlinearities can be collectively referred to as excitation nonlinearities (or external force nonlinearities). In practice, it is of vital importance to be able to accurately solve these nonlinear stochastic differential equations of motion in order to successfully design these deep-water offshore platforms. However, it is quite difficult to find accurate solutions for these nonlinear stochastic differential equations. Even for the deterministic nonlinear differential equations, except for some weakly nonlinear problems that have been analytically solved, most of the nonlinear deterministic problems can only be solved by some approximate or

10

1 Introduction

numerical methods [3]. The solutions obtained by these approximate or numerical methods often do not provide a satisfactory and exact interpretation of the problem. For example, when solving by numerical methods, given two initial conditions with very small differences, two quite different results can happen in a very short period of time. Multiple solutions, bifurcations, jumps, and chaotic motions can also occur when solving by these approximate or numerical methods. The difficulty is even greater when the problem is stochastic. Currently, for nonlinear stochastic dynamic systems, analytical solutions are available only if the stochastic excitation term is a Gaussian white noise, and if the system parameters satisfy certain demanding relationships with the value of the excitation spectrum [4]. These strict conditions rarely exist in the real world. Thus, only approximate methods are available now to solve the vast majority of nonlinear stochastic dynamic problems. Presently, the commonly used approximate approaches include the statistical linearization method, the perturbation method, the Monte Carlo simulation method, the numerical path integration method, etc. The following is an overview of the existing research on these methods, a brief description of the basic principles of these methods, and some advantages and disadvantages of each of these methods.

1.1.2 Statistical Linearization Method The control equation for a single degree of freedom nonlinear system with system nonlinearity and external force nonlinearity can be written as [5]: M x¨ + C x˙ + K x+h (x, x) ˙ = f L ( v ) + f N ( v, x˙ )

(1.26)

where M, C and K represent the mass coefficient, damping coefficient and stiffness coefficient of the system, which are generally assumed to be constants. x, x˙ and x¨ are the structural response displacement, response velocity and response accelera˙ represents a tion, respectively. f L (v) represents a linear external force term. h(x, x) ˙ represents a nonlinear external force term. system nonlinearity term, and f N (v, x) If the structure under consideration is an offshore platform subjected to Morison ˙ can be regarded as a nonlinear Morison drag force, where v forces, then f N (v, x) represents the wave water particle velocity. Because the wave water particle velocity ˙ is a stochastic excitation term, and Eq. (1.26) is a stochastic process, then f N (v, x) is a nonlinear stochastic differential equation. It is very difficult to obtain the analytical solution of Eq. (1.26) directly. In fact, it is impossible to obtain the analytical solution of this equation in the vast majority of practical situations. To seek approx˙ imate solutions, the so-called statistical linearization method is to linearize h(x, x) and f N (v, x) ˙ by minimizing the mean square value of the error. The error due to linearizing the stochastic processes is assumed to be ε 1 = h (x, x) ˙ − β 1 x − β 2 x˙

(1.27)

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

11

and ε 2 = f N (v, x) ˙ − α 1 (v − x) ˙

(1.28)

By minimizing the mean square value of the error, the coefficients β 1 ,β 2 and α 1 in the above two equations can be determined. { } { } { } ∂ E ε12 ∂ E ε12 ∂ E ε22 = 0, = 0, =0 ∂β1 ∂β2 ∂α1

(1.29)

In the above equation E{·} represents the mathematical expectation. Note that the above process is an iterative one, since the coefficients β 1 , β 2 and α 1 depend on the response displacement and the response velocity. The linearized stochastic differential equation M x¨ + C x˙ + K x + β 1 x + β 2 x˙ = f L (v) + α 1 (v − x) ˙

(1.30)

can then be easily solved. In order to elucidate the detailed procedures for carrying out the statistical linearization method, let us consider the following dynamic system only possessing ˙ but without a nonlinear excitation term: a system nonlinearity term h(x, x) M x¨ + C x˙ + K x+ h (x, x) ˙ = f (t)

(1.31)

Dividing each term of the above equation by M leads to the following equation of motion: x¨ + α x˙ + ωn2 x+ h 0 (x, x) ˙ = f 0 (t)

(1.32)

where α = C/M √ = linear damping. ωn = K /M = natural frequency of the undamped oscillation. h 0 (x, x) ˙ = per unit mass (or moment of inertia) nonlinear force (or moment). f 0 (t) = f (t)/M = external force (or moment) per unit mass (or moment of inertia). The equation of motion of a linear system that is equivalent to that of the nonlinear system given in Eq. (1.32) can be written as: x¨ + αe x˙ + ωe2 x = f 0 (t) + ε(t)

(1.33)

In Eq. (1.33) ε(t) is the error associated with the statistical linearization, and it is a stochastic process. By minimizing the mean square value of the error ε(t), the coefficients αe , ωe2 in Eq. (1.33) can be determined. From Eqs. (1.32) and (1.33), the error ε(t) can be written as:

12

1 Introduction

) ( ε(t)=(αe − α)x˙ + ωe2 − ωn2 x − h 0 (x, x) ˙

(1.34)

Then the mathematical expectation of (ε(t))2 is given as {[ } ) { ( ]2 } = E (ε(t))2 =E (αe − α)x˙ + ωe2 − ωn2 x − h 0 (x, x) ˙ { ) ) ( ( 2 E (αe − α)2 x˙ 2 + ωe2 − ωn2 x 2 + (h 0 (x, x)) ˙ 2 + 2(αe − α) ωe2 − ωn2 x x˙ ( } ) −2(αe − α)h 0 (x, x) ˙ x˙ −2 ωe2 − ωn2 h 0 (x, x)x ˙ = ) [ 2] [ ] [ 2] ( 2 2 2 2 ˙ 2 (αe − α) E x˙ + ωe − ωn E x + E (h 0 (x, x)) ) ( ˙ − 2(αe − α)E[h 0 (x, x) ˙ x] ˙ + 2(αe − α) ωe2 − ωn2 E[x x] ( 2 ) 2 − 2 ωe − ωn E[h 0 (x, x)x] ˙ (1.35) In order for the following discussion to be reasonably concise but still retain the essence of the statistical linearization method, we may assume that the excitation term f 0 (t) in Eq. (1.33) is a stationary Gaussian stochastic process with a zero expectation. Because Eq. (1.33) is a linear equation, the response x(t) will also be a stationary Gaussian stochastic process with a zero expectation. A stationary stochastic process can be defined in a number of different ways. We choose the option that is the simplest and best suited to our needs, i.e., if E[x(t)] and E[x(t) x(t + τ )] are both independent of t, then a stochastic process x(t) is said to be (weakly) stationary. This means that E[x(t)] is a constant (in this case E[x(t)] = 0) and the autocorrelation function E[x(t) x(t + τ )] only depends on τ . According to the classical stochastic process theory, a weakly stationary stochastic process x is orthogonal to its derivative x˙ when ˙ = 0. Therefore, evaluated at the same time, and orthogonality is defined as E[x x] the mathematical expectation of (ε(t))2 in Eq. (1.35) can be rewritten as: } )2 [ ] [ ] { [ ] ( E (ε(t))2 =(αe − α)2 E x˙ 2 + ωe2 − ωn2 E x 2 + E (h 0 (x, x)) ˙ 2 ) ( − 2(αe − α)E[h 0 (x, x) ˙ x] ˙ − 2 ωe2 − ωn2 E[h 0 (x, x)x] ˙

(1.36)

} { In order to determine whether the minimization of E (ε(t))2 is possible or not, } { we evaluate the first and second order derivatives of E (ε(t))2 with respect to the parameters α e and ωe2 as follows: } [ ] ∂ { E (ε(t))2 = 2(αe − α)E x˙ 2 − 2E[h 0 (x, x) ˙ x] ˙ ∂αe

(1.37)

} ( ) [ ] { ∂ E (ε(t))2 = 2 ωe2 − ωn2 E x 2 − 2E[h 0 (x, x)x] ˙ ∂ωe2

(1.38)

} [ ] ∂2 { E (ε(t))2 = 2E x˙ 2 > 0 2 ∂αe

(1.39)

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

} [ 2] { ∂ 2 ( )2 E (ε(t)) = 2E x > 0 ∂ ωe2

13

(1.40)

{ } Therefore, it is possible to minimize E (ε(t))2 by setting } [ ] ∂ { E (ε(t))2 = 2(αe − α)E x˙ 2 − 2E[h 0 (x, x) ˙ x] ˙ =0 ∂αe

(1.41)

and } ( ) [ ] { ∂ E (ε(t))2 = 2 ωe2 − ωn2 E x 2 − 2E[h 0 (x, x)x] ˙ =0 2 ∂ωe

(1.42)

The parameters α e and ωe2 can then be derived from Eqs. (1.41) and (1.42) as follows: αe =α + ωe2 =ωn2 +

˙ x] ˙ E[h 0 (x, x) [ ] 2 E x˙ ˙ E[h 0 (x, x)x] [ ] 2 E x

(1.43) (1.44)

When the nonlinearities in the damping and restoring forces (or moments) are separable, we can write: h 0 (x, x) ˙ = h 1 (x) + h 2 (x) ˙

(1.45)

From the definition of autocorrelation function we know that the relationship E[x x] ˙ = 0 means that the stochastic process x and its derivative x˙ are uncorrelated. Because for Gaussian process being uncorrelated means being independent, the Gaussian process x and its derivative x˙ are therefore also independent. Note that ˙ the following relationship exists: by independence of x and x, E[h 1 (x) x] ˙ = E[h 1 (x)]E[x] ˙

(1.46)

Furthermore, because previously we have mentioned that for the present case E[x(t)] = 0, we can thus derive the following relations: E[h 0 (x, x) ˙ x] ˙ = E[(h 1 (x) + h 2 (x)) ˙ x] ˙ = E[h 1 (x)x] ˙ + E[h 2 (x) ˙ x] ˙ ] [ d = E[h 1 (x)]E[x] ˙ x] ˙ ˙ + E[h 2 (x) ˙ x] ˙ = E[h 1 (x)]E (x(t)) + E[h 2 (x) (1.47) dt } { d E[x(t)] + E[h 2 (x) ˙ x] ˙ = E[h 2 (x) ˙ x] ˙ = E[h 1 (x)] dt

14

1 Introduction

E[h 0 (x, x)x] ˙ = E[(h 1 (x) + h 2 (x))x] ˙ = E[h 1 (x)x] + E[h 2 (x)x] ˙ ˙ = E[h 1 (x)x] = E[h 1 (x)x] + E[h 2 (x)]E[x]

(1.48)

Then, from Eqs. (1.43), (1.44), (1.47) and (1.48) we can derive the parameters α e and ωe2 as follows: αe =α +

˙ x] ˙ E[h 2 (x) [ ] 2 E x˙

(1.49)

ωe2 =ωn2 +

E[h 1 (x)x] [ ] E x2

(1.50)

We now provide an example of the application of the statistical linearization method. The system considered by us has nonlinearities only in the damping term, ˙ = β x| ˙ x|. ˙ The differential which is expressed as a Morison-type drag force h 2 (x) equation of motion of the offshore system can then be written as: x¨ + α x˙ + ωn2 x+β x| ˙ x| ˙ = f 0 (t)

(1.51)

The equation of motion of a linear system that is equivalent to that of the nonlinear system given in Eq. (1.51) can be written as: x¨ + αe x˙ + ωe2 x = f 0 (t)+ε(t)

(1.52)

Because in Eq. (1.51) the excitation term f 0 (t) is a Gaussian random process with zero expectation, the response velocity x˙ will also be normally distributed with zero expectation. In the following we write the unknown variance of x˙ as σx˙2 . [ ] E x˙ 2 = σx˙2

(1.53)

Then we have {∞ E[h 2 (x) ˙ x] ˙ =

√ ( 2 )] 2 2 ˙ √ ex p −x˙ / 2σx˙ d x = √ βσx˙ 3 β x˙ |x| π 2π σx˙ 2

−∞

1

[

2

(1.54)

Substituting the above result into Eq. (1.49) leads to: ˙ x] ˙ E[h 2 (x) [ ] =α+ αe =α + 2 E x˙

√ 2√ 2 βσx˙3 π σx˙2

√ 2 2 = α + √ βσx˙ π

(1.55)

If the frequency response function of the equivalent linear system is denoted by G e (ω), then from Eq. (1.7) the magnitude of G e (ω) can be written as:

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

1 |G e (ω)| = /( )2 ωn2 − ω2 + (αe ω)2

15

(1.56)

Consequently, the square of |G e (ω)| can be written as: 1 |G e (ω)|2 = ( )2 2 2 ωn − ω + (αe ω)2

(1.57)

When αe = α we have the following relationship |G(ω)|2 = (

ωn2 − ω2

1 )2

+ (αω)2

(1.58)

We next expand the function |G e (ω)|2 into a Taylor series in the neighborhood of α using the relationship given in Eq. (1.55): √ ∂ 2 2 |G e (ω)|2 = |G(ω)|2 + √ βσx˙ |G(ω)|2 ∂α π

(1.59)

Equation (1.59) can be further written as: √ ∂ 2 2 |G e (ω)|2 = |G(ω)|2 + √ βσx˙ |G(ω)|2 ∂α π ( ) √ √ ( ) 2 2 4 2 2 4 2 2 2 2 = |G(ω)| + √ βσx˙ −2αω |G(ω)| = |G(ω)| 1 − √ αβσx˙ ω |G(ω)| π π (1.60) By applying Eq. (1.60) we can then obtain the following relationship between the output and input spectral density functions of the equivalent linear system (1.52): ( Sx (ω) = S f0 (ω)|G(ω)|2

) √ 4 2 1 − √ αβσx˙ ω2 |G(ω)|2 π

(1.61)

The response standard deviation σx˙ in Eq. (1.61) is unknown and can be determined iteratively as follows: 1. First assign a value to σx˙ and then calculate the output spectral density function Sx (ω) in Eq. (1.61) for a given input spectral density function S f0 (ω). 2. Calculate the second moment of Sx (ω), which is equal to σx˙2 (Because σx˙2 = {∞ {∞ 2 Sx˙ (ω)dω = ω Sx (ω)dω). Compare the σx˙ value thus obtained with the −∞

−∞

assigned value in the first step.

16

1 Introduction

3. Repeat the procedure until the evaluated standard deviation (σx˙ ) value agrees with the assigned value. Among the first to adopt the statistical linearization method to solve stochastic oscillation problems were Booton [6] and Caughey [7, 8]. Iwan and Yang [9] extended the statistical linearization method to solve multi-degree-of-freedom nonlinear stochastic systems in 1972. Subsequent contributors to the development of the statistical linearization method include Spanos [10] and Roberts and Spanos [11]. In the field of Naval Architecture, Kaplan [12] and Vassilopoulos [13] applied the statistical linearization method to solve the rolling of ships on random waves. In offshore engineering, Borgman [14, 15] first applied the statistical linearization method to deal with nonlinear Morison drag forces. Spanos et al. [16] carried out an effective dynamic analysis with the statistical linearization method in the preliminary design decision regarding a Spar–Risers–Mooring lines system. The three stochastic differential equations for the surge, the heave, and the pitch motions of the Spar platform and the two stochastic differential equations for the surge and the heave motions of the risers were originally nonlinear. They linearized the original five differential equations through expressing the energy dissipation due to the friction on the Spar–Risers contact surface by introducing an equivalent linear damping term, through expressing the squared damping term in the original equation by introducing an equivalent linear damping term, and through expressing the nonlinearity of the mooring line forcedeflection curve by introducing an equivalent linear stiffness term. By solving the resulting five coupled linear differential equations of motion and an iterative solution process, they obtained the response statistics of the surge, the pitch and the heave motions of the Spar platform and demonstrated the practicality of their method in the preliminary design stage. However, the statistical linearization method has two main drawbacks: first, the response statistics of linear systems subject to Gaussian random excitations are always Gaussian, while the response statistics of nonlinear systems subject to Gaussian random excitations are always non-Gaussian. Consequently, the motion response statistics obtained by the statistical linearization method cannot reflect the non-Gaussian characteristics of the original system response. What’s worse, the non-Gaussian characteristics of the structural response have a great impact on the structural fatigue and reliability analysis [17]. The external excitations of deepwater platforms can be considered as Gaussian, but the differential equations of motion of deep-water platforms must be nonlinear, so it is not appropriate to conduct structural analysis of these platforms based on the motion responses of these deepwater platforms solved by the statistical linearization method. Second, the statistical linearization method cannot predict the responses in the frequency domain outside the excitation frequency range, and thus some response statistics may be underpredicted. The well-known subharmonic and super-harmonic resonance phenomena of nonlinear systems subjected to external excitations cannot be predicted by the statistical linearization method. For example, Spanos et al. [16] found that the statistical linearization method predicted lower heave responses in the low frequency range by comparing the heave responses of a Spar platform obtained using Monte Carlo

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

17

simulations. Another common method to deal with nonlinear stochastic dynamic problems, the perturbation method, is presented below.

1.1.3 Perturbation Method Another method to deal with the response of nonlinear marine structures under the action of random environmental loads is the perturbation method. For illustration purpose, let us consider the nonlinear system represented by Eq. (1.32) and write the nonlinear term in this equation as εh 0 ( x, x˙ ), where the parameter ε is taken to be a small number. That is: x¨ + α x˙ + ωn2 x+ εh 0 ( x, x˙ ) = f 0 ( t )

(1.62)

The perturbation method assumes that the solution of Eq. (1.62) is given as a power series of ε as follows: x (t ) = x 0 ( t ) + ε x 1 (t ) + ε2 x 2 ( t ) + · · ·

(1.63)

x 0 ( t ) in Eq. (1.63) is the solution of the linear equation. Expanding the term εh 0 ( x, x˙ ) in Eq. (1.62) into a Taylor series leads to: ( ) ∂ h 0 ( x 0 , x˙ 0 ) εh 0 ( x , x˙ ) = εh 0 ( x 0 , x˙ 0 ) + ε ε x 1 + ε2 x 2 + · · · ∂x ( ) ∂ h 0 ( x 0 , x˙ 0 ) + · · · + ε ε x˙ 1 + ε2 x˙ 2 + · · · ∂ x˙ (1.64) Substituting Eqs. (1.63) and (1.64) into Eq. (1.62) and by equating terms containing the same power of ε, we obtain the following set of linear equations for x 0 , x 1 , and x 2 , etc. x¨ 0 + α x˙ 0 + ωn2 x 0 = f 0 ( t )

(1.65)

x¨ 1 + α x˙ 1 + ωn2 x 1 = −h 0 ( x 0 , x˙ 0 )

(1.66)

x¨2 + α x˙2 + ωn2 x2 = −x 1

∂ ∂ h 0 ( x 0 , x˙ 0 ) − x˙ 1 h 0 ( x 0 , x˙ 0 ) · ·· ∂x ∂ x˙

(1.67)

It can be noted from Eqs. (1.65), (1.66) and (1.67) that x i+1 (t) can be obtained as the linear response to an excitation which is a nonlinear function of the previously determined x i (t). Unfortunately, it is extremely complicated to evaluate x i (t) for

18

1 Introduction

i ≥ 2; therefore, in practice, only the first two terms are usually taken into account. Thus, we limit our following discussion to this case. From Eq. (1.65), the solution x 0 ( t ) can be expressed as a convolution integral as follows: {∞ x 0 (t ) =

f 0 (t − θ )g(θ ) dθ

(1.68)

−∞

in which g(θ ) is the linear system’s impulse response function, which is equal to the inverse Fourier transform of the complex frequency response function G(ω ). Similarly, from Eq. (1.66), the solution x 1 ( t ) can also be expressed as a convolution integral as follows: {∞ x 1 (t ) = −

h 0 ( x 0 (t − θ ) , x˙ 0 (t − θ ))g(θ ) dθ

(1.69)

−∞

Consequently, the solution of the nonlinear system represented by Eq. (1.62) can be approximately obtained as follows: {∞ x (t ) = x 0 (t ) + ε x 1 (t ) =

f 0 (t − θ )g(θ ) dθ −∞

{∞ −ε

(1.70)

h 0 ( x 0 (t − θ ) , x˙ 0 (t − θ ))g(θ ) dθ

−∞

We next take the first order of the parameter ε in Eq. (1.63) and calculate the autocorrelation function of the response x (t ) as follows R x x (θ ) = E[x ( t )x ( t + θ )] = E[(x 0 ( t ) + ε x 1 ( t ))(x 0 ( t + θ ) + ε x 1 ( t + θ ))] ≈ E[x 0 ( t )x 0 ( t + θ )] + ε{E[x 0 (t )x 1 ( t + θ )] + E[x 1 (t )x 0 (t + θ )]} { } = R x 0 x 0 (θ ) + ε R x 0 x 1 (θ ) + R x 1 x 0 (θ ) (1.71) In Eq. (1.71), the autocorrelation function of the linear part of the response, R x 0 x 0 (θ ), can be calculated as

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

19

R x 0 x 0 (θ ) = E[x 0 (t )x 0 (t + θ )] {∞ {∞ = E[ f 0 (t − θ 1 ) f 0 (t + θ − θ 2 )] g(θ1 ) g(θ2 )dθ1 dθ2 (1.72)

−∞ −∞ {∞ {∞

=

R 0 0 (θ + θ 1 − θ 2 ) g(θ1 ) g(θ2 )dθ1 dθ2 −∞ −∞

In which R 0 0 (θ + θ 1 − θ 2 ) is the input autocorrelation function. Similarly, the cross-correlation functions R x 0 x 1 (θ ) and R x 1 x 0 (θ ) in Eq. (1.71) can be calculated as R x 0 x 1 (θ ) = E[x 0 ( t )x 1 (t + θ )] {∞ {∞ =− E[ f 0 (t − θ 1 )h 0 (t + θ − θ 2 )] g(θ1 ) g(θ2 )dθ1 dθ2

(1.73)

−∞ −∞

R

(θ ) = E[x 1 ( t )x 0 ( t + θ )] {∞ {∞ =− E[ f 0 (t + θ − θ 1 )h 0 (t − θ 2 )] g(θ1 ) g(θ2 )dθ1 dθ2

x1x0

(1.74)

−∞ −∞

In Eqs. (1.73) and (1.74) h 0 ( ) is the abbreviation of h 0 ( x 0 (t), x˙ 0 (t)). Finally, the response autocorrelation function R x x (θ ) can be calculated utilizing Eqs. (1.71), (1.72), (1.73) and (1.74). The response’s spectral density can then be determined by applying the Fourier transform to Eq. (1.71), that is, { } S x x (ω ) = S x 0 x 0 (ω ) + ε S x 0 x 1 (ω ) + S x 1 x 0 (ω )

(1.75)

In order to further obtain a more concise expression for S x x (ω ) (i.e., the spectral density of the response), we here review some properties of the cross-correlation function and the cross-spectral density function. From the classical stochastic process theory, we have the following relationship between the cross-correlation functions of two stochastic processes x ( t ) and y ( t ): R y x (θ ) = R x y (−θ )

(1.76)

For weakly stationary ergodic stochastic processes x 0 (t ) and x 1 ( t ), the crossspectral density function S x 0 x 1 (ω ) and the cross-correlation functions R x 0 x 1 (θ ) are a Fourier transform pair, i.e.,

20

1 Introduction

1 S x 0 x 1 (ω ) = π

{∞

R x 0 x 1 (θ )e−iωθ dθ

(1.77)

S x 0 x 1 (ω )eiωθ dω

(1.78)

−∞

1 R x 0 x 1 (θ ) = 2

{∞ −∞

Please note that the cross-spectral density function is a complex-valued function in contrast to a real-valued function for the auto-spectral density function. We next evaluate the cross-spectral density function S x 0 x 1 (ω ) in detail in order to show how the imaginary part of the function is introduced into S x 0 x 1 (ω ): {∞ 1 S x 0 x 1 (ω ) = R x 0 x 1 (θ )e−i ωθ dθ π −∞ } { 0 { {∞ 1 −iωθ −i ωθ =π R x 0 x 1 (θ )e dθ + R x 0 x 1 (θ )e dθ −∞

(1.79)

0

Utilizing the relationship in Eq. (1.76), we can obtain the following relationship: {0 R x 0 x 1 (θ )e −∞

−iωθ

{∞ dθ =

R x 1 x 0 (θ )eiωθ dθ

(1.80)

0

Therefore, Eq. (1.79) can be re-written as: ⎫ ⎧ 0 { {∞ ⎬ 1⎨ R x 0 x 1 (θ )e−i ωθ dθ + R x 0 x 1 (θ )e−iωθ dθ S x 0 x 1 (ω ) = ⎭ π⎩ −∞ 0 ⎫ ⎧∞ { {∞ ⎬ 1⎨ i ωθ = R x 1 x 0 (θ )e dθ + R x 0 x 1 (θ )e−iωθ dθ ⎭ π⎩ 0

=

1 π

{∞

0

{

} R x 1 x 0 (θ ) + R x 0 x 1 (θ ) cos ω θ dθ

0

{∞ { } 1 +i R x 1 x 0 (θ ) − R x 0 x 1 (θ ) sin ω θ dθ π 0 [ ] [ ] =Re S x 0 x 1 (ω ) + iImg S x 0 x 1 (ω ) where

(1.81)

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

[

]

1 Re S x 0 x 1 (ω ) = π [

{∞

{

} R x 1 x 0 (θ ) + R x 0 x 1 (θ ) cos ω θ dθ

21

(1.82)

0

]

1 Img S x 0 x 1 (ω ) = π

{∞

{

} R x 1 x 0 (θ ) − R x 0 x 1 (θ ) sin ω θ dθ

(1.83)

0

[ ] The real part Re S x 0 x 1 (ω ) given in Eq. (1.81) is called the co-spectrum, while [ ] the imaginary part Img S x 0 x 1 (ω ) is called the quadrature spectrum. Similarly, S x 1 x 0 (ω ) in Eq. (1.75) can be evaluated as: ⎫ ⎧ 0 { {∞ ⎬ 1⎨ R x 1 x 0 (θ )e−iωθ dθ + R x 1 x 0 (θ )e−i ωθ dθ S x 1 x 0 (ω ) = ⎭ π⎩ −∞ 0 ⎫ ⎧∞ { {∞ ⎬ 1⎨ = R x 0 x 1 (θ )ei ωθ dθ + R x 1 x 0 (θ )e−iωθ dθ ⎭ π⎩ 0

=

1 π

{∞

0

{

} R x 1 x 0 (θ ) + R x 0 x 1 (θ ) cos ω θ dθ

0

{∞ { } 1 +i R x 0 x 1 (θ ) − R x 1 x 0 (θ ) sin ω θ dθ π 0 [ ] [ ] =Re S x 1 x 0 (ω ) + iImg S x 1 x 0 (ω )

(1.84)

By carefully studying Eqs. (1.81) and (1.84), we can find: [ ] [ ] Re S x 0 x 1 (ω ) = Re S x 1 x 0 (ω )

(1.85)

[ ] [ ] Img S x 0 x 1 (ω ) = − Img S x 1 x 0 (ω )

(1.86)

Combining Eqs. (1.75), (1.85) and (1.86) altogether we obtain: [ ] S x x (ω ) = S x 0 x 0 (ω ) + 2 ε Re S x 0 x 1 (ω )

(1.87)

As an application of the perturbation technique developed above, let us take the nonlinear term h 0 ( x, x˙ ) in Eq. (1.62) to be h 0 ( x, x˙ ) = x 3 . Then Eq. (1.62) can be re-written as x¨ + α x˙ + ωn2 x+ εx 3 = f 0 ( t )

(1.88)

22

1 Introduction

In the field of Naval Architecture and Ocean Engineering, the rolling motion of a ship in random seas can sometimes be approximately described by the above equation. In the following, we assume that f 0 ( t ) in Eq. (1.88) is a steady state Gaussian stochastic process with zero mean. We also define the spectral density function of f 0 ( t ) to be S 0 0 (ω ). Because the complex frequency response function G(ω ) and the impulse response function g(θ ) of the linear system (ε = 0) are all known, we can obtain the linear system response autocorrelation function R x 0 x 0 (θ ) as 1 R x 0 x 0 (θ ) = 2

{∞ S x 0 x 0 (ω )e

iωθ

−∞

1 dω = 2

{∞ S 0 0 (ω )|G(ω)|2 eiωθ dω

(1.89)

−∞

where {∞ G(ω) =

g (θ )e−iωθ dθ

(1.90)

−∞

Obviously, the response spectral density of the linear system can be obtained as S x 0 x 0 (ω ) = S 0 0 (ω )|G(ω)|2

(1.91)

In the present case the cross-correlation function R x 0 x 1 (θ ) can be calculated as R x 0 x 1 (θ ) = E[x 0 (t )x 1 ( t + θ )] {∞ {∞ =− E[ f 0 (t − θ 1 )h 0 (t + θ − θ 2 )] g(θ1 ) g(θ2 )dθ1 dθ2 −∞ −∞ {∞ {∞

=−

(1.92) [ ] E f 0 (t − θ 1 )x03 (t + θ − θ 2 ) g(θ1 ) g(θ2 )dθ1 dθ2

−∞ −∞

Because we have assumed that f 0 ( t ) in Eq. (1.88) is a steady state Gaussian stochastic process with zero mean, then the linear system response x0 (t) will also be a Gaussian stochastic process with zero mean. According to the classical stochastic process theory, for Gaussian stochastic process with zero mean we have the following relationship:

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

23

[ ] E f 0 (t − θ 1 )x03 (t + θ − θ 2 ) = E[ f 0 (t − θ 1 )x0 (t + θ − θ 2 )]E[x0 (t + θ − θ 2 )x0 (t + θ − θ 2 )] + E[x0 (t + θ − θ 2 )x0 (t + θ − θ 2 )]E[ f 0 (t − θ 1 )x0 (t + θ − θ 2 )]

(1.93)

+ E[ f 0 (t − θ 1 )x0 (t + θ − θ 2 )]E[x0 (t + θ − θ 2 )x0 (t + θ − θ 2 )] = 3E[ f 0 (t − θ 1 )x0 (t + θ − θ 2 )]E[x0 (t + θ − θ 2 )x0 (t + θ − θ 2 )] In addition, from Eq. (1.68), we can derive E[ f 0 (t − θ 1 )x0 (t + θ − θ 2 )] {∞ = E[ f 0 (t − θ 1 ) f 0 (t + θ − θ 2 − θ 3 )] g(θ 3 )dθ 3 (1.94)

−∞ {∞

=

R 0 0 (θ + θ 1 − θ 2 − θ 3 ) g(θ 3 )dθ 3 −∞

and E[x0 (t + θ − θ 2 )x0 (t + θ − θ 2 )] = σx20

(1.95)

where σx20 is the variance of the response of the linear system. Combing Eqs. (1.93), (1.94) and (1.95) altogether leads to the following expression [ ] E f 0 (t − θ 1 )x03 (t + θ − θ 2 ) {∞ R 0 0 (θ + θ 1 − θ 2 − θ 3 ) g(θ 3 )dθ 3 = 3σx20 −∞ {∞

=

3 2 σ 2 x0

(1.96) {∞

S0 0 (ω)ei ω (θ +θ 1 −θ 2 −θ 3 ) g(θ 3 )dθ 3 dω

−∞ −∞

The cross-correlation function R x 0 x 1 (θ ) in Eq. (1.92) can then be calculated as {∞ { R x 0 x 1 (θ ) = − = − 23 σx20 = − 23 σx20

∞

−∞ −∞ {∞ {∞ {∞ {∞

[ ] E f 0 (t − θ 1 )x03 (t + θ − θ 2 ) g(θ1 ) g(θ2 )dθ1 dθ2

−∞ −∞ −∞ −∞ {∞ −∞

S0 0 (ω)ei ω (θ +θ 1 −θ 2 −θ 3 ) g(θ1 ) g(θ2 )g(θ 3 ) dθ1 dθ2 dθ 3 dω

S0 0 (ω)|G (ω)|2 G (ω)ei ω θ dω (1.97)

24

1 Introduction

The cross-spectral density function S x 0 x 1 (ω ) in Eq. (1.87) can then be obtained as S x 0 x 1 (ω ) = −3σx20 S0 0 (ω)|G (ω)|2 G (ω)

(1.98)

Combining Eqs. (1.87), (1.91) and (1.98) altogether we can obtain: S x x (ω ) = S 0 0 (ω )|G(ω)|2 − 6 ε σx20 S0 0 (ω)|G (ω)|2 Re[G (ω)] { } = S 0 0 (ω )|G(ω)|2 1 − 6 ε σx20 Re[G (ω)]

(1.99)

The relationship between the power spectral densities of the excitation and the response of the nonlinear system (1.88) is given by this Eq. (1.99). From the response spectrum S x x (ω ), the time history of the nonlinear system responses, response statistics, and motion extrema can subsequently be computed. Equations (1.62) through (1.99) are the mathematical procedures of the so-called perturbation method in the field of nonlinear stochastic dynamics. In the field of ocean engineering, the perturbation method had once been applied to study the stochastic responses of a deep-water jacket platform under the action of Morison-type nonlinear wave forces. However, it is well-known that the solutions obtained by the perturbation method are reliable only when the nonlinearity is very weak (i. e., when the parameter ε in Eq. (1.62) is very small). Another common method to deal with nonlinear stochastic dynamic problems, the Monte Carlo simulation method, is presented in the next sub-section.

1.1.4 Monte Carlo Simulation (MCS) Method Another common method to deal with the response of nonlinear ocean engineering systems under the action of stochastic environmental loads is the Monte Carlo Simulation (MCS) method. The main advantage of Monte Carlo Simulation is that if any deterministic problem has a solution (either analytical or numerical), then the exact solution of its corresponding stochastic problem can also be obtained. Another major advantage of the Monte Carlo Simulation method is that it can easily handle different forms of nonlinear terms. Shinozuka [18], Shinozuka and Jan [19], Shinozuka and Deodatis [20, 21] provided the basic theoretical background of the Monte Carlo Simulation method and the corresponding simulation formulations for one- and multidimensional Gaussian stochastic processes. An important part of the Monte Carlo Simulation method is the generation of sample functions for the stochastic processes in the problem. The generated sample functions must precisely express the probabilistic characteristics of their corresponding stochastic processes, which in the field of ocean engineering means that the wave water particle velocity and acceleration stochastic processes’ sample functions should be generated based on various existing wave spectral density functions [22, 23]. In this regard, Prof. Masanobu Shinozuka [24] has pioneered the theory of simulating stochastic waves by superimposing a

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

25

finite number of trigonometric functions with different amplitudes, frequencies and random phase angles. For example, the sample function of the water particle velocity of a wave can be generated by the following equation. N −1 √ ∑ { }1/2 Sv0 v0 [n (Δω)]Δω v(t) = 2 cos[n(Δω)t + Φn ]

(1.100)

n=0

where Δω is the frequency spacing, Sv0 v0 [n (Δω)] is the one-sided power spectral density function of wave water particle velocity, which can be derived from the one-sided power spectral density function of wave water particle displacement [25]. Φ0 , Φ1 , Φ2 , …, Φ N −1 are independent random phase angles uniformly distributed between 0 and 2π . After Eq. (1.100) is transformed by a transfer function, it is then substituted into v in the f N ( v, x˙ ) term in Eq. (1.26). Next, the fourth-order Runge–Kutta method can be used to numerically integrate the differential Eq. (1.26) to obtain a time history of the displacement response of the marine structure, and a time history of the first-order derivative of the displacement response of the marine structure can be obtained in the same way. The probability density curve of the displacement responses of the marine structure can be obtained by taking the average values of a sufficiently large number of time histories of the structure displacement responses. The probability density curve of the first order derivative of the displacement responses of the marine structure can be obtained in the same way. Based on the information from the probability density curves of these individual random variables, the joint probability density of the response values and the first-order derivatives of the response values can be derived. Next, the level up-crossing rates of the responses of the structural system can be obtained by utilizing the Rice’s formula [26, 27]. The above is the basic principle of the Monte Carlo Simulation method and the simulation procedures. To study the response of a simplified Tension Leg Platform model under wave forces, Spanos and Agarwal [28] employed a Monte Carlo Simulation method. In their study, they developed a single-degree-of-freedom differential equation of surge motion for this ocean platform. The elastic forces in their equation of motion are assumed to be linear and the excitation forces in their equation of motion include linear Morison inertia forces and nonlinear Morison drag forces when the presence of currents is not considered. Kareem et al. [5] have also used the Monte Carlo Simulation method in their study of the response of a Tension Leg Platform. They made an improvement to the equation of motion by adding a current term to the Morison drag force formula, however, the elastic forces in their equation of motion were still assumed to be linear. Tognarelli et al. [29] modeled an offshore system with a single degree of freedom equation of motion incorporating system nonlinearity and external force nonlinearity, with the system nonlinearity represented by the Duffing stiffness and the external force nonlinearity represented by the nonlinear Morison drag force. They calculated the response statistics of this offshore system using the Monte Carlo Simulation method. However, they did not consider the effect of currents when calculating the Morison drag force.

26

1 Introduction

The stochastic process generated by Monte Carlo Simulation can only be called “pseudo-random” and does not exactly have the stochastic characteristics in the rigorous sense of mathematics. Another disadvantage of Monte Carlo Simulation is that it is too time-consuming to compute. However, because Monte Carlo Simulation can easily handle nonlinear stochastic dynamic problems, it is still often used to verify the validity of other approximate solutions. Information on the whole response probability density curve is sometimes required for the design of the various marine structures described earlier in this chapter, for example, to improve the success of helicopter landings on the helicopter landing pads, information on the whole response probability density curve for these marine structures is required. This is where the use of Monte Carlo Simulation for the dynamic analysis of these marine structures shows its superiority. In some other cases, however, more attention is paid to the extreme responses [5] of these marine structures, i.e., the information in the tail region of the response probability density curve of these marine structures (as shown in the example in Fig. 1.2). The extreme response values of marine structures are generally those response values where the mean level up-crossing rate is below 10–4 -10–5 for a short period of time (3–6 h), which roughly corresponds to an annual exceedance probability below 10–2 (which we will explain in detail in Chap. 3). For semi-submersible platforms and floating production storage and offloading systems with catenary moorings, when the extreme response of these structures is too large, the forces within the mooring lines will exceed the critical values, which may lead to the failure of the mooring system. For example, a strong storm in the North Sea in December 1990 caused the mooring lines of some semi-submersible platforms to fail. In addition, when the extreme response of these marine structures is too large it can lead to unexpected decoupling of the riser. The riser is a key component of these marine structures, it may be used to transport oil directly from the subsea pipeline to its platform, or it may be used as a jacket for the drill string, or some risers may be used as a jacket for the backbone electrical cables, so if an unanticipated decoupling of the riser occurs, the economic loss will be significant. Also, when the extreme response of a floating production storage and offloading system (FPSO) is too large it can also affect the offloading operations. It can be seen that it is of great practical importance to predict accurately the extreme responses of these marine structures, but one of the main drawbacks of the Monte Carlo Simulation method is the low accuracy of the probability density values of the extreme responses it predicts. Another efficient and accurate method for statistical analysis and extreme value prediction of the stochastic response of nonlinear systems, the numerical path integration method, is presented below.

1.1.5 Numerical Path Integration Method Einstein first studied theoretically the general problem of physical systems subject to stochastic excitations in 1905, namely the statistical theory of Brownian motion [30].

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

27

Response probability density

0.8

0.6

0.4

extreme response

extreme response

0.2

0.0

-2

-1

0

1

2

Response displacement (m)

Fig. 1.2 An example of a response probability density curve and the extreme responses

In 1931, Andrey Nikolayevich Kolmogorov derived an exact mathematical equation controlling the probability density of Markov stochastic processes, namely the Chapman–Kolmogorov integral equation [4]. This equation is given by the following formula. { p (x3 , t3 |x1 , t1 ) = p (x3 , t3 |x2 , t2 ) p (x2 , t2 |x1 , t1 ) d x2 , t1 ≤ t2 ≤ t3 (1.101) where p (x3 ,t3 |x2 ,t2 ) and p (x2 ,t2 |x1 ,t 1 ) are called transition probability densities and x 1 , x 2 , x 3 are the response values of the system at different moments t 1 , t 2 , t 3 . By repeated use of the Chapman–Kolmogorov integral equation, we can express the evolution of the probability density p(x, t) from the initial probability density p (x0 , t0 ) in terms of the transition probability density. Dividing the time difference t-t 0 into N equal parts, we have {

{ p(x, t) = {

d x N −1

d x N −2 ...

d x0 p (x, t|x N −1 , t N −1 ) p (x N −1 , t N −1 |x N −2 , t N −2 )... p (x1 , t1 |x0 , t0 ) p (x0 , t0 )

(1.102) Equation (1.102) is the expression of the path integration method. If the initial probability density and each transition probability densities are known, the probability density at any time can be obtained from Eq. (1.102). Various interpolation methods are required for the actual calculations, and thus this method is also known as the numerical path integration method.

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The basic idea of path integration is to discretize in space and time respectively, and replace the integral with a path sum, i.e., first pick an initial probability density function arbitrarily, and then use this initial probability density function to multiply each short time transition probability density function in the integration space step by step to form the global transition probability density, and obtain the joint probability density of the state vector. Each short time transition probability density function being multiplied is set to be Gaussian, and it is derived by solving the set of moment equations derived from the original set of equations of motion, so that the short time transition probability density function being multiplied has the essential characteristics of the original equations of motion. Thus, in the process of successive multiplication step by step, the effect of each short-time transition probability density function on the final exact steady solution becomes more and more significant, while the effect of the arbitrarily chosen initial probability density function on the final exact steady solution becomes weaker and weaker. As time advances to a certain point in the distant future, the intermediate process solution of the problem will converge to a final exact steady solution, at which point the effect of the initial probability density function disappears. It can be seen that even if two different initial probability density functions are chosen, the intermediate process solution of the problem will always converge to a unique final exact steady solution after successive multiplications of path integrals, the only difference being the number of computational steps and the computational time required to reach this final unique exact steady solution will be different. The path integration method is based on Markov diffusion process theory and numerical interpolation procedures, and it is an efficient and accurate method for statistical analysis and extreme value prediction of nonlinear systems under stochastic excitations. Zhu [30] developed the theory of obtaining the response probability density of a system by solving the forward Kolmogorov equation by the path integration method starting from the averaging equation. Huang et al. [32] developed a stochastic averaging procedure to tackle strongly nonlinear oscillations with external and parametric harmonic and white noise excitations by using the generalized harmonic functions. They used the path integration method to solve the averaged Fokker-Plank (FPK) equation. Based on the stationary joint probability densities of amplitude and phase obtained by applying the stochastic averaging and path integration methods, they examined for the first time the stochastic jumps of a strongly nonlinear oscillatory system under harmonic and white noise excitations and the bifurcation of the system due to the variation of the system parameters. The system nonlinearities of the ideal structural systems they studied are similar to those of the offshore structural systems, but the external random excitation terms of the ideal structural systems they studied are independent, whereas the external random excitation terms of some offshore structures (e.g., the random terms in the Morison drag force) are not independent. Therefore, these external random excitation terms should be treated specially before applying the path integration method to analyze these types of offshore structures. Wehner and Wolfer [33] were the first to use the path integration method to solve the stochastic dynamic problems. In applying the path integration method they used the piece-wise constant interpolation procedure. The peaks of the numerical solutions

1.1 Introduction to Analysis Methods for Stochastic Dynamic Response …

29

of the probability density they obtained were on the low side, and the values at both tails they obtained were on the high side. Wehner and Wolfer [33] suggested later researchers should use a better interpolation procedure to improve the accuracy of the numerical solutions in the two tail regions, a suggestion later confirmed by Naess and Johnson [34]. For a one-dimensional problem, Naess and Johnson [34] used a cubic B spline interpolation procedure to improve the accuracy of the solution for the values at both tails of the probability density curve to 10–6 , a result that is significant because the values at both tails of the probability density curve are critical for system reliability analysis. Naess Arvid [35] has been working since 1993 in the area of predicting the response statistics of offshore structures by the path integration methods. The equations of motion of the offshore structures he has studied can be modeled by the Itô stochastic differential equation, the state vector process of which satisfies the FokkerPlank-Kolmogorov (FPK) equation. Path integration techniques together with an appropriate numerical procedure constitute a powerful tool for solving FPK equations with natural boundary conditions. Naess Arvid used the cubic B spline interpolation method in applying the path integration method. Karlsen and Naess [36] used the path integration method to study the response statistics of a nonlinear large volume structure moored in random waves. They investigated the performance of the path integration method in calculating the surge displacement responses of the structure in long-crested head seas. In particular, they focused on the modeling of nonlinear wave loads and nonlinear mooring properties. The restoring force characteristics of the mooring line system are weakly nonlinear, and this nonlinearity is approximated by employing a quadratic term. In order to investigate the effect of this nonlinear term on the response statistics, the case with a linear restoring force term is also calculated. It is shown that when the path integration method is used, the results will be in good agreement with the results obtained by Monte Carlo Simulation as long as the mesh is set to be fine enough. A significant advantage of using the path integration method is that the joint probability density of response displacement and response velocity can be obtained concurrently in the calculation results. This makes it easy to estimate, for example, the exceedance probability for a specific response level in a given time period, and the computational effort is largely independent of the time period corresponding to the required exceedance probability. The path integration method is based on the fact that the response of a dynamical system excited by Gaussian white noise can be modeled as a continuous multidimensional Markov process. This means that the stochastic excitation term of the equations of motion dealt with by the path integration method must be a Gaussian white noise (or at least sufficiently broad-banded). However, the real-world wave spectra are always narrow-banded, i.e., the first-order wave stochastic excitation is narrow-banded, and thus when applying the path integration method to solve the response of marine structures subjected to first-order wave stochastic excitations, a suitable nonlinear filter should be designed first in order to turn the Gaussian white noise into a specific narrow-band spectrum. This will undoubtedly increase the complexity of the problem-solving process.

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1.1.6 Other Methods Donley and Spanos [37, 38] studied the stochastic response of a Tension-Leg Platform using statistical quadratization. Quek et al. [39] studied the stochastic response of a deep-water jack-up platform under nonlinear Morison drag forces using statistical quadratization. Tognarelli et al. [29] studied the stochastic response of an offshore system containing system nonlinearity and external force nonlinearity using the statistical cubicization method. Li et al. [17] investigated the stochastic response of a deep-water jack-up platform under a Morison-type nonlinear wave force using the statistical cubicization method. The above methods can be collectively referred to as the Volterra series method, which belongs to the same system of methods as the statistical linearization method. The Volterra series method was developed to overcome the drawback of the statistical linearization method, i.e., the recognition of the fact that a single-frequency excitation can generate a multi-frequency response. In the Volterra series method, an impulse response function with multiple excitation times and a corresponding frequency response function with multiple excitation frequencies are introduced, and the system response is represented by a Volterra functional series, which is then truncated to obtain an approximate response spectral density. It has been shown that for weakly nonlinear systems, the response results obtained by retaining only two to three terms of the truncated Volterra functional series will be very accurate. However, for moderately nonlinear or strongly nonlinear systems, retaining only two to three terms is not sufficient [40]. However, if more terms are retained, the analysis will be extremely complicated, and for multi-degree-of-freedom systems the analysis will also be extremely complicated.

1.2 Introduction to Analysis Methods for Stochastic Dynamic Stability of Ships and Offshore Platforms The stability of an ocean engineering system and the motion response of the ocean engineering system are closely linked. Take an example of our most common ocean engineering system, a ship: when the amplitude of the rolling motion response of a ship (i.e., the rolling angle) exceeds the stability vanishing angle of the ship or the water inlet angle (when the water inlet angle is smaller than the stability vanishing angle), the ship will lose its stability and capsize. This example illustrates that the motion response and stability of an ocean engineering system are integrated in the actual physical process. Therefore, when we analyze the motions of an ocean engineering system, we cannot separate the response analysis and stability analysis, i.e., the study of the stability of the ocean engineering system should be carried out on the basis of the work of the response study, and the influence on the stability of the ocean engineering system should be considered concurrently in the response study. The problem of the stability of nonlinear ocean engineering systems (especially ships) under the action of stochastic wave loads has been an active area of research

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31

in the global ocean engineering community. Ludwig et al. [41] have reviewed the stability and capsizing of ships in random waves. The current codes and standards (IMO regulations) for ship stability are empirical and based on the ship’s restoring arm, which only considers the action of hydrostatic forces. These static criteria, which ignore the ship motions and wave effects, obviously cannot guarantee the overall stability. Therefore, researchers agree that these static criteria should be modified by using a ship-wave hydrodynamics model, by describing the sea surface as a random field and by analyzing the ship as a six-degree-of-freedom rigid body using nonlinear stochastic dynamical system theory. The extreme response of marine structures mentioned earlier in this chapter generally refers to the slow drift surge response. When a marine structure is in another degree of freedom—rolling motion, if the extreme rolling response of the marine structure exceeds a critical value, the structure will lose its stability and capsize. As one of the most common marine structures, the capsizing behavior of ships is familiar to the naval architects and ocean engineers. In the current practice, people continue to analyze the capsizing behavior of marine structures based on some empirical stability codes. However, when the type or scale characteristics of the analyzed marine structure are not within the scope of the code, then a direct computational analysis of the stability should be performed. At this point some people are often willing to use the Monte Carlo Simulation (MCS) method as well. Although some researchers have pointed out the low efficiency weakness of the MCS method to analyze the capsizing behavior of marine structures, some people still continue to use it, the reason for this is probably because the principle of the MCS method is very simple, the implementation of the MCS method is very convenient, and the easy-to-use method tends to be easily accepted and adopted. However, at the initial stage of design, one needs a reasonable method to quickly check the stability of the designed marine structure, for example, to roughly estimate whether the ship has enough damping to overcome the wave excitation and save it from capsizing, so that the design can be improved and then advanced along the design spiral. It is not advisable to conduct expensive and time-consuming model tests at this time, and the computationally intensive MCS method is also not recommendable at this design stage, so it is necessary to investigate a rational approach to analyze the stability of marine structures. Thus, the study of the stability of marine structures under random loads is also of great practical importance. It is worth noting that researchers worldwide have made unremitting efforts to develop a more efficient method for rational analysis of the ship stability. A geometric method for rational analysis of ship dynamic stability has been developed based on modern nonlinear stochastic dynamics theory, and the rationality of the geometric method has been demonstrated through real-world ship applications and ship model experiments. A brief description of this method is given below.

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1.2.1 Geometric Method One of the main approaches to the study of ship stability and capsizing in random waves is the geometric method, which does not directly solve the nonlinear stochastic differential equations of the ship’s rolling motion; instead, the geometric method focuses on the behavior of the qualitative aspects of the system (or, more precisely, the transitions between different behaviors of the qualitative aspects). An important analytical result of the geometric method is the Melnikov function, and an exposition of Melnikov theory can be found in the respective monographs of Guckenheimer and Holmes [42], Wiggins [43] and Moon [44]. The Melnikov function can predict the generation of chaos in certain kinds of systems, and in the presence of chaos, phase space domains with qualitatively different aspects of behavior (e.g., ship rolling motion safe domain and ship capsizing domain) can be transported from one domain (e.g., ship rolling motion safe domain) to another domain (e.g., ship capsizing domain). Note that the fact that the geometric method does not directly solve the equations of motion of the system does not mean that the study of stability can be separated from the study of the response, rather it is through Monte Carlo simulation that the validity of the geometric method has been verified. In the field of Naval Architecture, Jeffery et al. [45] first utilized a global geometric method to study the transient rolling motion of a small ship subjected to periodic wave excitation. Their analysis was based on determining the criteria that can predict the properties of the invariant manifolds representing the boundaries of safe and unsafe initial conditions. These criteria are dependent on the system parameters for a particular ship, and they use the data of a real ship in their analysis. Hsieh et al. [46] used a single-degree-of-freedom rolling model to study the capsizing of a ship in random waves. They considered several factors such as wave spectrum, nonlinear restoring moment characteristics and nonlinear damping in their analysis. A nonlinear probabilistic method is developed by combining the Melnikov function, the phase-space area flux and the random vibration. The conditions for ship capsizing expressed in terms of significant wave height, wave characteristic period, damping and stiffness coefficients were obtained. They demonstrated the validity of the analytical solution with extensive numerical simulations. Jiang et al. [47] investigated the strongly nonlinear rolling motion and capsizing of a ship in random waves using the stochastic Melnikov method. They considered the effect of the ship having an initial bias. Lin and Solomon [48] developed a mean-square based stochastic Melnikov method to study the capsizing of ships under periodic excitation with random noise. However, the restoring moments in their equations of motion are of symmetric type. Another approach to study the stability and capsizing of ships in random waves is to use the first passage theory in stochastic dynamics, which is briefly described below.

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33

1.2.2 First Passage Theory The first passage problem is recognized as the most difficult problem within the field of stochastic dynamics research. Zhu and Wu [49] studied the first passage time of a nonlinear oscillatory system driven by a combination of harmonic and white noise excitations. They first reduced the system equations of motion to a set of Itô stochastic differential equations using the stochastic averaging method. Then the backward Kolmogorov equation controlling the conditional reliability function and a set of generalized Pontryagin equation controlling the first passage time conditional moments were established. Finally, the conditional reliability function and the conditional probability density and conditional moments of the first passage time were obtained by solving the backward Kolmogorov equation and generalized Pontryagin equation with appropriate initial and boundary conditions. They verified their analytical solutions with numerical simulations. Their results of research can be directly utilized to forecast the conditional probability density of the first passage time of ship capsizing in irregular waves. The late British scholar John Brian Roberts [50–54] not only contributed a lot to the development of the first passage theory, but also was the first to apply the first passage theory to the study of ship stability and capsizing in random waves. For example, Roberts [55] and Roberts & Vasta [56] used a stochastic averaging method based on energy envelopes to predict the mean first passage time of the rolling response of ships and transportation barges excited by irregular waves. These authors first established the second-order differential equation for the rolling motion of ships, and the equation of motion they developed contained linear and squared damping terms as well as linear and cubic restoring force terms. They then transformed the second-order differential equation for the ship’s rolling motion into a standard form. By introducing an energy envelope process for the rolling response and its associated phase angle process, they transformed the single differential equation of ship rolling motion into two coupled first-order differential equations expressed in terms of the response energy envelope and phase angle after a series of transformations. After applying the stochastic averaging based on the Stratonovitch-Khasminskii limit theorem, the above coupled first-order differential equations were written as a set of first-order Itô stochastic differential equations, and the closed form expressions of the drift and diffusion coefficients in the two Itô stochastic differential equations were obtained by stochastic averaging and Fourier series expansions (both expressions for these coefficients contain wave excitation spectral density functions.). Moreover, the Itô stochastic differential equation controlling the response energy envelope process is decoupled from the Itô stochastic differential equation controlling the phase angle process, i.e., the energy envelope process of the ship’s rolling motion response converges to a one-dimensional Markov diffusion process. They then solved the above first-order stochastic differential equation to obtain a simple expression for the probability distribution of the energy envelope process of the ship’s rolling motion response, and they also obtained an appropriate expression for the joint probability density of the ship’s rolling angle and rolling angular

34

1 Introduction

velocity. Also included in their papers is an equation to calculate the average first passage time for the response energy envelope to reach a critical value. In principle, the probability of a ship capsizing in a specific time period can be estimated by this method. However, it should be noted that the studies in the literature Roberts [55] and Roberts & Vasta [56] focused on the development of a stochastic theory of the ship’s rolling motion, and the first passage theory concerning ship capsizing has not been fully and systematically elaborated, and the examples provided for the calculations are also too simplified. Cai et al. [57] and Cai and Lin [58] used a stochastic averaging method based on energy envelopes to study the first passage problem of a ship subjected to combined external and parametric excitations in random waves, but they did not calculate the first passage probability values of the ship’s rolling motion responses. Solomon et al. [59] studied the nonlinear coupled motions of a barge in 2005, and they developed a quasi-two-degree-of-freedom stochastic model and carried out a stability analysis in random waves. Since the coupling effect of nonlinear rolling and heaving motions is evident, by using the observed strong dependence between the experimental measurements of the heaving amplitudes and the wave elevations, they expressed the heaving value in the two-degree-of-freedom rolling and heaving model as a function of wave height, and thus proposed and developed an accurate and efficient quasi-two-degree-of-freedom model. This quasi-two-degreeof-freedom model was used to perform a stochastic stability analysis of this barge based on the first passage time equation. Based on the operating and survival sea states (Class 1 to 9) developed by the U.S. Navy, the results show that the reliability of this barge is significantly reduced when operating in sea states above Class 7. However, they did not systematically investigate the effect of nonlinear damping coefficients on the mean first passage time. Langtangen et al. [60] applied different methods to study the first passage time probability of the displacement response of a moored offshore structure and they showed the effect of non-Gaussian features of the response on the extreme value statistics.

1.3 The Contents of This Book and the Key Issues to Be Addressed The first chapter of this book gives a brief introduction to the analysis methods for stochastic dynamic response and stability of ships and offshore platforms. The second chapter of this book deals with the treatment of random loads acting on marine structures. The next four chapters, Chaps. 3, 4, 5 and 6, are the main part of this book, in which the author of this book will carry out a series of research studies with the objective of applying the latest theoretical findings in nonlinear stochastic structural dynamics to the field of Naval Architecture and Ocean Engineering in order to find the best methods for the analysis of the response and stability of ships

1.3 The Contents of This Book and the Key Issues to Be Addressed

35

and offshore platforms in random waves. At the same time, these research studies will also make innovations and developments to some existing theories in nonlinear stochastic structural dynamics. The main elements of these research studies and the key issues to be addressed include:

1.3.1 Monte Carlo Simulation Method for Analyzing Response Problems of Marine Structures In ocean engineering, the surge oscillatory motion of a compliant offshore structure is usually modeled by a single-degree-of-freedom differential equation containing a flow-induced nonlinear term. In the past decades, researchers in this field have used different force terms in the equations of motion in order to study the response characteristics of this type of compliant structures. To study the response of a simplified model of a Tension Leg Platform under wave forces, Spanos and Agarwal [28] used a Monte Carlo Simulation method. In their study, they developed a single-degree-offreedom differential equation of surge motion for this offshore platform. The elastic forces in their equation of motion are assumed to be linear, and the excitation forces in their equation of motion include the linear Morison inertial force and the nonlinear Morison drag force when the presence of currents is not considered. An improvement was made by Kareem et al. [5] when studying the response of a Tension Leg Platform in the frequency domain by adding a current term to the Morison drag force expression. However, the elastic forces in their equation of motion were still assumed to be linear. Tognarelli et al. [29] modeled an offshore system with a singledegree-of-freedom equation of motion that incorporates system nonlinearity and external force nonlinearity, with the system nonlinearity represented by the Duffing stiffness and the external force nonlinearity represented by the nonlinear Morison drag force. However, they did not consider the effect of currents when calculating the Morison drag force. Because tidal currents and wind-excited currents exist both in the remote ocean and the coastal waters, the effect of currents on the motion response of offshore structures cannot be neglected. Thus, it is physically reasonable and mathematically rigorous to model an offshore system with a single-degree-offreedom equation of motion that includes the Duffing stiffness nonlinear term and the Morison drag nonlinear term that takes into account the effects of currents. A major part of the research studies in this book is to calculate the response values, the response probability density values, the response central moment values, and the level up-crossing rates of the response values for a first ever proposed offshore system with a Duffing stiffness nonlinear term and a Morison drag nonlinear term that takes into account the effects of currents. The key issues to be solved are the transformation between the wave displacement spectrum and the wave horizontal velocity spectrum, the transformation between the wave displacement spectrum and the wave horizontal acceleration spectrum, the selection of the upper cut-off frequency and

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1 Introduction

the simulation time step, the developing of the simulation program and the analysis of the simulation results. The research studies in this book will then go on to point out the shortcomings of traditional MCS simulations to analyze the extreme response of slow-drift offshore structures. In order to study the slow-drift extreme response of an offshore structure under random excitations, a dynamic analysis of the offshore structure is required. In the dynamic analysis of the offshore structure, the added mass, damping and restoring force of the offshore structure, and the external excitation force per unit wave amplitude on the offshore structure can all be calculated by some mature commercial hydrodynamic software. The researcher must first get familiar with the theoretical backgrounds of these software and be proficient in their use. The software selected for use in the research studies in this book is the Calc/Hydro module of the TRIBON system from KCS, Sweden. With the information of the added mass, damping and restoring force of the offshore structure, and the external excitation force per unit wave amplitude on the offshore structure, the rigid body differential equation of motion of the offshore structure can subsequently be established [61]. Because the wave excitations are random, the differential equation of motion of the offshore structure is therefore a stochastic differential equation, which in reality has no analytical solutions and can only be solved by some numerical methods, of which the MCS simulation is one of the methods frequently used [62]. This book will provide an in-depth study on the performance of using the MCS method to analyze the slow-drift extreme response of marine structures. We will first consider an example of slow drift surge motion of a moored floating cylinder. Naess and Johnson [34] pointed out that the differential equation for the slow drift surge motion of this moored floating cylinder is identical to that for the slow drift surge motion of a semi-submersible platform or a Tension Leg Platform. Thus, the study of the slow drift surge motion of this moored floating cylinder is deemed representative. In the particular case when the equation of motion of this offshore structure has exact stationary analytical solutions, we will use this as a benchmark to compare the marginal probability density values of the extreme response of the slow-drift displacements of this offshore structure obtained from 1000 Monte Carlo simulations to see if the MCS accuracy is high. If the accuracy of the MCS results is not high, this study will increase the number of Monte Carlo simulations to 2000 and obtain another set of results to determine whether multiplying the number of Monte Carlo simulations will improve the prediction accuracy of the extreme response. If it is a slight improvement, we will continue to increase the number of Monte Carlo simulations and analyze the results obtained. If it is difficult to continue to improve the prediction accuracy of the extreme response by increasing the simulation number of times substantially after reaching a certain accuracy, it is then necessary to investigate other methods to analyze the slow drift extreme response of marine structures. In our study we will next carry out a quantitative study regarding the performance of using the MCS method to analyze the stability of marine structures (ships). Analyzing the stability of marine structures by the MCS method means to carry out the numerical integration of the differential equation of the rolling motion. We all

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37

know that each time when numerically integrating a differential equation of motion, a rolling angle and a corresponding rolling angular velocity at time t = 0 should be given, i.e., the initial condition of the motion should be given. In addition, the time duration of the integration should also be given. If for a specific stability analysis project the initial conditions used contain 61 different initial rolling angles and 61 different corresponding initial rolling angular velocities, and the time duration of integration is specified as 50 times the characteristic period of the excitation force during each simulation, then 3721 times Monte Carlo simulations (i. e., 3721 times numerical integrations) should be carried out in order to finish the stability analysis. We will count the time required to numerically integrate 3721 times based on a computer program of the same nature that we have developed. Based on the results of the above simulations, we can obtain an integrity number. Taking another different value of wave height and repeating another round of 3721 times Monte Carlo simulations, another integrity number can be obtained. In general, when analyzing the transient behavior of capsizing by the MCS method, we need to take approximately 15 wave height values, so that we can get 15 pairs of wave height-integrity number values. Using the integrity number value as the ordinate and the wave height value as the abscissa, the above 15 pairs of wave height- integrity number values can be traced into a curve, which is called the integrity curve. At a specific point this integrity curve will drop sharply, then the wave height corresponding to this specific point is considered as the critical wave height for transient capsizing. We will count the time required to obtain the entire integrity curve by Monte Carlo simulations, and if it takes a lot of time, we can then prove that the efficiency of analyzing the transient capsizing behavior by the MCS method is extremely low. It is therefore necessary to pursue other methods to rationally analyze the stability of marine structures. The efficiency of using the MCS method for an exhaustive study of the phase space of the initial conditions of the ship rolling has been studied as mentioned above. We must also study the performance of using the MCS method for analyzing the ship capsizing behavior when the initial conditions of motion are fixed and when the excitation frequency and excitation amplitude are varied, i.e., when the capsizing behavior of the ship is studied in the control parameter space. In this case, the excitation frequency can be simplified to the frequency at which the ship is most likely to capsize, i.e., the frequency at which the wave frequency is equal to the ship’s natural frequency of rolling, because even in random sea conditions, this is the most dangerous situation (worst-case scenario) that can occur. This way, it is only necessary to take different values of the significant wave height during the Monte Carlo simulations. If the range of the significant wave heights is chosen to be 0.0m-6.5m, and the spacing of the significant wave heights is chosen to be 0.05m, then 130 rounds are required for the entire study. The total time required for 130 rounds of simulations can be calculated after precisely counting the time required for each round of simulation. After the time required for recording the results and plotting the curves is added, the total time required for the entire study can be obtained. If a large amount of time is required, it can then be demonstrated that it is also inefficient to use the MCS method to study the capsizing behavior of a ship in the control parameter

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1 Introduction

space. It is therefore necessary to investigate other methods to rationally analyze the dynamic stability of marine structures.

1.3.2 Numerical Path Integration Method for Analyzing Response Problems of Marine Structures In this book we will study an efficient and accurate method for performing stochastic dynamic analysis and extreme value prediction of nonlinear systems—the path integration method. The basic idea of path integration is to discretize in space and time respectively, and to replace the integral by the path sum, i.e., to obtain the joint probability density of the state vector by connecting the short-time transition probability densities to form the global transition probability density. The path integration method is based on the theory of Markov diffusion process and the numerical interpolation scheme. Because of this, we will first delve into the theory of Markov diffusion processes and introduce an efficient numerical interpolation scheme, the Gauss–Legendre interpolation scheme. Since the invention of the path integration method based on the Gauss–Legendre scheme, all problems treated by this method have been limited to the weakly nonlinear cases. In our study, we will try to use the path integration method based on the Gauss– Legendre scheme to deal with the strongly nonlinear stochastic oscillation problems. We will utilize the same equation of motion previously used when predicting the slow drift extreme response of an offshore structure by the MCS method. This is a twodimensional nonlinear system. In the case where the response of this system has a known exact stationary probability density solution, we will compare the degree of agreement between the path integration solution and the exact solution of the response of this system, and in particular, we will investigate the degree of agreement between the path integration solution and the exact solution in the tail region of the response probability density curve. We will also continue to enhance the nonlinearity of the above nonlinear system to do further research. In terms of applying the method to the field of Naval Architecture and Ocean Engineering, a continuous Markov process will be used to model the response of a moored ship under non-Gaussian slow-drift wave force excitations. A numerical path integration solution based on the Gauss– Legendre scheme will be applied for the first time in the calculation so that the efficiency and accuracy of the numerical path integration can be improved. The key problems to be solved in this study include establishing the equations of motion of the moored ship in irregular waves, deriving the non-Gaussian slow-drift wave force spectral density function, approximating the slow-drift wave force spectral density function using a white noise, and applying the Gaussian closure principle when applying the path integration method to treat a two-dimensional nonlinear stochastic dynamic system. Next, we will analyze the extreme slow drift response of an offshore structure subjected to random excitations by the path integration method. For comparison, we

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39

will use the same offshore structure example that has been analyzed by the MCS method, i.e., the example of slow drift surge motions of a moored floating cylinder. First, we will compare the accuracy of the path integration solution with that of the MCS solution in the special case where the system equation of the structure has exact stationary analytical solutions. We will calculate the relative error of the marginal probability density value of the extreme response of the slow drift displacement of the offshore structure obtained from 5100 Monte Carlo simulations, and also calculate the relative error of the marginal probability density value of the extreme response of the slow drift displacement of the offshore structure obtained from 25 path integration runs. With this information, and measuring the time taken for 5100 runs of the Monte Carlo simulations and 25 runs of the path integration, respectively, it is possible to compare the efficiency and accuracy of the path integration method and the MCS method in predicting the slow drift extreme response of an offshore structure. We will plot the curves to visualize the above calculation results. Next, we will use the path integration method to obtain accurate predictions of the extreme slow-drift response of this moored floating cylinder in the general case where there is no analytical solution to the system equation of the structure. In the analysis of the extreme response problem of the slow drift surge motion of the above moored floating cylinder by the path integration method, the joint probability density of the displacement response and velocity response of the structure is first obtained. Because the level up-crossing rate of the extreme response of the offshore structure is related to the above joint probability density, and because the exceedance probability of the extreme response of the offshore structure is related to the level up-crossing rate of the extreme response of the structure, we can concurrently obtain the exceedance probability of the extreme response of the offshore structures when applying the path integration method, which is a further advantage of using the path integration method. We will develop a computer program to calculate the 3-h exceedance probabilities of the slow-drift surge displacement responses of the above moored floating cylinder, and before doing these calculations we need to learn the theories and the computational principles regarding the exceedance probability. This part of the study will conclude with an evaluation of the performance of the path integration method for analyzing the slow drift extreme response of marine structures subjected to stochastic excitations. The superiority of the path integration method over the MCS method in terms of efficiency and accuracy will be pointed out, as well as the problems that should be noted when applying the path integration method. The key problems to be solved in this study are as follows: to analyze the extreme slow-drift response of marine structures subjected to stochastic excitations by the path integration method, in view of the computational example we have chosen, there are two key problems, or difficulties, to be solved, in addition to the need of learning the theory of stochastic oscillations of moored offshore structures under the action of second-order slow-drift wave forces. The first one is to achieve a thorough understanding of the theoretical background of the path integration method, and the second one is to develop the computer program for implementing the path integration method.

40

1 Introduction

The path integration method is based on the theory of Markov diffusion process, but it is not enough to study the theory of Markov diffusion process only, because the theory of Markov diffusion process is under a large framework of nonlinear stochastic dynamics. Only with a clear understanding of the general framework of nonlinear stochastic dynamics is it possible to figure out the principles of Markov diffusion process theory and the path integration method. One of the most important tasks in this study is to focus time and effort on figuring out the principles of the path integration method. The second key problem to be solved in the process of analyzing the slow-drift extreme response of marine structures subjected to random excitations by the path integration method is to develop the computer program for implementing the path integration method. Based on the specific nature of the problem we are dealing with, and also on this author’s programming experience, we chose the Mathematica programming language, a new symbolic computer language that is able to manipulate a very wide range of objects involved in scientific computing using only few basic elements. Mathematica enables users to build many different types of programs with a fairly small but powerful symbolic programming approach. As we will see in this study, it is convenient to use Mathematica to generate the locations of the Gaussian quadrature points in the path integration method; it is convenient to use Mathematica to solve the moment equations derived from the original equations of motion to obtain the short-time transition probability densities; and it is convenient to use Mathematica to obtain the probability densities at any time based on the initial probability density and each transition probability density. In this way, the programming problem of the path integration method can be solved readily.

1.3.3 Global Geometric Method for Analyzing Stability Problems of Marine Structures In this part of the study, we will first learn the existing research results on the analysis of the stability of marine structures by the Melnikov method, specifically to learn some Melnikov criteria and to understand how these Melnikov criteria are derived. This requires an in-depth study of the theory of the motions of marine structures (ships). The key to this is to understand the concept of “pseudo-rolling-center” and how the differential equation of the ship’s rolling motion can be reduced to a dimensionless form by re-taking a scaled time. The next step is to understand the mechanism of unexpected capsizing of a ship from the view point of the Melnikov theory. The key to this is again to clarify some concepts in the theory of nonlinear stochastic dynamics, such as invariant manifolds (stable and unstable manifolds), heteroclinic tangle and diffeomorphism, etc. Figure out how the phase space transport theory and the lobe dynamics technique can be utilized to explain: In the presence of chaos, phase space domains with qualitatively different behaviors (e.g., ship rolling motion safety domain and ship

1.3 The Contents of This Book and the Key Issues to Be Addressed

41

capsizing domain) can be transported from one domain (e.g., ship rolling motion safety domain) to another domain (e.g., ship capsizing domain). That is, how the unexpected capsizing of the ship occurs. Then, a Melnikov criteria formula for analyzing the stability of a ship and another stochastic Melnikov criteria formula can be understood. In the application of the geometric method to the study of ship stability, based on the real ship data of a fishing vessel that capsized twice, we will calculate the Melnikov criteria for the capsizing of the vessel under harmonic excitations. The encountered wave frequency and the wave amplitude will be included directly in this Melnikov criteria. This study will next analyze the dynamic stability of a barge (i.e., calculate the damping required for the barge to prevent capsizing) using the existing Melnikov method. For this purpose, we will first calculate the hydrodynamic parameters in the differential equation of the vessel’s rolling motion using the Calc/Hydro module of the TRIBON system from KCS, Sweden. Next, the relevant hydrodynamic parameters will be nondimensionalized in order to analyze the stability of the barge using the Melnikov criteria, i.e., to obtain the damping values required to overcome the critical wave rolling excitation moments that cause the ship to capsize. The elapsed time of the calculation process will to be recorded and compared with the elapsed time of the MCS method. We will then analyze the stability of a ship with initial bias by the Melnikov method, and this study will simultaneously develop a mean-square based stochastic Melnikov method to study the capsizing of a ship in perturbed regular waves. The restoring moments in the ship’s equation of rolling motion are of asymmetric type, i.e., the effect of the ship having initial bias is considered. The key problems to be solved in this study include proposing the generalized Helmholtz-Thompson equation for the first time, the derivation of the analytical solutions of the heteroclinic orbits parameterized by time, and the derivation of the generalized Melnikov criteria. Given a value of bias, we will derive values of the critical wave excitation moment corresponding to different values of damping. We will compare simultaneously with the MCS method the efficiency in the exhaustive study of the phase space of the initial conditions of the ship rolling. There are two key problems, or difficulties, that need to be solved in the analysis of ship stability by the Melnikov method. One is to calculate each relevant hydrodynamic coefficient in the differential equation of rolling motion of the selected barge and to nondimensionalize this differential equation of motion. For this purpose it should be firstly clarified why the rolling and sway motions can be decoupled if a pseudo-rolling-center is assumed to exist. That is, to derive the equation to find the distance of the pseudo-rolling-center above the ship’s center of gravity. The next task is to investigate how to define the strips of the barge under study in the Calc/Hydro module of the TRIBON system, and how many strips should be defined. After doing this, the Calc/Hydro module automatically cuts the ship model along the longitudinal direction to generate strips, which are used to first generate two-dimensional added mass and damping coefficients. To calculate the two-dimensional added mass and damping coefficients on each strip, we have two options. If we choose the boundary integral method, we again need to divide half of the cross-sectional line on each strip

42

1 Introduction

of the ship into several straight-line segments, and then distribute the fluid source with constant (but unknown) intensity on each straight-line segment. How many straight line segments should each half cross-sectional line on each strip be divided into to be appropriate? What is the reason? What is the principle of the boundary integral method to finally calculate the two-dimensional added mass coefficients and damping coefficients on each strip? These are the questions that we should study. Next, we should also study the principles of calculating the Froude Krylov “forces” and the diffraction “forces” of the two-dimensional strips by the numerical integration method. After calculating the two-dimensional added mass coefficients, damping coefficients, Froude Krylov “forces” and the diffraction “forces” for each strip, and integrating along the ship’s longitudinal direction, the total generalized added mass coefficients, damping coefficients and wave excitation moment per unit wave amplitude will be obtained. Having obtained these relevant hydrodynamic parameters and substituting them into the differential equation of the rolling motion of the selected barge, we are faced with another problem: how can we nondimensionalize the differential equation of the rolling motion of the vessel in order to use the Melnikov method? To analyze the dynamic stability of the barge by the Melnikov method, the second key issue to be addressed is to clarify some concepts in the theory of nonlinear stochastic dynamics, such as invariant manifolds, etc. An important task in this part of the study is to focus time and effort on clarifying these concepts. Only when these concepts are clarified, it is possible to understand the existing Melnikov criteria formulas and to correctly apply them in the analysis of specific engineering examples.

1.3.4 The First Passage Theory for Analyzing Stability Problems of Marine Structures We should first systematically study the first passage theory in stochastic structural dynamics. The first passage failure problem is the most difficult problem in stochastic structural dynamics. Presently, mathematically exact solutions are only obtainable when the stochastic phenomena in the problem can be treated as a Markov diffusion process, and, again, the known solutions are limited to the one-dimensional case. Because the state space of the ship rolling model is two-dimensional, reducing the number of dimensions is necessary so that the results of the Markov theory can be adopted. Fortunately, this can be done by using the stochastic averaging method. In our study, the second-order differential equation of the ship’s rolling motion will first be established. By introducing an amplitude process of the rolling response and its associated phase angle process, and by introducing two generalized harmonic function terms, a single differential equation of the ship rolling motion can be transformed into two coupled first-order differential equations expressed in terms of the amplitude and the phase angle after a series of transformations. After applying a

1.3 The Contents of This Book and the Key Issues to Be Addressed

43

stochastic averaging method based on the Stratonovitch-Khasminskii limit theorem, the above coupled first-order differential equations can be written as a set of firstorder Itô stochastic differential equations. The closed-form expressions for the drift and diffusion coefficients in both Itô stochastic differential equations are obtained by stochastic averaging (both expressions for these coefficients contain the wave excitation spectral density functions). The Itô stochastic differential equation that controls the amplitude process will be decoupled from the Itô stochastic differential equation that controls the phase angle process. Consequently, the ship rolling motion amplitude process converges to a one-dimensional Markov diffusion process. The conditional reliability function of the rolling motion amplitude process satisfies the backward Kolmogorov equation, and the conditional moments of the first passage time of the rolling motion amplitude process satisfy the generalized Pontryagin equation. The coefficient of the first-order derivative term in the Pontryagin equation is equal to the drift coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process, and the coefficient of the second-order derivative term in the Pontryagin equation is equal to the diffusion coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process. The Pontryagin equation and its two boundary conditions form a boundary value problem, and the first-order conditional moments of the first passage time can be obtained by numerically integrating this boundary value problem. Knowing the first-order conditional moments of the first passage time is sufficient to estimate the first passage probability density of the rolling motion amplitude process. The effect of hydrodynamic damping on the mean first passage time can be studied by taking different values of the nonlinear damping coefficients in the ship’s rolling equation of motion. The effect of external excitation level on the mean first passage time can be studied by taking different levels of external excitation in the ship’s rolling equation of motion. In this way, the relationship between the hydrodynamic damping (or the wave external excitation level) and the ship capsizing can be expressed quantitatively, thus providing a rigorous theoretical method to study the stability of ships in random waves. The key problems to be solved in this part of the study are: how to transform the second-order differential equation of the ship’s rolling into a “standard” form; how to derive the drift and diffusion coefficients in the Itô’s stochastic differential equation controlling the amplitude process of ship’s rolling motion; how to derive the two boundary conditions of Pontryagin’s equation; how to solve the boundary value problem consisting of Pontryagin’s equation and its two boundary conditions by numerical integration, i.e., the developing of the computational codes.

44

1 Introduction

Exercises Exercise 1.1 Suppose the equation of motion of a dynamical system is: M x¨ + C x˙ + K x = 0

(E.1.1)

Equation (E.1.1) can also be written in terms of the natural frequency of the undamped oscillation of the system, denoted by ωn , as x¨ + 2 ζ ωn x˙ + ωn2 x = 0

(E.1.2)

where √ ωn = K /M = natural frequency of the undamped oscillation. ζ = C/(2Mωn ) = damping factor. The solution of equation (E.1.2) is expressed as follows: x(t) = J1 e−ζ ωn t e

(√ ) ζ 2 −1 ωn t

+ J2 e−ζ ωn t e

(√ ) − ζ 2 −1 ωn t

(E.1.3)

In the above equation J 1 and J 2 are determined from the initial conditions. Show that the oscillation expressed by Equation (E.1.3) damps out with time under each of the three possible conditions of damping (underdamping (ζ < 1), critical damping (ζ = 1), and overdamping (ζ > 1)) and under all possible initial conditions. Exercise 1.2 We change the free vibration system (E.1.2) to a forced oscillation system by adding a unit impulse δ(t) as an outside forcing term as follows: x¨ + 2 ζ ωn x˙ + ωn2 x = δ(t)

(E.1.4)

If g(t) denotes the impulse response function of the dynamic system (E.1.4), show that g(t) =

( √ ) e−ζ ωn t √ sin ωn 1 − ζ 2 t ωn 1 − ζ 2

(E.1.5)

Exercise 1.3 For a weakly stationary ergodic stochastic process x ( t ), denote its auto-correlation function as R x x (θ ), show that R x x (θ ) is an even function. That is: R x x (θ ) = R x x (−θ )

(E.1.6)

Exercise 1.4 For a weakly stationary ergodic stochastic process x ( t ), denote its auto-correlation function as R x x (θ ), show that R x x (θ ) is maximum at an θ = 0. That is:

Exercises

45

R x x (0 ) ≥ R x x (θ )

(E.1.7)

Exercise 1.5 For two weakly stationary ergodic stochastic processes x ( t ) and y (t ), denote the cross-correlation functions between them as R x y (θ ) and R yx (θ ). Show that: R yx (θ ) = R x y (−θ )

(E.1.8)

Exercise 1.6 For two weakly stationary ergodic stochastic processes x ( t ) and y (t ), denote the cross-spectral density function between them as S x y (ω ). In the text in this chapter we have derived an expression for S x y (ω ) as follows: S x y (ω ) = = =

1 π 1 π

{∞ {

⎧ 0 { 1⎨ π⎩

R x y (θ )e−i ωθ dθ +

−∞

{∞

R x y (θ )e−iωθ dθ

0

R yx (θ )ei ωθ dθ +

0

{∞

⎫ ⎬ ⎭

}

R x y (θ )e−i ωθ dθ

0

{∞ {

} R yx (θ ) + R x y (θ ) cos ω θ dθ

(E.1.9)

0

+ i π1

{∞ {

} R yx (θ ) − R x y (θ ) sin ω θ dθ

0

= C x y (ω ) + i Q x y (ω ) Denote the complex conjugate of S x y (ω ) to be S ∗x y (ω ). That is: S x y ∗ (ω ) = C x y (ω ) − i Q x y (ω )

(E.1.10)

S x y ∗ (ω ) = S yx (ω )

(E.1.11)

Show that:

Exercise 1.7 In the text in this chapter we have stated that when implementing the Monte Carlo simulation method, the sample function of the water particle velocity of a wave can be generated by the following equation. v(t) =

N −1 √ ∑ { }1/2 Sv0 v0 [n (Δω)]Δω 2 cos[n(Δω)t + Φn ]

(E.1.12)

n=0

where Δω is the frequency spacing, Sv0 v0 [n (Δω)] is the one-sided power spectral density function of wave water particle velocity. Φ0 , Φ1 , Φ2 , …, Φ N −1 are independent random phase angles uniformly distributed between 0 and 2π .

46

1 Introduction

The MATLAB rand function outputs uniformly distributed floating-point numbers between 0 and 1. The “rand” function in MATLAB is used to produce pseudorandom and pseudo-independent numbers. Although these numbers don’t exactly meet the mathematical definitions of randomness and independence, they still pass various statistical tests to demonstrate their randomness and independence, and their calculation can be repeated for diagnostic or testing purposes. In this Exercise, please use the “rand” function in MATLAB to generate 10,000 independent random phase angles (Φ0 , Φ1 , Φ2 , …, Φ9999 ) uniformly distributed between 0 and 2π . Exercise 1.8 Based on the 10,000 generated random phase angles (Φ0 , Φ1 , Φ2 , …, Φ9999 ) in Exercise 1.7, please use the “histogram” function in MATLAB to plot a figure and verify that the histogram of these 10,000 values is approximately flat, which indicates a fairly uniform sampling of numbers. Exercise 1.9 Equation (E.1.6) in the text in this chapter gives the complex frequency– response function G(ω) as follows: 1 ) G(ω) = ( 2 ωn − ω2 + 2iζ ωn ω

(E.1.13)

Equation (E.1.7) in the text in this chapter specifies that the magnitude of G(ω) becomes: |G(ω)| = /(

ωn2 − ω2

1 )2

(E.1.14) + (2ζ ωn ω)2

Denote the complex conjugate of G(ω ) to be G ∗ (ω ), show that: G (−ω ) = G ∗ (ω )

(E.1.15)

G (ω )G ∗ (ω ) = |G(ω)|2

(E.1.16)

References 1. Tognarelli MA (1998) Modeling nonlinear load effects on structures. Dissertation, Department of Civil Engineering and Geological Sciencies, University of Notre Dame 2. Ibrahim RA (2004) Nonlinear vibrations of suspended cables-Part III: Random excitation and interaction with fluid flow. Appl Mech Rev 57 (6): 515-549 3. Guckenheimer J, Holmes P (1997) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York 4. Lin YK, Cai GQ (2004) Probabilistic structural dynamics. Advanced theory and applications. McGraw-Hill Professional, New York 5. Kareem A, Zhao J, Tognarelli MA (1995) Surge response statistics of tension leg platforms under wind and wave loads: a statistical quadratization approach. Probab Eng Mech 10: 225-240

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6. Booton RC (1954) Nonlinear control systems with random inputs. IRE trans. circuit theory CT-1 1: 9–19 7. Caughey TK (1959a) Response of a nonlinear string to random loading. J Appl Mech-Trans ASME 26(3): 341.344 8. Caughey TK (1959b). Response of a van der Pol oscillator to random excitation. J Appl MechTrans ASME 26(3): 345-348 9. Iwan WD, Yang IM (1972). Application of statistical linearization technique to nonlinear multi-degree-of-freedom system. J Appl Mech-Trans ASME 39: 545-550 10. Spanos PD (1981) Stochastic linearization in structural dynamics. Appl Mech Rev 34(1): 1.8. 11. Roberts JB, Spanos PD (2003) Random vibration and statistical linearization. Dover Publications, New York 12. Kaplan R (1966) Lecture notes on nonlinear theory of ship roll motion in a random seaway. Proceedings, 11th international towing tank conference, Tokyo, Japan, 1966 13. Vassilopoulos LA (1971) Ship rolling at zero speed in random beam seas with nonlinear damping and restoration. Journal of Ship Research, J Ship Res 15(4): 289- 294 14. Borgman LE (1967) Spectral analysis of ocean wave forces on piling. J waterway harbor-Trans ASCE 93 No. WW2: 557-583 15. Borgman LE (1969) Ocean wave simulation for engineering design J waterway harbor-Trans ASCE 95 No. WW4: 129-156 16. Spanos PD, Ghosh R, Finn LD, Halkyard J (2005) Coupled analysis of a spar structure: Monte Carlo and statistical linearization solutions. J Offshore Mech Arct Eng Trans ASME 127 (11): 11.16 17. Li XM, Quek ST, Koh CG (1995) Stochastic response of offshore platforms by statistical cubicization. J Eng Mech-ASCE 121 (10): 1056-1068 18. Shinozuka M (1972) Monte Carlo solution of structural dynamics. Comput Struct 2(5,6): 855–874 19. Shinozuka M, Jan CM (1972) Digital simulation of random processes and its applications. J Sound Vibr 25(1): 111.128 20. Shinozuka M, Deodatis G (1991) Simulation of stochastic processes by spectral representation. Appl Mech Rev 1991: 191.204 21. Shinozuka M, Deodatis G (1996) Simulation of multi-dimensional Gaussian stochastic fields by spectral representation. Appl Mech Rev 49(1): 29-53 22. Chakrabarti SK (1987) Hydrodynamics of offshore structures. Computational Mechanics Publications Inc, Berlin 23. Price WG, Bishop RED (1974) Probabilistic theory of ship dynamics. Chapman and Hall, London 24. Shinozuka M, Wai P (1979) Digital simulation of short-crested sea surface elevations. J Ship Res 23(1): 855-874 25. Ochi MK (1998) Ocean waves, the stochastic approach. Cambridge University Press, Cambridge 26. Naess A (2001) Crossing rate statistics of quadratic transformations of Gaussian processes. Probab Eng Mech 16: 209-217 27. Naess A, Hans CK (2004) Numerical calculation of the level crossing rate of second order stochastic Volterra systems. Probab Eng Mech 19: 155-160 28. Spanos PD, Agarwal VK (1984) Response of a simple TLP model to wave forces calculated at displaced position. J Energy Resour Technol-Trans ASME 106: 437-443 29. Tognarelli MA, Zhao J, Kareem A (1997) Equivalent statistical cubicization for system and forcing nonlinearities. J Eng Mech-ASCE 123 (8): 890-893. 30. Einstein A, Furth R, Cowper AD (1956) Investigations on the theory of the Brownian movement. Dover Publications, New York 31. Zhu WQ (2003) Nonlinear stochastic dynamics and control. Science Press, Beijing 32. Huang ZL, Zhu WQ, Suzuki Y (2000) Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations. J Sound Vibr 238(2): 233–256

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1 Introduction

33. Wehner MF, Wolfer WG (1983) Numerical evaluation of path-integral solution to Fokker-Plank equations. Phys Rev A 27(S): 2663–70 34. Naess A, Johnson JM (1993) Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Prob Eng Mech 8: 91.106 35. Naess A (1994) Prediction of extreme response of nonlinear oscillators subjected to random loading using the path integral solution technique. J Res Natl Inst Stand Technol Vol. 99, No. 4 36. Karlsen HC, Naess A (2005) Statistical response predictions for a nonlinearly moored large volume structure in random seas. Appl Ocean Res 27: 273-280 37. Donley MG, Spanos PD (1991) Stochastic response of a tension leg platform to viscous drift forces J Offshore Mech Arct Eng Trans ASME 113(2): 148-155 38. Donley MG, Spanos PD (1992) Stochastic response of a tension leg platform to viscous and potential drifts. Proceedings of the International Offshore Mechanics and Arctic Engineering Symposium, v 2, Safety and Reliability, 1992, p 325–334 39. Quek S, Li X, Koh C (1994) Stochastic response of jack-up platform by the method of statistical quadratization. Appl Ocean Res 16 (2): 113-122 40. Cai GQ, Lin YK (1997) Response spectral densities of strongly nonlinear systems under random excitation. Prob Engng Mech 12(1): 41 -47 41. Ludwig A, Igor C, Gunter O (2004) Stability and capsizing of ships in random sea - a survey. Nonlinear Dyn 36: 135 -179 42. Guckenheimer J, Holmes, P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences (Springer-Verlag New York Inc.); v. 42. 1983 43. Wiggins S (1988) Global bifurcation and chaos. Springer, New York 44. Moon FC (1992) Chaotic and fractal dynamics. Wiley, New York 45. Falzarano JM, Shaw SW, Troesch AW (1992) Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. Int. J. Bifurcation Chaos 2(1): 101.115 46. Hsieh SR, Troesch AW, Shaw SW (1994) Nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc R Soc London Ser A-Math Phys Eng Sci 446:195-211 47. Jiang CB, Troesch AW, Shaw SW (1996) Highly nonlinear rolling motion of biased ships in random beam seas, J Ship Res 40(2): 125-135 48. Lin H, Solomon CSY (1995) Chaotic roll motion and capsize of ships under periodic excitation with random noise. Appl Ocean Res 17(3):185-204 49. Zhu WQ, Wu YJ (2003) First-passage time of Duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dyn 32(3): 291 -305. 50. Roberts JB (1976) First-passage probabilities for non-linear oscillators, J Eng Mech-ASCE 1976, 102:851.866 51. Roberts JB (1978) First-passage time for oscillators with non-linear restoring force. J. Sound Vibr 56: 71.86 52. Roberts JB (1978) First-passage time for oscillators with non-linear damping. J Appl MechTrans ASME 45: 175-180 53. Roberts JB (1986) First-passage probabilities for randomly excited systems: Diffusion methods. Probab Eng Mech 1(2):66-81 54. Roberts JB, Spanos PD (1986) Stochastic averaging: an approximate method of solving random vibration problems. Int J Non-linear Mech 21:111.134 55. Roberts JB (1986) Response of an oscillator with non-linear damping and a softening spring to non-white random excitation. Probab Eng Mech 1(1): 40-48 56. Roberts JB, Vasta M (2000). Markov modelling and stochastic identification for nonlinear ship rolling in random waves. Proc R Soc London Ser A-Math Phys Eng Sci 358(1771):1917-1941 57. Cai GQ, Yu SJ, Lin YK (1994) Ship rolling in random sea, Stochastic dynamics and reliability of nonlinear ocean systems. ASME DE, 77:81.88. 58. Cai GQ, Lin YK (1998) Failures of stochastically excited systems. In: Shlesinger MF, Swean T (ed) Stochastically excited nonlinear ocean structures, World Scientific, Singapore, p 129-155 59. Solomon CSY, Tongchate N, Erick TH (2005) Coupled nonlinear barge motions, part two: stochastic models and stability analysis, J Offshore Mech Arct Eng Trans ASME 127: 83

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60. Langtangen HP, Marthinsen T, Mathiesen J (1998) Comparison of methods for the statistics of slow-drift oscillations. Probab Eng Mech (2):97-106 61. Wang YG (2020) Predicting absorbed power of a wave energy converter in a nonlinear mixed sea. Renew Energy 153: 362-374 62. Wang YG (2023) Robust adaptive analysis of dynamic responses of offshore sustainable energy systems. Ocean Eng 273: 114022

Chapter 2

Random Loads Acting on Marine Structures

2.1 Introduction By marine structures we mean here ships, offshore platforms and floating production storage and offloading systems, etc. These types of structures are subjected to severe marine environmental loads during their life cycles. The environmental loads to which marine structures are subjected include wind, wave and current loads. Because this book focuses on the comparison between various approaches, we introduce here the following assumptions: (i) the total response of a marine structure is equal to the sum of the response due to wind loads alone and the response due to wave (including current) loads alone; and (ii) wind loads are small compared to wave (including current) loads. This assumption has also been adopted by other researchers [1]. This allows us to focus on the action of waves (including currents) on marine structures. Since no two wave trains are recorded as precisely similar in actual sea conditions, mathematical statistics is the only valid way to describe waves. As a result, the deterministic approach is increasingly being replaced by the probabilistic approach when analyzing problems of waves and dynamics of marine structures. The research within the field of stochastic waves can be divided into two main categories: one is how to represent stochastic waves, where the key is the creation of the wave spectral density function, and the other is the simulation of stochastic waves based on a specific wave spectrum. This chapter will first provide explanations on these two types of problems. Next, this chapter will introduce the Morison formulation for calculating the hydrodynamic loads on slender marine structures, and we will see that the wave water particle velocity and acceleration stochastic processes are directly included in the Morison formulation. Subsequently, this chapter also briefly describes the treatment of other types of random wave loads on marine structures, including examples of the treatment of second-order slow-drift random excitations on compliant offshore structures, as well as formulas for calculating the spectral density function of random wave loads on marine structures with known response amplitude operators. Finally, some methods to simplify the treatment of random wave loads will be presented. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5_2

51

52

2 Random Loads Acting on Marine Structures

2.2 Random Waves in the Ocean 2.2.1 Randomness of Sea Waves and Their Probabilistic Characteristics Our understanding of the motions of marine structures, and our ability to predict the behavior of these structures in the design phase, begins with our study of the nature of the waves that are acting on these marine structures. The most striking characteristic of the waves over an open ocean is their irregularity, or randomness. Irregular waves are not only found in stormy seas, but also in relatively calm seas. The ideal regular waves only exist in the testing tanks. The wind-generated wave profile that is seen in the ocean varies randomly over time and is not repetitive in either space or time. In reality, wave period and wave height vary erratically from one cycle to the following, and the following Fig. 2.1 shows a partial record of the wave profile in rough sea conditions. Figure 2.1 shows part of a record of waves measured from 17.00 pm to 21.20 pm on 24th December 1989 at the Gullfaks C platform in the North Sea. It is clear from this record that it is nearly impossible to assess the characteristics of random waves using a wave-by-wave method in the time domain. However, it is possible to assess the statistical characteristics of the waves in the frequency and probability domains if we think of a randomly varying wave as a stochastic process. Waves in deep water are treated as a Gaussian stochastic process when the stochastic process approach is used, meaning that the probability distribution of the waves’ displacement from its mean position obeys a normal distribution law. Rudnick, P. (1951) made the initial discovery of this law through the analysis of empirical data collected in the Pacific Ocean [2]. The roughness of the sea surface and the depth of the water are two factors that generally affect the Gaussian properties of waves. It is safe to say that regardless of the choppy sea conditions, including the choppy sea conditions brought on by hurricanes, the waves can be regarded as Gaussian random processes if the water is deep enough. If the sea conditions are very mild, waves in relatively shallow water can also be thought of as Gaussian stochastic processes [2]. Figure 2.2 shows a histogram plotted based on the 39,000 wave elevation points measured from 17.00 pm to 21.20 pm on 24th December 1989 at the Gullfaks C platform in the North Sea. The water depth at this measuring site is 218 m, and this water depth is regarded as deep water for the most important wave components. From Fig. 2.2 we can obviously find that the distribution of the elevations of these measured waves is approximately Gaussian. The results in Fig. 2.2 clearly show the beauty and symmetry of our mother nature.

2.2 Random Waves in the Ocean

53

Fig. 2.1 A measured wave profile in rough sea conditions

Fig. 2.2 A histogram plotted based on the 39,000 wave elevation points measured at the Gullfaks C platform in the North Sea

2.2.2 Spectral Analysis of Random Ocean Waves The wave spectral density function, which is crucial in determining the statistical characteristics of random waves, represents the potential and kinetic energy of random waves. The wave spectral density function is described below, and since the autocorrelation function and the spectral density function are closely related, the definition and properties of the autocorrelation function will also be presented before the spectral density function is introduced. The mathematical definition of a stochastic process, as adapted for our purposes in this book, is as follows: A stochastic process, x(t), is defined as a family of random variables. Strictly speaking, x(t) is a function of time t and sample space [2]. Let’s use a set of n wave recorders (1 x, 2 x, 3 x, …, n x) scattered across a specific sea area, as shown in the left portion of Fig. 2.3 [2], to further explain what is meant by

54

2 Random Loads Acting on Marine Structures

1

1

x

x(t)

2

3

x

x k

x

2

x(t)

3

x(t)

k

x(t)

k

k n

x

n

x(tj )

x( tj )

x(t) tj

tj

Fig. 2.3 Definition of ensemble of random waves: the left part shows a set of wave records; the right part shows the ensemble at time t j and t j + τ

a stochastic process. Let us consider a set of n time histories {1 x(t), 2 x(t), 3 x(t), …,n x(t)} for these wave recorders, as shown in the right part of Fig. 2.3 [2]. We understand that x(t j ) is a random variable at any given time t j and that the set consisting of {1 x(t j ), 2 x(t j ), 3 x(tj ), …,n x(t j )} can be viewed as a random sample with n elements. These simultaneously collected wave data observed at a particular moment in time are called an ensemble. In principle the statistical properties of the random waves x(t) must be obtained from its ensemble. In practice, however, an ensemble of wave records has never been considered; instead, the statistical properties of random waves are typically evaluated using analysis of a single wave record. This is acceptable if the waves are assumed to have the ergodic property [2]. The ergodic theorem tells us that a stochastic process x(t) is ergodic if all statistics associated with the ensemble can also be derived from a single time history x(t). In this book, it is assumed that random waves possess the ergodic property. The autocorrelation function and the spectral density function of a random wave record can now be defined based on this property of the random waves. First, the auto-correlation function, Rxx (τ ), of the random waves x(t) is defined as [2]. 1 R x x (τ ) = lim T →∞ 2T

{T x(t)x(t + τ )dt

(2.1)

−T

As seen in the above equation, the product of two readings taken from the same record and separated by a time shift τ is used to calculate the auto-correlation function (see Fig. 2.4) [2]. The autocorrelation function has the following properties [2].

2.2 Random Waves in the Ocean

55

Fig. 2.4 Definition of the auto-correlation function

x(t) x( t+ ) Time

1. The function Rxx (τ ) is even. 2. When τ = 0, Rxx (τ ) has the maximum value. The wave data’s second order moment, E[x 2 (t)], is equal to Rxx (0). Rxx (0) represents the variance of the wave record because the mean of x(t) is zero. 3. The time average of the wave energy P is represented by Rxx (0). In other words, from (1) we can obtain 1 Rx x (0) = lim T →∞ 2T

{T {x(t)}2 dt = P

(2.2)

−T

Figure 2.5 depicts an example of the dimensionless auto-correlation function obtained based on the 39,000 wave elevation points measured from 17.00 pm to 21.20 pm on 24th December 1989 at the Gullfaks C platform in the North Sea. We next introduce the following Parseval’s theorem [2], which can be utilized to express the average energy P in terms of the wave frequency ω in radians per second. {∞ {x(t)}2 dt = −∞

1 2π

{∞ |X (ω)|2 dω

(2.3)

−∞

in which X(ω) is the Fourier transform of x(t), i.e. [2]. {∞ X (ω) =

x(t)e−i ω t dt

−∞

The Parseval’s theorem’s proof is given below: x(t) is X(ω)’s Fourier transform and can be calculated as

(2.4)

56

2 Random Loads Acting on Marine Structures

Fig. 2.5 Example of the dimensionless auto-correlation function evaluated from the measured waves at the Gullfaks C platform in the North Sea

1 x(t) = 2π

{∞

X (ω)ei ω t dω

(2.5)

−∞

Therefore, we have {∞ −∞

⎧ ⎫ {∞ ⎨( / ⎬ ) {x(t)}2 dt = x(t) 1 (2π ) X (ω)ei ω t dω dt ⎩ ⎭ −∞ −∞ ⎫ ∞ ⎧ ∞ { ⎬{ 1 ⎨ = x(t)ei ω t dt X (ω)dω ⎭ 2π ⎩ {∞

−∞

(2.6)

−∞

Using Eq. (2.4) we can derive the following relationship {∞ X (−ω) =

x(t)ei ω t dt

(2.7)

−∞

Let us denote the complex conjugate of X (ω) to be X ∗ (ω). Then, from the classical Fourier transform theory we have X ∗ (ω) = X (−ω) When Eqs. (2.6), (2.7), and (2.8) are combined, we obtain

(2.8)

2.2 Random Waves in the Ocean

{∞

57

1 {x(t)} dt = 2π

{∞

2

−∞

−∞

1 X (ω)X (ω)dω = 2π

{∞

∗

|X (ω)|2 dω

(2.9)

−∞

With the help of the Parseval’s theorem (i. e., Eq. (2.3)), the average energy P in Eq. (2.2) can be calculated as [2]. 1 P = lim T →∞ 4π T

{T |X T (ω)|2 dω

(2.10)

−T

The definition of the spectral density function (i.e., Sx x (ω)) of the random waves x(t) is specified as follows [2] Sx x (ω) = lim

T →∞

1 |X T (ω)|2 2π T

(2.11)

Because the spectral density function is an even function, the time average of the wave energy P can be derived from Eq. (2.10) and Eq. (2.11) [2]. 1 P= 2

{

{∞ Sx x (ω)dω =

∞

Sx x (ω)dω

(2.12)

0

−∞

According to the aforementioned equations, the random waves’ average energy with respect to time is represented by the area under the spectral density function. Additionally, based on the characteristics of the auto-correlation function, the variance of the waves x(t) is equal to the area under the spectral density function. In dealing with practical problems, it is important to derive the wave velocity spectral density function (in the vertical direction) and wave acceleration spectral density function (in the vertical direction) from the wave displacement (elevation) spectral density function. Let Rx x (τ ) represents the auto-correlation function of a wave record x(t), and let Sx x (ω) be the corresponding spectral density function of the same wave record x(t). Then the auto-correlation functions of the wave velocity and acceleration can be calculated as follows [2]. Rx˙ x˙ (τ ) = −

d2 Rx x (τ ) dτ 2

(2.13)

and Rx¨ x¨ (τ ) =

d4 Rx x (τ ) dτ 4

The proof of Eq. (2.13) is given as follows:

(2.14)

58

2 Random Loads Acting on Marine Structures

In this case we consider the wave record stochastic process x(t) to be a weakly stationary stochastic process with finite second moments. Recall that in Chap. 1 we have already given the definition of a weakly stationary stochastic process as follows: If E[x(t)] and E[x(t) x(t + τ )] are both independent of t, then a stochastic process x(t) is said to be weakly stationary. This means that E[x(t)] is a constant and the auto-correlation function E[x(t) x(t + τ )] only depends on τ . In the present case, because E[x(t)] is a constant, and because the operations of expectation and differentiation commute, we have the following result:

d d x(t) = E[x(t)] = 0 E[x(t)] ˙ =E dt dt

(2.15)

We next introduce a variable s = t + τ and calculate the autocovariance function Cov[x(t), ˙ x(t ˙ + τ )] of the derivative process as follows:

d d Cov[x(t), ˙ x(t ˙ + τ )] = Cov[x(t), ˙ x(s)] ˙ = Cov x(t), x(s) dt ds



] d d d d x(t) E x(s) x(s) − E = E dt x(t) ds dt ds d d d = dtd ds E[x(t) x(s)]− E[x(t)] E[ x(s)] dt ds d d d d E[x(t) x(s)] = dt ds E[x(t) x(s)]− 0 = dt ds

(2.16)

Because we have mentioned that E[x(t) x(s)] is independent of t and s, we have the following relations. d d d = =− dt d(s − τ ) dτ

(2.17)

d d d = = ds d(t + τ ) dτ

(2.18)

Combining Eqs. (2.16), (2.17) and (2.18) altogether leads to. Cov[x(t), ˙ x(t ˙ + τ )] = −

d2 d2 E[x(t) x(s)] = − E[x(t) x(t + τ )] dτ 2 dτ 2

(2.19)

On the left side of the above equation Cov[x(t), ˙ x(t ˙ + τ )] can be written as Rx˙ x˙ (τ ). Furthermore, on the right side of the above equation E[x(t) x(t + τ )] can be written as R x x (τ ). Therefore, we have derived the following equation: Rx˙ x˙ (τ ) = −

d2 Rx x (τ ) dτ 2

(2.20)

2.2 Random Waves in the Ocean

59

Obviously, Eq. (2.14) can be derived using a similar derivation process as specified from Eq. (2.15) to Eq. (2.20). In order to obtain the wave velocity spectral density function and the wave acceleration spectral density function from the wave displacement (elevation) spectral density function Sx x (ω), we need to utilize the property regarding the Fourier transform of the time derivative of a function, i.e., if we write the Fourier transform of a stochastic process x(t) as F{x(t)} = X (ω)

(2.21)

We have the following relationships: {∞ X (ω) =

x(t)e −∞

−i ω t

{∞ dt =

{∞ x(t) cos (ω t)dt − i

x(t) sin (ω t)dt

(2.22)

x(t) cos (ω t)dt = −i ωX (ω)

(2.23)

−∞

−∞

{∞

X˙ (ω) = −ω

{∞ x(t) sin (ω t)dt − i ω

−∞

X¨ (ω) = −ω2

−∞

{∞

{∞ x(t) cos(ω t)dt + i ω

2

−∞

x(t) sin (ω t)dt = −ω2 X (ω) (2.24)

−∞

Recall that the definition of the spectral density function (i.e., Sx x (ω)) of the random waves x(t) is as follows [2]. 1 |X (ω)|2 T →∞ 2π T

Sx x (ω) = lim

(2.25)

We can write the wave velocity spectral density function (i. e., Sx˙ x˙ (ω)) and the wave acceleration spectral density function (i. e., Sx¨ x¨ (ω)) in the same form as the wave displacement spectral density function [2]. |2 1 || ˙ X T (ω)| T →∞ 2π T

(2.26)

|2 1 || ¨ X T (ω)| T →∞ 2π T

(2.27)

Sx˙ x˙ (ω) = lim Sx¨ x¨ (ω) = lim

Substituting Eq. (2.23) into Eq. (2.26), the spectral density functions of the wave velocity (i. e., Sx˙ x˙ (ω)) can be calculated as follows: |2 1 1 || ˙ |−i ωX (ω)|2 = ω 2 Sx x (ω) X (ω)| = lim T →∞ 2π T T →∞ 2π T

Sx˙ x˙ (ω) = lim

(2.28)

60

2 Random Loads Acting on Marine Structures

Analogously, the spectral density function of the wave acceleration (i. e., Sx¨ x¨ (ω)) can be calculated as follows: Sx¨ x¨ (ω) = lim

T →∞

|2 |2 1 || 1 || ¨ X (ω)| = lim −ω 2 X (ω)| = ω 4 Sx x (ω) T →∞ 2π T 2π T

(2.29)

2.2.3 Wave Spectrum Mathematical Formulations The calculation principles of the spectral density functions of the wave displacement, velocity and acceleration are given in the previous section. In this section, specific formulas for wave spectral density functions with specific constant parameters will be given. There are many forms of specific formulas for wave displacement (elevation) spectral density functions, which have been derived by meteorologists and other researchers through statistical analysis of wave data collected from visual or actual measurements in different parts of the world. For example, the Pierson-Moskowitz (P-M) displacement spectrum formulation was created through analysis of measured wave data obtained in the North Atlantic by Tucker wave recorders mounted on weather ships. Only those wave data that are considered to be on fully developed seas were selected for the analysis. The one-sided Pierson-Moskowitz displacement spectrum is given by the following equation [2]. { ) } ( g2 g/U 4 Sx x (ω) = A 5 exp −B ω ω

(2.30)

where g is the acceleration of gravity, ω is the wave frequency, A = 8.10 × 10–3 , B = 0.74, and U is the wind speed (m/s) measured on the deck of the ship at 19.5 m above the sea surface. In practice, it would be more convenient to express the wave displacement spectrum as a function of the wave’s significant wave height HS (instead of the wind speed U). For this purpose we integrate the spectral density function in Eq. (2.30) to obtain [2]). {∞ S(ω) dω =

A U4 4B g 2

(2.31)

0

We also assume that the wave displacement spectrum is narrow-banded, the area under the curve of the wave spectrum is equal to (HS /4)2 [2]. {∞ S(ω) dω = (HS /4)2 0

(2.32)

2.2 Random Waves in the Ocean

61

Thus, for fully developed seas, we can obtain the following relationship between the wind speed and the significant wave height from Eqs. (2.31) and (2.32) [2]. √ ( ) ( ) HS = 2 A/B U 2 /g = 0.21 U 2 /g

(2.33)

where A = 8.10 × 10–3 , B = 0.74. Then from Eq. (2.33) and Eq. (2.30) an alternative expression for the one-sided Pierson-Moskowitz displacement spectrum can be derived [2]. Sx x (ω) =

8.10 g 2 −0.032(g/HS )2 /ω4 e 103 ω5

(2.34)

where g is the gravitational acceleration, ω is the wave frequency, and HS is the significant wave height. Figure 2.6 below shows the tendency of the wave displacement spectral density function varying with the wave frequency when the significant wave height is equal to 12 m, and Fig. 2.7 below shows the tendency of the wave displacement spectral density function varying with the wave frequency when the significant wave height is equal to 3 m. We see that the frequency domain for wave energy decreases with the increase of the significant wave height, and the wave frequency value corresponding to the peak value of the spectral density function also decreases with the increase of the significant wave height. Using the principle of Eq. (2.28), we can derive the expression for the one-sided Pierson-Moskowitz velocity spectrum as. 40

Significant height = 12m

35 30

Spectral density

25 20 15 10 5 0 -5 0

1

2

Frequency

Fig. 2.6 Wave displacement spectrum when H S = 12 m

3

4

62

2 Random Loads Acting on Marine Structures Significant height = 3m 1.2

1.0

Spectral density

0.8

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Frequency

Fig. 2.7 Displacement spectrum when H S = 3 m

Sx˙ x˙ (ω) = ω2 Sx x (ω) =

8.10 g 2 −0.032(g/Hs )2 /ω4 e 103 ω3

(2.35)

where g is the acceleration of gravity, ω is the wave frequency, and HS is the significant wave height. Figure 2.8 below shows the tendency of the wave velocity spectral density function varying with the wave frequency when the significant wave height is equal to 12 m, and Fig. 2.9 below shows the tendency of the wave velocity spectral density function varying with the wave frequency when the significant wave height is equal to 3 m. We see that the frequency domain corresponding to the wave velocity spectrum shrinks as the significant wave height increases, and the wave frequency value corresponding to the peak value of the velocity spectral density function decreases with the increase of the significant wave height. Using Eq. (2.29) we can derive the expression for the one-sided P-M acceleration spectrum as. Sx¨ x¨ (ω) = ω4 Sx x (ω) =

8.10 g 2 −0.032(g/HS )2 /ω4 e 103 ω

(2.36)

where g is the acceleration of gravity, ω is the wave frequency, and HS is the significant wave height. Figure 2.10 below shows the tendency of the wave acceleration spectral density function varying with the wave frequency when the significant wave height is equal to 12 m, and Fig. 2.11 below shows the tendency of the wave acceleration spectral density function varying with the wave frequency when the significant wave height is equal to 3 m. We see that the frequency domain corresponding to the wave acceleration spectrum shrinks as the significant wave height increases, and the wave

2.2 Random Waves in the Ocean

63

Significant height = 12m

6

5

Spectral density

4

3

2

1

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

3.5

4.0

4.5

Frequency Fig. 2.8 Wave velocity spectrum when H S = 12 m Significant height = 3m 0.7 0.6

Spectral density

0.5 0.4 0.3 0.2 0.1 0.0 -0.1 0.0

0.5

1.0

1.5

2.0

2.5

Frequency

Fig. 2.9 Wave velocity spectrum when H S = 3 m

3.0

64

2 Random Loads Acting on Marine Structures

Significant height = 12m

1.2

Spectral density

1.0

0.8

0.6

0.4

0.2

0.0 0

2

4

6

8

10

Frequency Fig. 2.10 Wave acceleration spectrum when H S = 12 m

frequency value corresponding to the peak value of the acceleration spectral density function decreases with the increase of the significant wave height. There are other mathematical formulations for the wave spectral density functions, such as ITTC spectrum, ISSC spectrum, etc. An introduction to the various wave spectral density function formulations can be found in the respective monographs of S. K. Chakrabarti [3], Michel K. Ochi [2] and O. F. Hughes [4].

2.2.4 Numerical Simulation of Random Ocean Waves In the previous sections, the principles of the mathematical representation of random ocean waves are described, and we see that the key task is the establishment of the wave spectral density function, and the theory in this area can now be deemed to be well established. In practice, naval architects and ocean engineers are often faced with another type of task: the simulation of stochastic ocean waves based on a known wave spectral density function. For this task, oceanographers have found that the irregularity of the sea surface can be represented by the superposition of an infinite number of regular sine waves with different wave heights, wave lengths and random phase angle differences [3]. This model is based on an integral expression of Fourier-Stieltjes type and the concept of an infinitesimal random wave [5]

2.2 Random Waves in the Ocean

65

0.6

Significant height = 3m

Spectral density

0.5

0.4

0.3

0.2

0.1

0.0 0

2

4

6

8

10

frequency Fig. 2.11 Acceleration spectrum when H S = 3 m

{∞ η(t) =

d B(ω)e−i ω t

(2.37)

−∞

In order to be suitable for practical applications, the above integral is generally replaced by an infinite summation, which is often truncated to further become a finite summation. There are two variations of this model, the first form contains deterministic amplitudes and random phase angles and is known as the random phase angle model; the second form contains non-deterministic amplitudes and is called the random amplitude model, with the random phase angle model being more popular in the offshore industry. In the random phase angle model, the amplitude of each regular sine wave is considered to be fully deterministic and is calculated based on the value of the spectral density function at discrete frequency points. Thus, the wave displacement (elevation) process at a point on a long-crested irregular sea surface can be calculated by the following equation [5]. η(t) =

N −1 ∑

Cn cos[n(Δω)t + Φn ]

(2.38)

n=0

where the deterministic amplitude is determined by the following equation [5].

66

2 Random Loads Acting on Marine Structures

Cn =

√

2Sη0 η0 [n(Δω)]Δω

(2.39)

In Eqs. (2.38) and (2.39), N is a large positive number that can be accepted in practical engineering problems, Δω is the frequency spacing, Sη0 η0 [n(Δω)] is the one-sided power spectral density function of the wave water particle elevation, Φ0 , Φ1 , Φ2 ,.., Φ N −1 are independent random phase angles uniformly distributed between 0 and 2π. Equations (2.37) to (2.39) are the steps to simulate the wave surface displacements (elevations) based on a known wave displacement spectrum. Equations for simulating the wave water particle horizontal velocity u and horizontal acceleration u˙ can be derived as follows: Assuming that each of the above regular sine waves conforms to the linear gravity wave theory, let z be the vertical coordinate in the given coordinate system, here is the height of the wave water particle from the mean still water surface, and set the water depth as d, then the horizontal velocity u n of the wave water particle is [5]. u n = ωn

cosh k(z + d) ηn sinh kd

(2.40)

From Eqs. (2.38–2.40) it can be deduced that u n = n(Δω)

cosh k(z + d) √ 2Sη0 η0 [n(Δω)]Δω cos[n(Δω)t + Φn ] sinh kd

(2.41)

In Eqs. (2.40) and (2.41) k is the wave number, and satisfies [5]. k tanh kd =

ω2 g

(2.42)

where ω is the wave frequency and g is the gravitational acceleration. In deep water, i.e., when d → ∞ Eq. (2.42) can be simplified as k=

ω2 g

(2.43)

Also, when d → ∞. cosh k(z + d) 2 → e|k|z = eω z/g sinh kd

(2.44)

Finally, the horizontal velocity u of the wave water particle in deep water irregular waves can be simulated as [5] u(t) =

N −1 ∑ n=0

√ 2 [n(Δω)] 2Sη0 η0 [n(Δω)]Δω e[n(Δω)] z/g cos[n(Δω)t + Φn ]

(2.45)

2.3 Fluid Forces on Slender Offshore Structures

67

Similarly, the horizontal acceleration of the wave water particle in deep water irregular waves can be simulated as [5]. u(t) ˙ =−

N −1 ∑

√ 2 [n(Δω)]2 2Sη0 η0 [n(Δω)]Δω e[n(Δω)] z/g sin[n(Δω)t + Φn ] (2.46)

n =0

The above explains the random phase angle model for simulating stochastic ocean waves. In the following we describe the random amplitude model for simulating stochastic ocean waves. In the random amplitude model, the wave surface displacement (elevation) process at a point on a long-crested irregular sea surface can be represented by the following equation [5]. η(t) =

N −1 ∑

{An cos[n(Δω)t] + Bn sin[n(Δω)t]}

(2.47)

n =0

where the coefficients An and Bn are two independent random variables with zero mean and their standard deviations are given by the following equation [5]. σ An = σ Bn =

√

2Sη0 η0 [n(Δω)]Δω

(2.48)

This model was shown to better represent the wave group effect. These simulation procedures are the main steps in the Monte Carlo method for calculating the motion response of marine structures, because the horizontal velocity and acceleration processes of the wave water particles are directly used in the calculation of the fluid loads on the marine structures, as will be seen in the next section.

2.3 Fluid Forces on Slender Offshore Structures 2.3.1 Morison Formula Some types of offshore platforms contain many slender structural members, such as the vast majority of underwater structural members on a jacket platform, most structural members underwater on a jack-up platform, and various structural members underwater on a Tension Leg Platform. When the ratio of the characteristic dimensions of these slender structural members (e.g., the pipe diameter, etc.) to the wavelength is less than a critical value, the fluid loads acting on these slender structural members can be calculated using the Morison formula. Through an experimental study of the wave force on a pile and a heuristic approximation of the measured force, Morison and co-workers obtained the Morison formula [5]. The Morison formula is now widely used by the industry to calculate fluid loads on submerged slender

68

2 Random Loads Acting on Marine Structures

structural members on offshore platforms. A joint industry study was conducted by Standard Oil, Shell R&D and Exxon Production R&D to verify the validity of the Morison formula [5]. The joint study measured the actual forces on structures in the 30- and 100-foot water depths in the Gulf of Mexico during several periods of high hurricane and wave activity. The measured data indicate that this method of calculating fluid loads on slender structures using the Morison formula is correct and reasonable. The measured data also provide a calibration of the coefficients of the Morison formula for random waves. The findings of this study were published in a series of papers [5]. The Morison formula takes into account both viscous and inertial forces in an unsteady flow. According to this formula, the combined force acting on the slender structural member is equal to the algebraic sum of the drag force and the inertia force.

2.3.2 Morison Drag Force The Morison drag force is the part of the viscous contribution to the combined force that incorporates boundary layer and flow separation effects. The Morison drag force depends on the fluid velocity in a squared way and on the projected surface area of the structure in a linear way, and the drag force applied on each unit length of the slender structural body is [5]. Fd =

1 ρ C D D|u|u 2

(2.49)

where ρ is the fluid density, D is the diameter (or other characteristic size) of the slender structural member, u is the undisturbed fluid velocity, and C D is the drag force coefficient determined by the test.

2.3.3 Morison Inertia Force The Morison inertia force is the part of contribution to the combined force due to the fluid acceleration. This inertia force incorporates the added mass effect and the Froude Krylov force. The Morison inertia force depends on the fluid acceleration and the cross-sectional area of the structural member, and the Morison inertia force applied to each unit length of the slender structural body is [5]. Fi = ρ C M A S u˙

(2.50)

where ρ is the fluid density, A S is the cross-sectional area of the slender structural member, u˙ is the fluid acceleration, and C M is the inertia force coefficient associated with the geometry of the structural member. The inertia force coefficient is generally

2.3 Fluid Forces on Slender Offshore Structures

69

determined experimentally, and the coefficient can be derived theoretically only in some very special cases.

2.3.4 Combined Forces According to the Morison model, the combined force acting on the structure is equal to the sum of the drag force and the inertial force, and thus the total hydrodynamic force acting on each unit length of the slender structural member in an unsteady flow is [5]. FT = Fd + Fi =

1 ρC D D|u|u + ρC M A S u˙ 2

(2.51)

To perform the calculations in the above equation, the velocity and acceleration values in this equation must first be known, and these values should be taken as the velocity and acceleration values of the flow field assuming that the structural member is not present. This simplification implies that the structural element does not significantly affect the flow field, which is obviously only valid for slender structural elements.

2.3.5 Generalized Form of the Morison Formula The Morison formula is derived from fixed piles. When analyzing the forces on a moving structure, the absolute values of the velocity and acceleration of the wave water particle should not be considered, but the relative values of the velocity and acceleration between the wave water particle and the moving structure should be considered. Thus, the Morison formula becomes [5]. FT = Fd + Fi =

1 ρ C D D|u − x|(u ˙ − x) ˙ + ρ C M A S u˙ − ρ C A A S x¨ 2

(2.52)

where x represents the displacement of the moving structure. C A is the added mass coefficient, C A = C M − 1. The symbol (˙) represents the derivative with respect to time, and the same Morison drag force coefficient and inertia force coefficient are used for the calculation. We see that the wave water particle velocity u and acceleration u˙ in the above equation are two stochastic processes, and they can be simulated using Eqs. (2.45) and (2.46).When currents are considered, the current values should also be reflected in the expressions used to calculate the Morison drag force [5].

70

2 Random Loads Acting on Marine Structures

FT = Fd + Fi =

1 ρ C D D|u + U − x|(u ˙ + U − x) ˙ + ρ C M A S u˙ − ρ C A A S x¨ 2 (2.53)

In the above equation, U represents the current value. Finally, it should be noted that the Morison formula is only applicable when the ratio of the slender structural member’s characteristic dimension D to the wavelength λ is less than or equal to 0.2 [5]. After the fluid load is calculated by Morison formula, the damping and restoring forces on an offshore structure are also calculated to establish the differential equation of the rigid body motion of the offshore structure. By solving this equation, the motion response of the offshore structure can be obtained.

2.4 Other Types of Random Wave Loads on Marine Structures 2.4.1 Second-Order Slow-Drift Stochastic Excitations on Compliant Offshore Structures In order to meet the needs of offshore oil development to the deep sea, some types of compliant offshore structures have been developed and designed. The representative types of compliant structures include semi-submersible platforms and floating production storage and offloading systems with catenary moorings, and Tension Leg Platforms with Tension Leg moorings. By designing the mooring system of these structures with low stiffness, the natural frequency of these compliant offshore structures will be much lower than the frequency of first-order wave forces. However, the natural frequencies of these structures are difficult to avoid coinciding with the second-order slow-drift wave force frequencies, so that they tend to resonate harmonically under the second-order slow-drift wave forces resulting in large motion amplitudes. We now give an example to illustrate the treatment of second-order slow-drift wave forces. Naess A et al. [6] analyzed the extreme surge response of a Tension Leg Platform. The spectral density function value of the wave drift force (F(t)) per unit mass of the platform they used is S0 = 2.82 × 10–4 m2 s−3 , and thus F(t) is a random excitation term. Let us first elucidate the procedures for generating the time history of the second-order slow-drift wave forces F(t). Let us consider a 1D-1V stationary stochastic process f 0 (t) with zero mean, autocorrelation function R f0 f0 (τ ) and two-sided power spectral density function S f0 f0 (ω). When N → ∞, the stochastic process f 0 (t) can be simulated by the following series (Shinozuka and Deodatis [7]). f (t) =

N −1 √ ∑ 2 An cos(ωn t + Φn ), n=0

(2.54)

2.4 Other Types of Random Wave Loads on Marine Structures

71

in which An = (2S f0 f0 (ωn )Δω)1/2 , n = 0, 1, 2, ... N − 1

(2.55)

ωn = nΔω, n = 0, 1, 2, ... N − 1,

(2.56)

Δω = ωu /N

(2.57)

In Eq. (2.57), ωu stands for an upper cutoff frequency beyond which it can be assumed, either mathematically or physically, that the power spectral density function S f0 f0 (ω) is zero. In Eq. (2.54), Φ0 , Φ1 , Φ2 , …, Φ N −1 are independent random phase angles uniformly distributed between 0 and 2π. Replacing the sequence of random phase angles Φ0 , Φ1 , Φ2 , …, Φ N −1 with their respective i-th realizations ϕ0(i) , ϕ1(i) , ϕ2(i) ,.., ϕ N(i)−1 , A sample function f i (t) of the simulated stochastic process f (t) can be obtained (Shinozuka and Deodatis [7]). f i (t) =

N −1 √ ∑ 2 An cos(ωn t + ϕn(i) )

(2.58)

n=0

It should also be noted here that when the sample functions for the simulated stochastic process are generated according to Eq. (2.58), the time step Δt separating the generated f i (t) values in the time domain should satisfy the following requirement (Shinozuka and Deodatis [7]). Δt ≤ 2π/2ωu

(2.59)

Equations (2.54–2.59) describe the whole procedures of simulating the stochastic process f 0 (t). We generated a series of random numbers using a uniform [0, 1] random number generator, and then multiplied each of the generated random numbers by 2π to obtain the random phase angle ϕn in Eq. (2.58). The afore-mentioned procedures were then used in the simulation to generate a time history (as shown in Fig. 2.12) of the wave drift force F(t) with spectral density function value S0 = 2.82 × 10–4 m2 s−3 . The time histories (as shown in Figs. 2.12, 2.13, 2.14,2.15) of the wave drift force F(t) with the spectral density function value S0 = 2.82 × 10–4 m2 s−3 were generated similarly. With the above time histories of stochastic excitations of second-order slow-drift wave forces, the extreme response of the slow-drift surge motions of compliant offshore structures can be studied next.

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2 Random Loads Acting on Marine Structures

0.3

wave drift force (N)

0.2

0.1

0.0

-0.1

-0.2

20

30

40

50

60

70

time (sec)

Fig. 2.12 Time history 1 of the second order slow drift wave force random excitation

0.3

Wave drift force (N)

0.2

0.1

0.0

-0.1

-0.2

-0.3

20

30

40

50

60

70

Time (sec)

Fig. 2.13 Time history 2 of the second order slow drift wave force random excitation

2.4.2 Spectral Density Function of Random Wave Loads on Marine Structures The calculation of the spectral density function for random wave loads on general purpose marine structures (both large volumed or slender shaped) is presented below. For this purpose, the response amplitude operator (RAO) of the marine structure should be calculated first. There exists a rigorous hydrodynamic theory about the response amplitude operator (RAO), but it is beyond the scope of this book, and

2.4 Other Types of Random Wave Loads on Marine Structures

73

0.3

wave drift force (N)

0.2 0.1 0.0 -0.1 -0.2 -0.3 20

30

40

50

60

70

time (sec)

Fig. 2.14 Time history 3 of the second order slow drift wave force random excitation

0.3

wave drfit force (N)

0.2 0.1 0.0 -0.1 -0.2 -0.3 20

30

40

50

60

70

time (sec) Fig. 2.15 Time history 4 of the second order slow drift wave force random excitation

it is assumed here that suitable commercial hydrodynamic software is available to calculate the response amplitude operator (RAO) of the marine structure, or model tests can be conducted to derive the response amplitude operator (RAO) of the marine structure. Then, there is a concise expression between the spectral density function of the random wave loads on the marine structure and the spectral density function of the wave displacements as follows [3].

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2 Random Loads Acting on Marine Structures

SL (ω) = [R AO(ω)]2 Sx x (ω)

(2.60)

where SL (ω) is the spectral density function of the random wave loads on the marine structure and Sx x (ω) is the spectral density function of the wave displacements. ω represents the encountered wave frequency.

2.4.3 Simplified Procedure In some cases researchers have modeled the wave external excitations as harmonic excitations superposing with a Gaussian white noise [8]. This is allowed during the initial design phase of the marine structure, when the detailed design parameters of the marine structure have not been finalized, the response amplitude operator of the marine structure has not been obtained by detailed calculations, or the response amplitude operator of the marine structure has not been obtained by model tests in wave tanks. Adding a Gaussian white noise to approximate the actual random wave excitations allows for a quick evaluation of the response and stability performance of the marine structure in order to devise refinement proposals and move to the next cycle of the design. If the random wave loads acting on a marine structure is denoted by L, then we have [8]. L(t) = γ cos (ω t)+η(t)

(2.61)

where γ is the amplitude of the harmonic excitation and ω represents the encountered wave frequency. η (τ ) is an ideal, zero-mean and δ-related Gaussian white noise. This additional Gaussian white noise approximates ⟩ the perturbation to the external ⟨ harmonic excitation forces. ⟨η(τ )⟩ = 0. η(τ , )η(τ ) = kδ(τ , − τ ). δ(...) is the Dirac δ function and k represents the Gaussian white noise intensity. Please note that a Gaussian white noise is defined as a stochastic process with a constant value spectral density function.

2.5 Summary This chapter introduces the mathematical formulations of stochastic ocean waves, which focuses on the theoretical backgrounds of the wave displacement, velocity and acceleration spectral density functions. A wave spectrum model is presented, and details of the principles for simulating stochastic ocean waves have been elucidated. Then, this chapter briefly introduces the principles of calculating the fluid forces on slender offshore structures. Next, this chapter briefly describes how other kinds of random wave loads on marine structures can be treated, giving an example of simulating the second-order slow-drift wave forces on compliant offshore structures,

Exercises

75

which is a prerequisite for our study of the extreme response of slow-drift surge motions of compliant offshore structures. Next, this chapter presents an equation to calculate the spectral density function of random wave loads on marine structures when the response amplitude operator is known, and finally a simplified procedure for calculating random wave loads is presented.

Exercises Exercise 2.1 Recall that in the text in this Chapter the auto-correlation function, Rxx (τ ), of the random waves x(t) is defined as: 1 R x x (τ ) = lim T →∞ 2T

{T x(t)x(t + τ )dt

(E.2.1)

−T

As seen in the above equation, the product of two readings taken from a same record and separated by a time shift τ is used to calculate the auto-correlation function. Based on this definition, prove that: 1. The function Rx x (τ ) is even, i.e., Rx x (τ ) = Rx x (−τ ). 2. When τ = 0, Rx x (τ ) has the maximum value, i.e., Rx x (0) ≥ Rx x (τ ). Exercise 2.2 Recall that in the text in this Chapter we have mentioned: X(ω) is the Fourier transform of x(t), i. e.: {∞ X (ω) =

x(t)e−i ω t dt

(E.2.2)

−∞

Let us denote the complex conjugate of X (ω) to be X ∗ (ω). Utilizing the classical Fourier transform theory to prove that X ∗ (ω) = X (−ω)

(E.2.3)

Exercise 2.3 In dealing with practical problems, it is important to derive the wave acceleration spectral density function (in the vertical direction) from the wave displacement (elevation) spectral density function. Let R x x (τ ) represents the autocorrelation function of a wave elevation record x(t), show that the auto-correlation function of the wave acceleration can be calculated as follows: R x¨ x¨ (τ ) =

d4 Rx x (τ ) dτ 4

(E.2.4)

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2 Random Loads Acting on Marine Structures

Exercise 2.4 δ(...) is the Dirac δ function whose value is zero everywhere except at t = 0, where the value of δ(0) is infinite. {∞ δ(x) d x = 1

(E.2.5)

−∞

Let f (x) be a continuous function at x = a, show that {∞ δ(x − a) f (x) d x = f (a)

(E.2.6)

−∞

δ(ax) =

1 δ(x) |a|

(E.2.7)

δ(−x) = δ(x)

(E.2.8)

F{δ(x − a)} = e−i ωa

(E.2.9)

F{δ(x)} = 1

(E.2.10)

F −1 {δ(ω − ω0 )} =

1 i ω0 t e 2π

F −1 {2π δ(ω)} = 1

(E.2.11) (E.2.12)

References 1. Kareem A, Zhao J, Tognarelli MA (1995) Surge response statistics of tension leg platforms under wind and wave loads: a statistical quadratization approach. Probab Eng Mech 10: 225-240 2. Ochi MK (1998) Ocean waves, the stochastic approach. Cambridge University Press, Cambridge 3. Chakrabarti, SK (1987) Hydrodynamics of offshore structures, Springer-Verlag, Berlin. 1987. 4. Hughes OF (1983) Ship structural design, a rational-based, computer-aided, optimization approach. John Wiley and Sons, New York 5. Bhattacharjee S (1990) Filter approaches to stochastic dynamic analysis of compliant offshore platforms, PhD dissertation, Department of Mechanical Engineering, Rice University 6. Naess A, Galeazzi F, Dogliani M (1992) Extreme response predictions of nonlinear compliant offshore structures by stochastic linearization. Appl Ocean Res 14(2): 71-81 7. Shinozuka M, Deodatis G (1991) Simulation of stochastic processes by spectral representation. Appl Mech Rev 44(4): 191- 204 8. Lin H and Yim CS (1995) Chaotic roll motion and capsize of ships under periodic excitation with random noise. Appl Ocean Res 17: 185-204

Chapter 3

Monte Carlo Simulation Method for Dynamic Response of Marine Structures

3.1 Introduction Having obtained the added mass of an offshore structure as well as the damping, restoring forces and external excitations on the offshore structure, the rigid body differential equation of motion of the offshore structure can then be established. Because the wave excitations are random, the differential equation of motion of the offshore structure is a stochastic differential equation, which in general has no analytical solutions. In this chapter, a commonly used numerical method for solving the differential equations of motion of offshore structures, Monte Carlo Simulation (MCS), is introduced and used to calculate the motion responses, response statistics and level up-crossing rates of the motion responses of an offshore structure. This chapter will also present the deficiencies of the Monte Carlo Simulation method for analyzing the slow-drift extreme response and stability of marine structures under stochastic excitations. In order to study the slow drift extreme response of a marine structure under random excitations, a dynamic analysis of the marine structure is required. In the dynamic analysis of the marine structure, the added mass of the marine structure, the damping and restoring forces on the marine structure, and the wave excitation force (or moment) per unit wave amplitude on the marine structure can all be calculated by some well-established commercial hydrodynamic software. With all these information, the rigid body stochastic differential equations of motion of the marine structure can then be established and solved by some numerical methods, of which the MCS is a method frequently used by naval architects and offshore engineers. The principles of Monte Carlo Simulation are very simple and straightforward— i.e., to perform numerical integration of the differential equations of motion and then to statistically process the dynamic response results. One can use the MCS method to derive the entire probability density curve of the responses together with the extreme slow drift responses of an offshore structure. Many studies have been reported on the use of Monte Carlo Simulation to find the slow-drift response statistics of marine structures under random excitations, and only a few typical examples © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5_3

77

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3 Monte Carlo Simulation Method for Dynamic Response of Marine …

are given below. Langley [1] used a finite number of superimposed trigonometric functions with different amplitudes, frequencies and random phase angles to generate random waves, and studied the response statistics of a moored ship under secondorder slow-drift wave forces using Monte Carlo simulations. Winterstein et al. [2] used Monte Carlo simulations to analyze the slow drift response of a Tension Leg Platform under second-order wave forces in long-crested heading sea conditions. They obtained the level up-crossing rates of the slow drift surge responses of the Tension Leg Platform for a given sea state, and also the extreme values of the secondorder slow drift surge responses of the Tension Leg Platform for a given sea state corresponding to different time periods. Emmerhoff and Sclavounos [3] studied the large-amplitude slow-drift motions of an offshore structure under random excitations using Monte Carlo simulations. They generated random wave signals by filtering Gaussian white noise and used the Runge–Kutta method to numerically integrate the differential equations of motion to obtain statistical information on the responses of the offshore structure. However, it has been mentioned in their study [3] that the Monte Carlo simulations are very time consuming. Garrett [4] studied the dynamic responses (including second-order low-frequency motion responses) of a large semisubmersible platform moored in 1800 m deep water in the Gulf of Mexico under random wave excitations using Monte Carlo simulations, in which he pointed out that the Monte Carlo simulations are very time-consuming. Low [5] used a hybrid method to forecast the response extremes of a catenary moored floating structure, in which the stochastic wave environment was described by a three-parameter longcrested JONSWAP wave spectrum. An important strategy of this hybrid method is to use the Monte Carlo Simulation to analyze and forecast the second-order lowfrequency responses of the floating structure and to take relatively large time steps in the numerical integration to improve the computational efficiency. However, the computational time cost of this hybrid method is still on the high side [5], and a large error occurs when a simplified synthetic formulation is used to predict some dynamic response extremes of the floating structure. Proppe et al. [6] also pointed out that direct Monte Carlo Simulation is not suitable for providing values in the tail region of the response probability distribution (i.e., response extremes) required in reliability analysis. It is worth noting that although some of the above studies [3] have mentioned that the Monte Carlo Simulation method is time consuming, none of them have further investigated the effect of simulation efficiency on the accuracy of the extreme responses, i.e., none of them have so far considered the two issues of simulation efficiency and the accuracy of the extreme response prediction in an integrated manner. Obviously, if the accuracy of extreme response prediction is to be improved, the number of simulations should continue to increase, which will further reduce the efficiency of the simulation. So far this problem has not attracted enough attention, probably because the principle of the Monte Carlo Simulation method is very simple and the implementation of the Monte Carlo Simulation method is very straight-forward, and the easy-to-use method is easy to be accepted and adopted. In this chapter, we will conduct an in-depth study on this issue. This chapter will first show by examples that the accuracy of the simulation prediction of the extreme responses of the surge oscillations of marine structures is not high, and in order to

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

79

improve the accuracy of the extreme response prediction, the number of simulations will be increased in this study. This study will show that after the accuracy of the extreme response is improved to a certain level, it is almost useless to continue to increase the number of simulations, while the accuracy of the extreme response value has not yet reached a satisfactory level. We know that when a marine structure is in the rolling motion, if the extreme rolling response of the marine structure exceeds a critical value, the marine structure will lose its stability and capsize. As one of the most common marine structures, the capsizing behavior of ships is familiar to the naval architects and marine engineers. In order to rationally analyze the stability of marine structures (ships), some people are often willing to use the Monte Carlo Simulation method as well. Some researchers have pointed out the weakness of the low efficiency of the MCS method to analyze the capsizing behavior of marine structures, but people still continue to use it today. The reason for this is probably because the principle of the Monte Carlo Simulation method is very simple and the implementation of the MCS method is very convenient, and the simple and easy-to-use method is easy to be accepted and popularized. In this chapter, it will be quantitatively shown that it is inefficient to analyze the stability of marine structures by the Monte Carlo Simulation method. It will also be analytically shown that it is generally difficult to draw conclusions about the stability of marine structures based on the results of Monte Carlo simulations unless the initial conditions of the motion and the parameter space of the marine structure system are exhaustively studied.

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo Simulation Method 3.2.1 Nonlinearities Handled by the Monte Carlo Simulation Method A method widely used in industry to deal with the response of marine structures under random loads is spectral analysis in the frequency domain, which is simple to understand and easy to implement in engineering practice. However, spectral analysis in the frequency domain is based on the assumption of linearity, while the rigid body equations of motion of deep-water marine structures are generally nonlinear in a strictly physical sense. These nonlinearities come from two sources: on the one hand from the nonlinear material properties of the structure, the nonlinear interactions between the structure and the sea floor rock and soils, the nonlinear behaviors of the catenary moorings and tension legs, etc. These nonlinearities can be collectively referred to as structural nonlinearities (or system nonlinearities). On the other hand, the nonlinearities also arise from nonlinear external excitations. For example, from the introduction of the previous chapter, it can be seen that the Morison drag force is a nonlinear function of the wave water particle velocity. These nonlinearities can be

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3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Fig. 3.1 A Tension Leg Platform (TLP)

collectively referred to as excitation nonlinearities (or external force nonlinearities). It is of practical importance to be able to solve these nonlinear stochastic differential equations of motion for deepwater marine structures directly in the time domain. The Monte Carlo Simulation is an effective method for solving nonlinear stochastic dynamic problems in the time domain, and this method has been used to analyze the motion response of deepwater marine structures in a number of applications. Spanos and Agarwal [7] studied the response of a simplified Tension Leg Platform (TLP) under wave forces using the Monte Carlo Simulation method. A TLP is a compliant vertically moored offshore structure. In the field of ocean engineering, a compliant offshore structure is defined as the one that is flexibly connected to the seafloor and can move freely with the waves. Figure 3.1 depicts a TLP under the influence of a unidirectional random wave field propagating along its longitudinal axis. A TLP’s mooring is provided by a number of vertical steel tubes (tendons) that extend to pile-anchored templates on the ocean floor and are located at each of the TLP’s corner columns. The platform’s excess buoyancy keeps the mooring system taut so that the vertical tendons never become slack. The tendons do a good job of controlling the TLP’s heave, pitch, and roll motions. However, the surge, sway, and yaw motions in the horizontal plane are very compliant with the motion of the waves. In the study of Spanos and Agarwal [7], the Tension Leg Platform is modeled as a single-degree-of-freedom oscillator driven by wave loads. The elastic force in their

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

81

vibration equation is linear and equal to the product of the mooring line stiffness and the surge displacement. The wave forces in their equations are calculated using the Morison formulation without the current term. The time history of a typical steady-state responses when the system is subjected to deterministic excitations is obtained. For the case subjected to random wave excitations, they carried out a series of Monte Carlo simulations to obtain a typical mean and standard deviation record of the structural responses. Kareem et al. [8] have also used the Monte Carlo Simulation method in their study of the response of a Tension Leg Platform. They applied a standard single-degree-offreedom control equation to find the surge oscillatory responses of this Tension-Leg Platform under wave and current actions. The Morison drag force equation they used in their study included a current term, however, the elastic force in their control equation was still assumed to be linear. The above are two examples that contain only excitation nonlinearities. Tognarelli et al. [9] studied a single-degree-of-freedom offshore system containing a statistically symmetric nonlinear stiffness term and a statistically symmetric nonlinear excitation term, and they had also used Monte Carlo simulations in the course of their research. However, their formulation for calculating the Morison wave drag force did not include the sea current term. In this chapter, a single-degree-of-freedom equation of motion is used to model a catenary moored offshore platform exposed to viscous hydrodynamic loads. The platform, column, and pontoon that make up the hull of this type of offshore platform can be thought of as a rigid body when analyzing it. The centroid of the offshore platform’s mass is assumed to be the point at which all forces act in order to study only the surge (x direction) motion. It is then possible to model the offshore platform as a nonlinear, single-degree-of-freedom system that is subject to wave and current loads. In our study, the offshore platform equation will contain a Duffing stiffnesstype nonlinearity and a Morison drag-type nonlinearity with a current term. To the best knowledge of the author of this book, this is the first time that Monte Carlo Simulation has been used to deal with such a pair of nonlinearities. The time history of the surge displacement responses, the response probability density curve and the level up-crossing rates of the response values for this offshore platform will be calculated. The effect of the currents on the response central moments up to the fourth order will also be investigated in the calculation of the probability density curves. Some new findings will be pointed out after analyzing the simulation results.

3.2.2 Differential Equation of Motion for an Offshore Platform The equation of motion used by Tognarelli et al. [9] in the analysis of the surge oscillatory response of a catenary moored offshore platform exposed to viscous hydrodynamic loads is

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3 Monte Carlo Simulation Method for Dynamic Response of Marine …

) ( M x¨ + C x¨ + K x + εx 3 = K M u˙ + K D |u − x|(u ˙ − x) ˙

(3.1)

In Eq. (3.1), M is the mass of the platform, x is the horizontal displacement response of the platform, C is the damping of the platform, K is the stiffness coefficient of the mooring system, εx 3 represents the Duffing stiffness, and the symbol (˙) represents the derivative with respect to time. K M = ρC M V E , K D = 0.5ρC D AE , where ρ is the sea water density, C M is the Morison inertia force coefficient, C D is the Morison viscous drag force coefficient. V E represents the effective volume of the submerged portion of the platform structure and AE represents the effective area of the submerged part of the platform structure. u and u˙ are the velocity and acceleration in the horizontal direction of the wave water particle, respectively. The spectral density function of the velocity u in the horizontal direction of the wave water particle is linked to the wave elevation spectrum (e.g., P-M spectrum) by a linear transfer function. Because the wave water particle velocity u and acceleration u˙ are two stochastic processes, Eq. (3.1) is a stochastic differential equation. Because offshore tidal currents often reach a maximum velocity of 1–2 knots (0.5144–1.0288 m/s), and even 10 knots (5.144 m/s) at some locations [10], it is not reasonable to exclude the current term from the Eq. (3.1) for calculating the Morison drag forces. Therefore, the correct differential equation of motion for the offshore platform should be ˙ M x¨ + C x˙ + K (x + εx 3 ) = K M u˙ + K D |u + U − x|(u + U − x) ˙

(3.2)

where U represents the value of the sea current velocity. The stochastic differential Eq. (3.2) has no analytical solutions, and currently it can only be solved by some approximate or numerical solutions. The principle of solving the Eq. (3.2) numerically by the Monte Carlo Simulation method is given below. Consider a 1D-1V stationary stochastic process f 0 (t) with zero mean, an autocorrelation function R f0 f0 (τ ) and a one-sided power spectral density function S f0 f0 (ω). When N → ∞, the stochastic process f 0 (t) can be simulated by the following series (Shinozuka and Deodatis [11]): f (t) =

N −1 √ ∑ 2 An cos(ωn t + Φn ),

(3.3)

n=0

among which An = (S f0 f0 (ωn )Δω)1/2 , n = 0, 1, 2, ... N − 1

(3.4)

ωn = nΔω, n = 0, 1, 2, ... N − 1,

(3.5)

Δω = ωu /N

(3.6)

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83

In Eq. (3.6), ωu represents an upper cutoff frequency above which the power spectrum S f0 f0 (ω) can be assumed to be zero for either physical or mathematical reasons. Typically, the value of ωu is estimated using the criterion listed below: ωu

∞ S f0

f 0 (ω)

dω = (1− ∈)

0

S f0

f 0 (ω)

dω

(3.7)

0

in which ∈ ≤ 1 (e. g., ∈ = 0.001). In Eq. (3.3), Φ0 , Φ1 , Φ2 , …, Φ N −1 are independent random phase angles uniformly distributed between 0 and 2π. Replacing the ith realization ϕ0(i) , ϕ1(i) , ϕ2(i ) , . . . , ϕ N(i )−1 of each of the random phase angle sequences Φ0 , Φ1 , Φ2 , …, Φ N −1 by themselves yields a sample function f i (t) of the simulated random process f (t) (Shinozuka and Deodatis [11]). f i (t) =

N −1 √ ∑ 2 An cos (ωn t + ϕn(i ) )

(3.8)

n=0

It should also be noted here that when the sample functions for the simulated stochastic process are generated according to Eq. (3.8), the time steps separating the generated f i (t) values in the time domain should satisfy the following requirements (Shinozuka and Deodatis [11]). Δt ≤ 2π/2ωu

(3.9)

The process of Eqs. (3.3)–(3.9) is the process of using Monte Carlo simulations to generate a time history of a stochastic process. The Monte Carlo Simulation method can provide exact solutions for nonlinear stochastic dynamic problems and can evaluate the accuracy of other approximation methods such as the perturbation method. Using the above Monte Carlo Simulation method, and using Eqs. (2.43) and (2.44), the horizontal velocity u of the wave water particle in deep water irregular waves can be simulated as u(t) =

N −1 ∑

√ [n(Δω)] 2Sη0

η0 [n(Δω)]Δω

e[n(Δω)]

2

z/g

cos[n(Δω)t + ϕn ]

(3.10)

n=0

Similarly, it can be derived that the horizontal acceleration of the wave water particle in deep water irregular waves can be simulated as u(t) ˙ =

N −1 ∑ n=0

√ [n(Δω)]2 2Sη0

η0 [n(Δω)]Δω

2

e[n(Δω)]

z/g

sin[n(Δω)t + ϕn ] (3.11)

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3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Substituting Eqs. (3.6), (3.10) and (3.11) into (3.2), we finally obtain the differential equation for the surge oscillatory motion of the offshore platform as   M x¨ + C x˙ + K x + εx 3 = ⎫ ⎧   2 / −1  ⎨ N∑  ω  ω

⎬ ω  ωu  2 z/g n ωNu u u u n e t + ϕn 2Sη0 η0 n sin n Km − ⎭ ⎩ N N N N n=0 | |   2 | | N −1   /  ω  ω

ω  ωu | |∑ ω z/g n u u u u N n e t + ϕn + U − x˙ || cos n 2Sη0 η0 n + K d || N N N N | | n=0 ⎧ ⎫   / 2 −1  ⎨ N∑ ⎬ ω 

 ω  ω ωu  z/g n ωNu u u u × e t + ϕn + U − x˙ 2Sη0 η0 n cos n n ⎩ ⎭ N N N N n=0

(3.12) For the motion response analysis of this offshore platform, the fourth-order Runge– Kutta method will be used to numerically integrate the differential Eq. (3.12), and the detailed simulation procedures will be explained in Sect. 3.2.4. However, before the detailed simulations are performed, the mathematical background of the fourth-order Runge–Kutta method will first be elucidated in next section.

3.2.3 The Fourth-Order Runge–Kutta Method for Numerical Integration A numerical method should be used for the dynamic analysis when the differential equation governing the dynamics of a system cannot be integrated in closed form. This section introduces the fourth-order Runge–Kutta method, which is based on the approximation of the derivatives found in the equation of motion and the boundary conditions. Methods for numerical integration share two key traits. First of all, they are only meant to satisfy the governing differential equation(s) at discrete time intervals Δt apart rather than at all time t. Second, it is assumed that there will be a ˙ and acceleration x¨ over suitable type of variation in the displacement x, velocity x, each time interval Δt. Depending on the kind of variation assumed for the displacement, velocity, and acceleration within each time interval Δt, various numerical integration methods can be obtained. We’ll assume that at time t = 0, the values of x and x˙ are known to be x0 and x˙0 , respectively, and that the problem needs to be solved between t = 0 and t = T . The time duration T is divided into n equal steps Δt in the following so that Δt = T /n and the solution is sought at t0 = 0, t1 = Δt, t2 = 2Δt, …, tn = nΔt = T . Utilizing the fourth-order Runge–Kutta method, we will derive formulas for determining the solution at ti = iΔt from the known solution at ti−1 = (i − 1)Δt. The Runge–Kutta method involves expressing the current displacement (solution) in terms of the previously determined values of displacement, velocity, and acceleration, and then solving the resulting equations to

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

85

determine the current displacement. This method falls under the category of explicit integration methods. In the fourth-order Runge–Kutta method, we try to find the values of the desired solution at a finite number of mesh or grid points that replace the solution domain (over which the solution of the given differential equation is required). Typically, it is assumed that the grid points are evenly spaced along each of the independent coordinates. The approximate formula for deriving xi+1 from xi in the Runge–Kutta method is made to coincide with the Taylor’s series expansion of x at xi+1 up to terms of order (Δt)n : x(t + Δt) = x(t) + xΔt ˙ + x¨

(Δt)2 ... (Δt)3 .... (Δt)4 +x + x +··· 2! 3! 4!

(3.13)

The Runge–Kutta method does not explicitly require derivatives beyond the first, in contrast to Eq. (3.13), which calls for higher-order derivatives. Let us consider a standard second-order differential equation as follows: M x¨ + C x˙ + K x = F(t)

(3.14)

By defining x1 = x and x2 = x, ˙ we first reduce the above second-order differential equation to two first-order equations as follows: x˙1 = x2 x˙2 = x¨ =

1 1 [F(t) − C x˙ − K x] = [F(t) − C x2 − K x1 ] = f (x1 , x2 , t) M M (3.15)

By specifying U→ (t) =



x1 (t) x2 (t)



and V→ (t) =



x2 f (x1 , x2 t)

 (3.16)

the fourth-order Runge–Kutta method is used to determine the values of U→ (t) at various grid points ti using the recurrence formula shown below

1 → → → → U→i+1 = U→i + W 1 + 2 W2 + 2 W3 + W4 6

(3.17)

→ 1 = (Δt)V→ (Ui , ti ) W

(3.18)

  1 1 → → → W2 = (Δt)V Ui + W1 , ti + (Δt) 2 2

(3.19)

in which

86

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

  → 3 = (Δt)V→ Ui + 1 W → 2 , ti + 1 (Δt) W 2 2   → 3 , ti+1 → 4 = (Δt)V→ Ui + W W

(3.20) (3.21)

3.2.4 Motion Response Analysis of the Platform by Monte Carlo Simulation Method The same data as used by Tognarelli et al. [9] are used for the numerical integration of the differential Eq. (3.12) and these are: M = 71,286,000 kg, C = 447,910 N-s/ m, K = 281,430 N/m, ε = 0.2 m−2 , K m = 40,000,000 kg, K d = 1,500,000 N-s2 /m2 . Note that the value of M here already includes the added mass term on the right side of Eq. (2.51). Offshore tidal currents often reach a maximum velocity of 1–2 knots (0.5144–1.0288 m/s) and in some locations even 10 knots (5.144 m/s) [10]. Thus, six values of current velocity U = 0.0, U = 0.5, U = 1.0, U = 2, U = 3 and U = 4 m/s are taken for numerical integration of the differential Eq. (3.12). The significant wave height in the P-M elevation spectrum is taken as H S = 12 m, and thus the one-sided power spectral density function of the wave water particle displacement is 0.778 e Sx x (ω) = ω5



− 0.0213 ω4



(3.22)

An upper cut-off frequency value ωu = 4π ≃ 12.566 rad/sec is obtained by using the following criterion [11]: ωu

∞ Sx x (ω)dω = (1− ∈)

0

Sx x (ω)dω

(3.23)

0

where ∈ = 1.04 × 10–3 . Also, the N value in Eq. (3.12) is taken as 128. The simulation process obtained from Eq. (3.3) is periodic with a period of [11]: T0 =

2π 2π N 2π = = = 64 sec Δω (ωu /N ) ωu

(3.24)

A uniform [0, 1] random number generator is utilized to generate a series of random numbers, and then each random number generated is multiplied by 2π to obtain the random phase angle ϕn in Eqs. (3.10) and (3.11). The time interval Δt ˙ is taken to be 0.1 s as required by Δt ≤ separating the generated u(t) and u(t) (2π)/(2 ωu ). Then the fourth-order Runge–Kutta method is used to numerically integrate the differential Eq. (3.12) with a time step satisfying Δt = 0.1s. During the process of integration, assumption has been made that the Morison resultant force

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

87

U=0 m/s

6

Response displacement (m)

4

2

0

-2

-4

-6 200

300

400

500

600

Time (sec)

Fig. 3.2 Steady-state displacement response (U = 0 m/s)

center is at the still water level. Therefore, the exponential type transfer functions in Eq. (3.12) become unity. One time history of the displacement responses of this offshore structure is obtained, which is achieved over 6400 equally spaced time points in 10 cycles. To remove transients, the response values within the first few cycles on the time history curve are truncated. The time histories of the steady-state displacement responses of this offshore structure for different values of current velocity U = 0.0, U = 0.5, U = 1.0, U = 2, U = 3 and U = 4 m/s are shown in Figs. 3.2, 3.3, 3.4, 3.5, 3.6 and 3.7. After careful analysis of the above graphs, the following conclusions can be drawn: (1) In all six cases (U = 0.0, U = 0.5, U = 1.0, U = 2, U = 3 and U = 4 m/s) the time histories of the structural displacement responses exhibit some regular vibration patterns. The positive response amplitude peak increases with the increase of the sea current value from 0.0 to 4 m/s. (2) The absolute value of the peak negative response amplitude at U = 0.5 and U = 1.0 m/s is smaller than that at U = 0.0 m/s. However, as the current velocity continues to increase, the absolute value of the peak negative response amplitude increases dramatically at U = 2, U = 3.0 and U = 4 m/s. This shows that the absolute value of the peak negative response amplitude does not simply decrease with the increase of the current velocity value.

88

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

U=0.5m/s

6

Response displacement (m)

4

2

0

-2

-4 200

300

400

500

600

Time (sec)

Fig. 3.3 Steady-state displacement response (U = 0.5 m/s)

8

U=1m/s

Response displacement (m)

6

4

2

0

-2

-4 200

300

400

500

600

Time (sec)

Fig. 3.4 Steady-state displacement response (U = 1 m/s)

3.2.5 Response Statistics Analysis of the Offshore Structure by the Monte Carlo Simulation Method After obtaining the time histories of the surge oscillation displacement responses of this offshore structure, the response statistics of the platform structure are further investigated. From a design point of view, the information in the region of the response

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo … 12

89

U=2m/s

10

Response displacement (m)

8 6 4 2 0 -2 -4 -6 200

300

400

500

600

Time (sec)

Fig. 3.5 Steady-state displacement response (U = 2 m/s)

U=3m/s

Response displacement (m)

12 10 8 6 4 2 0 -2 -4 -6 200

300

400

500

600

Time (sec)

Fig. 3.6 Steady-state displacement response (U = 3 m/s)

probability density curve near the response mean is important for fatigue analysis, and the information in the region of the tails of the response probability density curve is important for reliability analysis. Also, the non-Gaussian nature of the structural responses has an important effect on the probability of failure of the structure, which

90

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

U=4m/s

Response displacement (m)

15

10

5

0

-5

200

300

400

500

600

Time (sec) Fig. 3.7 Steady-state displacement response (U = 4 m/s)

includes failure due to fatigue and failure due to extreme value responses. In this study, the probability density values of the structural responses at different values of sea current velocity are derived, and the following Table 3.1 gives the probability density values at typical response displacements. To reflect all the structural response probability density values at different values of the current velocity, the structural response probability density curves at different values of the current velocity are shown in Figs. 3.8, 3.9, 3.10, 3.11, 3.12 and 3.13, where each curve is obtained by averaging the total of 400 response records. Each response record, in turn, contains 6400 points, of which the points within the first 4 cycles are truncated in order to delete transient values. The probability density curves in Fig. 3.8 through Fig. 3.13 are clearly nonGaussian. Increasing the number of records for which the overall mean is taken does not change the statistical characteristics of the responses obtained from the simulations. These plots indicate that the response of a nonlinear offshore structural system subject to Gaussian random excitations is non-Gaussian. Further observations reveal that the probability density curves become skewed with increasing values of the current velocity, a fact that suggests that further investigation of the higher order central moments of the system responses at different values of the current velocity is necessary. If the system response mean is denoted by the symbol μ and the standard deviation of the responses is denoted by σ , the second-order central moment (variance) of the response is σ 2 . The skewness and kurtosis represent numerically the asymmetry and peakedness of the response probability density distribution, respectively. The skewness is equal to the third-order central moment

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

91

Table 3.1 The platform response probability density values at different ocean current velocities Current value (m/s) Response value (m)

0

0.5

1

2

3

4

−8.9

0

0

0

0

0.000041

0

−8.1

0

0

0

0

0.000134

0

−7.3

0

0

0.0000006

0.000484

0.000367

0.000184

−6.5

0

0

0.000142

0.001508

0.000446

0.000311

−5.7

0.0003428

0.000155

0.000406

0.002250

0.001068

0.000808

−4.9

0.001777

0.000481

0.001194

0.003294

0.0016896

0.001134

−4.1

0.004545

0.002021

0.002527

0.004660

0.002413

0.001575

−3.3

0.012190

0.005586

0.004735

0.005217

0.0033498

0.002155

−2.5

0.022730

0.010843

0.008072

0.005718

0.004684

0.002909

−1.7

0.028974

0.017598

0.013654

0.006886

0.005539

0.003584

−0.9

0.033381

0.025286

0.019531

0.009020

0.006229

0.004759

−0.1

0.033424

0.029591

0.023177

0.010268

0.007074

0.005947

0.7

0.032044

0.035349

0.02625

0.013342

0.008900

0.007207

1.5

0.030183

0.035417

0.029505

0.016585

0.011658

0.008541

2.3

0.024446

0.035540

0.032209

0.022712

0.014450

0.010282

3.1

0.015462

0.028326

0.031082

0.027082

0.016025

0.012011

3.9

0.006106

0.016991

0.025725

0.028738

0.019138

0.014273

4.7

0.002224

0.005999

0.016319

0.027229

0.022208

0.017333

5.5

0.000644

0.000841

0.008557

0.021764

0.024636

0.01954

6.3

0

0

0.004190

0.015862

0.02508

0.021087

7.1

0

0

0.001608

0.011943

0.024033

0.022366

7.9

0

0

0.000424

0.008170

0.020612

0.022606

8.7

0

0

0.000055

0.005253

0.014818

0.021547

9.5

0

0

0

0.002567

0.008497

0.018945

10.3

0

0

0

0.000610

0.004341

0.013991

11.1

0

0

0

0.000057

0.001793

0.009472

11.9

0

0

0

0

0.000435

0.005540

12.7

0

0

0

0

0.00014

0.00199

13.5

0

0

0

0

0

0.0009

14.3

0

0

0

0

0

0.000123

14.9

0

0

0

0

0

0.000023

92

3 Monte Carlo Simulation Method for Dynamic Response of Marine … U=0 m/s

0.040 0.035

Probability density

0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 -6

-4

-2

0

2

4

6

4

6

Surge displacement (m)

Fig. 3.8 Response probability density curve (U = 0 m/s)

U=0.5 m/s 0.040 0.035

Probability density

0.030 0.025 0.020 0.015 0.010 0.005 0.000 -0.005 -6

-4

-2

0

2

Surge displacement (m)

Fig. 3.9 Response probability density curve (U = 0.5 m/s)

divided by σ 3 , and thus the skewness is calculated by the following equation: 

(x − μ)3 f (x)d x σ3

(3.25)

The kurtosis is equal to the fourth-order central moment divided by σ 4 , and thus the kurtosis is calculated by the following equation:

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

93

U=1m/s

0.035 0.030

Probability density

0.025 0.020 0.015 0.010 0.005 0.000 -0.005 -15

-10

-5

0

5

10

Surge displacement (m)

Fig. 3.10 Response probability density curve (U = 1 m/s)

U=2m/s 0.030

Probability density

0.025

0.020

0.015

0.010

0.005

0.000 -10

-5

0

5

10

Surge response (m)

Fig. 3.11 Response probability density curve (U = 2 m/s)



(x − μ)4 f (x)d x σ4

(3.26)

94

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

U=3m/s

0.030

0.025

Probability density

0.020

0.015

0.010

0.005

0.000 -10

-5

0

5

10

Surge displacement (m)

Fig. 3.12 Response probability density curve (U = 3 m/s)

U=4m/s

0.025

Probability density

0.020

0.015

0.010

0.005

0.000

-10

-5

0

5

10

15

Surge displacement (m)

Fig. 3.13 Response probability density curve (U = 4 m/s)

In the above equation f (x) is the probability density function of the system responses. In this study, the mean, variance, skewness, and kurtosis of the system responses at a particular current velocity are obtained by averaging the total of 400 surge oscillatory response time histories, respectively. Each record (time history) contains 6400 points, where the points within the first four periods are truncated in

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo …

95

order to remove transient values. The results of the calculations are included in the following Table 3.2. The mean, variance, skewness, and kurtosis of the system responses at current velocities ranging from U = 0.5 to U = 4.0 m/s are shown in Figs. 3.14 through 3.17. Figures 3.14 and 3.15 show that the mean and variance of the system responses increase monotonically with the increasing values of the current velocity. The skewness is a measure of the asymmetry of the response probability density distribution. If the skewness is equal to 0, the shape of the probability density curve is symmetric with respect to the mean value. If the skewness is less than 0, then the mode of the probability density curve is located at a value larger than the mean. Figure 3.16 shows that the skewness of the responses of the offshore platform is always less than 0 when the current velocity U > 0.0 m/s, which indicates that the shape of the probability density curve of the platform responses is always right skewed when U > 0.0 m/s. Table 3.2 The platform response mean, variance, skewness and Kutosis at different ocean current velocities Items Current velocity ( m/s)

Mean

Variance

Skewness

Kutosis

0.5

0.944946

3.95481

−0.261476

2.52206

1

1.60899

5.78043

−0.243582

2.72485

2

2.96058

11.1669

−0.588401

3.12816

3

4.48151

12.5493

−0.698104

3.18855

4

5.83097

14.293

−0.643744

3.10919

Fig. 3.14 Response mean-current velocity

6

Response mean (m)

5

4

3

2

1 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Current velocity (m/s)

3.5

4.0

96

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Fig. 3.15 Response variance-current velocity

16

Response variance

14

12

10

8

6

4 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

3.5

4.0

Current velocity (m/s)

Fig. 3.16 Response skewness-current velocity

-0.2

Response skewness

-0.3

-0.4

-0.5

-0.6

-0.7

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Current velocity (m/s)

Using the Gaussian distribution as the reference metric, the kurtosis represents a degree of peakedness of the response probability density distribution. The kurtosis of a Gaussian distribution is equal to 3. If the kurtosis is 3, the distribution is said to be leptokurtic (sharp peak). Figure 3.17 shows that the kurtosis of the response of the offshore platform is always >3 when the current velocity ranges from U = 2.0 to U = 4.0 m/s. This indicates that the shape of the probability density curve of the platform responses is sharp peaked at this time (relative to the Gaussian distribution). Figure 3.17 shows that the kurtosis of the responses of the offshore platform is always less than 3 when the current velocity is from U = 0.0 to U = 1.0 m/s, which indicates

3.2 Dynamic Analysis of an Offshore Structure by the Monte Carlo … Fig. 3.17 Response kurtosis-current velocity

97

3.2

Response kurtosis

3.1 3.0 2.9 2.8 2.7 2.6 2.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Current velocity (m/s)

that the shape of the probability density curve of the platform responses is mild peaked (relative to the Gaussian distribution) at this time.

3.2.6 Level Up-Crossing Rates of the Structural Motion Responses by the Monte Carlo Simulation Method In this study, the level up-crossing rates of the response values of the above platform are then calculated, and these up-crossing rate values are the key to derive many important data regarding the response statistics and reliability applications. Let X(t) be the dynamic responses at a critical point of the above compliant offshore structure, and let X(t) be a stochastic process with continuous values, which is also continuous for its parameter t. In structural reliability analysis, it is useful to obtain a probabilistic characterization of the number of times that this stochastic process X(t) crosses a particular threshold value ξ in a given time period. This event is referred to as the threshold crossing problem. Mathematically, level up-crossing rate is found by the following equation: νx+ (ξ )

∞ =

x˙ p X X˙ (ξ, x) ˙ d x˙

(3.27)

0

This result is the expected number of times per unit time that the level ξ is crossed with a positive slope. It is derived by Stephen O. Rice. The detailed derivation process of this formula can be found in Sect. 4.5.4. This formula is applicable to any stationary process; the process does not have to be Gaussian or narrow-banded. In the ˙ above equation, x˙ is the first-order derivative of the response value, and p X X˙ (ξ, x)

98

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

is the joint probability density of this stochastic process and its first-order derivative. The probability density curve of the displacement response of the structure can be obtained by averaging the total of 500 records of the displacement responses of the offshore platform as described above, and the probability density curve of the first order derivative of the displacement responses of the offshore structure can be obtained by the same method. Based on the information from the probability density ˙ curves of these individual random variables, the joint probability density PX, X˙ (ξ, x) of the response values and the first-order derivatives of the response values can be derived. Next, the level up-crossing rates of the responses of the structural system can be found by Rice’s Eq. (3.27). The following Table 3.3 gives the values of the level up-crossing rates at typical response displacements. To reflect the values of the level up-crossing rates for all structural responses at different values of current velocity, the level up-crossing rate curves for structural responses at different values of current velocity are shown in Figs. 3.18, 3.19, 3.20, 3.21, 3.22 and 3.23 (the horizontal coordinates are the response displacements and the vertical coordinates are the level up-crossing rate values).

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift Extreme Responses of Marine Structures Under Random Excitations 3.3.1 Concept of the Slow Drift Extreme Responses of Marine Structures Semi-submersible platforms, floating production storage and offloading systems, and Tension Leg Platforms can be collectively referred to as compliant offshore structures. The natural frequencies of these compliant offshore structures are much lower than the frequencies of the first-order wave forces. However, the natural frequencies of these compliant offshore structures are difficult to avoid coinciding with the second-order slow-drift wave force frequencies, so that they tend to resonate harmonically under the second-order slow-drift wave forces, resulting in very large motion amplitudes. Therefore, the design of these types of structures should be subjected to dynamic analysis. Because the external loads on these marine structures are random, the responses of these structures obtained from the dynamic analysis is not deterministic, but can only be expressed in the form of response probability density curves. In the design process of the above mentioned compliant offshore structures, one of the measures widely used in industry today to ensure their safety level is to ensure that their load values or their dynamic response values have a specified mean return period, e.g., 100 years. The above requirement can also be expressed in the form of an equivalent annual exceedance probability, e.g., 10–2 . Practical experience shows that

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift …

99

Table 3.3 The platform response level up-crossing rates at different ocean current velocities Current value (m/ s) Response value 0 (m)

0.5

−7.9

0.000002 0.0000003 0.000019 0.000008 0.000003

0.000005

1

2

3

4

−7.7

0.000005

0.000003 0.0000003 0.000023 0.000009 0.000005

−7.5

0.000006

0.000003 0.000002

0.000024 0.000027 0.000006

−7.3

0.000006

0.000003 0.000002

0.000036 0.000031 0.000014

−6.5

0.000006

0.000006 0.00001

0.000091 0.000055 0.000031

−5.7

0.000028

0.000007 0.000032

0.000206 0.000132 0.000048

−4.9

0.000110

0.000022 0.00008

0.000286 0.000182 0.000106

−4.1

0.000338

0.000138 0.000172

0.000405 0.000292 0.000162

−3.3

0.00096

0.000512 0.000354

0.000496 0.00039

0.000246

−2.5

0.001602

0.000836 0.000644

0.000596 0.000491 0.000366

−1.7

0.00218

0.001292 0.001024

0.000645 0.000519 0.000466

−0.9

0.002414

0.001848 0.001429

0.00083

0.000591 0.000567

−0.1

0.002434

0.002236 0.001737

0.001001 0.000739 0.000733

0.7

0.002359

0.002631 0.002056

0.001245 0.000952 0.000869

1.5

0.002256

0.002728 0.002369

0.001568 0.001177 0.001108

2.3

0.001808

0.002624 0.002589

0.001933 0.001363 0.001347

3.1

0.001123

0.002068 0.002543

0.002516 0.001768 0.001497

3.9

0.000465

0.001280 0.00206

0.002583 0.002034 0.001786

4.7

0.000142

0.000411 0.001305

0.002504 0.002432 0.001982

5.5

0.000033

0.000081 0.000645

0.002

6.3

0.000007

0.000012 0.000316

0.001513 0.002643 0.002371

0.002617 0.002175

7.1

0.000005

0.000007 0.000099

0.001074 0.002567 0.002659

7.9

0.000002

0.000003 0.000023

0.000765 0.002153 0.002770

8.7

0.000002

0.000002 0.000002

0.000493 0.001489 0.002524

9.5

0.000001

0.000001 0

0.000232 0.000901 0.002270

10.3

0.000001

0

0

0.000053 0.000455 0.00165

11.1

0.000001

0

0

0.000003 0.000165 0.001091

11.9

0

0

0

0

0.000041 0.000593

12.7

0

0

0

0

0

0.000261

13.5

0

0

0

0

0

0.000089

14.3

0

0

0

0

0

0.000009

14.5

0

0

0

0

0

0.000009

14.7

0

0

0

0

0

0.000004

14.9

0

0

0

0

0

0.000004

15.1

0

0

0

0

0

0.000001

100

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Fig. 3.18 Level up-crossing rates curve (U = 0.0 m/s)

U = 0 m/s 0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000

-10

-5

0

5

10

Surge response (m)

Fig. 3.19 Level up-crossing rates curve (U = 0.5 m/s)

0.0030

U=0.5m/s

0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000 -10

-5

0

5

10

Surge response (m)

the results obtained by splitting a storm with a 100-year return period into storms with 3-h intervals for design checks are very good, where the 100-year storm is considered a non-stationary random process and each 3-h storm is considered a stationary random process. A stochastic process is said to be stationary if its entire probability structure is independent of the shifting of the origin of the time parameter. Naess [12, 13] and Naess et al. [14] proved that a key function required to obtain a good approximate solution to the extreme value distribution, whether for a stationary stochastic process or a non-stationary stochastic process, is the mean rate of level up-crossings. Let us first consider the stationary stochastic process, and denote the number of times the stationary stochastic process X (t) crosses the level ξ upwards in

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift … Fig. 3.20 Level up-crossing rates curve (U = 1 m/s)

101

U=1 m/s

0.0030

0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000 -10

-5

0

5

10

Surge response (m)

Fig. 3.21 Level up-crossing rates curve (U = 2 m/s)

U=2 m/s

0.0030

0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000 -10

-5

0

5

10

Surge response (m)

a time period T as N + (ξ, T ) [14]. Because the stochastic process X (t) is stationary, we have [14]     E N + (ξ, T ) = E N + (ξ, 1) T

(3.28)

  where E N + (ξ, 1) is called the level up-crossing rate of the random process X (t). We use the following notation [14].   ν + (ξ ) = E N + (ξ, 1)

(3.29)

102

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Fig. 3.22 Level up-crossing rates curve (U = 3 m/s)

U= 3 m/s

0.0030

0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000 -10

-5

0

5

10

Surge response (m)

Fig. 3.23 Level up-crossing rates curve (U = 4 m/s)

U=4 m/s

0.0030

0.0025

Up-crossing rate

0.0020

0.0015

0.0010

0.0005

0.0000 -10

-5

0

5

10

15

20

Surge response (m)

Let the extremum of the stochastic process X (t) of the dynamic response of a compliant offshore structure over a time period T be [14]. M(T ) = max{M(T ) : 0 ≤ t ≤ T }

(3.30)

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift …

103

For a stationary short-term sea state, the cumulative distribution function of M(T ) under Poisson assumptions is given by the following relation with the level upcrossing rate term [14].   Prob(M(T ) ≤ ξ ) = exp −ν + (ξ )T

(3.31)

where ν + (ξ ) denotes the level up-crossing rate of the dynamic response stochastic process X (t) against the level ξ for this compliant offshore structure. Equations (3.31) show that a key function needed to obtain a good approximate solution for the extreme value distribution is the level up-crossing rate. For a non-stationary long-term sea state, Eq. (3.31) can be rewritten as [14]. ⎧ T ⎫ ⎨  ⎬ Prob(M(T ) ≤ ξ ) = exp − ν + (ξ, t)dt ⎩ ⎭

(3.32)

0

where ν + (ξ, t) represents the average level up-crossing rate of the dynamic response stochastic process X (t) against the level ξ at the moment t for this compliant offshore structure. T equals the long-term time period under consideration. For engineering purposes, the above equation can be written as [14]: ⎧ ⎨

Prob(M(T ) ≤ ξ ) = exp −T ⎩



ν + (ξ, w) f W (w)dw

w

⎫ ⎬ ⎭

(3.33)

where f W (w) represents the probability density function of the parameters of interest for the considered long-term time period sea state, and its values can be found by referring to the data in some wave scatter diagrams [4]. Note that the long-term time period sea state under consideration is still an ergodic stochastic process. In the general case: ) ( W = Hs , T p

(3.34)

where Hs denotes the significant wave height of a stationary short-time sea state, T p denotes the spectral peak period of this stationary short-time sea state, and Eq. (3.33) can thus be written as: ⎧ ⎫ ⎪ ⎪   ⎨ ⎬ ( | ) ( ) + | (3.35) Prob(M(T ) ≤ ξ ) = exp −T ν ξ, h s , t p f Hs Tp h s , t p dh s dt p ⎪ ⎪ ⎩ ⎭ hs t p

( ) where Hs = h s , T p = t p . With the joint probability density function f Hs Tp h s , t p of the relevant parameters for the considered long-term time period, we can transform this joint model into a space containing independent, standard Gaussian random

104

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

Fig. 3.24 A 100-year environmental contour line based on measured data at NDBC46050

variables, and then use the inverse first-order reliability method to obtain circular contour lines corresponding to the 10–2 annual exceedance probabilities. These circular contour lines are then transformed back into the physical parameter space to obtain environmental contour lines corresponding to the 10–2 annual exceedance probabilities. Figure 3.24 shows an example of a 100-year environmental contour line based on a wave dataset obtained from the U.S. National Data Buoy Center (NDBC) Station 46,050. The location of this data buoy is in Stonewall Bank- 20 nautical miles west of Newport, Oregon, the Unite States. This wave dataset, which was collected hourly between January 1, 1996 and December 31, 2022 has 194,475 significant wave height (HS ) values and 194,475 corresponding energy period (Te ) values. The role played by the moderate sea states is insignificant for the extreme response of the slow drift motions of marine structures. The most important environmental conditions that need to be considered should be the extremely harsh sea states. Using the obtained environmental contour line, we can limit the analysis to a few harshest short-term 3-h sea states along the contour line, such as those few sea states near the maximum significant wave height on the 100-year environmental contour line in Fig. 3.24. In the dynamic analysis based on the selected sea states, the quantity we need is the extreme response value ξ with a level up-crossing rate below 10–4 –10–5 , and it is obvious that ξ is in the region at the tails of the response probability density curve.

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift …

105

3.3.2 Method for Treating External Random Excitations During Monte Carlo Simulations The equation of motion used by Naess and Johnson [15] in the analysis of the slow drift surge response of a moored floating cylinder is M x¨ + D(t)x˙ + ρC D R x| ˙ x| ˙ + K 1 x + K 3 x 3 = F(t)

(3.36)

Naess and Johnson [15] pointed out that the differential equation for the slow drift surge motion of the above moored floating cylinder is identical to that for the slow drift surge motion of a semi-submersible platform or a Tension Leg Platform. Thus, the study of the slow drift surge motion of this moored floating cylinder has a representative significance. The above Eq. (3.36) takes into account the nonlinearities in the time-varying wave drift force damping, vortex-generation damping, and restoring forces, and is thus more realistic than some linear time-invariant response models. In the above equation, M is the total mass of the structure (including the added mass), x is the surge displacement response of the structure, and the symbol (˙) represents the derivative with respect to time. D(t) is the time-varying wave drift force damping, ρ is the density of seawater, C D is the Morison drag force type damping coefficient, R is the characteristic size of the structure, K 1 is the linear restoring force coefficient, and K 3 is the nonlinear restoring force coefficient. In the analysis of the extreme response of the slow drift surge oscillations of a Tension Leg Platform, Naess et al. [16] took F(t) to be the wave drift force with a spectral density function value S0 , and thus F(t) is a random excitation term. In the following we first describe the principle of calculating the random excitation term F(t). Let η(t) and F(t) represent the wave surface elevation and the second-order slowdrift wave force, respectively, and let x(t) be the slow-drift surge response displacement of this compliant offshore structure. x(t), η(t) and F(t) will be assumed to be stationary stochastic processes throughout this chapter. Most of the wave energy spectra have a distinct peak, i.e., their energy components are concentrated in a narrow frequency range near the peak. It is therefore reasonable to treat the wave elevation process η(t) as a narrow band process. This type of process can be expressed by the following equation η(t) = h [a(t) cos ω0 t − b(t) sin ω0 t]

(3.37)

where h is a height value, and a(t) and b(t) are two stationary Gaussian random processes whose mean values are ⟨a(t)⟩ = 0 and ⟨b(t)⟩ = 0, respectively. ω0 is the frequency at which the spectral density of η(t) reaches its peak. a(t) and b(t) will both be assumed to be slowly varying so that they do not vary considerably over a period of 2π/ω0 . Then, it can be deduced that η(t) is a Gaussian random process with zero mean. Equation (3.37) can also be written as: η(t) = H (t) cos[ω0 t + ψ (t)]

(3.38)

106

3 Monte Carlo Simulation Method for Dynamic Response of Marine …

in which H (t) = h A(t) = h[a 2 (t) + b2 (t)]1/2

(3.39)

ψ(t) = arctan[b(t)/a(t)]

(3.40)

and

In the irregular sea state, η(t) is sufficiently narrow-banded, and from the above analysis, H (t) is also slowly varying, so we can calculate the slowly varying drift force F(t) by the following equation: F(t) = α(ω0 )H 2 (t)

(3.41)

where α(ω0 ) is a coefficient that depends on the peak frequency ω0 of the spectral density function. Because H (t) is a random process, F(t) is thus a random excitation term. In Roberts [17], it was demonstrated by an example that F(t) can also be calculated exactly as follows F(t) = cW (t)

(3.42)

where W (t) is a random excitation term with spectral density function value S0 = (1/2π ) m2 s−3 , c is a constant coefficient, and the ratio of c to the peak frequency spectral level of the slowly varying drift force F(t) is 1/2π . From the above two equations, it is obvious that c is related to the wave height and the peak frequency of the wave spectral density function. The random excitation term W (t) can be simulated based on the theories as elucidated in Eqs. (2.52)–(2.57) in Chap. 2. Figures 3.25, 3.26, 3.27 and 3.28 show the Monte Carlo simulated time histories of the wave drift force W (t) with a spectral density function value S0 = (1/2π ) m2 s−3 . The afore-mentioned simulated time history 1 (or time history 2, or 3, or 4…) of the slow drift wave force random excitations is substituted into Eq. (3.36). Then, the fourth-order Runge–Kutta method can be used to numerically integrate the differential equation of motion (3.36) to obtain a time history of the slow drift displacement responses of the offshore structure. In the same way, a time history of the first-order derivative of the slow-drift displacement responses of the offshore structure can be obtained. The probability density curve of the slow-drift displacement responses of the offshore structure can be obtained by averaging the total of enough times of the time histories of the slow-drift displacement responses of the offshore structure, and the probability density curve of the first-order derivative of the slow-drift displacement responses of the offshore structure can be obtained in the same way. The joint probability density of the response value and the first-order derivative of the response value can be obtained based on the information of the probability density curves of these individual random variables. The next section presents a specific example of calculating the extreme response statistics of the slow-drift surge oscillations of the aforementioned moored floating cylinders.

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift …

107

Wave drift force (N)

1.5

1.0

0.5

0.0

-0.5

-1.0

2400

2600

2800

3000

3200

Time (sec) Fig. 3.25 Simulated time history 1 of the slow drift wave force random excitations

Wave drift force (N)

1.0

0.5

0.0

-0.5

-1.0

-1.5 2400

2600

2800

3000

3200

Time (sec) Fig. 3.26 Simulated time history 2 of the slow drift wave force random excitations

3.3.3 Analysis of the Slow Drift Extreme Response of a Moored Floating Cylinder by Monte Carlo Simulation As can be seen from the explanations in Sect. 3.2, the principle of Monte Carlo Simulation is very straightforward-i.e., numerical integration of the differential equation of motion. Monte Carlo simulations can be used to derive the entire response probability

108

3 Monte Carlo Simulation Method for Dynamic Response of Marine … 1.5

Wave drift force (N)

1.0

0.5

0.0

-0.5

-1.0

-1.5 2400

2600

2800

3000

3200

Time (sec)

Fig. 3.27 Simulated time history 3 of the slow drift wave force random excitations

1.5

Wave drift force (N)

1.0

0.5

0.0

-0.5

-1.0

-1.5 2400

2600

2800

3000

3200

Time (sec)

Fig. 3.28 Simulated time history 4 of the slow drift wave force random excitations

density curve of the offshore structure together with the slow drift extreme response of the offshore structure. In order to elucidate the specific calculation process, let us go back to Eq. (3.36) ˙ x| ˙ + K 1 x + K 3 x 3 = F(t) M x¨ + D(t) x˙ + ρC D R x| Utilizing Eq. (3.42) we have

(3.43)

3.3 Shortcomings of Monte Carlo Simulation to Analyze the Slow Drift …

109

M x¨ + D(t)x˙ + ρC D R x| ˙ x| ˙ + K 1 x + K 3 x 3 = cW (t)

(3.44)

Because the velocity of the slow drift surge motion of this moored floating cylinder will be much 0 such that g(x0 ) /= 0 and V (x0 ) > 0; (iii) a point x l < 0 can be located such that g(xl ) /= 0 and V (xl ) =V (x0 ); (iv) for all x ∈ (xl , x0 ), V (x) t - t0 |z(t0 ) = z0 ] zl

(6.34)

z l ≤ z 0 < z c ; t ≥ t0 where zl is the left boundary of the Markov diffusion process Z(t), and R(t, zc ; t 0 , z0 ) represents the probability that zl ≤ Z(t) ≤ zc at time t. We assume that zl is not a critical point. The critical state zc corresponds to an absorbing boundary, because when a sample function reaches the boundary it will be removed from the population of sample functions. With an absorbing boundary, the Z(t) process will never reach stationarity. Further, the total probability in the range zl ≤ Z(t) ≤ zc is no longer conserved, otherwise, the integral in Eq. (6.34) would be equal to 1. It is clear that this reliability function R(t, zc ; t 0 , z0 ) satisfies the same backward Kolmogorov equation as q(z, t|z0 , t 0 ). ∂ ∂ 1 ∂2 R + m(z0 ) R + j(z0 ) 2 R = 0 ∂t0 ∂z0 2 ∂z0

(6.35)

where m(z0 ) = m(z0 ) and j(z0 ) = σ 2 (z0 ) are the first- and second derivate moments, respectively. In Eq. (6.35) R is treated as a function of t 0 and z0 , while t and zc are treated as parameters. Another option is to treat R as a function of τ = t-t 0 and z0 , i.e., R(τ, zc , z0 ), and Eq. (6.35) can be rewritten as [5]. −

∂ ∂ 1 ∂2 R + m(z0 ) R + j(z0 ) 2 R = 0 ∂τ ∂z0 2 ∂z0

(6.36)

6.3 First Passage Forecasting of Ship Capsizing

301

The boundary conditions for solving Eq. (6.36) are as follows [R(τ, zc , z0 )]τ =0 = 1, zl ≤ z0 < zc

(6.37)

[R(τ, zc , z0 )]z0 = zc = 0

(6.38)

[R(τ, zc , z0 )]z0 = zl = finite

(6.39)

The physical meanings of Eqs. (6.37) and (6.38) are clear, and Eq. (6.39) is obtained from the fact that the transition probability density q(z, t|z0 , t 0 ) in Eq. (6.34) is always integrable. Equations (6.36) through (6.39) define an eigenvalue problem, yet only in rare cases do solutions of closed-form exist. Here we concentrate on solving the statistical moment problem for the first passage time, which can be easily mastered. It is clear that the probability distribution function of the first passage time T is given by the following equation [5] FT (τ, zc , z0 ) = Pr ob[T < τ |Z(t0 ) = z0 ] = 1−R(τ, zc , z0 )

(6.40)

The probability density function of the first passage time T is given by the following equation [5] pT (τ, zc , z0 ) =

∂ ∂ FT (τ, zc , z0 ) = − R(τ, zc , z0 ) ∂τ ∂τ

(6.41)

Thus, the nth order moment of the random time T can be derived as follows [5] ∞ μn (zc , z0 ) = E[ T ] = − n

∂ R(τ, zc , z0 )d τ =n τ ∂τ

∞ τ n−1 R(τ, zc , z0 )d τ (6.42)

n

0

0

where we assume that as τ → ∞, τ n R(τ, zc , z0 ) tends to zero. Now multiplying both sides of Eq. (6.36) by τ n and integrating with respect to τ , we obtain [5] (n + 1)μn + m(z0 )

d 1 d2 μn+1 + j(z0 ) 2 μn+1 = 0 dz0 2 dz0

(6.43)

The above equation is known as the generalized Pontryagin equation and its two boundary conditions are [5] [μn+1 (zc , z0 )]z0 =zc = 0

(6.44)

[μn+1 (zc , z0 )]z0 =zl = finite

(6.45)

302

6 First Passage Theory for Ship Stability Analysis

The boundary condition (6.44) shows that if the initial state is already critical, then the first passage time must be zero. The boundary condition (6.45) is based on the assumption that the first passage event will occur sooner or later, and this assumption is also used to derive the generalized Pontryagin equation, without which it should be noted that the generalized Pontryagin equation would not hold. The boundary condition (6.45) is a qualitative boundary condition, not a quantitative boundary condition. If Eq. (6.43) needs to be solved numerically, then a quantitative boundary condition (6.45) is needed, and in many practical cases, b(zl ) = 0, we obtain directly from Eq. (6.43) that [μ n+1 (zc , z0 )]z0 =zl = −

(n + 1){[μn (zc , z0 )]z0 =zl } a(zl )

(6.46)

If it is valid, the boundary condition (6.46) can be used in place of the boundary condition (6.45). The well-known Pontryagin-Vitt equation corresponds to the special case of n = 0 in Eq. (6.43). Equation (6.43) and its two boundary conditions (6.44) and (6.46) form a boundary value problem, and the statistical moments of the first passage time of the stochastic process Z(t) can be obtained by solving this boundary value problem using the fourth-order Runge–Kutta method. In the following, the above theory will be applied to study the capsizing of a ship in random waves.

6.3.2 Stochastic Averaging of the Ship’s Rolling Equation The port-starboard symmetry inherent in a ship results in the absence of first-order couplings from surge, heave and pitch to roll. However, the couplings from sway and yaw to roll are not nullified. The present investigation postulates that the yaw coupling with roll and sway is of negligible magnitude. Consequently, solely the interdependence between sway and roll is taken into account. Typically, due to damping, it is not feasible to separate the roll motion and sway. In certain instances, specifically in undamped or proportionally damped systems, it has been demonstrated that the vessel undergoes rolling movements around a roll center akin to that of a pendulum, and the rolling motion can be decoupled from sway. If we suppose that there is a pseudo roll center when there is general damping, then we get the following single-degree-of-freedom rolling equation with quadratic roll damping [6] | | I φ¨ + Bφ˙ + C φ˙ |φ˙ | + ΔGM (φ − φ 3 /φR2 ) = K(t)

(6.47)

where φ is the rolling angle (−φR ≤ φ ≤ φR ), I is the moment of inertia (including added moment of inertia) of the ship about its rolling axis. In Eq. (6.47), B is the linear rolling damping coefficient, C denotes the quadratic viscous damping coefficient, Δ represents the displacement of the ship, GM is the initial stability height of the ship, φR denotes the positive angle of vanishing stability, and K(t) represents the external wave excitation moment. Note that the above equation is essentially the same as the

6.3 First Passage Forecasting of Ship Capsizing

303

nonlinear rolling differential equation we used in Chap. 5, except that here we have taken a different expression for the restoring moment. Let us retake a scaled rolling angle in terms of φR and retake a scaled time in terms of the ship’s resonant frequency ωR / ψ = φ/φR ,

T = ωR t =

ΔGM t I

(6.48)

Note that in the above equation, because the unit of ωR is rad/sec, the unit of T becomes rad. Then, the ship’s rolling equation becomes | | ψ¨ + α ψ˙ + β ψ˙ |ψ˙ |+ψ − ψ 3 = w(T )

(6.49)

Note that the differentiation in the above equation is now with respect to T. w(T ) is the normalized effective wave slope excitation, and the unit of w(T ) is φR . In the above equation α=

BωR , ΔGM

β=

CωR2 φR ΔGM

(6.50)

Note that in the above equation, because the unit of B is kg-m2 /sec, the unit of ωR is rad/sec, the unit of Δ is N, the unit of GM is m, thus the unit of α is (kg-m2 / sec)(rad/sec) / (Nm), and finally the unit of α becomes rad. In the above equation, because the unit of C is kg-m2 , the unit of ωR is rad/sec, the unit of Δ is N, the unit of GM is m, and the unit of φR is rad, thus the unit of β is(kg-m2 ) (rad 2 /sec2 ) rad/(Nm), and finally the unit of β becomes rad 3 . The quadratic damping term in Eq. (6.49) can be rewritten as /   | | 1 3 2 | | ˙ ˙ ˙ ˙ ˙ ψ˙ 3 ψ + ψ = 0.8β ψ+0.266β (6.51) β ψ ψ =β π 3 Substituting Eq. (6.51) into Eq. (6.49), the following equation of motion in standard form is obtained after rearranging ˙ + e(ψ, ψ)w(T ˙ ψ¨ + g(ψ) = f (ψ, ψ) )

(6.52)

g(ψ)=ψ − ψ 3 ,

(6.53)

˙ ˙ f (ψ, ψ)= − (α + 0.8β)ψ−0.266β ψ˙ 3 ,

(6.54)

˙ e(ψ, ψ)=1

(6.55)

where

304

6 First Passage Theory for Ship Stability Analysis

˙ is small compared to the restoring In the usual case, the damping term f (ψ, ψ) moment term g(ψ), and when the excitation level is not very high, the motion of the system (6.52) will be approximately periodic and can be written as ψ(t)=A cos Φ(t) + B,

(6.56)

˙ ψ(t)= − AΩ(A, Φ) sin Φ(t),

(6.57)

Φ(t)=ψ(t) + Θ(t),

(6.58)

where cosΦ(t) and sin Φ(t) are referred to as generalized harmonic functions. Ω(A, Φ) is referred to as an instantaneous frequency of the oscillation and can be calculated as / 2[V (A + B) − V (A cos Φ + B)] dψ = (6.59) Ω(A, Φ) = dt A2 sin2 Φ In the above equation, V is the potential energy, which is related to the restoring moment term as follows ψ V (ψ) =

g(u)du

(6.60)

0

In Eqs. (6.56) through (6.59), A, B, Φ, ψ and Θ are stochastic processes, and A and B are related to the total energy H of the free undamped oscillation by the following relationship V (A + B) = V (−A + B) = H

(6.61)

After the transformations in Eqs. (6.56–6.59) are accomplished, Eq. (6.6–6.52) becomes dA = m1 (A, Θ) + σ11 (A, Θ)w(t) dt

(6.62)

dΘ = m2 (A, Θ) + σ21 (A, Θ)w(t) dt

(6.63)

where m1 (A, Θ) =

−A f (A cos Φ + B, −AΩ(A, Φ) sin Φ)Ω(A, Φ) sin Φ g(A + B)(1 + h)

6.3 First Passage Forecasting of Ship Capsizing

=

−A2 [(α + 0.8β)Ω2 (A, Φ) sin2 Φ + 0.266βA2 Ω4 (A, Φ) sin4 Φ] g(A) (6.64) −A e(A cos Φ + B, −AΩ(A, Φ) sin Φ)Ω(A, Φ) sin Φ g(A + B)(1 + h)

σ11 (A, Θ) = =

305

−A [Ω(A, Φ) sin Φ] g(A)

(6.65) σ21 (A, Θ) = =

−1 e(A cos Φ + B, −AΩ(A, Φ) sin Φ)Ω(A, Φ) cos Φ g(A + B)(1 + h)

−1 [Ω(A, Φ) cos Φ] g(A)

(6.66) h=

g(−A + B) + g(A + B) dB = =0 dA g(−A + B) − g(A + B)

(6.67)

The last equation is obtained based on the fact that in the present case B = 0. According to the Stratonovitch-Khasminskii limit theorem, when the bandwidth of the input wave spectrum is sufficiently larger than the bandwidth of the rolling response (which is true in the present case), the amplitude process in Eq. (6.62) will weakly converge to a Markov diffusion process governed by the following Itô stochastic differential equation dA = b(A)dt + σ (A)dB(t)

(6.68)

in which B(t) is a Brownian motion process, and b(A) and σ (A) are respectively the drift and diffusion coefficients, and /

0 

b(A) = m1 + −∞

\ | |

∂σ11 || ∂σ11 || σ11 |t+τ + σ21 |t+τ R11 (τ )d τ ∂A |t ∂Θ |t

(6.69)

t

/ ∞ σ 2 (A) =



 σ11 |t σ11 |t+τ R11 (τ )d τ

\

−∞

(6.70) t

where ⟨·⟩t represents the following time averaging operation 1 ⟨[·]⟩t = lim T →∞ 2T

T [·]dt −T

(6.71)

306

6 First Passage Theory for Ship Stability Analysis

In order to obtain the averaged drift and diffusion coefficients in Eqs. (6.69) and (6.70), m1 and σ i1 are first expanded into Fourier series with respect to Φ as follows m1 (A, Θ) = m10 (A) +

∞ ∑

(s) m(c) 1n cos nΦ + m1n sin nΦ

(6.72)

n=1

σi1 (A, Θ) = σi10 (A) +

∞ ∑

(c) (s) σi1n cos nΦ + σi1n sin nΦ, i = 1, 2

(6.73)

n=1

After completing the integration with respect to τ and the time averaging with respect to t in Eqs. (6.69) and (6.70), we obtain the following explicit expressions for the average drift and diffusion coefficients d σ110 σ110 S11 (0)+ b(A) = m10 (A) + π dA   ∞ (c) (s) d σ11n π∑ d σ11n (c) (s) (s) (c) (c) (s) σ + σ + n(σ11n σ21n − σ11n σ21n ) S11 (nω(A)) (6.74) 2 n=1 dA 11n dA 11n    (c) (s) d σ11n d σ (c) (s) (s) (c) (c) σ (s) − 11n σ11n + + n(σ11n σ21n + σ11n σ21n ) I11 (nω(A)) dA 11n dA  ∑  (c) (c) (s) (s) σ 2 (A) = 2π σ110 σ110 S11 (0) + π σ11n σ11n + σ11n σ11n S11 (nω(A))    (6.75) (c) (s) (s) (c) + σ11n σ11n − σ11n σ11n I11 (nω(A)) where 1 S11 (ω) = π 1 I11 (ω) = π

0 R11 (τ ) cos ωτ d τ,

(6.76)

R11 (τ ) sin ωτ d τ

(6.77)

−∞

0 −∞

The averaged Fokker-Plank -Kolmogrov equation associated with the Itô equation has the following form ∂ 1 ∂2 2 ∂p = − [b(A)p] + [σ (A)p] ∂t ∂A 2 ∂A2

(6.78)

In which p = p(A, t|A0 ) is the transition probability density of the rolling displacement amplitude, and for the purpose of performing the calculations in Eqs. (6.74) and (6.75), the instantaneous frequency Ω(A, Φ) is first calculated as

6.3 First Passage Forecasting of Ship Capsizing

Ω(A, Φ) = =

√ √

1 − 0.75A2 − 0.25A2 cos 2Φ √ 1 − 0.75A2 1 − λ cos 2Φ = b0 (A) + b2 (A) cos 2Φ

307

(6.79)

in which λ = 0.25A2 /(1 − 0.75A2 ), 

λ2 √ b0 (A) ≈ 1 − (1 − 0.75A2 ) , 16

 λ 3λ2 √ (1 − 0.75A2 ). b2 (A) ≈ − − 2 64

(6.80)

(6.81)

(6.82)

Based on the information from Eqs. (6.74–6.82), we end up with closed-form expressions for the averaged drift and diffusion coefficients for the amplitude process −A2 [(α + 0.8β)(1 − 0.75A2 )(1 + 0.5λ)] 2(A − A3 ) 0.266βA4 [(1 − 0.75A2 )2 (3/8 + 0.5λ + 7λ2 /32)] − (A − A3 )    −A π d + (b0 (A) − 0.5b2 (A)) 2 dA A − A3   −A (b0 (A) − 0.5b2 (A)) S11 (ω(A)) A − A3     −1 −A (0.5b2 (A)) S11 (ω(A)) (b0 (A) − 0.5b2 (A)) + A − A3 A − A3     −A −A d (0.5b2 (A)) S11 (3ω(A)) (0.5b2 (A)) + dA A − A3 A − A3     −1 −A (0.5b (A)) S (3ω(A)) (0.5b (A)) (6.83) +3 2 11 2 A − A3 A − A3   A2 2 S11 (ω(A)) σ 2 (A) = π (b (A) − 0.5b (A)) 0 2 (A − A3 )2 (6.84)    A2 2 S (0.5b (A)) (3ω(A)) + 2 11 (A − A3 )2

b(A) =

308

6 First Passage Theory for Ship Stability Analysis

6.3.3 First Passage Time of Ship Capsizing In order to calculate the b(A) and σ 2 (A) functions in Eqs. (6.83) and (6.84), the solutions for the following free undamped oscillating system must be found. ¨ ψ+ψ − ψ3 = 0

(6.85)

The solution of the above equation is (see Appendix B for the derivation process) ψ = A sn(ω1 τ |m)

(6.86)

in which A is the oscillation amplitude and sn is the Jacobi elliptic function. Further / ω1 =

(1 −

A2 ) 2

(6.87)

and m=

A2 2 − A2

(6.88)

The natural period T of the oscillation is T =√

4K(m) 1 − 0.5A2

(6.89)

where π/2 √

k(m) = 0

dζ 1 − m sin2 ζ

(6.90)

In Eq. (6.90), K(m) is a complete elliptic integral of the first kind, and thus the natural oscillation frequency ω(A) in Eqs. (6.83) and (6.84) can be computed as follows ! √ √ 1 − 0.5A2 π 1 − 0.5A2 (6.91) ω(A) = 2π/T = 2π = 4K(m) 2K(m) In order to calculate b(A) and σ 2 (A) in Eqs. (6.83) and (6.84), we take the expression for the wave excitation spectral density function for wave processes 3 in Dalzel [6]. For wave process 3, the wave excitation spectral density is Sx (ω) = P

 

 1 + π/8 π ω2 1 3π 1 exp − − − + ω 4ω4 16 4 32

(6.92)

6.3 First Passage Forecasting of Ship Capsizing

309

In the above equation P is the peak value of the excitation spectrum, and with the information from Eqs. (6.86–6.92), b(A) and σ 2 (A) in Eqs. (6.83) and (6.84) can be derived, and with the information from b(A) and σ 2 (A), m(z0 ) and j(z0 ) in Eq. (6.43) can be derived as follows m(z0 ) = b(A)

(6.93)

j(z0 ) = σ 2 (A)

(6.94)

Taking n = 0 in Eq. (6.43), we obtain the well-known Pontryagin-Vitt equation 1 + b(A)

d 1 d2 μ1 + σ 2 (A) 2 μ1 = 0 dA 2 dA

(6.95)

Its two boundary conditions are [μ1 (Ac , A0 )]A0 =Ac = 0 [μ 1 (Ac , A0 )]A0 =Al = −

{[μ1 (Ac , A0 )]A0 =Al } b(Al )

(6.96) (6.97)

Note that in Eq. (6.95) μ1 is the first order statistical moment of the first passage time of the rolling amplitude process A, while what we really require is the first order statistical moment of the first passage time of the rolling displacement of the ship (i.e. the rolling angle ψ). In the literature [2] J. B. Roberts has shown that if the rolling response displacement process is sufficiently narrow-banded (and in general the ship’s rolling response process is indeed sufficiently narrow-banded), the firstorder statistical moment of the first passage time of the rolling amplitude process will be a good approximation of the first-order statistical moment of the first passage time of the rolling angle ψ. Thus, here we take the critical state of the rolling amplitude process A to be Ac = 1, and from Eq. (6.56) we can derive that the critical state of the rolling angle ψ should be 1. Because the rolling angle ψ is a proportional angle, the critical state of the true ship rolling angle from Eq. (6.48) is φ=ψφR = φR

(6.98)

That is, the real critical state of the ship rolling angle is the stability vanishing angle of the ship, and it is obvious that we are correct to take the critical state of the rolling amplitude process A to be Ac = 1. Equation (6.95) and its two boundary conditions (6.96) and (6.97) together form a boundary value problem, and the firstorder statistical moments of the first passage time of the stochastic process A(t) can be obtained by solving this boundary value problem using the fourth-order Runge– Kutta method. Tables 6.1 and 6.2 show the calculated mean first passage time values for different damping coefficients and for different excitation spectrum peak values.

310

6 First Passage Theory for Ship Stability Analysis

Note that an excitation spectrum peak value in Tables 6.1 and 6.2 is a proportional value corresponding to the actual external wave excitation moment K(t) in Eq. (6.47). Note also that the mean first passage time T is a proportional time, T = ωR t. Because the unit of ωR is rad/sec, the unit of T becomes rad. Physically, this mean first passage time represents the cycles of the ship’s rolling motions. Similarly, we have explained earlier that the unit of the nonlinear damping coefficient is rad 3 . We can see from Tables 6.1 and 6.2 that the effect of the excitation spectrum peak value on the mean first passage time is quite significant. Under a certain damping condition, when the excitation spectrum peak value is smaller than a certain value, the mean first passage time of the ship rolling will be very large, which explains why large ships seldom capsize in less severe sea conditions (due to the large rolling mass moment of inertia and added mass moment of inertia of large ships, their proportional excitation spectrum peak value will be small under the same sea conditions). To investigate in depth the effect of nonlinear damping coefficients on the mean first passage time, we have taken different combinations of damping coefficients and excitation spectrum peak values in the calculation of the mean first passage time. Figures 6.1 and 6.2 show the results of our calculations. In these two figures, we have taken a linear damping coefficient of α = 0.1 and we have taken three nonlinear damping coefficients of β = 0.0, β = 0.3 and β = 1.0, which are realistic combinations of values of α and β for ship’s rolling motions, established by Dalzel [6] after analyzing the experimental data obtained from ship rolling decay tests. Analysis of these two plots shows that the effect of the nonlinear damping coefficient on the mean first passage time is strong, and this is more significant at low levels of the excitation spectrum peak values. After the mean first passage time T is found, the first passage probability density can be approximated by the following equation [2]   f (A0 , τ ) = (1/T ) exp −τ/T

(6.99)

Table 6.1 The mean first passage time at different nonlinear damping coefficients and excitation spectrum peak values Damping coefficient β value Excitation spectral peak value 0.002 N2 m2 s 0.003 N2 m2 s 0.004 N2 m2 s 0.005 N2 m2 s 0

rad3

0.3 rad3

142316

11197.7

3095.37

1403.73

1528030000

1770700000

19090500

1213050

Table 6.2 The mean first passage time at different nonlinear damping coefficients and excitation spectrum peak values (continued) Damping coefficient β value Excitation spectral peak value 0.006 N2 m2 s 0.007 N2 m2 s 0.008 N2 m2 s 0.009 N2 m2 s 0 rad3

813.659

542.609

395.213

305.499

0.3 rad3

192912

51922.5

19403.9

9019.23

6.3 First Passage Forecasting of Ship Capsizing

311

1200

Mean first passage time (cycles)

1000

800

600

400

200

0

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

Peak value of wave excitation spectral density

Fig. 6.1 Mean first passage time corresponding to different damping coefficients (wave process 3)

300

Mean first passage time(cycles)

250

200

150

100

50

0 0.02

0.04

0.06

0.08

0.10

Peak value of wave excitation spectral density Fig. 6.2 Mean first passage time corresponding to different damping coefficients (wave process 3)

312

6 First Passage Theory for Ship Stability Analysis P=0.04, P=0.08, P=0.10,

0.08

First passage probability density

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 0

20

40

60

80

100

Time (cycles)

Fig. 6.3 The first passage probability density function for A(t)

Figures 6.3 and 6.4 show the results of our calculations regarding the first passage probability densities. Figure 6.3 shows the first passage probability density versus time at the same damping coefficient values for the cases P = 0.04, 0.08 and 0.10. The damping coefficient values are kept at α = 0.1 and β = 0.3. It can be noticed that as the wave excitation level increases, the ship has a larger probability to capsize when the rolling time is not very long. Figure 6.4 shows the first passage probability density versus time at the same excitation level for the cases β = 0, 0.3, and 1.0. The other parameters are kept at α = 0.1 and P = 0.08. It can be noticed that as the nonlinear damping level increases, the ship has a smaller probability to capsize when the rolling time is not very long.

6.4 Summary In this chapter, the applicability of the first passage theory in stochastic dynamics to the study of ship stability is explored by first establishing a second-order differential equation for the ship’s rolling motion, which contains linear and quadratic damping terms and a linear plus cubic restoring moment term. The solution in terms of the Jacobi elliptic function for the free undamped rolling motion of the ship is found first, and then the natural frequency of the rolling motion is obtained by using the complete

6.4 Summary

313 P=0.08, P=0.08, P=0.08,

0.08

First passage probability density

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -0.01 0

20

40

60

80

100

Time (cycles)

Fig. 6.4 The first passage probability density function for A(t)

elliptic integral of the first kind. By introducing an amplitude process of the rolling response and its associated phase angle process, and by introducing two generalized harmonic function terms, the single differential equation of the ship’s rolling motion is transformed into two coupled first-order differential equations in terms of an amplitude and a phase angle after a series of transformations. The coupled first-order differential equations are written as a set of first-order Itô stochastic differential equations by firstly time-averaging the terms without external excitation over one period and then applying stochastic averaging based on the Stratonovitch-Khasminskii limit theorem to the terms with external excitation. The closed-form expressions for the drift and diffusion coefficients in Ito’s stochastic differential equation are obtained (both expressions for these coefficients contain the wave excitation spectral density function). It is also found that the Itô stochastic differential equation controlling the amplitude process is decoupled from the Itô stochastic differential equation controlling the phase angle process, which indicates that the ship’s rolling motion amplitude process converges to a one-dimensional Markov diffusion process. The first-order conditional moment of the first passage time of this rolling motion amplitude process satisfies the Pontryagin-Vitt equation. The coefficient of the firstorder derivative term in the Pontryagin-Vitt equation is equal to the drift coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process, and the coefficient of the second-order derivative term in the Pontryagin-Vitt equation is equal to the diffusion coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process. The Pontryagin-Vitt equation together with its two boundary conditions form a boundary value problem, and the mean first passage

314

6 First Passage Theory for Ship Stability Analysis

time of the ship’s rolling motion can be obtained by numerically integrating this boundary value problem. We find that the effect of the excitation spectrum peak value on the mean first passage time is quite significant. The effect of the nonlinear damping coefficient on the mean first passage time is also found to be strong, which is especially significant at low levels of the excitation spectrum peak value. We then use an approximate formula to calculate the first passage probability density values for the ship rolling-induced capsizing, and the results show that as the value of the nonlinear damping coefficient (β) increases, the probability of ship capsizing will decrease when the rolling time is not too long.

Exercises Exercise 6.1 Please derive the backward Kolmogorov equation for the Ornstein– Uhlenbeck process, which is governed by the following Itô stochastic differential equation dX = −aXdt +

√ 2aσ 2 dB(t),

in which a > 0 and σ are constants, and B(t) denotes a unit Brownian motion. Exercise 6.2 Please derive the Pontryagin–Vitt equations for the Ornstein–Uhlenbeck process in Exercise 6.1.

References 1. Denis M, Pierson WJ (1953) On the motions of ships in confused seas. Trans. SNAME 61: 280–357 2. Roberts JB (1986) Response of an oscillator with non-linear damping and a softening spring to non-white random excitation. Probab Eng Mech 1(1): 40–48 3. Xu Z, Cheung YK (1994) Averaging method using generalized harmonic functions for strongly non-linear oscillators. J Sound Vibr 174: 563–576 4. Zhu WQ, Huang ZL, Suzuki Y (2001) Response and stability of strongly nonlinear oscillators under wide-band random excitation. Int J Non-Linear Mech 36:1135–1250 5. Lin YK, Cai GQ (2004) Probabilistic structural dynamics. advanced theory and applications. McGraw-Hill Professional, New York 6. Dalzel JF (1973) A study of the distribution of maxima of non-linear ship rolling in a seaway. J Ship Res 17(4): 217–226

Chapter 7

Summary and Outlook

In the previous chapters, various methods used to perform the response and stability analysis of ships and marine structures under the action of random loads are described. Among them, the Monte Carlo Simulation method and the numerical path integration method for analyzing the response problems of marine structures are described. The shortcomings of the traditional Monte Carlo Simulation method for analyzing the slow-drift extreme response and stability of marine structures are pointed out. The performance when analyzing the slow drift extreme response of marine structures subjected to random excitations by the path integration method is investigated. The global geometry method and the first passage theory for analyzing the stability problems of ships are also described, and the performance of the Melnikov method for analyzing the dynamic stability of ships subject to random excitations is also investigated. The following sections summarize the contents of this book and also points out some directions for further research in this area.

7.1 Main Contents of This Book (1) In analyzing the response problem of marine structures by the Monte Carlo Simulation method, a single-degree-of-freedom equation of motion with a nonlinear Duffing stiffness term and a nonlinear Morison drag force term taking into account the effect of currents was developed for an offshore system. In this equation, the Duffing stiffness is a statistically symmetric nonlinear term and the Morison drag force term is a statistically asymmetric nonlinear term. In the study, the fourth-order Runge–Kutta method was used to numerically integrate the single-degree-of-freedom equation of motion and to obtain the time histories of the surge oscillatory displacement responses of the offshore structure at different current velocities. The analysis of these simulation results led to some conclusions about the response oscillation patterns and the positive and negative peaks of the response amplitudes at different current speeds, which © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5_7

315

316

7 Summary and Outlook

are useful in engineering practice. Subsequently, the probability density curves of the system responses at different current velocities were also derived from the Monte Carlo simulations, and these curves showed that the responses of a nonlinear structural system subjected to Gaussian random excitations are nonGaussian. Further studies on the mean and variance of the system responses were also performed by using Monte Carlo simulations, and the results showed that these values increase monotonically with the increasing values of the sea current velocity. A study of the skewness and kurtosis of the system responses showed that the shape of the probability density curve of the platform responses is always right skewed with respect to the mean when U > 0.0 m/s; while the shape of the probability density curve of the platform response is sometimes sharp peaked and sometimes mild peaked. This part of the study concluded with the calculations of the level up-crossing rates of the response values for the above offshore platform under different current velocities by using Rice’s formula, and these level up-crossing rate values are the key to derive many important data for response statistics and reliability applications. In order to study the slow drift extreme response and stability of marine structures under random excitations, dynamic analysis of the marine structures is required. In the dynamic analysis of a marine structure, the added mass of the marine structure, the damping and restoring forces of the marine structure, and the external excitation force per unit wave amplitude of the marine structure can all be calculated by some mature commercial hydrodynamic software, so that the nonlinear stochastic differential equation of the motion of the marine structure can be established. The main content of this book is the study of efficient and accurate methods for solving these nonlinear stochastic differential equations. In the study of slow drift extreme responses of marine structures, the differential equation of motion of a marine structure is a stochastic differential equation, in the real situation the stochastic differential equation has no analytical solutions, and can only be solved with the help of some numerical methods, of which the Monte Carlo Simulation is one of the methods often used. Performing Monte Carlo simulations essentially means to conduct numerical integration of the differential equation of motion and then statistically processing the response time histories. In order to evaluate the performance of the Monte Carlo Simulation method in analyzing the slow drift extreme response of marine structures, this book integrates the two issues of the efficiency of simulation and the accuracy of the slow drift extreme response predicted by simulation. We first considered the case of a moored floating cylinder. In the special case where the equation of motion of this floating cylinder has an exact stationary analytical solution, as a comparative benchmark, we compared the marginal probability density values of the displacement response of this marine structure obtained from 1000 Monte Carlo simulations and found that the accuracy of the extreme response of the marine structure predicted by the simulation is not high. To improve the accuracy of the extreme response prediction, we increased the number of simulations to 2000 and obtained another set of results. We saw that multiplying the number of simulations did improve the accuracy of the extreme

7.1 Main Contents of This Book

317

response prediction to some extent. For this reason, we continued to increase the number of simulations to 5100, and we saw an improvement in the accuracy of the extreme response prediction when we increased the number of simulations to 5100. However, it is difficult to continue to improve the prediction accuracy of the extreme response by continuing to increase the simulation runs substantially, when the accuracy of the extreme response values has not yet reached a satisfactory level. Based on the recorded simulation elapsed time, we also judged that the efficiency of predicting the extreme response of marine structures under stochastic excitations using the Monte Carlo Simulation method is on the low side. We therefore need to investigate alternative methods to analyze the slow drift extreme responses of marine structures under random excitations. In this book, a quantitative study of the performance of Monte Carlo simulations to analyze the stability of marine structures (ships) was also done. It was found that the efficiency of Monte Carlo simulations for the exhaustive study of the phase space of the initial conditions of the ship rolling is extremely low, i. e., the efficiency of Monte Carlo simulations for the study of the transient behavior of ship capsizing is extremely low. The book then investigated the performance of Monte Carlo simulations to analyze the stability of a ship when the initial conditions of motion are fixed and when the frequency of the excitation force and the amplitude of the excitation force are varied, and it was found that the efficiency of Monte Carlo simulations to study the ship’s stability in the control parameter space is also low. A comprehensive analysis of the above results shows that it is generally difficult to draw conclusions about the dynamic stability of a ship based on the results of Monte Carlo simulations unless the initial conditions of motion and the parameter space of the ship system are studied in detail. It is therefore necessary to investigate alternative methods to rationally analyze the stability of ships under stochastic excitations. (2) In analyzing the response problems of marine structures by the numerical path integration method, this book first systematically elaborates the theory of Markov diffusion processes. This book also presents a comparative study and analysis of various numerical interpolation procedures used to perform path integration, and focuses on an efficient numerical interpolation procedure, the Gauss–Legendre integration scheme. Although the path integration method based on the Gauss–Legendre integration scheme was first proposed by other authors, the author of this book utilized a more efficient programming language to apply this method in the case studies in this book. In addition, since the development of the path integration method based on the Gauss–Legendre integration scheme, the problems handled by this method have been limited to the weakly nonlinear cases. In this book, the author has successfully used the path integration method based on the Gauss–Legendre integration scheme to deal with a strongly nonlinear stochastic oscillation problem. The restoring force in this problem is a combination of a linear component and a cubic component, and the system is a strongly nonlinear Duffing oscillator. Next, the author of this book investigated the response of a moored ship under non-Gaussian slow-drift wave force excitations using the numerical path integration method. To this end, the

318

7 Summary and Outlook

author first derived the spectral density function of the second-order slow-drift wave forces on the moored vessel and determined that it is appropriate to assume the slow-drift wave force excitations as a Gaussian white noise. This proves that the joint process of the ship response displacement and response velocity is a Markov diffusion process, and the equation with such stochastic processes is suitable to be solved by the path integration method. The author of this book then used the path integration method based on the Gauss–Legendre integration scheme to solve the stochastic differential equation for the slow drift motion of the ship and obtained the probability density values of the responses. This study shows that the path integration method can provide a powerful tool for naval architects and ocean engineers to predict the slow drift oscillation responses of moored ships in irregular waves. Because of the importance of improving the prediction accuracy of slow drift extreme response in the design process of marine structures, it is necessary to explore other methods to predict the slow drift extreme response of marine structures efficiently and accurately. For this purpose, we have investigated the performance of the path integration method when using it to analyze the slow drift extreme response of marine structures subjected to random excitations. In a case study in this book, we analyzed the extreme slow drift response of a specific marine structure subjected to random excitations by the path integration method. For comparison, we used the same marine structure example that has been analyzed by the Monte Carlo Simulation method, i.e., the slow drift surge motion of a moored floating cylinder. First, we compared the accuracy of the path integration solution with that of the Monte Carlo Simulation in the special case where the system equation of the structure has an analytical solution. It was found that the results obtained by the path integration method are much more accurate than those obtained by the MCS method when forecasting the extreme responses of the marine structure under random excitations. It was also found that the path integration method is more efficient than the MCS method in forecasting the extreme responses of the marine structure. Next, in the general case where the system equation of the structure does not have an analytical solution, we obtained accurate forecasts of the slow drift extreme responses of this moored floating cylinder using the path integration method. We found that another major advantage of using the path integration method is that the exceedance probabilities of the slow drift extreme responses of this moored floating cylinder can be obtained concurrently and with little additional computing time costs required. When the path integration method is used to find the slow drift extreme response of a marine structure, the initial probability density of the slow drift response of the marine structure is arbitrarily selected by the researcher himself (or herself), and the good or bad selection of the initial probability density has a great influence on the running time costs of the path integration method. In order to find a better way to further improve the efficiency of the path integration method, a compound path integration method is proposed in this book. The idea is to first roughly estimate an initial probability density by using a small amount of Monte Carlo simulations and

7.1 Main Contents of This Book

319

combining the 3σ principle of the normal distribution, so that the obtained initial probability density will be close to the final result. Then, the obtained initial probability density is substituted into the computer source program of the original path integration method, and the number of path integration runs can be saved. We compared the efficiency and accuracy of the compound path integration method with the original path integration method in the special case where the system equation of the structure has an exact analytical solution, and the accuracy and efficiency of the compound path integration method have been substantiated. Next, in the general case where the system equation of the structure does not have an analytical solution, we utilized the compound path integration method to calculate the slow drift extreme response of a moored floating cylinder and obtained accurate prediction results. This part of the study shows that the path integration method can provide the naval architects and ocean engineers with a powerful tool to predict the slow drift extreme responses of moored marine structures under stochastic excitations, if used in conjunction with some suitable commercial hydrodynamic software. (3) In applying the global geometric method to the study of ship stability, this book first reviewed the history of the development of the current IMO intact stability codes and examined the basis on which these intact stability codes were developed. Some of these codes are empirically based statistical codes, which are not based on a realistic physical model and do not mention sea states in the codes. Some of these codes are semi-empirical codes, which are also not based on an actual physical model, and the randomness of the wind and wave actions, the nonlinearity of ship motions and the coupling effects between the motions of each degree of freedom are hardly taken into account. The book also examined the long-term trend in the development of the IMO’s intact stability codes, i. e., the new codes should be based on “first-principles analysis”. The book went on to provide an introduction to the global geometric method for assessing the stability of ships under harmonic excitations. Phase space transport theory and lobe dynamics were used to show how motions starting from initial conditions in domains confined by intersected manifolds evolve, and also to explain how unexpected capsizing can occur. The book went on to calculate the Melnikov criterion for the capsizing of a fishing ship under harmonic excitations based on the real ship data of this vessel that capsized twice. The encountered wave frequency and the wave amplitude are directly included in this criterion. Next, this book analyzed the dynamic stability of a barge (i.e., the damping required to prevent the barge from capsizing) using the Melnikov method. To this end, we first calculated the relevant hydrodynamic parameters in the differential equation of the ship’s rolling motion, then we nondimensionalized the relevant hydrodynamic parameters. Next, we calculated the nonlinear damping coefficient required to overcome the critical wave excitation moment based on the existing Melnikov criterion, and then we calculated the actual nonlinear damping coefficient of the barge using empirical formulas. We found that solving some very simple analytical expressions in just a few minutes can lead to some important conclusions about ship stability, and these conclusions are based on a first-principles analysis

320

7 Summary and Outlook

that takes into account some of the ship’s design parameters, such as the shape of the restoring arm curve and the ship’s damping values, and also relates these parameters to the external environment. The Melnikov criteria relates the critical wave excitation to the damping value of the ship, which allows the designer to adjust the damping value of the ship according to the critical wave excitation value, for example, by increasing the bilge keel size, or by adjusting some other major parameters of the ship. This is particularly important in the initial design phase of the ship when the design requires repeated modifications. We also analyzed the stability of a ship with a cubic viscous damping term using the global geometry method, and the resulting conclusions were verified by using the maximum Lyapunov exponents. We also compared the efficiency of the Monte Carlo Simulation method when studying the dynamic stability of a ship in the control parameter space and find that Melnikov analysis has advantages in terms of both efficiency and ease of reaching conclusion over the Monte Carlo Simulation method. We next investigated the nonlinear rolling and capsizing behavior of a ship with an initial bias in random waves using the global geometry method. The randomness of the waves was represented by sinusoidal waves plus white noise disturbances. Thompson’s α-parameterized family of restoring functions were used to represent the bias in the ship’s equations of motion, and the generalized Helmholtz-Thompson equation was proposed for the first time. To take into account the randomness presented in the excitation and response, a stochastic Melnikov method was developed and a mean square criterion was obtained to provide an upper bound to the potential chaotic rolling motion domain. It was shown that increasing the bias value in the restoring function can reduce the threshold level of the chaotic rolling motion of the ship, thus expanding the chaotic domain in the parameter space. It was also shown that Melnikov analysis can take into account the global dynamic characteristics of the system and can easily include the combined effects of both the deterministic and stochastic excitations. Given a value of bias, we derived values of critical wave excitation moments corresponding to different values of damping. We also compared the efficiency of an exhaustive study of the phase space of the initial conditions of the ship rolling using Monte Carlo simulations. It was found to be efficient to analyze the capsizing behavior of a ship with an initial bias by the Melnikov method at the initial design stage of the ship. (4) In the application of the first passage theory to the study of ship stability, this book first provided a systematic account of the first passage theory in stochastic dynamics, focusing on the stochastic averaging method based on generalized harmonic functions and the first passage theory for the one-dimensional Markov diffusion process. In this book, the second-order differential equation of the ship’s rolling motion was then established, which contains linear and quadratic damping terms and a linear plus cubic restoring moment term. We found the solution in terms of the Jacobi elliptic function for the free undamped rolling motion of the ship, and we proceeded to use the complete elliptic integral of the first kind to find the rolling motion natural frequency, which was used later in the application of the stochastic averaging procedure based on the generalized

7.1 Main Contents of This Book

321

harmonic functions. The study in this book proceeded to transform the single differential equation of ship’s rolling motion into two coupled first-order differential equations expressed in terms of amplitude and phase angle by introducing a rolling response amplitude process and its associated phase angle process, by introducing two generalized harmonic function terms and after a series of transformations. We then wrote the above coupled first-order differential equations into a set of first-order Itô stochastic differential equations by firstly conducting time averaging on the terms without external excitation in the above coupled first-order differential equations and then by applying stochastic averaging based on the Stratonovitch-Khasminskii limit theorem to the terms with external excitation in the above coupled equations. The closed-form expressions for the drift and diffusion coefficients in both the Itô stochastic differential equations were obtained by stochastic averaging (both expressions for these coefficients contain wave excitation spectral density functions). Moreover, it was found that the Itô stochastic differential equation controlling the amplitude process is decoupled from the Itô stochastic differential equation controlling the phase angle process, i.e., the ship rolling motion amplitude process converges to a one-dimensional Markov diffusion process. The conditional reliability function of this rolling motion amplitude process satisfies the backward Kolmogorov equation, and the conditional moments of the first passage time of this rolling motion amplitude process satisfy the generalized Pontryagin equation. The coefficient of the first-order derivative term in the Pontryagin equation is equal to the drift coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process, and the coefficient of the second-order derivative term in the Pontryagin equation is equal to the diffusion coefficient of the above-mentioned Itô stochastic differential equation for the amplitude process. The PontryaginVitt equation together with its two boundary conditions form a boundary value problem, and the first-order conditional moments of the first passage time were obtained by solving this boundary value problem numerically with the fourthorder Runge–Kutta method. The first-order conditional moments of the first passage time were then used to calculate the first passage probability density of the rolling motion amplitude process. The effect of the hydrodynamic damping on the mean first passage time was investigated by taking different values of nonlinear damping coefficients in the ship’s rolling equation of motion. The effect of the external excitation level on the mean first passage time was investigated by taking different levels of the external wave excitation in the ship’s rolling equation of motion. In this way, the relationship between the hydrodynamic damping (or the external wave excitation level) and the ship capsizing is expressed quantitatively, which provides a rigorous theoretical approach to study the stability of ships in random waves.

322

7 Summary and Outlook

7.2 The Novel Aspects in This Book (1) In applying the Monte Carlo Simulation method to analyze the response problem of marine structures: • In this book, the differential equation of motion with a nonlinear Duffing stiffness term and a nonlinear Morison drag force term considering the effect of currents was first developed for an offshore system. The Monte Carlo simulations were used to calculate the time history, probability density and central moments up to the fourth order for the response values of this offshore system at different values of current velocity. The analysis of these simulation results led to some useful conclusions for engineering purposes. • In this book, the level up-crossing rates of the response values of an offshore system were first calculated at different values of current velocity by using the Rice’s formula, and these level up-crossing rate values are important in the reliability analysis of the system. • Monte Carlo simulations are often used to analyze and forecast the extreme behaviors (extreme responses and capsizing behavior) of marine structures under random loads. The principles of the Monte Carlo Simulation are very concise, and the implementation of the MCS method is also very convenient. Although the inefficiency of simulation has been mentioned by some researchers in some studies, none of the studies so far has considered the two issues of the efficiency of simulation and the accuracy of the extreme responses predicted by simulation in an integrated manner. In this book, for the first time, an in-depth study has been made to address this issue, and a compliant offshore structure example has been used to illustrate that the accuracy of the simulation prediction of the slow drift extreme responses of marine structures is not high. To improve the accuracy of the predicted slow drift extreme responses, one would like to increase the number of runs of simulations. In this book, we have shown for the first time with a case study that after the accuracy of the slow drift extreme responses is improved to a certain level, it is almost useless to continue increasing the number of runs of simulations, while the accuracy of the slow drift extreme response values has not yet reached a satisfactory level. This raises the need for exploring alternative methods to predict the extreme responses of marine structures under random loads. • This book has also performed a quantitative study of using Monte Carlo simulations to analyze the stability performance of marine structures (ships). The results of the analysis have demonstrated that it is generally difficult to draw conclusions about the stability of marine structures based on the results of Monte Carlo simulations unless the initial conditions of motion and the parameter space of the marine structure system are studied exhaustively. This raises the need to investigate alternative methods to rationally analyze the stability of marine structures.

7.2 The Novel Aspects in This Book

323

(2) In the analysis of the response problem of marine structures by the numerical path integration method: • In applying the path integration method to analyze the extreme response of nonlinear stochastic systems, this book has conducted a comparative study of various numerical interpolation procedures used to perform the path integration, with a focus on an efficient numerical interpolation procedure, the Gauss–Legendre integration scheme. Since the development of the path integration method based on the Gauss–Legendre integration scheme, the problems dealt with by this method have been limited to the weakly nonlinear cases. In this book, we have successfully used the path integration method based on the Gauss–Legendre integration scheme to tackle a strongly nonlinear stochastic oscillation problem and found that the forecasting accuracy of the extreme response is also high in the case of strong nonlinearity. • In this book, the path integration method based on Gauss–Legendre integration scheme was used for the first time to analyze the extreme response of a compliant offshore structure subject to stochastic excitations, and it was found that the results obtained by the path integration method are much more accurate than those obtained by the Monte Carlo Simulation method. It was also found that the efficiency of the path integration method for forecasting the extreme response of the structure is higher than that of the Monte Carlo Simulation method. We also found that another major advantage of using the path integration method is that the exceedance probabilities of the slow drift extreme responses of this marine structure can be obtained concurrently and there is little extra time consumed while the computer program is running. • In this book, we have proposed for the first time a compound path integration method, which is based on the 3 σ principle of the normal distribution and combines the respective superiority of the Monte Carlo Simulation method and the path integration method to find an efficient way to further improve the efficiency of the original path integration method by optimizing the selection of the initial probability density function. We compared the efficiency and accuracy of the compound path integration method with the original path integration method in the special case where the system equation of the structure has an analytical solution, and demonstrated the accuracy and efficiency of the compound path integration method. In the general case where the system equation of the structure does not have an analytical solution, we have calculated the extreme slow drift surge responses of a moored floating cylinder by utilizing the compound path integration method. (3) In the application of the global geometry method to study the stability of ships: • In this book, we have provided an in-depth review and some insightful thoughts on the current IMO intact stability codes.

324

7 Summary and Outlook

• In this book, we proceeded to calculate the Melnikov criterion for the capsizing of a ship under harmonic excitations based this ship’s real data, and the obtained Melnikov criterion directly contains the encountered wave frequency and the wave amplitude. • By applying the Melnikov method to a barge example and by comparing the efficiency of Monte Carlo simulations to study the stability of the barge in the control parameter space, we found that Melnikov analysis is both more efficient and easier to reach conclusions than the Monte Carlo Simulation method. This book also provides an additional case study to verify the Melnikov criteria by using the maximum Lyapunov exponent. • In this book, we have proposed the generalized Helmholtz-Thompson equation for the first time. • In this book, we have for the first time derived a mean square value based stochastic Melnikov criterion for ship capsizing. We have used the stochastic Melnikov method to analyze the stability of a ship with an initial bias, and we have also compared the efficiency when the phase space of the initial conditions of the ship rolling was studied exhaustively using Monte Carlo simulations. It has been found to be efficient to analyze the stability of a ship with an initial bias by the Melnikov method at the initial design stage of the ship. • In the light of our findings, we concluded that the Melnikov method can be utilized as an efficient tool to analyze the dynamic stability of a ship in the initial design phase of the ship if some suitable commercial hydrodynamic software is available for use with it. (4) In the application of the first passage theory to study the stability of ships: • In order to adopt the Markov diffusion process theory for the first passage problem, the dimensionality of the state space of the ship’s rolling motion must first be reduced. To achieve the above purpose, a new set of stochastic averaging procedure based on generalized harmonic functions have been adopted in this book for the first time. • In applying the stochastic averaging procedure based on generalized harmonic functions, we have for the first time in this book utilized the complete elliptic integral of the first kind to find the natural frequency of the rolling motion. • In this book, we have for the first time calculated the first passage probability density values of the ship rolling motion amplitude process. • In this book, we have for the first time studied the effects of the nonlinear damping coefficients of the ship rolling motion on the mean first passage time.

7.3 Directions for Further Research

325

7.3 Directions for Further Research (1) In applying the Monte Carlo Simulation method to analyze the response of marine structures, one of the most straightforward solutions to the disadvantage that the Monte Carlo Simulation method is too computationally intensive is to use a large-scale parallel computing platform. To address the drawback that the Monte Carlo Simulation method is not very accurate in forecasting the probability density values of extreme responses, some variance reduction techniques, controlled-simulation techniques and genetic algorithms should be further developed. It is obvious that the existing variance reduction techniques, controlled- simulation techniques and genetic algorithms cannot be applied to large-scale problems, but it is possible to make breakthroughs in these areas with further developments. (2) In the analysis of the response problem of marine structures by the numerical path integration method, the random excitation term in the equation of motion dealt with by the path integration method must be a Gaussian white noise. When applying the path integration method to solve the response problem of marine structures subjected to first-order narrow-band wave random excitations, a suitable nonlinear filter should be designed first in order to turn the Gaussian white noise into a specific narrow-band spectrum. The existing research in this area is only with respect to first-order shaping filters. However, the peak value of the wave velocity spectrum generated by the first-order shaping filter appears at the zero frequency. However, for real ocean systems the peak value of the wave velocity spectrum does not occur at the zero frequency. Therefore, higher-order shaping filters should be designed, for example, the Pierson-Moskowitz wave spectrum should be fitted by a fourth-order shaping filter. Applying the path integration method to such problems would be of more realistic significance. In this book we have investigated the use of a path integration method based on the Gauss–Legendre integration scheme to analyze the slow-drift extreme response of compliant offshore structures subjected to random excitations, verified the efficiency and accuracy of the path integration method, proposed a compound path integration method, and also demonstrated the accuracy and efficiency of the compound path integration method. Further research is suggested to be carried out to use the path integration method or the compound path integration method to tackle some more challenging problems. For example, there is a dry friction damping (Coulomb damping) in the surge oscillation equation of a deep draft Spar platform. If one tries to analyze its dynamic responses by the compound path integration method, it will be difficult to derive the moment equations from the original equation of motion because of the requirement to solve the mathematical expectation of a special class of functions or their composite functions, and because of the need to derive analytic expressions. Even if this difficulty can be overcome, solving the resulting 5 joint moment equations is not an easy task, because the set of equations contains several more special classes of functions. Only when this set of moment equations is solved, it is possible to find the short-time transition probability densities and eventually

326

7 Summary and Outlook

derive the joint probability density function of the responses using path integration. The author believes that the successful solution of such problems by the compound path integration method will attract much attention from international colleagues. On the other hand, with the rapid development of the world economy, the power supply in some countries is also increasingly tight, which has threatened the economic security. As most countries in the world are strictly limiting the construction of new thermal power plants, the geographical constraints on nuclear power development are not small, hydropower developments in most countries in the world have been nearly finished, solar energy is difficult to popularize because the cost is too high, and thus some countries’ decision makers have turned their attention to the wind power industry. Wind power is a kind of green and clean renewable energy, but because land resources in most countries in the world are very scarce, seeking clean wind energy from the sea has become a strategic choice. The world’s offshore areas, especially offshore deep-water areas, are vast and have huge wind farm potential. In deep offshore waters, floating offshore wind turbines will be the most economical solutions, e.g., Spar-platform floating wind turbines [1, 2] and semi-submersible floating wind turbines [3, 4], etc. It is noted here that if the wind turbine is considered as a part of the overall support platform, then the motion response analysis of a deepwater floating wind turbine system is exactly the same as that of a deep-water floating structure (Spar platform, semi-submersible platform, etc.) in the traditional field of ocean engineering. Thus, it will be of great engineering importance to apply the compound path integration method proposed in this book to forecast the slow drift response extremes of these new deep-water floating wind turbine systems. (3) In the application of the global geometry method to study the stability of the ship, only a single degree of freedom nonlinear differential equation of motion for rolling was used in this book without considering the coupling effects with other degrees of freedom. Further studies should naturally consider the Melnikov criteria for the multi-degree of freedom cases, such as the case of two-degree of freedom motions with coupled rolling and swaying, and the case of threedegree of freedom motions with coupled rolling, swaying and heaving motions. In addition, the stability problem will be a prominent issue in the development of new ship concepts, especially in the development and design of some military high-speed ships, because the current stability codes are reliable only when applied to common ship types. In this regard, Melnikov analysis, together with Monte Carlo simulations and ship model experiments, will provide powerful tools for analyzing the stability problems of these special ship types. The author believes that it is worthwhile to conduct such kind of research on the application of the Melnikov method. In addition, on some problems that have limitations when treated with Melnikov analytical analysis, such as the stability of a ship when following the sea, is it possible to use the Melnikov method in conjunction with the Monte Carlo simulations? In other words, is it worthwhile to first utilize Melnikov analysis to narrow down the range of the initial conditions and the control parameters of the ship that should be studied, and then to conduct Monte Carlo simulations in a targeted manner, which will greatly improve the efficiency

References

327

of the simulation? The author believes that further research is also worthwhile in this area. (4) In applying the first passage theory to study the stability of a ship, this book only studies the first passage of the ship’s rolling amplitude, i.e., the transverse stability problem. Now some major international classification societies have been formulating the regulations about the longitudinal stability of container ships, and it is of strong practical significance to further apply the first passage theory to the study of the longitudinal stability of ships.

References 1. Laface V, Alotta G, Failla G, Ruzzo C, Arena F (2022) A two-degree-of-freedom tuned mass damper for offshore wind turbines on floating spar supports. Mar Struct 83: 103146 2. Fenu B, Bonfanti M, Bardazzi A, Pilloton C, Lucarelli A, Mattiazzo G (2023) Experimental investigation of a Multi-OWC wind turbine floating platform. Ocean Eng 281: 114619 3. Yu Z, Amdahl J, Rypestøl M, Cheng Z (2022) Numerical modelling and dynamic response analysis of a 10 MW semi-submersible floating offshore wind turbine subjected to ship collision loads. Renew Energy 184: 677-69 4. Hawari Q, Kim T, Ward C, Fleming J (2023) LQG control for hydrodynamic compensation on large floating wind turbines. Renew Energy 205: 1–9

Appendix A

Derivation of the Kolmogorov Equation

Equation (4.7) in Chap. 4 is referred to as the Chapman-Kolmogorov-Smoluchowski equation. Equation (4.7) can be considered as an integral equation controlling the transition probability density of a Markov stochastic process, and this integral equation can be transformed into an equivalent differential equation called the FokkerPlank-Kolmogorov equation. We provide the detailed transformation procedures below with reference to the monograph of Lin and Cai [1], and we also refer to Ochi [2], Lin [3] and Risken [4] in the derivation process. Let X(t) be a vector Markov process defined on a m-dimensional space, and let us consider the following integral  I =

 ···

R( y)

∂ q( y, t|x 0 , t0 ) d y1 dy2 · · · dym ∂t

(A.1)

Note that in the above equation a boldface letter represents a vector. y is also a vector defined on a m-dimensional space and its elements are y j , j = 1, 2, . . . , m. R( y) is an arbitrary function of y j , and R( y) will rapidly converge to zero as any y j converges to an upper or lower limit of the integration. We also assume that for any s = k + l + · · · + r , when any y j tends to an upper or lower limit of the integration we have ∂s R( y) → 0 · · · ymr

∂ y1k y2l

(A.2)

Now the integral (A.1) can be written as 



 1  q( y, t + Δt|x 0 , t0 ) − q( y, t|x 0 , t0 ) d y1 dy2 · · · dym R( y) lim Δt→0 Δt     1 = lim · · · R( y) q( y, t + Δt|x 0 , t0 ) − q( y, t|x 0 , t0 ) d y1 dy2 · · · dym Δt→0 Δt (A.3)

I =

···

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5

329

330

Appendix A: Derivation of the Kolmogorov Equation

The above equation is derived because the order of the integral and differential operations in the above equation is interchangeable as long as the expression on the right side of the second equality sign in the above equation converges uniformly in the neighborhood of t. Using the Chapman-Kolmogorov-Smoluchowski equation,  q(x2 , t2 |x1 , t1 ) =

q(x2 , t2 |y, t) q(y, t|x1 , t1 ) dy

(A.4)

The afore-mentioned Eq. (A.3) can be written as 

  dy1 dy2 · · · dym R( y) · · · [q( y, t + Δt|x, t) q(x, t|x 0 , t0 ) ]    d x1 d x2 · · · d xm − · · · R( y)q( y, t|x, t) dy1 dy2 · · · dym (A.5) 1 Δt→0 Δt

I = lim



···

We further assume that any function R( y) can be expanded as a Taylor series with respect to the point x R( y) = R(x) + (y j − x j ) +

∂ R(x) 1 ∂ 2 R(x) + (y j − x j )(yk − xk ) ∂x j 2! ∂ x j ∂ xk

∂ 3 R(x) 1 (y j − x j )(yk − xk )(yl − xl ) + ··· 3! ∂ x j ∂ x k ∂ xl

(A.6)

Substituting Eq. (A.6) into Eq. (A.5), we first integrate with respect to y to obtain   ∂ R(x) 1 ∂ R2 (x) a j (x, t) + b jk (x, t) ∂x j 2! ∂ x j ∂ xk  3 ∂ R (x) 1 + · · · q(x, t|x 0 , t0 ) d x1 d x2 · · · d xm + c jkl (x, t) 3! ∂ x j ∂ x k ∂ xl 

I =

···

(A.7)

where   1 a j (x, t) = lim · · · (y j − x j )q( y, t + Δt|x, t) dy1 dy2 · · · dym (A.8) Δt→0 Δt   1 b j (x, t) = lim · · · (y j − x j )(yk − xk )q( y, t + Δt|x, t) dy1 dy2 · · · dym Δt→0 Δt (A.9) 1 Δt→0 Δt

c j (x, t) = lim





···

(y j − x j )(yk − xk )(yl − xl )q( y, t + Δt|x, t) dy1 dy2 · · · dym · ·

(A10)

Appendix A: Derivation of the Kolmogorov Equation

331

For deriving Eq. (A.7), we have made use of the following fact that, for arbitrary x, t and Δt   · · · q( y, t + Δt|x, t) dy1 dy2 · · · dym = 1 (A.11) Equations (A.8)–(A.10) can be respectively written with a more explicit meaning as follows:  1  E X j (t + Δt) − X j (t)|X(t) = x Δt→0 Δt

a j (x, t) = lim b j (x, t) = lim

Δt→0

(A.12)

 1  E X j (t + Δt) − X j (t) [X k (t + Δt) − X k (t)]|X(t) = x Δt (A.13)  1  E X j (t + Δt) − X j (t) [X k (t + Δt) − X k (t)] Δt→0 Δt (A.14) ×[X l (t + Δt) − X l (t)]|X(t) = x }

c j (x, t) = lim

The functions defined in Eqs. (A.8) through (A.10) give the rates of various moments of the increment in X(t) conditional on X(t) = x. They are referred to as the derivate moments. The integration in Eq. (A.7) can now be carried out by parts for each term. By use of the assumption (A.2), the integration leads to    ∂  1 ∂2  a j (x, t)q(x, t|x 0 , t0 ) + b jk (x, t)q(x, t|x 0 , t0 ) R(x) − ∂x j 2! ∂ x j ∂ xk  3   ∂ 1 − c jkl (x, t)q(x, t|x 0 , t0 ) + · · · d x1 d x2 · · · d xm (A.15) 3! ∂ x j ∂ xk ∂ xl 

I =



···

Combining the Eqs. (A.1) and (A.15), we obtain: 





| |   ∂ 1 ∂2 ∂ | | a j (x, t)q(x, t |x 0 , t0 ) − b jk (x, t)q(x, t |x 0 , t0 ) q+ ∂t ∂x j 2! ∂ x j ∂ xk  |  ∂3 1 | c jkl (x, t)q(x, t |x 0 , t0 ) − · · · d x1 d x2 · · · d xm = 0 (A.16) + 3! ∂ x j ∂ xk ∂ xl

···

R(x)

Because the above equation must be valid for any chosen R(x) that satisfies the general restriction (A.2) and is expandable into a Taylor series, then the following relationship must exist:  1 ∂2   ∂  ∂ q+ a j (x, t)q(x, t|x 0 , t0 ) − b jk (x, t)q(x, t|x 0 , t0 ) ∂t ∂x j 2! ∂ x j ∂ xk 3   1 ∂ + c jkl (x, t)q(x, t|x 0 , t0 ) − · · · = 0 (A.17) 3! ∂ x j ∂ xk ∂ xl

332

Appendix A: Derivation of the Kolmogorov Equation

Equation (A.17) is known as the Fokker-Plank-Kolmogorov equation for the transition probability density q(x, t|x 0 , t0 ) of the vector Markov process X(t), and for the one-dimensional scalar Markov process X (t), we have ∂ ∂ 1 ∂2 q(x, t|x0 , t0 ) + [b(x, t)q(x, t|x0 , t0 ) ] [a(x, t)q(x, t|x0 , t0 ) ] − ∂t ∂x 2 ∂x2 1 ∂3 + (A.18) [c(x, t)q(x, t|x0 , t0 ) ] − · · · = 0 3! ∂ x 3 When the above theory is applied to the practical problem of stochastic dynamics, the derivate moments a j (x, t), b j (x, t) and c j (x, t) are determined by the equations of motion, and the Fokker-Plank-Kolmogorov equation associated with them can then be solved, taking into account the boundary conditions and the following initial condition: q(x, t0 |x 0 , t0 ) = δ(x − x 0 ) =

m ∏

δ(x j − x j0 )

(A.19)

j=1

Equation (A.18) is also known as the Kolmogorov forward equation because ∂/∂t[q( · )] is a derivative with respect to a later time. We then refer here to the Kolmogorov backward equation, where the unknown quantity q(x, t|x 0 , t0 ) in Eq. (A.17) is taken as a function of t and x j , j = 1, 2, · · · m. When the boundary condition (A.19) is imposed, the result of the solution q(x, t|x 0 , t0 ) will contain the parameters t0 and x j0 , j = 1, 2, · · · m.. Conversely, q(x, t|x 0 , t0 ) can also be treated as a function of t0 and x j0 , j = 1, 2, · · · m., while t and x j , j = 1, 2, · · · m. are considered as parameters. We do have the following equation for q  1 ∂ 2q   ∂ ∂q  a j (x 0 , t0 ) + b jk (x 0 , t0 ) q+ ∂t0 ∂ x j0 2! ∂ x j0 ∂ xk0 3   1 ∂ q + c jkl (x 0 , t0 ) + · · · = 0 3! ∂ x j0 ∂ xk0 ∂ xl0

(A.20)

where a j (x 0 , t0 ), b jk (x 0 , t0 ) and c jkl (x 0 , t0 ) are also derivate moments, but they are functions of t0 and x 0 . The proof of the Kolmogorov backward equation (A.20) is given as follows. We write the following equation: 1 ∂q = lim [q(x, t|x 0 , t0 ) − q(x, t|x 0 , t0 − Δt0 ) ] Δt0 →0 Δt0 ∂t0

(A.21)

Using the Chapman-Kolmogorov-Smoluchowski equation  q(x2 , t2 |x1 , t1 ) =

q(x2 , t2 |y, t) q(y, t|x1 , t1 ) dy,

(A.22)

Appendix A: Derivation of the Kolmogorov Equation

333

we obtain  q(x, t|x 0 , t0 − Δt0 ) =

 ···

  q(x, t| y, t0 ) q( y, t0 |x 0 , t0 − Δt0 ) d y1 dy2 · · · dym . (A.23)

Because 

 ···

  q( y, t0 |x 0 , t0 − Δt0 ) d y1 dy2 · · · dym = 1,

(A.24)

so we can also write the following equation  q(x, t|x 0 , t0 ) =

 ···

  q(x, t|x 0 , t0 ) q( y, t0 |x 0 , t0 − Δt0 ) d y1 dy2 · · · dym . (A.25)

Combining the five Eqs. (A.21)–(A.25) yields     1 ∂q = lim ··· q(x, t|x 0 , t0 ) − q(x, t| y, t0 ) Δt0 →0 Δt0 ∂t0 × q( y, t0 |x 0 , t0 − Δt0 ) dy1 dy2 · · · dym

(A.26)

Expanding q(x, t| y, t0 ) into a Taylor series: q(x, t| y, t0 ) = q(x, t|x 0 , t0 ) + (y j − x j0 ) + +

∂ q(x, t|x 0 , t0 ) ∂ x j0

1 ∂2 (y j − x j0 )(yk − xk0 ) q(x, t|x 0 , t0 ) 2! ∂ x j0 ∂ xk0 1 ∂3 (y j − x j0 )(yk − xk0 )(yl − xl0 ) q(x, t|x 0 , t0 ) + · · · 3! ∂ x j0 ∂ xk0 ∂ xl0 (A.27)

Substituting Eq. (A.27) into Eq. (A.26) yields ∂q 1 = lim ∂t0 Δt0 →0 Δt0

 

 ···

−(y j − x j0 )

| | ∂ 1 q(x, t || x 0 , t0 ) − (y j − x j0 )(yk − xk0 ) ∂ x j0 2!

∂2 1 q(x, t|x 0 , t0 ) − (y j − x j0 )(yk − xk0 )(yl − xl0 ) ∂ x j0 ∂ xk0 3!  ∂3 q(x, t|x 0 , t0 ) − · · · q( y, t0 |x 0 , t0 − Δt0 ) dy1 dy2 · · · dym × ∂ x j0 ∂ xk0 ∂ xl0

×

(A.28)

334

Appendix A: Derivation of the Kolmogorov Equation

Each derivate moment can now be rewritten as   1 a j (x 0 , t0 ) = lim · · · (y j − x j0 )q( y, t0 |x 0 , t0 − Δt0 ) dy1 dy2 · · · dym Δt0 →0 Δt0 (A.29)   1 b j (x 0 , t0 ) = lim · · · (y j − x j0 )(yk − xk0 )q( y, t0 |x 0 , t0 − Δt0 ) Δt0 →0 Δt0 (A.30) dy1 dy2 · · · dym   1 · · · (y j − x j0 )(yk − xk0 )(yl − xl0 ) Δt→0 Δt0 × q( y, t0 |x 0 , t0 − Δt0 ) dy1 dy2 · · · dym · ·

c j (x 0 , t0 ) = lim

(A.31)

Substituting Eqs. (A.29)–(A.31) into Eq. (A.28) gives Eq. (A.20), which is called the Kolmogorov backward equation. The variables such as x j0 are called backward variables because they are associated with a previous time t0 . For a Markov diffusion process, Eq. (A.20) can be simplified as  1 ∂ 2q   ∂ ∂q  q+ a j (x 0 , t0 ) + b jk (x 0 , t0 ) = 0 ∂t0 ∂ x j0 2! ∂ x j0 ∂ xk0

(A.32)

When the above theory is applied to the practical problem of stochastic dynamics, the derivate moments a j (x 0 , t0 ), b j (x 0 , t0 ) and c j (x 0 , t0 ) are also determined by the equations of motion, and the Kolmogorov backward equation associated with them can then be solved, taking into account the boundary conditions and the following initial condition: q(x, t0 |x 0 , t0 ) = δ(x − x 0 ) =

m ∏

δ(x j − x j0 ).

(A.33)

j=1

References

1. Lin YK, Cai GQ (2004) Probabilistic structural dynamics. advanced theory and applications. McGraw-Hill Professional, New York. 2. Ochi MK (1990) Applied probability and stochastic processes: in engineering and physical sciences. Wiley-Interscience, New York. 3. Lin YK (1967) Probabilistic theory of structural dynamics, McGraw-Hill, Inc., New York. 4. Risken H (1989) The Fokker-Plank equation, methods of solution and application, Springer, Berlin.

Appendix B

Derivation of Eq. (5.44)

The derivation of Eq. (5.44) in Chap. 5 is now given, and Eq. (5.42) in Chap. 5 is written in the following form d2x = −x + αx 3 dt 2

(B.1)

Utilizing the following variable transformation [1] t=

√

 1 + m · t1 ,

(B.2)

Eq. (B.1) then becomes d2x = −(1 + m)x + α(1 + m)x 3 . dt12

(B.3)

Next, utilizing the following variable transformation [1] / x=

2m α(1 + m)

 · x1 ,

(B.4)

Eq. (B.3) then becomes d 2 x1 = −(1 + m)x1 + (2m)x13 . dt12

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5

(B.5)

335

336

Appendix B: Derivation of Eq. (5.44)

To continue the derivation process, the concepts and properties of the Jacobi elliptic functions are needed. Consider the following Jacobi elliptic functions sn t, cn t and dn t, which are related to each other as follows [2] √ 1 − sn 2 t

(B.6)

√ 1 − msn 2 t

(B.7)

cn t = dn t =

where m is the parameter of the Jacobi elliptic function. At the same time, we have [2]: d (sn t) = cn t · dn t dt

(B.8)

d (cn t) = −sn t · dn t dt

(B.9)

d (dn t) = −msn t · cn t dt

(B.10)

Differentiating Eq. (B.8) one more time yields [1]: d d2 d (sn t) = (cn t) · dn t + cn t · (dn t) dt 2 dt dt = (−sn t · dn t) · dn t + cn t · (−msn t · cn t) = (−sn t) · (1 − msn 2 t) + (−msn t) · (1 − sn 2 t) = −(1 + m) · sn t + 2msn 3 t

(B.11)

From Eqs. (B.5) and (B.11) it follows that x1 = sn t1

(B.12)

From Eqs. (B.12), (B.4) and (B.2) it follows that / x=

2m α(1 + m)



/ · sn

1 t 1+m

 (B.13)

Because the following relationship exists between the parameter of the Jacobi elliptic function m and the modulus of the Jacobi elliptic function m = k2,

(B.14)

Appendix B: Derivation of Eq. (5.44)

337

Then the Eq. (5.44) in Chap. 5 is proved / x(τ ) =

  t 2k 2 sn √ ,k . α(1 + k 2 ) (1 + k 2 )

(B.15)

References

1. Jiang CB (1995) Highly nonlinear rolling motion leading to capsize, Ph.D Dissertation, Naval Architecture ang Marine Engineering in the University of Michigan, Ann Arbor. 2. Bowman F (1953) Introduction to elliptic functions with applications, English Universities Press Ltd. London.

Appendix C

Gravity Center Movement Principle

We first examine a simple figure shown as Fig. C.1. The center of gravity of the whole figure moves as the position of one part of its area changes. Figure ACDF with area S shown in the figure consists of figure ABEF with area s and figure BCDE with area s, , namely S = s + s,

Fig. C.1 The movements of gravity centers

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5

339

340

Appendix C: Gravity Center Movement Principle

Let the gravity center of the graph ABEF be at point g, the gravity center of the graph BCDE be at point g, , and the gravity center of the whole graph ACDF be at point G. Then, by taking static moment with respect the point G s × gG = s , × Gg , or gG

s, s

=

Gg ,

(C.1)

That is, the segment gg , is divided by the point G according to the inverse ratio of the areas. Now move the area associated with the graph BCDE to FEKH, and the gravity center of the graph FEKH is at point g,, . In this way, the whole figure becomes the shape of ABKH. The gravity center of the new shape ABKH is at the point G 1 on the line segment gg ,, , then gG 1 G 1 g ,,

gG s, = s Gg ,,

=

(C.2)

According to the geometric principle, it is known that the line segment GG 1 is parallel to g , g ,, , i.e., the moving direction of the gravity center of the whole graph is parallel to the moving direction of the gravity center of one part of the graph. At the same time, we have GG 1 g , g ,,

=

gG gg ,

.

Calculating static moments with respect to the point g, we have s , × gg , = S × gG, or gG gg ,

=

s, . S

Therefore, we have: GG 1 g , g ,,

=

s, S

(C.3)

The above equation shows that the distance that the gravity center moves is inversely proportional to the area associated with it.

Appendix D

Derivation of the Melnikov Function

A fundamental understanding of the principles underlying nonlinear dynamical systems is a necessary foundation for the subsequent development of the Melnikov theory. In this Appendix we will provide the detailed derivation processes for obtaining the Melnikov function, largely following the discussion in Wiggins [1]. Let us consider a system of first-order ordinary differential equations with continuous time variable x˙ = f(x)

(D.1)

The given expression in (D.1) involves a vector function denoted by x = x(t) = (x1 , x2 . . . , xn )T ∈ Rn , which is dependent on the independent variable t. Additionally, there exists a smooth vector function f = ( f 1 , f 2 . . . , f n )T referred to as the vector field and defined on a certain subset E ⊆ Rn . The over-dot notation in Eq. (D.1) is utilized to represent differentiation with respect to the time t (i. e., “ dtd ”). The expression represented by Eq. (D.1) can be written in scalar form x˙1 = f 1 (x1 , x2 , . . . , xn ), x˙2 = f 2 (x1 , · · · x2 , . . . , xn ), .. .

(D.2)

x˙n = f n (x1 , x2 , . . . , xn ).

The space of the variables x1 , x2 . . . , xn is referred to as a phase space or, for n = 2, the phase plane. The representation of the motion in the phase plane is known as a phase plane diagram. Ordinary differential equations that do not depend explicitly on time (such as x˙ = f(x)) are called autonomous or time independent ordinary differential equations, or vector fields. It may be helpful to differentiate a solution curve by a specific point in phase space that it passes through at a starting time, i.e., for a solution x(t), we have x(t0 ) = x0 . This is because a solution curve can be distinguished by a particular point in phase space that it passes through at © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 Y. Wang, Stochastic Dynamic Response and Stability of Ships and Offshore Platforms, Ocean Engineering & Oceanography 27, https://doi.org/10.1007/978-981-99-5853-5

341

342

Appendix D: Derivation of the Melnikov Function

a starting time. Whenever this step is taken, we are said to be specifying an initial condition. This is frequently written down as x(t, t0 , x0 ) when it is incorporated into an expression for a solution. There are a number of other “notions” that might be considered to be synonymous with the term “solution of the system (D.1)”. The expression x(t, t0 , x0 ) is also sometimes referred to as the trajectory or phase curve that passes through the point x0 when time is equal to t0 . Let’s assume that a point in the phase space of the system (D.1) is denoted by x0 . We refer to the collection of points in phase space that lie on a trajectory that goes through x0 as the orbit through x0 , which is symbolized by the symbol O(x0 ). To be more precisely, for x0 ∈ U ⊂ Rn where U is an open set in Rn , the orbit through x0 is described by the equation O(x0 ) = {x ∈ Rn |x = x(t, t0 , x0 ), t ∈ I }, where I is the time interval of existence. Finally, a fixed point of the system (D.1) is a point x ∈ Rn such that f(x) = 0, i.e., a solution which does not change in time. Let x(t) be any solution of the system (D.1). In essence, the stability of x(t) can be determined by observing whether solutions that begin in close proximity to x(t) at a specific time remain in close proximity to x(t) for all subsequent times. The system can be deemed asymptotically stable if the solutions in its vicinity not only maintain proximity, but also exhibit convergence towards x(t) as t approaches infinity. To determine the stability of x(t), it is imperative to comprehend the characteristics of solutions in the vicinity of x(t). Let x = x(t) + y.

(D.3)

Upon substitution of Eq. (D.3) into Eq. (D.1) and subsequent Taylor expansion around x(t), the resulting expression is obtained as follows   ˙ + y˙ = f (x(t)) + D f (x(t))y + O |y|2 x˙ = x(t)

(D.4)

in which Df = [∂ f i /∂ x j ] is the Jacobian matrix f = of the first partial derivatives of the function ( f 1 (x1 , x2 , . . . , xn ), f 2 (x1 , · · · x2 , . . . , xn ), · · ·, f n (x1 , x2 , . . . , xn ))T (T denotes transpose), and |y| ≤ 1 . In Eq. (D.4), | · | denotes a norm on Rn . Utilizing the fact ˙ = f (x(t)), Eq. (D.4) becomes that x(t)   y˙ = D f (x(t))y + O |y|2

(D.5)

The evolution of orbits in the vicinity of x(t) is described by Eq. (D.5). The issue of stability pertains to the characteristics of solutions in close proximity to x(t). Therefore, it is plausible to address this matter by examining the associated linear system y˙ = D f (x(t))y

(D.6)

As a result, the question of whether or not x(t) is stable requires two stages to be completed in this order: 1. Determine whether the y = 0 solution of Eq. (D.6)

Appendix D: Derivation of the Melnikov Function

343

is stable; 2. Demonstrate that the stability (or instability) of the y = 0 solution of Eq. (D.6) implies the stability (or instability) of x(t). The above-mentioned stage 1 can pose a challenge comparable to our original problem, as there exists no universal analytical techniques for finding the solution of linear ordinary differential equations with time-dependent coefficients. However, if x(t) is a fixed point, that is, x(t) = x, then D f (x(t)) = D f (x) is a matrix with constant entries, and the solution of Eq. (D.6) through the point y0 ∈ Rn of t = 0 can immediately be written as y(t) = e D f (x)t y0

(D.7)

Therefore, y(t) is said to be asymptotically stable if all eigenvalues of D f (x) have negative real parts. The answer to stage 2 can be derived from the subsequent theorem: Assuming that the real parts of all the eigenvalues of D f (x) are negative, then the equilibrium solution x = x of the nonlinear vector field (D.1) is asymptotically stable. Consider a fixed point x = x of the system x˙ = f(x), where x belongs to the n-dimensional Euclidean space Rn . If none of the eigenvalues of D f (x) have zero real part, then the x is said to be a hyperbolic fixed point. It is important to note that the concept of “hyperbolicity of a fixed point” is defined in terms of the linearization about the fixed point. This is something that should be kept in mind. In addition to this, we should mention that hyperbolicity “persists under perturbations”. Throughout the evolution of dynamical systems theory, hyperbolicity has been a fundamental concept. A vector field’s hyperbolic fixed point is denoted as a saddle when a subset of the eigenvalues of the corresponding linearization exhibit real parts that are greater than zero, while the remaining eigenvalues exhibit real parts that are less than zero. In the event that the real part of all the eigenvalues is negative, the hyperbolic fixed point is denoted as a stable node or sink. Conversely, if the real part of all the eigenvalues is positive, the hyperbolic fixed point is referred to as an unstable node or source. A nonhyperbolic fixed point is classified as a center if its eigenvalues are both nonzero and purely imaginary. The analysis of dynamical systems heavily relies on the utilization of invariant manifolds, specifically stable, unstable, and center manifolds. Let us first give the definition of a invariant set. Let S ⊂ Rn be a set, then it is said that the set S is invariant under the vector field x˙ = f(x) if, for each x0 ∈ S, we have x(t, 0, x0 ) ∈ S for all t ∈ R (where x(0, 0, x0 ) = x0 ). When the values of time are limited to positive values (t ≥ 0), the set S is denoted as positively invariant. Conversely, when the values of time are negative, the set S is referred to as negatively invariant. In brief, it can be asserted that invariant sets possess the property that trajectories starting in the invariant set remain in the invariant set, for all of their future, and all of their past. If an invariant set S ⊂ Rn possesses the structure of a Cr differentiable manifold, then we say that S is a Cr (r ≥ 1) invariant manifold. Here, we remark that a function is said to be C r if it is r times differentiable and each derivative is continuous. Evidently, we must define the phrase “Cr differentiable

344

Appendix D: Derivation of the Melnikov Function

manifold” before proceeding. However, as this is a separate course’s topic, we won’t define the term “manifold” in its entirety; instead, we’ll focus on the specific parts of its wide theory that apply to our situation. A manifold is, roughly speaking, a set that locally has the structure of Euclidean space. Manifolds are most frequently encountered in applications as m-dimensional surfaces embedded in Rn . If the surface in question does not contain any singular points—that is, if the derivative of the function defining the surface possesses maximal rank—then, according to the implicit function theorem, the surface can be locally represented as a graph. A surface is classified as a Cr manifold if the local graphs that represent it are Cr . Let us return to our investigation of the orbit structure close to fixed points in order to determine how some significant invariant manifolds emerge. Consider x ∈ Rn to be a fixed point of x˙ = f(x),

x ∈ Rn

(D.8)

By the discussion mentioned previously, it is natural to think about the corresponding linear system y˙ = Ay,

y ∈ Rn

(D.9)

in which A ≡ D f (x) is a constant n × n matrix. The solution to Eq. (D.9) through the point y0 ∈ Rn at time t = 0 is provided by y(t) = eAt y0 ,

(D.10)

in which eAt = id + At +

1 1 2 2 A t + A3 t 3 + · · · 2! 3!

(D.11)

and “id”represents the n × n identity matrix. At this time, Rn can be expressed as the direct sum of three subspaces, namely E s , E u and E c . These subspaces are defined as E s = span{e1 , · · ·, es } E u = span{es+1 , · · ·, es+u } E = span{es+u+1 , · · ·, es+u+c } c

s+u+c =n (D.12)

in which {e1 , · · ·, es } represent the (generalized) eigenvectors of matrix A that correspond to the eigenvalues of A having negative real part. On the other hand, the set {es+1 , · · ·, es+u } represent the (generalized) eigenvectors of A that correspond to eigenvalues of A having positive real part. Lastly, {es+u+1 , · · ·, es+u+c } denote the (generalized) eigenvectors of A that correspond to the eigenvalues of A having zero real part. E s , E u and E c are respectively called the stable, unstable and center subspaces. E s , E u and E c are also examples of invariant subspaces (or manifolds),

Appendix D: Derivation of the Melnikov Function

345

due to the fact that solutions of Eq. (D.9) with initial conditions that are exclusively confined within either E s , E u or E c must forever remain in that specific subspace for an indefinite duration. In addition, it can be observed that solutions starting in E s approach y = 0 asymptotically as the time t tends towards positive infinity, while solutions starting in E u approach y = 0 asymptotically as the time t tends towards negative infinity. It is important to note that the initial motivation for examining the linear system y˙ = Ay,

y ∈ Rn

(D.13)

in which A ≡ D f (x), was to gain insight into the characteristics of solutions in the vicinity of the fixed point x = x in the non-linear equation x˙ = f(x),

x ∈ Rn

(D.14)

A specific theorem regarding stable, unstable, and center manifold offers an answer to this question. Prior to delving into the details, it is necessary to convert Eq. (D.14) into a more convenient format. First, we use the translation y = x − x to move the fixed point x = x of Eq. (D.14) to the origin. When this occurs, Eq. (D.14) becomes y˙ = f(x + y),

y ∈ Rn

(D.15)

Expanding the function f(x + y) about the point x = x using Taylor series expansion yields   y˙ = Df(x)y + R(y) = Df(x)y + O |y|2 ,

y ∈ Rn

(D.16)

In obtaining the above equation we have utilized the relation f(x) = 0. The linear equation (D.13) can be transformed into block diagonal form by utilizing elementary linear algebra techniques. This transformation can be achieved through the use of a linear transformation T ⎛ ⎞ ⎛ ⎞⎛ ⎞ u˙ u As 0 0 ⎝ v˙ ⎠ = ⎝ 0 Au 0 ⎠⎝ v ⎠ (D.17) ˙ 0 0 Ac w w in which T −1 y ≡ (u, v, w)T ∈ Rs ×Ru ×Rc , s +u +c = n. As is a square matrix of size s ×s, whose eigenvalues have negative real parts. Au is a square matrix of size u × u, whose eigenvalues have positive real parts. Ac is a square matrix of size c × c, , whose eigenvalues have zero real parts. Please note that we are drawing attention to the (hopefully) clear fact that the “0” in Eq. (D.17) do not represent scalar zeros but rather the suitably sized block which is composed of all zeros. The application of the identical linear transformation for the purpose of transforming the coordinates of the non-linear vector field (D.16) results in the derivation of an equation as follows

346

Appendix D: Derivation of the Melnikov Function

u˙ = As u + Rs (u, v, w), v˙ = Au v + Ru (u, v, w), ˙ = Ac w + Rc (u, v, w), w

(D.18)

in which Rs (u, v, w), Ru (u, v, w) and Rc (u, v, w) are respectively the first s, u and c components of the vector T −1 R(y). Consider now the linear vector field represented by the Eq. (D.17). According to our prior discussion, Eq. (D.17) possesses an s-dimensional invariant stable manifold, a u-dimensional invariant unstable manifold, and a c-dimensional invariant center manifold, all of which intersect in the origin. The subsequent theorem demonstrates how this structure changes when the nonlinear vector field (D.18) is taken into consideration. The Theorem of Local Stable, Unstable, and Center Manifolds of Fixed Points: Assuming that Eq. (D.18) satisfies the condition of being C r , where r is greater than or equal to 2. Then the fixed point (u, v, w) = 0 of Eq. (D.18) possesses a C r ss dimensional local, invariant stable manifold denoted as Wloc (0), a C r u-dimensional u local, invariant unstable manifold denoted as Wloc (0), and a C r c-dimensional local, c invariant center manifold denoted as Wloc (0). These manifolds intersect at the fixed point (u, v, w) = 0. These manifolds are all tangent to the corresponding invariant subspaces of the linear vector field (D.17) at the origin, and as a result, they are capable of being locally represented as graphs. To be more specific, we have | s Wloc (0) = (u, v, w)T ∈ Rs × Ru × Rc |v = h sv (u), w = h sw (u); Dh sv (0) = 0, Dh sw (0) = 0; |u| su f f iciently small}

(D.19)

| u Wloc (0) = (u, v, w)T ∈ Rs × Ru × Rc |u = h uu (v), w = h uw (v); Dh uu (0) = 0, Dh uw (0) = 0; |u| su f f iciently small}

(D.20)

| c Wloc (0) = (u, v, w)T ∈ Rs × Ru × Rc |u = h cu (w), v = h cv (w); Dh cu (0) = 0, Dh cv (0) = 0; |u| su f f iciently small}

(D.21)

in which v = h sv (u), w = h sw (u),u = h uu (v), w = h uw (v), u = h cu (w), and v = h cv (w) are C r functions (recall that a function is said to be C r if it is r times s differentiable and each derivative is continuous). In addition, trajectories in Wloc (0) u and Wloc (0) respectively have the identical asymptotic properties as the trajectories s in E s and E u . Namely, the trajectories of (D.18) with initial conditions in Wloc (0) approach the origin at an exponential rate asymptotically as the time t approaches u positive infinity. The trajectories of (D.18) with initial conditions in Wloc (0) approach the origin at an exponential rate asymptotically as the time t approaches negative infinity. Finally, the conditions Dh sv (0) = 0, Dh sw (0) = 0, etc., indicate that the nonlinear manifolds are tangent to the associated linear manifolds at the point of origin.

Appendix D: Derivation of the Melnikov Function

347

We now provide some comments regarding the above-mentioned significant theorem. It is very common to hear the terms “stable manifold,” “unstable manifold,” or “center manifold” used alone; yet, these terms are insufficient to convey the dynamical situation when used alone. Take note that the title of the above-mentioned theorem is “The Theorem of Local Stable, Unstable, and Center Manifolds of Fixed Points”. The term “of Fixed Points” holds significant importance as it is imperative to specify the stable, unstable, or center manifold of a particular object in order to make sense. The “particular objects” that have been investigated up to this point have been fixed points. In addition, assuming that the fixed point is hyperbolic, meaning that E c = ∅. In this instance, one can interpret the theorem as stating that the behavior of trajectories of the nonlinear vector field within a sufficiently small neighborhood of the fixed point is the same as that of trajectories of the associated linear vector field. We now consider the special case of planar flows (i. e., n = 2), under the assumption that a C 2 function (a function is said to be C 2 if it is two times differentiable and each derivative is continuous) H (x1 , x2 ) exists such that f 1 = ∂ H/∂ x2 and f 2 = −∂ H/∂ x1 . The function H (x1 , x2 ) is called as a Hamiltonian. The system (D.2) then becomes x˙1 = ∂ H/∂ x2 , x˙2 = −∂ H/∂ x1 .

(D.22)

The system denoted as (D.22) pertains to the category of Hamiltonian systems. In Chap. 5 of this book, to facilitate the application of the Melnikov method for the analysis of ship rolling-induced capsizing, we have written the rolling equation in the following non-dimensional form: x(τ ¨ ) + εδ x(τ ˙ ) + εδ3 x˙ 3 (τ ) + x(τ ) − kx 3 (τ ) = εγ cos(Ωτ + ψ)

(D.23)

We have also written Eq. (D.23) in the following first-order form: x(τ ˙ ) = y(τ )

(D.24)

  AFr oll cos(θ + ψ) (D.25) y˙ (τ ) = −x(τ ) + kx 3 (τ ) + ε −δy(τ ) − δ3 (y(τ ))3 + C1 Δ θ˙ = Ω

(D.26)

Equations (D.24)–(D.26) together represent a perturbed integrable Hamiltonian system. When the third part to the right of the equal sign of Eq. (D.25) (i.e., the value in the bracket) equals zero, the unperturbed Hamiltonian system becomes x(τ ˙ ) = y(τ )

(D.27)

348

Appendix D: Derivation of the Melnikov Function

y˙ (τ ) = −x(τ ) + kx 3 (τ )

(D.28)

The unperturbed Hamiltonian system is energy conserving and it carries a Hamiltonian operator H (x, y) =

x2 kx 4 y2 + − 2 2 4

(D.29)

The Melnikov approach is put to use in many different situations in order to provide predictions regarding the development of chaotic orbits in non-autonomous smooth nonlinear systems that are subjected to periodic perturbation. It is possible, in accordance with the method, to build a function that is referred to as the “Melnikov function”. This function has the capability of predicting either the regular or the chaotic behavior of a dynamical system. Therefore, a measure of the distance between stable and unstable manifolds in the Poincaré map will be determined with the assistance of the Melnikov function. In addition, according to the approach, those manifolds will have crossed each other transversally when this measure is equal to zero; as a result of this crossing, the system will become chaotic. In the following, we shall proceed to construct a perturbation technique aimed at demonstrating the presence of transverse homoclinic orbits to hyperbolic periodic orbits within a specific category of two-dimensional, time-periodic vector fields. Let us consider the following class of systems: ∂H (x, y) + εg1 (x, y, t, ε), ∂y ∂H (x, y) + εg2 (x, y, t, ε), (x, y) ∈ R2 ; y˙ = − ∂x

x˙ =

(D.30)

Equation (D.30) can also be written in the vector form as follows q˙ = KDH(q) + εg(q, t, ε),

(D.31)

T  in which q = (x, y)T , DH = ∂∂Hx , ∂∂Hy , g = (g1 , g2 )T , and K =   0 1 . −1 0 Let us consider a scenario where system (D.30) exhibits smoothness within the relevant region. Additionally, we assume that ε represents a small perturbation parameter, while g is a vector function that is periodic in t, with a period of T = 2π/ω. When ε = 0, the unperturbed system corresponding to system (D.30) is written as x˙ = ∂∂Hy (x, y) y˙ = − ∂∂Hx (x, y)

(D.32)

Appendix D: Derivation of the Melnikov Function

y

349

Γp0 qα(t)

x

p0

Fig. D.1 Phase space representing the assumptions 1 & 2 with respect to the system (D.32)

Equation (D.32) can also be written in the vector form as follows q˙ = KDH(q)

(D.33)

In addition, we made the following assumptions concerning the structure of the phase space of the unperturbed system (see Fig. D.1). 1. It is assumed that the system in its unperturbed state has a hyperbolic fixed point denoted by p0 , which is connected to itself via a homoclinic orbit q0 (t) ≡ (x0 (t), y0 (t)); 2. The system contains a continuous family of periodic orbits denoted by q α (t) α with period α ∈ (−1, 0), which are located within p0 . p0 = | T , 2| q ∈ R q = q0 (t), t ∈ R ∪ { p0 } = W s ( p0 ) ∩ W u ( p0 ) ∪ { p0 }. We assume that q α (t) satisfies limα→0 q α (t) = q0 (t) and limα→0 T α = ∞. The homoclinic Melnikov method will be developed through a series of steps, which will be concisely outlined as follows: The first step involves the development of a parametrization for the homoclinic “manifold” of the unperturbed system. The second step involves the development of a measure of the “splitting” of the manifolds in the perturbed system, utilizing the unperturbed “homoclinic coordinates.” In the third step of the analysis, the Melnikov function will be derived and its connection to the separation between the manifolds will be demonstrated. We want to recast (D.30) as an autonomous three-dimensional system as follows before moving on to the first step. ∂H (x, y) + εg1 (x, y, φ, ε), ∂y ∂H (x, y) + εg2 (x, y, φ, ε), (x, y, φ) ∈ R1 × R1 × S 1 ; y˙ = − ∂x φ˙ = ω x˙ =

(D.34)

350 Fig. D.2 The homoclinic manifold, γ . The lines on γ represent a typical trajectory

Appendix D: Derivation of the Melnikov Function

=2π y x γ(t) Γγ identify

=0 Equation (D.34) can also be written in the vector form as follows q˙ = KDH(q) + εg(q, t, ε), φ˙ = ω

(D.35)

The unperturbed system is obtained from (D.35) by setting ε = 0, that is: q˙ = KDH(q), φ˙ = ω

(D.36)

Recall that the first step involves the development of a parametrization for the homoclinic “manifold” of the unperturbed system. Upon observation within the threedimensional phase space R2 × S 1 , it can be observed that the hyperbolic fixed point p0 of the q component of the unperturbed system (D.35) transforms into a periodic orbit (see Fig. D.2) γ (t) = ( p0 , φ(t) = ωt + φ0 )

(D.37)

The designations W s (γ (t)) and W u (γ (t)) are utilized to represent the stable and unstable manifolds of γ (t) in two dimensions. As a result of the aforementioned Assumption 1, it can be observed that the functions W s (γ (t)) and W u (γ (t)) coincide along a two-dimensional homoclinic manifold. This homoclinic manifold is denoted by γ , as depicted in Fig. D.2. We would like to point out that the structure of Fig. D.2 should not come as a surprise; it is a reflection of the fact that the unperturbed phase space is independent of time (φ). Our objective is to figure out how the γ “breaks up” when it is subjected to the perturbation. In order to accomplish this, we will work on developing a measurement that can determine the amount that the perturbed stable and unstable manifolds of γ (t) deviate from γ . This will include taking a measurement in the direction that is normal to γ to find out how far apart the

Appendix D: Derivation of the Melnikov Function

351

=2π

t0

y x

Γγ DH(q0(-t0)),0)≡π p

γ(t) p=(q0(-t0)),

0)

identify

=0 Fig. D.3 The homoclinic coordinates

perturbed stable and unstable manifolds are from one another. Because it is obvious that the results of this measurement will differ depending on where you take a point on γ , the first step is to specify how γ is parametrized. The present study concerns the parametrization of γ through the use of homoclinic coordinates. Each individual point located on γ can be expressed or denoted by (q0 (−t0 ), φ0 ) ∈ γ

(D.38)

The time it takes to travel from the point q0 (−t0 ) to the point q0 (0) along the unperturbed homoclinic trajectory q0 (t) is what is meant to be understood by the value of the variable t0 . Because it can be shown that (q0 (−t0 ), φ0 ) corresponds to a unique point on γ , we have

γ = (q, φ) ∈ R2 × S 1 |q = q0 (−t0 ), t0 ∈ R1 ; φ = φ0 ∈ (0, 2π ]

(D.39)

Figure D.3 should make it clear what the parameters t0 and φ0 mean geometrically. At every point p ≡ (q0 (−t0 ), φ0 ) belonging to γ , we generate a vector denoted as πp , which is perpendicular to γ . The definition of this vector πp is as follows:  πp =

∂H ∂H (x0 (−t0 ), y0 (−t0 )), (x0 (−t0 ), y0 (−t0 )), 0 ∂x ∂y

 (D.40)

Equation (D.40) can also be written in vector form, π p ≡ (D H (q0 (−t0 )), 0)

(D.41)

Therefore, adjusting t0 and φ0 allows π p to be moved to each and every point on γ , see Fig. D.3. If ε /= 0 is small enough, the periodic orbit γ (t) of the unperturbed vector field (D.18) persists as a periodic orbit, denoted as γε (t) = γ (t)+O(ε), of the

352

Appendix D: Derivation of the Melnikov Function

γ(t)∩Σ

0

=p 0

Γ γ ∩Σ

0

=Γp0

y

=2π x Σ

0

=

0

identify

=0 Fig. D.4 A cross-section of the phase space that is parallel to the q-plane s u perturbed vector field (D.17). In addition, Wloc (γε (t)) and Wloc (γε (t)) are C r ε-close s u to Wloc (γ (t)) and Wloc (γ (t)), respectively. During certain geometrical arguments, a comparison will be made between the trajectories of the unperturbed vector field and those of the perturbed vector field. In many cases, it may be more convenient to analyze the trajectories by projecting them onto the q-plane or a plane that is parallel to it. Let us demonstrate how to complete this task. Consider the following cross-section of the phase space (it is advised to refrain from contemplating Poincaré maps at this point).



 φ0 = (q, φ) ∈ R2 |φ = φ0 .

(D.42)

It is evident that the parallelism between  φ0 and the q-plane is established, with both planes coinciding when φ0 equals zero, as depicted in Fig. D.4. At this point in time, we have arrived at a position where we are able to specify the splitting of W s (γε (t)) and W u (γε (t)). Select an arbitrary point p that belongs to γ . The intersection of π p and W s (γ (t)) and W u (γ (t)) occurs transversely at point p. Therefore, due to the persistence of transversal intersections and the fact that W s (γε (t)) and W u (γε (t)) are C r in ε, given a sufficiently small ε, W s (γε (t)) and W u (γε (t)) will intersect π p in a transverse manner at the respective points pεs and pεu . Therefore, it is natural to define the distance between W s (γε (t)) and W u (γε (t)) at the point p, which will be given by the notation d( p, ε), to be | | d( p, ε) ≡ | pεu − pεs |.

(D.43)

Appendix D: Derivation of the Melnikov Function

353

W s(γ (t)) W u(γ (t))

pu =2π

ps

y p

x

πp

γ (t) identify

=0 Fig. D.5 The distance between W s (γε (t)) and W u (γε (t)) at the point p

Please refer to Fig. D.5. In the subsequent step, we will find that it is to our advantage to redefine Eq. (D.43) in an equivalent, albeit somewhat less natural manner as described below  u  pε − pεs · (D H (q0 (−t0 )), 0) (D.44) d( p, ε) ≡ ||D H (q0 (−t0 ))|| where the symbol “ · ” represents the vector scalar product and / ||D H (q0 (−t0 ))|| =

∂H (q0 (−t0 )) ∂x

2

 +

∂H (q0 (−t0 )) ∂y

2 (D.45)

Because pεs and pεu are constrained to fall on the vector (D H (q0 (−t0 )), 0), it ought to be abundantly evident that the magnitudes of (D.44) and (D.43) are identical. This is because pεs and pεu are chosen to lie on (D H (q0 (−t0 )), 0). Please take note that because pεs and pεu lie on π p , we are able to write   pεu = qεu , φ0 ,

  pεs = qεs , φ0

(D.46)

That is to say, pεs and pεu have an identical value for their φ0 coordinate. Therefore, Eq. (D.44) is equivalent to   D H (q0 (−t0 )) · qεu − qεs d(t0 , φ0 , ε) ≡ ||D H (q0 (−t0 ))||

(D.47)

354

Appendix D: Derivation of the Melnikov Function

where we are now representing d( p, ε) by d(t0 , φ0 , ε), because each point p ∈ γ can be uniquely expressed through the parameters (t0 , φ0 ), where t0 is an element of the real numbers and φ0 is an element of the interval (0, 2π]. This is in accordance with the parametrization p = (q0 (−t0 ), φ0 ) outlined previously. Expanding Eq. (D.47) via Taylor series expansion with respect to ε = 0 yields: d(t0 , φ0 , ε) = d(t0 , φ0 , 0) + ε

  ∂d (t0 , φ0 , 0) + O ε2 ∂ε

(D.48)

in which d(t0 , φ0 , 0) = 0

(D.49)

and ∂d (t0 , φ0 , 0) = ∂ε

D H (q0 (−t0 )) ·



∂qεu | ∂ε ε=0

−

∂qεs | ∂ε ε=0



||D H (q0 (−t0 ))||

(D.50)

The definition of the Melnikov function is given as follows  M(t0 , φ0 ) ≡ D H (q0 (−t0 )) ·

∂q s ∂qεu |ε=0 − ε |ε=0 ∂ε ∂ε

 (D.51)

The objective now is to obtain a formula for M(t0 , φ0 ) that can be calculated without requiring knowledge of the solution of the perturbed vector field. This is achieved through the application of Melnikov’s initial technique. A time-dependent Melnikov function is defined by utilizing the flow generated by both the unperturbed and perturbed vector fields. Caution must be exercised in this scenario as there is a lack of prior knowledge regarding arbitrary orbits generated by the perturbed vector field. Following this, we arrive at an ordinary differential equation that the timedependent Melnikov function is required to satisfy. As it turns out, the ordinary differential equation is of the first order and linear; as a result, it can be solved in a straightforward manner. When the solution is analyzed at the right moment, the Melnikov function will be obtained. To get started, let’s define the time-dependent Melnikov function as follows   u ∂q s (t) ∂qε (t) |ε=0 − ε |ε=0 . (D.52) M(t; t0 , φ0 ) ≡ D H (q0 (t − t0 )) · ∂ε ∂ε where qεu (t) and qεs (t) are the trajectories starting at qεu = qεu (0) and qεs = qεs (0) respectively. Following this, we will proceed to derive an ordinary differential equation that M(t; t0 , φ0 ) has to satisfy. Differentiating Eq. (D.52) with respect to time enables certain simplifications. The derivative with respect to time of one of the terms in Eq. (D.52) is

Appendix D: Derivation of the Melnikov Function

355

    u  u d d ∂qε (t) ∂qε (t) |ε=0 |ε=0 = (D H (q0 (t − t0 ))) · D H (q0 (t − t0 )) · dt ∂ε dt ∂ε d ∂qεu (t) |ε=0 . + D H (q0 (t − t0 )) · dt ∂ε (D.53) The derivative with respect to time of the other term in Eq. (D.52) is     s  s d d ∂qε (t) ∂qε (t) |ε=0 |ε=0 = (D H (q0 (t − t0 )) ) · D H (q0 (t − t0 )) · dt ∂ε dt ∂ε s d ∂qε (t) |ε=0 . + D H (q0 (t − t0 )) · dt ∂ε (D.54) Recall that we have the equation of motion as follows: q˙ = KDH(q) + εg(q, φ, ε),

(D.55)

Therefore, we have the following equations of motion:     d u  qε (t) = KDH qεu (t) + εg qεu (t), φ(t), ε , dt

(D.56)

    d s  q (t) = KDH qεs (t) + εg qεs (t), φ(t), ε , dt ε

(D.57)

in which φ(t) = ωt + φ0 . The differentiation of Eqs. (D.56) and (D.57) can be respectively performed with respect to ε, and subsequently, the order of the ε and t differentiations can be interchanged to yield:  u   u  ∂qε (t) |ε=0 = K D H qε (t) |ε=0 ∂ε      u   d  u |ε=0 g qε (t), φ(t), ε + g qε (t), φ(t), ε |ε=0 + ε dε ∂q u (t) = K D 2 H (q0 (t − t0 )) ε |ε=0 + g(q0 (t − t0 ), φ(t), 0), ∂ε (D.58)   s   s       ∂qε (t) d ∂qε (t) |ε=0 = K D 2 H qεs (t) |ε=0 |ε=0 + g qεs (t), φ(t), ε |ε=0 dt ∂ε ∂ε     d  s |ε=0 g qε (t), φ(t), ε + ε dε ∂q s (t) = K D 2 H (q0 (t − t0 )) ε |ε=0 + g(q0 (t − t0 ), φ(t), 0), ∂ε (D.59) d dt



∂qεu (t) |ε=0 ∂ε





2

356

Appendix D: Derivation of the Melnikov Function

Equations (D.58) and (D.59) are called the first variational equations. We make ∂qεu (t) |ε=0 solves the Eq. (D.58) for t in the range (−∞, 0], but the observation that ∂ε s ∂qε (t) | solves the Eq. (D.59) for t in the range (0, ∞]. Substituting (D.58) into ∂ε ε=0 (D.53) leads to     u  u d d ∂qε (t) ∂qε (t) |ε=0 |ε=0 = (D H (q0 (t − t0 )) ) · D H (q0 (t − t0 )) · dt ∂ε dt ∂ε u ∂q (t) + D H (q0 (t − t0 )) · K D 2 H (q0 (t − t0 )) ε |ε=0 ∂ε + D H (q0 (t − t0 )) · g(q0 (t − t0 ), φ(t), 0). (D.60) Substituting (D.59) into (D.54) leads to     s  s d d ∂qε (t) ∂qε (t) |ε=0 |ε=0 = (D H (q0 (t − t0 ))) · D H (q0 (t − t0 )) · dt ∂ε dt ∂ε s ∂q (t) + D H (q0 (t − t0 )) · K D 2 H (q0 (t − t0 )) ε |ε=0 ∂ε + D H (q0 (t − t0 )) · g(q0 (t − t0 ), φ(t), 0). (D.61) By performing an explicit evaluation of the matrix multiplications and dot products, one may demonstrate that the sum of the first two terms on the right-hand side of Eq. (D.60) is equal to zero. The proof is given as follows. To begin, take note that the unperturbed system equation is: q˙0 (t − t0 ) = KDH(q0 (t − t0 ))

(D.62)

Therefore, we have d (D H (q0 (t − t0 )) ) = D 2 H (q0 (t − t0 ))q˙0 (t − t0 ) dt = D 2 H (q0 (t − t0 ) ) KDH (q0 (t − t0 )) Denote

∂qεu (t) | ∂ε ε=0

=



∂ xεu (t) | , ∂ε ε=0

∂ yεu (t) | ∂ε ε=0

T , then we can obtain

(D.63)

Appendix D: Derivation of the Melnikov Function





 ∂qεu (t) |ε=0 D 2 H (KDH) ∂ε



357

⎞ ⎞ ∂ 2 H ∂ 2 H ⎛ ∂ H ⎞ ⎛ ∂ xεu (t) |ε=0 ⎟ ⎜ ∂ x 2 ∂ x∂ y ⎟⎜ ∂ y ⎟ ⎜ ∂ε ⎟ =⎜ ⎠ ⎝ ∂ 2 H ∂ 2 H ⎠⎝ ∂ H ⎠ · ⎝ ∂ yεu (t) |ε=0 − ∂ε ∂x ∂ x∂ y ∂ y 2 ⎞ ⎛ ⎛ 2 ⎞ 2 u ∂ H ∂H ∂ H ∂H ∂ xε (t) |ε=0 ⎟ ⎜ ∂ x 2 ∂ y − ∂ x∂ y ∂ x ⎟ ⎜ ∂ε ⎟· =⎜ u ⎠ ⎝ ∂2 H ∂ H ∂ 2 H ∂ H ⎠ ⎝ ∂ yε (t) |ε=0 − ∂ε ∂ x∂ y ∂ y ∂ y2 ∂ x  2   u 2 ∂ H ∂H ∂ H ∂H ∂ xε (t) |ε=0 − = ∂ε ∂x2 ∂y ∂ x∂ y ∂ x  2   u ∂ H ∂H ∂2 H ∂ H ∂ yε (t) |ε=0 − + ∂ε ∂ x∂ y ∂ y ∂ y2 ∂ x (D.64) ⎛

in which for the sake of a less cumbersome notation, we have omitted the argument q0 (t − t0 ) = (x0 (t − t0 ), y0 (t − t0 )). We can also obtain ⎞⎛ ⎞ ⎞ ⎛ ∂2 H ∂2 H ∂H ∂ xεu (t) | ε=0 ⎟  ⎜ x ⎟ ⎜ ∂ x∂ y ∂ y2 ⎟ ⎟⎜ |ε=0 = ⎝ ∂∂H (DH) · K D 2 H u ⎝ ∂ y∂ε ⎠ ⎠·⎜ 2 2 ⎠ ⎝ (t) H H ∂ ∂ ε ∂ε |ε=0 − − ∂y ∂ε ∂x2 ∂ x∂ y   u  ⎞ ⎞ ⎛ ∂ 2 H  ∂ x u (t) ⎛ 2 ∂ H ∂ yε (t) ∂H ε |ε=0 + |ε=0 ⎟ ⎜ 2 ⎜ x ⎟ ⎜ ∂ x∂ y  ∂ε  ∂ y2  ∂εu ⎟ = ⎝ ∂∂H ⎠ · ⎝ ∂ 2 H ∂ x u (t) ⎠ ∂ H ∂ yε (t) ε |ε=0 − |ε=0 − 2 ∂y ∂x ∂ε ∂ x∂ y ∂ε  u  2    u 2 ∂ xε (t) ∂ H ∂H ∂ H ∂H ∂ yε (t) |ε=0 |ε=0 = − 2 + + ∂ε ∂x ∂y ∂ x∂ y ∂ x ∂ε   2 2 ∂ H ∂H ∂ H ∂H + (D.65) − ∂ x∂ y ∂ y ∂ y2 ∂ x   ∂qεu (t)



⎛

Adding Eqs. (D.64) and (D.65) gives the result that the sum of the first two terms on the right-hand side of Eq. (D.60) is equal to zero. Similarly, by performing an explicit evaluation of the matrix multiplications and dot products, one may also demonstrate that the sum of the first two terms on the right-hand side of Eq. (D.61) is equal to zero. T  s ∂ xε (t) ∂qεs (t) ∂ yεs (t) |ε=0 , ∂ε |ε=0 , then The proof is given as follows. Denote ∂ε |ε=0 = ∂ε we can obtain





 ∂qεs (t) |ε=0 D 2 H (KDH) ∂ε



⎛

∂2 H ⎜ ∂x2 =⎜ ⎝ ∂2 H ∂ x∂ y

⎞ ⎞ ∂ 2 H ⎛ ∂ H ⎞ ⎛ ∂ xεs (t) | ⎟ ε=0 ⎟ ∂ x∂ y ⎟⎜ ∂ y ⎟ · ⎜ ∂ε s ⎠ ∂ 2 H ⎠⎝ ∂ H ⎠ ⎝ ∂ yε (t) |ε=0 − ∂ε ∂x ∂ y2

358

Appendix D: Derivation of the Melnikov Function

⎞ ⎛ ⎞ ∂2 H ∂ H ∂2 H ∂ H ∂ xεs (t) |ε=0 ⎟ ⎜ ∂ x 2 ∂ y − ∂ x∂ y ∂ x ⎟ ⎜ ∂ε ⎟· =⎜ s ⎠ ⎝ ∂2 H ∂ H ∂ 2 H ∂ H ⎠ ⎝ ∂ yε (t) |ε=0 − ∂ε ∂ x∂ y ∂ y ∂ y2 ∂ x  s  2  2 ∂ xε (t) ∂ H ∂H ∂ H ∂H |ε=0 = − ∂ε ∂x2 ∂y ∂ x∂ y ∂ x  2   s ∂ H ∂H ∂ yε (t) ∂2 H ∂ H |ε=0 + − ∂ε ∂ x∂ y ∂ y ∂ y2 ∂ x (D.66) ⎛

in which for the sake of a less cumbersome notation, we have omitted the argument q0 (t − t0 ) = (x0 (t − t0 ), y0 (t − t0 )). We can also obtain 

(DH) · KD2 H

 ∂q s (t) ε

∂ε

 |ε=0

⎞⎛ ⎞ ⎞ ⎛ ∂2 H ∂2 H ∂H ∂ xεs (t) |ε=0 ⎟ 2 ⎟⎜ ⎜ ∂x ⎟ ⎜ ∂ x∂ y ∂ y ⎟⎝ ∂ε = ⎝ ∂H ⎠ · ⎜ ⎠ s ⎝ ∂2 H ∂ 2 H ⎠ ∂ yε (t) |ε=0 − − ∂y 2 ∂ε ∂x ∂ x∂ y    ⎞ ⎞ ⎛ ∂ 2 H  ∂ x s (t) ⎛ ∂ 2 H ∂ yεs (t) ∂H ε | | + ε=0 ε=0 ⎟ 2 ⎜ x ⎟ ⎜ ∂ x∂ y ∂ε  ∂ y2  ∂εs ⎟ = ⎝ ∂∂H ⎠·⎜ ⎠ ⎝ ∂ 2 H  ∂ xεs (t) ∂ H ∂ yε (t) |ε=0 − |ε=0 − 2 ∂y ∂x ∂ε ∂ x∂ y ∂ε  2   s ∂ H ∂H ∂ xε (t) ∂2 H ∂ H |ε=0 − 2 = + ∂ε ∂x ∂y ∂ x∂ y ∂ x    s ∂2 H ∂ H ∂ yε (t) ∂2 H ∂ H |ε=0 − (D.67) + + ∂ε ∂ x∂ y ∂ y ∂ y2 ∂ x ⎛

Adding Eqs. (D.66) and (D.67) gives the result that the sum of the first two terms on the right-hand side of Eq. (D.61) is equal to zero. Therefore, we can obtain the following two equations:    u d ∂qε (t) |ε=0 = D H (q0 (t − t0 )) · g(q0 (t − t0 ), φ(t), 0). D H (q0 (t − t0 )) · dt ∂ε (D.68)    s d ∂qε (t) |ε=0 = D H (q0 (t − t0 )) · g(q0 (t − t0 ), φ(t), 0). D H (q0 (t − t0 )) · dt ∂ε (D.69) By performing separate integrations of Eqs. (D.68) and (D.69) over the intervals from −τ to 0 and from 0 to τ (where τ is a positive value), we obtain the following results    u  u ∂qε (−τ ) ∂qε (0) |ε=0 − D H (q0 (−τ − t0 )) · |ε=0 D H (q0 (0 − t0 )) · ∂ε ∂ε

Appendix D: Derivation of the Melnikov Function

359

0 =

D H (q0 (t − t0 )) · g(q0 (t − t0 ), ωt + φ0 , 0)dt. −τ



D H (q0 (τ − t0 )) · τ =

∂qεs (τ ) |ε=0 ∂ε



(D.70) 

− D H (q0 (0 − t0 )) ·

∂qεs (0) |ε=0 ∂ε

D H (q0 (t − t0 )) · g(q0 (t − t0 ), ω t + φ0 , 0)dt.



(D.71)

0

Please note that we have substituted φ(t) = ω t + φ0 into the integrand in the above two equations. At this point, let us recall that our time dependent Melnikov function is   u ∂qεs (t) ∂qε (t) |ε=0 − |ε=0 . (D.72) M(t; t0 , φ0 ) ≡ D H (q0 (t − t0 )) · ∂ε ∂ε The Melnikov function can thus be calculated as   u ∂qεs (0) ∂qε (0) |ε=0 − |ε=0 . M(t0 , φ0 ) = M(0; t0 , φ0 ) ≡ D H (q0 (0 − t0 )) · ∂ε ∂ε (D.73) Combing Eqs. (D.70), (D.71) and (D.73) leads to τ M(t0 , φ0 ) =

D H (q0 (t − t0 )) · g(q0 (t − t0 ), ω t + φ0 , 0)dt −τ



∂qεu (−τ ) |ε=0 + D H (q0 (−τ − t0 )) · ∂ε   s ∂qε (τ ) |ε=0 . − D H (q0 (τ − t0 )) · ∂ε



(D.74)

In the following analysis, we aim to examine the limit of Eq. (D.74) as the parameter τ approaches infinity. In the limit as τ approaches infinity, the function D H (q0 (τ − t0 )) rapidly converges to zero at an exponential rate, because q0 (τ − t0 ) ∂qεs (τ ) |ε=0 is bounded as approaches a hyperbolic fixed point. In the meantime, ∂ε   s ∂qε (τ ) | τ approaches infinity. Therefore, D H (q0 (τ − t0 )) · goes to zero as ∂ε ε=0 τ approaches infinity. In the limit as τ approaches negative infinity, the function D H (q0 (τ − t0 )) rapidly converges to zero at an exponential rate, because q0 (τ − t0 ) ∂q u (−τ ) approaches a hyperbolic fixed point. In the meantime, ε∂ε |ε=0 is bounded as τ   u ∂qε (−τ ) | goes to approaches negative infinity. Therefore, D H (q0 (−τ − t0 )) · ε=0 ∂ε zero as τ approaches negative infinity. Consequently, the limit of Eq. (D.74) as the

360

Appendix D: Derivation of the Melnikov Function

parameter τ approaches infinity becomes ∞ M(t0 , φ0 ) =

D H (q0 (t − t0 )) · g(q0 (t − t0 ), ω t + φ0 , 0)dt

(D.75)

−∞

Prior to presenting a main theorem, it is pertinent to highlight a noteworthy characteristic of the Melnikov function. If the transformation t → t + t0 is made, the Eq. (D.75) becomes: ∞ M(t0 , φ0 ) =

D H (q0 (t)) · g(q0 (t), ω t + ω t0 + φ0 , 0)dt

(D.76)

−∞

It is important to keep in mind that the function g(q, · , 0) is periodic. This indicates that the function M(t0 , φ0 ) is periodic in t0 with the period 2π/ω and periodic in φ0 with the period 2π . It ought to be obvious from Eq. (D.76) that varying t0 and varying φ0 both have the same effect on the system. Furthermore, based on the Eq. (D.76) and the periodicity of g(q, · , 0), it can be deduced that ∂M 1 ∂M (t0 , φ0 ) = (t0 , φ0 ) ∂φ0 ω ∂t0

(D.77)

We now state a theorem as follows:     Suppose there is a point (t0 , φ0 ) = t 0 , φ 0 such that M t 0 , φ 0 = 0 and | ∂M | /= 0, then, when ε is sufficiently small, W s (γε (t)) and W u (γε (t)) intersect ∂t0 (t 0 , φ 0 )       transversely at q0 −t 0 + O(ε), φ 0 . Furthermore, if the condition M t 0 , φ 0 /= 0 holds for all (t0 , φ0 ) ∈ R1 × S 1 , then the intersection of W s (γε (t)) ∩ W u (γε (t)) = ∅.

Reference

1. Wiggins S (2003) Introduction to Applied Non-linear Dynamical Systems and Chaos. Springer, New York.