Nonlinear response of a moored vessel

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NONLINEAR RESPONSE OF A MOORED VESSEL

by

ARSHAD UMAR

DEPARTMENT OF CIVIL ENGINEERING

Submitted

in fulfillment of the requirements of the degree of

DOCTOR OF PHILOSOPHY

to the

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

HAUZ KHAS, NEW DELHI - 110016 AUGUST, 1999

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I. I. T. DELHI.

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CERTIFICATE

This is to certify that the thesis entitled, "Nonlinear Response of a Moored Vessel", being submitted by Arshad Umar, to the Indian Institute of Technology, New Delhi, for the award of the Degree of "DOCTOR OF PHILOSOPHY" in Civil Engineering is a record of the bonafide research work carried out by him under our supervision and guidance. He has fulfilled the requirements for submission of this thesis, which to the best of our knowledge, has reached the requisite standard.

The material contained in this thesis has not been submitted in part or full to any other University or Institute for the award of any degree or diploma.

(Prof. T.K.Datta) (Dr Suhail Ahmed)

Associate Professor Professor and Head

Applied Mechanics Department Civil Engineering Department Indian Institute Of Technology Indian Institute of Technology New Delhi — 110016, INDIA New Delhi — 110016, INDIA

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ACKNOWLEDGMENTS

I express all my gratitude to Almighty Allah — Lord of the heavens and the earth, for successful completion of this thesis under the able guidance of Prof. T.K. Dalt(' and Dr.

Slrhail Ahmad. 1 shall always be indebted to them for their invaluable guidance, help, inspiration and constant encouragement throughout this study. May Almighty Allah give them a better return in this life and life to come. I am also thankful to Dr. A K Jain for his valuable suggestions and guidance.

I also feel immense pleasure in expressing my profound regard, deep sense of gratitude, heartiest devotion to Prof V. P. Mittal, Chairman, Department of Civil Engineering, A.M.U, Aligarh and Prof. Tariq Aziz, Chairman, Department of Applied Mathematics, A.M.U. Aligarh for their enthusiasm and encouragement to complete this work.

I fail to find words to acknowledge Mr Nadeem Ahsan Siddiqui, Mr. Quamrill Hassan, Mr. Amit Agarwal, Mr. Md. Umair, Mr. ,S'ajid Amin and Hafiz Siffj)an Beg for their suggestions and timely help in completing the thesis. I also extend my sincere thanks to Mr. R.K. Gupta, Mr. Aslam, Dr. M. Ahmad, Mr. Zafar A. Khan, Mr. A.

Munaff, Mr S.I.Anwar and Mr. Ravi K. Sharma for their inspiration and time to time help.

My sincere thanks are due to Dr. (Mrs.) Karunes for rendering full help and support to make my stay in Delhi comfortable.

I am thankful to Mr R. Agarwal for tracing the figures. My thanks are also due to Bhasker and Mr Duli Chand for their cooperation.

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I acknowledge my heartfelt thanks to my friends, Mr A.K. Shukla, Mr M.M. Rao, and Ms Tabassum Naqvi.

The author attributes the successful completion of the thesis to the sincere prayers, continuous support, love and affection of his late mother, wife, sons, brothers, sisters, sisters in law and brother in law. They have always been a major source of motivation and strength for all time endeavors.

Finally, I would like to thank all those who helped me directly or indirectly in completion of this research work.

Lastly but not the least, I tender my grateful thanks to my wife Ghazala for her understanding, love and companionship and above all for the inspiration to initiate this project.

(Arshad Umar)

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ABSTRACT

Response of a multi-point mooring system is nonlinear because of the nonlinear force excursion relationship of the mooring lines. Under certain conditions, the nonlinearity of the system may produce complex response behaviour including many instability phenomena like period doubling, symmetry breaking bifurcation, 3T and 5T solution etc. There have been several studies on the development of the method of analysis of mooring systems and flexible or floating structures anchored to the seabed. Also, there have been studies in characterizing their responses in both regular and random waves. However, very few studies are reported on the stability analysis of such systems under different kinds of wave excitations. The present study specifically addresses this problem and investigates different types of instability phenomena that may occur in a mooring system. For this purpose, a moored buoy, consisting of a large diameter cylinder anchored to the seabed by means of six mooring lines in 150m depth of water, is considered in the analysis.

Wave forces on a large diameter cylinder are calculated by a number of methods and are compared for a parametric study, in order to determine the most suitable methods to be adopted for calculating the wave forces on the buoy.

The results of the parametric study indicate that (i) the methods for calculating first order and second order wave forces in regular sea proposed by Eatoc Taylor and Hung [1987] are quite satisfactory. (ii) Also, the same method provides good

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estimates of primary wave forces in random sea and (iii) Newman's method of calculating the second order drift force in random sea is quite suitable.

A procedure for the stability analysis of the moored system is presented for periodic wave excitation. It consists of finding the approximate response of the system by harmonic balance method. A two-term solution is used in the harmonic balance method and the restoring force of the nonlinear mooring system is approximated by a 5th order antisymmetric polynomial. The conditions for determining the local and global stability of the approximate solutions are established using Hill's variational approach and Floquet's theory. Further, the procedure for examining the possible instability phenomena and chaos of a nonlinear system are outlined. The computer programs are developed based on the procedures developed. The program is validated by solving a problem of taut mooring lines reported by Gottlieb and Yim [1992]. The results of the stability analysis of the moored system are compared with those obtained by Gottlieb and Yim. It is found that the results obtained by the proposed methods compared fairly well with those presented by Gottlieb and Yim for all the cases.

Stability analysis of the moored buoy is carried out for different frequencies of harmonic excitation in order to illustrate the possibility of instability phenomena that can occur in the system. It is shown that the system can have period doubling, symmetry breaking bifurcation of solutions. Also, the system can undergo chaotic behavior under certain conditions of excitation. The possibility of instability phenomena occurring under three regular sea-states namely, 8m/

5sec, 12m/10sec and 18m/15sec are also examined. It is shown that the system

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shows problems of stability when subjected to primary wave forces for 12m/

10sec and 18m/15sec waves. However, the system is found to be stable under second order wave forces for all the three sea-states. Under the action of combined first and second order wave force, the results are found to be similar to those for primary wave force.

The responses of the moored vessel for idealized narrow band sea states and two actual sea states are obtained and analyzed to study the response characteristics. The two states correspond to significant wave height and average time period as 5m/5sec and 18m/15sec. For the two idealized sea- states, the bifurcation of the solutions may occur. However, for the real states, the system remains stable. Probabilistic characteristics of the responses show that (i) the distribution of the responses of the system deviates considerably from the Gaussian distribution, (ii) the peak value distribution does not follow Gumbel Type I distribution. (iii) The power spectral density function (PSDF) of the second order wave force is relatively broad banded.

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CONTENTS

Page No.

CERTIFICATE

ACKNOWLEDGEMENTS ii

ABSTRACT iv

CONTENTS vii

LIST OF FIGURES xiii

LIST OF TABLES xxvii

NOMENCLATURE xxviii

CHAPTER — 1 INTRODUCTION 1

1.1 GENERAL 1

1.2 PROBLEMS ASSOCIATED WITH THE RESPONSE

ANALYSIS OF MOORED VESSELS 5

1.3 NEED FOR THE PRESENT WORK 7

1.4 OBJECTIVES OF THE PRESENT STUDY 8

1.5 ORGANISATION OF THE THESIS 9

CHAPTER — 2 LITERATURE REVIEW 14

2.1 GENERAL 14

2.2 SECOND ORDER FORCE 14

2.3 DRIFT FORCE 19

2.4 MOORING LINES AND MOORED VESSELS 23

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2.5 STUDIES ON DYNAMIC STABILITY, BIFURACATION

AND CHAOS IN FLEXIBLE SYSTEMS 33 CHAPTER — 3 WAVE FORCES IN REGULAR AND

IRREGULAR SEA 36

3.1 INTRODUCTION 36

3.2 FIRST AND SECOND ORDER FORCES IN

REGULAR SEA 38

3.3 NUMERICAL RESULTS FOR INFINITE DEPTH

OF WATER 39

3.4 CONCLUSIONS (FOR INFINITE WATER DEPTH) 43 3.5 NUMERICAL RESULTS FOR FINITE DEPTH OF WATER 45 3.6 CONCLUSIONS (FOR FINITE WATER DEPTH) 49

3.7 MEAN DRIFT FORCE ANALYSIS 51

3.7.1 Importance of Drift 51

3.8 NUMERICAL RESULTS AND DISCUSSIONS 53

3.9 CONCLUSIONS (DRIFT FORCE) 55

3.10 IMPLICATIONS OF THE STUDY CARRIED OUT IN THE

CHAPTER 56

TABLE 58

FIGURES 59

CHAPTER — 4 THEORETICAL DERIVATIONS OF THE STABILITY

ANALYSIS 81

4.1 GENERAL 81

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4.2 ASSUMPTIONS 82

4.3 SYSTEM MODEL 83

4.4 DERIVATION OF NONLINEAR EQUATIONS

BASED ON HBM 84

4.5 STABILITY, BIFURCATION AND CHAOS OF

NONLINEAR SYSTEMS 89

4.6 GLOBAL STABILTIY OF THE SYSTEM 90 4.7 LOCAL STABILITY ANALYSIS OF THE APPROXIMATE

RESPONSE OBTAINED BY HBM 93

4.7.1 Derivation of Equation 93 4.8 BIFURCATIONS AND THE ONSET OF CHAOS 97

4.9 VALIDATION OF PROGRAM 100

4.10 REPRESENTATION OF RESTORING FORCE AS 5th

ORDER CHEBYSHEV POLYNOMIAL OF FIRST KIND 102

4.11 RESULTS AND DISCUSSIONS 103

4.12 CONCLUSIONS 106

FIGURES 107

CHAPTER — 5 STABILITY ANALYSIS OF MOORED VESSEL

UNDER REGULAR SEA 115

5.1 INTRODUCTION 115

5.2 SYSTEM MODEL 116

5.3 FORCE EXCURSION RELATIONSHIP OF THE

MOORED BUOY 117

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5.3.1 Assumptions 118 5.4 NUMERICAL STUDY AND DISCUSSION OF RESULTS 118 5.4.1 Instability Phenomena for the System 120 5.4.2 Stability of the System under Regular Waves 123 5.4.2.1 Responses due to 5ml5sec Wave 124 5.4.2.2 Responses due to 12m/10sec Wave 127 5.4.2.3 Responses due to 18m/15sec Wave 131 5.4.3 Response of a Modified System 135

due to 12m/10sec Wave

5.5 CONCLUSIONS 136

FIGURES 138

CHAPTER — 6 RESPONSE CHARACTERISTICS OF MOORED

VESSEL UNDER RANDOM WAVES 182

6.1 INTRODUCTION 182

6.2 IRREGULAR WAVES, WAVE SPECTRUM AND

SIMULATION OF WAVES 183

6.3 SIMULATION OF FIRST ORDER IRREGULAR FORCE 186 6.4 FORMULATION OF DRIFT FORCE 187

6.5 TIME HISTORY OF RESPONSE 188

6.6 NUMERICAL STUDY 188

6.6.1 Response Characteristics of the System for

Primary Wave Force 189

6.6.2 Response Characteristics of the System for

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Second Order Drift Forces 192 6.6.3 Response Characteristics of the System for

Combined Primary and Second Order Drift Forces 193 6.6.4 Probabilistic Characteristics of Responses Due To

Primary Wave Force 194

6.6.5 Probabilistic Characteristics of Responses Due To

Drift Force 195

6.6.6 Probabilistic Characteristics of Response for

Combined Primary and Second Order Drift Forces 197

6.7 CONCLUSIONS 197

FIGURES 200

CHAPTER — 7 CONCLUSIONS AND RECOMMENDATIONS FOR

THE FUTURE WORK 227

7.1 CONCLUSIONS 227

7.2 CONTRIBUTIONS OF THE PRESENT STUDY 232 7.3 RECOMMENDATIONS FOR THE FUTURE WORK 233

REFERENCES 235

APPENDIX — I METHODS FOR DETERMINING WAVE FORCES

IN REGULAR SEA FOR INFINITE DEPTH 251 AI.1 LIGHTHILL'S APPROACH 251

AI.1.1 Assumptions 251

AI.2 EATOC TAYLOR APPROACH 253

A1.2.2 General Formulations 253 AI.3 HUNT AND BADDOUR METHOD 255

AI.3.2 General Formulations 256 AI.3.3 Second Order Force 258

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AI.3.4 Wave Forces On The Cylinder 260 APPENDIX — II METHODS FOR DETERMINING WAVE FORCES IN

REGULAR SEA FOR FINITE DEPTH 263 AII.1 METHODS DEALING WITH FINITE DEPTH 263 AII.2 M. RAHMAN'S APPROACH 263

AII.2.1Assumptions 263

AII.2.2Wave Forces Formulations 263 AII.2.3Second Order Force 264 AII.3 EATOC TAYLOR AND HUNG'S APPROACH 267 AII.4 KIM AND YUE'S METHOD 267

AII.4.1Assumptions 267

AII.4.2Second Order Wave Force 267

APPENDIX — III SLOWLY VARYING DRIFT FORCES IN

RANDOM WAVES 271

AIII.1 NEWMAN'S INDEX APPROXIMATION

METHOD 273

AIII.2 KIM AND YUE'S METHOD 274 AIII.3 MARTHINSEN METHOD 275 AIII.3.1General Formulation 275 APPENDIX — IV DERIVATION OF THE RESTORING FORCE IN

TERMS OF TWO TERM SOLUTION OF HBM 277 APPENDIX — V DERIVATION OF THE STABILITY (FREQUENCY)

BOUNDS 285

APPENDIX — VI CALCULATION OF TOTAL MOORING RESTORING

FORCES 290

FIGURE 294

APPENDIX — VII CALCULATION OF ADDED MASS AND DAMPING

OF THE SYSTEM (REGULAR SEA) 295

FIGURE 297

APPENDIX — VIII CALCULATION OF MASS, ADDED MASS AND

DAMPING OF THE SYSTEM (RANDOM SEA) 298 BIODATA

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