Abstract
This research addresses the mathematical solution of the elastic catenary, a fundamental problem in offshore mooring engineering. A novel exact solution in a non-Lagrangian form is developed through rigorous mathematical derivation, distinguishing it from classical Lagrangian solutions. The procedure is reported in detail, and the resulting expressions are applied to analyze the mooring system of a reference floating turbine. A general approach is introduced to solve the transcendental equations associated with catenary mooring problems. The newly derived formulae exhibit greater applicability to geometry-to-force problems compared to existing Lagrangian expressions, making them particularly useful for conceptual designs and front-end engineering. In summary, this work provides valuable new insights into the exact solution of the elastic catenary, enhancing understanding and enabling practical applications in the field of floating wind turbines.
Issue Section:
Ocean Renewable Energy
References
1.
Snyder
, B.
, and Kaiser
, M. J.
, 2009
, “Ecological and Economic Cost-Benefit Analysis of Offshore Wind Energy
,” Renew. Energy
, 34
(6
), pp. 1567
–1578
. 2.
Huang
, W.-H.
, and Yang
, R.-Y.
, 2021
, “Water Depth Variation Influence on the Mooring Line Design for FOWT Within Shallow Water Region
,” J. Marine Sci. Eng.
, 9
(4
), p. 409
. 3.
Cella
, P.
, 1999
, “Methodology for Exact Solution of Catenary
,” J. Structural Eng.
, 125
(12
), pp. 1451
–1453
. 4.
Faltinsen
, O.
, 1993
, Sea Loads on Ships and Offshore Structures
, Cambridge University Press
, Cambridge, UK
.5.
Ma
, K.-T.
, Luo
, Y.
, Kwan
, C.-T. T.
, and Wu
, Y.
, 2019
, Mooring System Engineering for Offshore Structures
, Gulf Professional Publishing
, Cambridge, MA
.6.
Gubesch
, E.
, Abdussamie
, N.
, Penesis
, I.
, and Chin
, C.
, 2022
, “Effects of Mooring Configurations on the Hydrodynamic Performance of a Floating Offshore Oscillating Water Column Wave Energy Converter
,” Renewable. Sustainable. Energy. Rev.
, 166
, p. 112643
. 7.
Mousavi
, M. E.
, Reza
, Z.
, Upadhye
, S.
, Vijayaraghavan
, V.
, and Haverty
, K.
, 2016
, “A Simplified Method for Quantitative Reliability and Integrity Analysis of Steel Catenary Risers
,” ASME J. Offshore Mech. Arct. Eng.
, 138
(1
), p. 011601
. 8.
Smith
, R. J.
, and MacFarlane
, C. J.
, 2001
, “Statics of a Three Component Mooring Line
,” Ocean. Eng.
, 28
(7
), pp. 899
–914
. 9.
Garza-Rios
, L. O.
, Bernitsas
, M. M.
, Nishimoto
, K.
, and Matsuura
, J. P. J.
, 2000
, “Dynamics of Spread Mooring Systems with Hybrid Mooring Lines
,” ASME J. Offshore Mech. Arct. Eng.
, 122
(4
), pp. 274
–281
. 10.
Yuan
, F.
, Guo
, Z.
, Li
, L.
, and Wang
, L.
, 2012
, “Numerical Model for Pipeline Laying During S-lay
,” ASME J. Offshore Mech. Arct. Eng.
, 134
(4
), p. 021703
. 11.
Amaechi
, C. V.
, Wang
, F.
, and Ye
, J.
, 2021
, “Mathematical Modelling of Bonded Marine Hoses for Single Point Mooring (SPM) Systems, with Catenary Anchor Leg Mooring (Calm) Buoy Application–a Review
,” J. Marine Sci. Eng.
, 9
(11
), p. 1179
. 12.
Riggs
, H.
, and Leraand
, T.
, 1990
, “Efficient Static Analysis and Design of Flexible Risers
,” ASME J. Offshore Mech. Arct. Eng.
, 113
(3
), pp. 235
–240
. 13.
Liu
, Y. J.
, Buchanan
, G. R.
, and Peddieson
, J.
, 2013
, “Application of Groebner Basis Methodology to Nonlinear Static Cable Analysis
,” ASME J. Offshore Mech. Arct. Eng.
, 135
(4
), p. 041601
. 14.
Pevrot
, A.
, and Goulois
, A.
, 1979
, “Analysis of Cable Structures
,” Computers Struct.
, 10
(5
), pp. 805
–813
. 15.
Ogawa
, Y.
, 1984
, “Fundamental Analysis of Deep Sea Mooring Line in Static Equilibrium
,” Appl. Ocean. Res.
, 6
(3
), pp. 140
–147
. 16.
Wichers
, J.
, 2013
, Guide to Single Point Moorings
, CreateSpace Independent Publishing Platform
.17.
Palomo
, M.
, 2017
, “Describing Reality: Bernoulliâs Challenge of the Catenary Curve and Its Mathematical Description by Leibniz and Huygens,” The Dialogue Between Sciences, Philosophy and Engineering: New Historical and Epistemological Insights. Homage to Gottfried W. Leibniz and 1646–2016
, College Publications Ltd
, Rickmansworth, UK
, p. 334
.18.
Bobillier
, Finck
, 1826
, “Questions résolues. Solution des deux problèmes de statique proposés à la page 296 du précédent volume
,” Ann. math. pures appl.
, 17
, pp. 59
–68
.19.
Irvine
, H.
, 1981
, Cable Structures
, MIT Press
, Cambridge, MA
.20.
Al-Solihat
, M. K.
, and Nahon
, M.
, 2016
, “Stiffness of Slack and Taut Moorings
,” Ships Offshore Struct.
, 11
(8
), pp. 890
–904
. 21.
Davidson
, J.
, and Ringwood
, J. V.
, 2017
, “Mathematical Modelling of Mooring Systems for Wave Energy Converters–A Review
,” Energies
, 10
(5
), p. 666
. 22.
Barltrop
, N.
, and Technology
, P.
, 1998
, “Floating Structures: A Guide for Design and Analysis,” Mooring Systems
, Vol. 2
, Chapter 9, Centre for Marine and Petroleum Technology.23.
NREL WISDEM Team
, 2016
, “Documentation for MAP
.”24.
Jonkman
, J. M.
, 2007
, “Dynamics Modeling and Loads Analysis of an Offshore Floating Wind Turbine
,” Tech. Rep., National Renewable Energy Lab. (NREL), Golden, CO
, 12.25.
Jonkman
, J.
, Butterfield
, S.
, Musial
, W.
, and Scott
, G.
, 2009
, “Definition of A 5-MW Reference Wind Turbine for Offshore System Development
,” Tech. Rep., National Renewable Energy Lab. (NREL)
, Golden, CO, 2.26.
Karimirad
, M.
, and Moan
, T.
, 2012
, “Feasibility of the Application of a Spar-Type Wind Turbine At a Moderate Water Depth
,” Energy Procedia
, 24
, pp. 340
–350
. Selected papers from Deep Sea Offshore Wind R&D Conference, Trondheim, Norway, Jan. 19–20. 27.
Pecher
, A.
, Foglia
, A.
, and Kofoed
, J. P.
, 2014
, “Comparison and Sensitivity Investigations of a Calm and Salm Type Mooring System for Wave Energy Converters
,” J. Marine Sci. Eng.
, 2
(1
), pp. 93
–122
. 28.
Munir
, H.
, Lee
, C. F.
, and Ong
, M. C.
, 2021
, “Global Analysis of Floating Offshore Wind Turbines with Shared Mooring System
,” IOP Conference Series: Materials Science and Engineering, Vol. 1201
, IOP Publishing
, p. 012024
.29.
Garza-Rios
, L. O.
, and Bernitsas
, M. M.
, 2001
, “Effect of Size and Position of Supporting Buoys on the Dynamics of Spread Mooring Systems
,” ASME J. Offshore Mech. Arct. Eng.
, 123
(2
), pp. 49
–56
. 30.
Guo
, Z.
, Wang
, L.
, and Yuan
, F.
, 2016
, “Quasi-Static Analysis of the Multicomponent Mooring Line for Deeply Embedded Anchors
,” ASME J. Offshore Mech. Arct. Eng.
, 138
(1
), p. 011302
. 31.
Hall
, M.
, Housner
, S.
, Sirnivas
, S.
, and Wilson
, S.
, 2021
, “MoorPy (Quasi-Static Mooring Analysis in Python)
,” National Renewable Energy Lab. (NREL)
, Golden, CO (United States), 7.32.
Hall
, M.
, Housner
, S.
, Zalkind
, D.
, Bortolotti
, P.
, Ogden
, D.
, and Barter
, G.
, 2022
, “An Open-Source Frequency-Domain Model for Floating Wind Turbine Design Optimization
,” J. Phys.: Conference Ser.
, 2265
(4
), p. 042020
.33.
Orcina
, 2022
, “The Online Help Documentation for Orcaflex (version 11.2d)
.”34.
MARINTEK
, 2009
, SIMO Theory Manual. SINTEF Ocean.35.
MARINTEK
, 2008
, RIFLEX Theory Manual. SINTEF Ocean.36.
NI
, Y.
, KO
, J.
, and ZHENG
, G.
, 2002
, “Dynamic Analysis of Large-Diameter Sagged Cables Taking Into Account Flexural Rigidity
,” J. Sound. Vib.
, 257
(2
), pp. 301
–319
. 37.
Orcina Ltd.
, 2022
, “K03 15MW semi-sub FOWT
,” https://www.orcina.com/wp-content/uploads/examples/k/k03/K03 15MW semi-sub FOWT.pdf.38.
Kheirabadi
, A. C.
, and Nagamune
, R.
, 2020
, “Real-Time Relocation of Floating Offshore Wind Turbine Platforms for Wind Farm Efficiency Maximization: An Assessment of Feasibility and Steady-State Potential
,” Ocean. Eng.
, 208
, p. 107445
. 39.
Mahfouz
, M. Y.
, Hall
, M.
, and Cheng
, P. W.
, 2022
, “A Parametric Study of the Mooring System Design Parameters to Reduce Wake Losses in a Floating Wind Farm
,” J. Phys.: Conference Ser.
, 2265
(4
), p. 042004
.40.
Froese
, G.
, Ku
, S. Y.
, Kheirabadi
, A. C.
, and Nagamune
, R.
, 2022
, “Optimal Layout Design of Floating Offshore Wind Farms
,” Renew. Energy
, 190
, pp. 94
–102
. 41.
Allen
, C.
, Viscelli
, A.
, Dagher
, H.
, Goupee
, A.
, Gaertner
, E.
, Abbas
, N.
, Hall
, M.
, and Barter
, G.
, 2020
, “Definition of The Umaine Volturnus-s Reference Platform Developed for the Iea Wind 15-megawatt Offshore Reference Wind Turbine
,” Tech. Rep., National Renewable Energy Lab. (NREL)
, Golden, CO; Univ. of Maine, Orono, 7.42.
Ypma
, T. J.
, 1995
, “Historical Development of the Newton–Raphson Method
,” SIAM Rev.
, 37
(4
), pp. 531
–551
. 43.
Brilliant.org
, 2022
, “Newton Raphson Method
.”44.
Meurer
, A.
, Smith
, C. P.
, Paprocki
, M.
, Čertík
, O.
, Kirpichev
, S. B.
, Rocklin
, M.
, Kumar
, A.
, et al., 2017
, “Sympy: Symbolic Computing in Python
,” PeerJ Computer Sci.
, 3
, p. e103
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