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Research Papers: Offshore Technology

Modeling of Response Amplitude Operator for Coupled Sway, Roll and Yaw Motions of a Floating Body in Sinusoidal Waves Using Frequency Based Analysis

[+] Author and Article Information
Samir K. Das

Professor
Department of Applied Mathematics,
Defence Institute of Advanced Technology,
Girinagar, Pune 411025, India
e-mails: samirkdas@diat.ac.in;
samirkumar_d@yahoo.com

Masoud Baghfalaki

Department of Mathematics,
Kermanshah Branch,
Islamic Azad University,
Kermanshah 6718997551, Iran
e-mail: masoudbaghfalaki@yahoo.com

1Corresponding author.

2Formerly, Senior Research Officer, Mathematical Modelling and Coastal Engineering Centre, Central Water and Power Research Station, Khadakwasla, Pune-411024, India.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received November 25, 2013; final manuscript received March 3, 2015; published online April 8, 2015. Assoc. Editor: M. H. (Moo-Hyun) Kim.

J. Offshore Mech. Arct. Eng 137(3), 031303 (Jun 01, 2015) (18 pages) Paper No: OMAE-13-1111; doi: 10.1115/1.4030019 History: Received November 25, 2013; Revised March 03, 2015; Online April 08, 2015

The paper investigates the characteristics of response amplitude operators (RAO) or transfer function of a floating body in frequency domain for coupled sway, roll and yaw motions in sinusoidal waves. The floating body is considered to be initially at rest and waves act as beam to the floating body with varying frequency (ω) between 0.3 rad/s and 1.2 rad/s. The hydrodynamic coefficients (HC) are computed using strip theory formulation and the general expression of RAO is derived. The behavior of RAO under coupled conditions is examined by considering two asymptotic cases, corresponding to ω0 and ω. For the intermediate frequency range, analytical expression for system frequency is derived. The effects of viscous damping for uncoupled and coupled transfers have been compared with the result of nonviscous case. A mathematical analogy with respect to Mathieu and Hill equations has been established using frequency based classifications of governing equations. This modeling approach can provide useful guidelines to determine RAO for coupled motions and computing of wave loads and sensitivity analysis with respect to initial conditions of a floating body for the wide range of frequencies.

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Figures

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Fig. 1

Schematic diagram of a floating body with sign convention

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Fig. 2

Schematic diagram of strips of a floating body

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Fig. 5

Roll amplitude for case C with and without damping: (a) w = 0.3 (IC-1), (b) w = 0.56 (IC-1), (c) w = 0.74 (IC-1), (d) w = 1.2 (IC-1) (linear damping), (e) w = 0.74 (IC-1), and (f) w = 1.2 (IC-1) (unbounded damping)

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Fig. 6

Comparison of roll amplitude of case C with the full form and approximation of Hill and Mathieu equations: (a) w = 0.3 (IC-1), (b) w = 0.56 (IC-1), (c) w = 0.74 (IC-1), (d) w = 1.2 (IC-1), (e) w = 0.56 (IC-2) Eq. (73), and (f) w = 0.74 (IC-2) Eq. (76)

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Fig. 7

Comparison plots of ɛ in δ–t plane indicating parametric excitation

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Fig. 8

Plots of X·4 with time for the frequencies ω = 0.56 (where b44 = 0) and 0.74 Eq. (80)

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Fig. 9

Sway and yaw amplitude for case C new: (a) w = 0.3 (IC-1), (b) w = 0.30 (IC-1), (c) w = 0.56 (IC-1), (d) w = 56 (IC-1), (e) w = 0.74 (IC-1), and (f) w = 0.74 (IC-1)

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Fig. 4

Real part, imaginary part, and norm of transfer functions for uncoupled (1DOF) and coupled (3DOF) sway, roll and yaw motions

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Fig. 3

Comparison of sway, roll and yaw exciting force/moments at time t = 7.85 (s) and at frequency, w = 0.8 (rad/s)

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