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Offshore and Structural Mechanics

Nonlinear Dynamics and Internal Resonances of a Ship With a Rectangular Cross-Section in Head Seas

[+] Author and Article Information
A. Kleiman

Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel

O. Gottlieb1

Department of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israeloded@technion.ac.il

1

Corresponding author.

J. Offshore Mech. Arct. Eng 131(4), 041301 (Sep 04, 2009) (12 pages) doi:10.1115/1.3160532 History: Received September 01, 2008; Revised May 01, 2009; Published September 04, 2009

Finite amplitude dynamics of ships in head seas due to parametric instabilities is a subject of renewed interest with an increasing demand of operation in severe and variable environmental conditions. In this current study we investigate the nonlinear dynamics and internal resonances of a ship with a rectangular cross-section in head seas. We employ an asymptotic averaging method to obtain the slowly varying system evolution dynamics for the weakly nonlinear response, complemented by numerical integration in the strongly nonlinear regime. A weakly nonlinear frequency response is obtained analytically for a principal parametric resonance and a 1:1 roll—pitch internal resonance. Comparison of results with three degrees of freedom numerical simulations reveals a good fit. A strongly nonlinear numerical analysis reveals that beyond the stability thresholds, the system’s responses included quasiperiodic dynamics. This combined approach resolves both parametric instabilities and internal resonances induced for both weak and finite nonlinear interactions, and culminates with criteria for orbital stability thresholds describing the onset of quasiperiodic response and magnification of energy transfer between coupled pitch-heave and ship roll.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 5

Exciting frequency (Ω), heave (ωh), pitch (ωθ), and natural frequencies as a function of the breadth parameter (γ)

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Figure 6

Trivial roll stability threshold for the exciting amplitude (ζ) as a function of frequency (Ω): solid line—Eq. 19, cross-numerical simulation of Eq. 9

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Figure 7

Roll magnitude as a function of exciting frequency (Ω). Solid—Eq. 20 (stable), dotted—Eq. 20 (unstable), and circle—numerical simulation of Eq. 9.

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Figure 13

State space and Poincaré map (left), and power spectra (right) for Ω=1.11 and ζ=0.004

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Figure 14

Time domain response for Ω=1.162 and ζ=0.0066

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Figure 15

Poincaré maps (left) and power spectra (right) analysis for Ω=1.162 and ζ=0.0066

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Figure 12

Time domain response for Ω=1.11, ζ=0.004, [ϕ0 θ0 h0 ϕτ,0 θτ,0 hτ,0]= (a) [0.100 0.055 0.004 0 0 0] and (b) [0.200 0.055 0.004 0 0 0]

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Figure 11

State space and Poincaré map (left) and power spectra (right) for Ω=1.2609 and ζ=0.012

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Figure 10

Time domain response for Ω=1.2609 and ζ=0.012, [ϕ0 θ0 h0 ϕτ,0 θτ,0 hτ,0]= (a) [0.100 0.023 0.012 0 0 0] and (b) [0.157 0.023 0.012 0 0 0]

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Figure 9

Roll and pitch magnitude versus Ω for ζ=0.004. Solid line—analytical stable solution, dotted line—analytical unstable solution, and circle—numerical 3 DOF simulation.

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Figure 8

Trivial roll stability threshold for the exciting amplitude (ζ) as function of frequency (Ω). Solid line—analytical and cross-numerical simulation of Eq. 9.

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Figure 3

Equilibrium points as function of γ solid line—stable solution, dotted line—unstable solution, and dashed dotted line—combined stable/unstable solution

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Figure 2

Sketch of a vessel with a rectangular cross-section

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Figure 4

Exciting frequency (Ω), heave (ωh), pitch (ωθ), and natural frequencies as a function of the breadth parameter (γ)

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Figure 1

(a) Body fixed and inertial reference frames and (b) definition sketch

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