Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

Brouwers, Jjh Bert

doi:10.1016/j.physd.2011.02.009

Cited by 23 publications

(15 citation statements)

References 13 publications

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“…Across the tests, a clear positive trend was found between the total horizontal motions and the amount of parametric excitation at twice the pendulum frequency. The data also broadly agrees with a stability criterion for irregular wave tests, which compares this parametric excitation with the linear damping (adapted from [27]).…”

Section: Resultssupporting

confidence: 72%

“…T p = 1.32s in passing that the estimated value is approximately an order of magnitude higher than the surge/sway radiation damping b 11 at f n1 , which is very small for this rather flat cylinder. The stability criterion of [27], reproduced in Equation 3.2, is an asymptotic solution for very large times. Here, the irregular wave runs analysed are of relatively short duration (≈ 120 cycles at f n1 ).…”

Section: (C) Irregular Wave Tests: Modelmentioning

confidence: 99%

“…In this theoretical work, both linearly and non-linearly damped systems are analysed, though here we refer only to the results for a system with linear damping. The non-dimensionalised equation considered by [27] is…”

Section: (C) Irregular Wave Tests: Modelmentioning

confidence: 99%

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Transverse motion instability of a submerged moored buoy

et al. 2019

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Wave energy converters and other offshore structures may exhibit instability, in which one mode of motion is excited parametrically by motion in another. Here, theoretical results for the transverse motion instability (large sway oscillations perpendicular to the incident wave direction) of a submerged wave energy converter buoy are compared to an extensive experimental dataset. The device is axi-symmetric (resembling a truncated vertical cylinder) and is taut-moored via a single tether. The system is approximately a damped elastic pendulum. Assuming linear hydrodynamics, but retaining nonlinear tether geometry, governing equations are derived in six degrees of freedom. The natural frequencies in surge/sway (the pendulum frequency), heave (the springing motion frequency) and pitch/roll are derived from the linearized equations. When terms of second order in the buoy motions are retained, the sway equation can be written as a Mathieu equation. Careful analysis of 80 regular wave tests reveals a good agreement with the predictions of sub-harmonic (period-doubling) sway instability using the Mathieu equation stability diagram. As wave energy converters operate in real seas, a large number of irregular wave runs is also analysed. The measurements broadly agree with a criterion (derived elsewhere) for determining the presence of the instability in irregular waves, which depends on the level of damping and the amount of parametric excitation at twice the natural frequency.

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Section: Resultssupporting

confidence: 72%

Section: (C) Irregular Wave Tests: Modelmentioning

confidence: 99%

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Transverse motion instability of a submerged moored buoy

et al. 2019

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“…The asymptotic solution of the Mathieu instability problem under stochastic parametric excitation and non-linear damping has been studied with the combination of linear and non-linear power-law damping. 11 Any possibly undesirable phenomena that occur in the transverse vibration of the riser structure caused by this fluctuation need to be investigated to develop a frequency domain method for linear systems with general time-varying parameters. 12 A suitable mathematical model to explore the stability of a submerged floating pipeline between two floating structures under vortex and parametric excitations has been presented and discussed.…”

Section: Introductionmentioning

confidence: 99%

Parametric instability analysis of the top-tensioned riser in consideration of complex pre-stress distribution

Chen

2018

Advances in Mechanical Engineering

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The parametric vibration instability of a riser is studied in consideration of a complex pre-stress distribution. Differential equations of the riser are derived according to Euler-Bernoulli beam theory, and a method to solve the differential equations is proposed. With the parametric vibration of a top-tensioned riser as an example, the effects of the amplitude and direction of complex pre-stress on frequency, mode shapes, and instability characteristics are investigated. Results show that welding residual stress influences the dynamic response of the riser structure. A new approach to eliminate the complex loading of the riser is obtained.

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“…This phenomenon is called parametric instability. Parametric instability of slender marine structures (such as cables, risers and tethers) is caused by fluctuation of its tension in time (Brouwers, 2011;Zhang et al, 2010;Wang and Xie, 2012). In oil production riser is installed between wellhead at the sea bed and floating platform.…”

Section: Introductionmentioning

confidence: 99%

Parametric instability analysis of deepwater top-tensioned risers considering variable tension along the length

Zhang

Tang

2015

J. Ocean Univ. China

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Parametric instability of a riser is caused by fluctuation of its tension in time due to the heave motion of floating platform. Many studies have tackled the problem of parametric instability of a riser with constant tension. However, tension in the riser actually varies linearly from the top to the bottom due to the effect of gravity. This paper presents the parametric instability analysis of deepwater top-tensioned risers (TTR) considering the linearly varying tension along the length. Firstly, the governing equation of transverse motion of TTR under parametric excitation is established. This equation is reduced to a system of ordinary differential equations by using the Galerkin method. Then the parametric instability of TTR for three calculation models are investigated by applying the Floquet theory. The results show that the natural frequencies of TTR with variable tension are evidently reduced, the parametric instability zones are significantly increased and the maximum allowable amplitude of platform heave is much smaller under the same damping; The nodes and antinodes of mode shape are no longer uniformly distributed along the axial direction and the amplitude also changes with depth, which leads to coupling between the modes. The combination resonance phenomenon occurs as a result of mode coupling, which causes more serious damage.

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Asymptotic solutions for Mathieu instability under random parametric excitation and nonlinear damping

Cited by 23 publications

References 13 publications

Transverse motion instability of a submerged moored buoy

Transverse motion instability of a submerged moored buoy

Parametric instability analysis of the top-tensioned riser in consideration of complex pre-stress distribution

Parametric instability analysis of deepwater top-tensioned risers considering variable tension along the length

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