The top tension riser (TTR) is one of the most frequently used equipment in deep-sea petroleum engineering. At present, the research methods of its vortex-induced vibration (VIV) are mainly focused on finite element analysis and experiment. The understanding of its various nonlinear mechanical mechanisms would be inadequate via limited numerical or experimental studies rather than nonlinear analysis of its rich dynamics. Based on the Van der Pol wake oscillator model, the nonlinear dynamic model of the TTR subject to shear flow VIV is established. The proposed model includes the fluid-structure interaction of the TTR under shear flow. Dynamic behavior of the TTR in association with the variation of flow velocity is investigated. The dynamic behavior is simulated by computing the local maximum displacement response via the fifth-order Galerkin discretization. The Poincare map is then utilized to quantify the dynamic property of TTR under each individual flow velocity, which helps identifying the bifurcation path of the nonlinear system. The time history, phase diagram, FFT spectrum, and envelope diagram about the riser VIV at typical flow velocity in different regions of the bifurcation diagram are then given. It is found that the VIV response of the TTR depicts the Hopf bifurcation phenomena with bistable characteristics. Together with the structural eigen-analysis and the three-dimensional spectrum contour, the main dynamic features of the TTR in shear flow are more comprehensively understood. Such understandings may provide new ideas and references for the design and optimization of the riser structural parameters.
Recycling noise is a kind of more common noise. The nonlinear dynamic system can be controlled by adjusting its parameters. However, so far, the effect of recycling noise on tri-stable dynamic system has not been reported. In this paper, stochastic P-bifurcations in tri-stable Duffing–Van der Pol oscillator induced by additive recycling noise are investigated. Firstly, the stationary probability density function is derived using stochastic averaging method. Then, the general expression of the critical parameter conditions of stochastic P-bifurcation is given by the singularity theory. The stationary probability density of response amplitude in different parameter areas are also shown, which is verified by Monte Carlo numerical simulation. Based on these results, the influence of related parameters of recycling noise and damping coefficient on the stochastic P-bifurcation is studied. The result shows that the critical parameter of bifurcation can be changed by adjusting the delay time and fraction coefficient of the recycling noise. It has also been found that the stationary probability density and stochastic bifurcation show a periodic dependence on the delay time.
The stochastic P-bifurcation behavior of tri-stability in a fractional-order van der Pol system with time-delay feedback under additive Gaussian white noise excitation is investigated. Firstly, according to the equivalent principle, the fractional derivative and the time-delay term can be equivalent to a linear combination of damping and restoring forces, so the original system can be simplified into an equivalent integer-order system. Secondly, the stationary probability density function of the system amplitude is obtained by the stochastic averaging, and based on the singularity theory, the critical parameters for stochastic P-bifurcation of the system are found. Finally, the properties of stationary probability density function curves of the system amplitude are qualitatively analyzed by choosing corresponding parameters in each sub-region divided by the transition set curves. The consistence between numerical results obtained by Monte-Carlo simulation and analytical solutions has verified the accuracy of the theoretical analysis. The method used in this paper has a direct guidance in the design of fractional-order controller to adjust the dynamic behavior of the system.
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