Stability Analysis of Two-Point Mooring System in Surge Oscillation

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Harmonic Wave ◽  
Period Doubling ◽  
Mooring System ◽  
Restoring Force ◽  
Incremental Harmonic Balance ◽  
Cubic Polynomial ◽  
Motion Characteristics ◽  
The Stability ◽  
Continuation Technique ◽  
Amplitude Versus

Nonlinear surge response behavior of a multipoint mooring system under harmonic wave excitation is analyzed to investigate various instability phenomena such as bifurcation, period-doubling, and subharmonic and chaotic responses. The nonlinearity of the system arises due to nonlinear restoring force, which is modeled as a cubic polynomial. In order to trace different branches at the bifurcation point on the response curve (amplitude versus frequency of excitation plot), an arc-length continuation technique along with the incremental harmonic balance (IHBC) method is employed. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme. The period-one and subharmonic solutions obtained by the IHBC method are compared with those obtained by the numerical integration of the equation of motion. Characteristics of solutions from stable to unstable zones, chaotic motion, nT solutions, etc., are identified with the help of phase plots and Poincaré map sections.

Stability Analysis of TLP Tethers Under Vortex-Induced Oscillations

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
The Stability ◽  
Lock In ◽  
Continuation Technique ◽  
Mode Response ◽  
Stability Of The Solution

The vortex-induced oscillation of TLP tether is investigated in the vicinity of lock-in condition. The vortex shedding is caused purely due to current, which may vary across the depth of the sea. The vibration of TLP is modeled as a SDOF problem by assuming that the first mode response of the tether dominates the motion. Nonlinearity in the equation of motion is produced due to the relative velocity squared drag force. In order to trace different branches of the response curve and investigate different instability phenomena that may exist, an arc-length continuation technique along with the incremental harmonic balance method (IHBC) is employed. A procedure for treating the nonlinear term using distribution theory is presented so that the equation of motion is transformed to a form amenable to the application of IHBC. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme.

Periodic and Nonperiodic Thermomechanical Responses of Shape-Memory Oscillators

2001 ◽  
Author(s):  
Walter Lacarbonara ◽  
Davide Bernardini ◽  
Fabrizio Vestroni
Keyword(s):  
Shape Memory ◽  
Numerical Procedure ◽  
Period Doubling ◽  
Hysteresis Loops ◽  
Response Curves ◽  
Restoring Force ◽  
Codimension One ◽  
Nonlinear Responses ◽  
Gradient Based ◽  
The Stability

Abstract Nonlinear responses of shape-memory oscillators are investigated systematically using a numerical procedure and a modified Ivshin-Pence model for the restoring force. Due to the discontinuities in the tangent stiffness, classical gradient-based shooting techniques for determining periodic responses are not applicable. Herein the implemented algorithm searches the periodic solutions as fixed points of the Poincaré map. The Jacobian of the map is calculated via a central finite-difference scheme and its eigenvalues are computed to ascertain the stability of the solutions and the associated codimension-one bifurcations. A number of frequency-response curves are constructed for some meaningful shape-memory oscillators (characterized by different hysteresis loops) and for various excitation levels in the primary and superharmonic frequency ranges. The investigations are conducted both in isothermal and non-isothermal conditions. A rich class of solutions and bifurcations — including jump phenomena, pitchfork, period doubling, complete or incomplete bubble structures with a variety of nonperiodic responses — is found and discussed.

Turning Dynamics: Part 1 — Experimental Analysis

2012 ◽  
Author(s):  
Eric B. Halfmann ◽  
C. Steve Suh ◽  
N. P. Hung
Keyword(s):  
Instantaneous Frequency ◽  
Period Doubling ◽  
Displacement Sensor ◽  
Laser Displacement Sensor ◽  
Turning Process ◽  
Time Frequency ◽  
Spectral Components ◽  
Tool Vibrations ◽  
Exact Moment ◽  
The Stability

The workpiece and tool vibrations in a lathe are experimentally studied to establish improved understanding of cutting dynamics that would support efforts in exceeding the current limits of the turning process. A Keyence laser displacement sensor is employed to monitor the workpiece and tool vibrations during chatter-free and chatter cutting. A procedure is developed that utilizes instantaneous frequency (IF) to identify the modes related to measurement noise and those innate of the cutting process. Instantaneous frequency is shown to thoroughly characterize the underlying turning dynamics and identify the exact moment in time when chatter fully developed. That IF provides the needed resolution for identifying the onset of chatter suggests that the stability of the process should be monitored in the time-frequency domain to effectively detect and characterize machining instability. It is determined that for the cutting tests performed chatters of the workpiece and tool are associated with the changing of the spectral components and more specifically period-doubling bifurcation. The analysis presented provides a view of the underlying dynamics of the lathe process which has not been experimentally observed before.

Motion Simulation of Underwater Mooring Platform with Ocean Current

2013 ◽  
Vol 321-324 ◽  
pp. 815-818
Author(s):  
Fang Ze Zhao ◽  
Bao Wei Song ◽  
Xiao Xu Du
Keyword(s):  
Ocean Current ◽  
Mooring System ◽  
Motion Simulation ◽  
Motion Model ◽  
Motion Characteristics

Underwater mooring platforms which anchored by the anchor and cable have a certain function of the platform mooring at a certain depth. In this paper, the motion model of underwater mooring system was built through analyseing the motion characteristics of the cable geometry and the force of the cable. And the motion simulation of underwater mooring platform with ocean current was done. The results show that the motion of underwater mooring platform is stable.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

scholarly journals Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
S. S. Askar
Keyword(s):  
Fixed Point ◽  
Equilibrium Point ◽  
Profit Maximization ◽  
Period Doubling ◽  
Speed Of Adjustment ◽  
Discrete Dynamic ◽  
Cobweb Model ◽  
Numerical Studies ◽  
Market Clearing Price ◽  
The Stability

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

scholarly journals Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

2021 ◽  
Vol 29 (3) ◽  
Author(s):  
Murtala Bello Aliyu ◽  
Mohd Hafiz Mohd ◽  
Mohd Salmi Md. Noorani
Keyword(s):  
Bifurcation Analysis ◽  
Species Coexistence ◽  
Real Life ◽  
Period Doubling ◽  
Type Model ◽  
Coexistence Mechanisms ◽  
Multiple Species ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Multiple Interactions

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

scholarly journals Dynamics of a Duopoly Game with Two Different Delay Structures

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Shumin Jiang ◽  
Fei Xu ◽  
Zhanwen Ding ◽  
Chen Yang ◽  
Huanhuan Liu
Keyword(s):  
Time Delay ◽  
Market Price ◽  
Period Doubling ◽  
System Stability ◽  
Delayed Feedback Control ◽  
System Control ◽  
Heterogeneous Players ◽  
Cournot Game ◽  
The Stability ◽  
Time Delayed Feedback

Two different time delay structures for the dynamical Cournot game with two heterogeneous players are considered in this paper, in which a player is assumed to make decision via his marginal profit with time delay and another is assumed to adjust strategy according to the delayed price. The dynamics of both players output adjustments are analyzed and simulated. The time delay for the marginal profit has more influence on the dynamical behaviors of the system while the market price delay has less effect, and an intermediate level of the delay weight for the marginal profit can expand the stability region and thus promote the system stability. It is also shown that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play different roles on the system controllability: the delay of the marginal profit has more influence on the system control than the delay of the market price.

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