Limit cycle behavior in three or higher dimensional nonlinear systems: the Lotka-Volterra example

2003 ◽  
Author(s):  
S.M. Shahruz
Keyword(s):  
Nonlinear Systems ◽  
Limit Cycle ◽  
Higher Dimensional ◽  
Limit Cycle Behavior

Modeling Unsteady Flow of a Lean Partially Premixed Flame on a Dry Low NOx Combustion System

2013 ◽  
Author(s):  
Salvatore Matarazzo ◽  
Hannes Laget ◽  
Evert Vanderhaegen ◽  
Jim B. W. Kok
Keyword(s):  
Gas Turbine ◽  
Limit Cycle ◽  
Gas Turbines ◽  
Premixed Flame ◽  
Computational Domain ◽  
Pressure Oscillations ◽  
Combustion Dynamics ◽  
Partially Premixed ◽  
Partially Premixed Flame ◽  
Limit Cycle Behavior

The phenomenon of combustion dynamics (CD) is one of the most important operational challenges facing the gas turbine (GT) industry today. The Limousine project, a Marie Curie Initial Training network funded by the European Commission, focuses on the understanding of the limit cycle behavior of unstable pressure oscillations in gas turbines, and on the resulting mechanical vibrations and materials fatigue. In the framework of this project, a full transient CFD analysis for a Dry Low NOx combustor in a heavy duty gas turbine has been performed. The goal is to gain insight on the thermo-acoustic instability development mechanisms and limit cycle oscillations. The possibility to use numerical codes for complex industrial cases involving fuel staging, fluid-structure interaction, fuel quality variation and flexible operations has been also addressed. The unsteady U-RANS approach used to describe the high-swirled lean partially premixed flame is presented and the results on the flow characteristics as vortex core generation, vortex shedding, flame pulsation are commented on with respect to monitored parameters during operations of the GT units at Electrabel/GDF-SUEZ sites. The time domain pressure oscillations show limit cycle behavior. By means of Fourier analysis, the coupling frequencies caused by the thermo-acoustic feedback between the acoustic resonances of the chamber and the flame heat release has been detected. The possibility to reduce the computational domain to speed up computations, as done in other works in literature, has been investigated.

Limit-Cycle Analysis of Three-Dimensional Flexible Shaft/Rigid Rotor/Autobalancer System With Symmetric Rigid Supports

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
DaeYi Jung ◽  
H. A. DeSmidt
Keyword(s):  
Limit Cycle ◽  
Three Dimensional ◽  
Design Parameters ◽  
Rigid Rotor ◽  
Modal Shape ◽  
Flexible Shaft ◽  
Flexible Mode ◽  
Automatic Balancing ◽  
Rigid Supports ◽  
Limit Cycle Behavior

In recent years, there has been much interest in the use of automatic balancing devices (ABD) in rotating machinery. Autobalancers consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance. This “automatic balancing” phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor wherein the balancer masses naturally synchronize with the rotor with appropriate phase to cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other undesirable nonsynchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced positions resulting in increased rotor vibration. To explore this nonsynchronous behavior of ABD, the unstable limit-cycle analysis of three-dimensional (3D) flexible shaft/rigid rotor/ABD/rigid supports described by the modal coordinates has been investigated here. Essentially, this paper presents an approximate harmonic analytical solution to describe the limit-cycle behavior of ABD–rotor system interacting with flexible shaft, which has not been fully considered by ABD researchers. The modal shape of flexible shaft is determined by using well-known fixed–fixed boundary condition due to symmetric rigid supports. Here, the whirl speed of the ABD balancer masses is determined via the solution of a nonlinear characteristic equation. Also, based upon the analytical limit-cycle solutions, the limit-cycle stability of three primary design parameters for ABD is assessed via a perturbation and Floquet analysis: the size of ABD balancer mass, the ABD viscous damping, and the relative axial location of ABD to the imbalance rotor along the shaft. The coexistence of the stable balanced synchronous condition and undesirable nonsynchronous limit-cycle is also studied. It is found that for certain combinations of ABD parameters and rotor speeds, the nonsynchronous limit-cycle can be made unstable, thus guaranteeing asymptotic stability of the synchronous balanced condition at the supercritical shaft speeds between each flexible mode. Finally, the analysis is validated through numerical simulation. The findings in this paper yield important insights for researchers wishing to utilize ABD in flexible shaft/rigid rotor systems and limit-cycle mitigation.

A modified Oregonator model exhibiting complicated limit cycle behavior in a flow system

1978 ◽  
Vol 69 (6) ◽  
pp. 2514 ◽  
Author(s):  
Kenneth Showalter ◽  
Richard M. Noyes ◽  
Kedma Bar-Eli
Keyword(s):  
Limit Cycle ◽  
Flow System ◽  
Oregonator Model ◽  
Limit Cycle Behavior

Limit cycle behavior of the bump-on-tail instability

1981 ◽  
Vol 24 (2) ◽  
pp. 268 ◽  
Author(s):  
Peter A. E. M. Janssen
Keyword(s):  
Limit Cycle ◽  
Limit Cycle Behavior
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