Rotor Balancing Via an Enhanced Automatic Dynamic Balancer With Inductively Coupled Shunt Circuit

2019 ◽  
Vol 141 (3) ◽  
Author(s):  
Xiaowen Su ◽  
Hans A. DeSmidt
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Lagrangian Description ◽  
Widespread Application ◽  
Inductively Coupled ◽  
Rlc Circuit ◽  
Automatic Balancing ◽  
Rotor Balancing ◽  
Balancing Principle

Despite the elegant nature of the automatic balancing principle for passive imbalance vibration control, the co-existence of undesired whirling limit-cycles is a major impediment to the more widespread application of automatic dynamic balancing devices also called automatic dynamic balancer (ADB) in industry. To enlarge the region of stable perfect balancing and to eliminate whirling limit-cycles, we develop an innovative enhanced ADB system. This new idea harnesses the automatic balancing principle via moving permanent magnet balancer masses which are inductively coupled to a parallel resistor–inductor–capacitor (RLC) circuit. It is found that the circuit parameters can be adjusted properly to suppress the whirling limit-cycle to enlarge the perfect balancing region. We start from a Lagrangian description of the system and get nonlinear autonomous equations-of-motion. We then solve two dominant steady-state solutions for the enhanced ADB system. One solution is for the perfect balancing equilibrium points (EPs), which can be solved analytically. While the other solution is for the whirling limit-cycle which is solved via a harmonic balance method. The stability of these solutions is then evaluated through eigenvalue analysis and Floquet theory. The newly involved electrical parameters, such as coupling coefficient, equivalent capacitance, and equivalent resistance, are designed via an arc-length continuation method to destabilize the limit-cycle solutions to then guarantee stable rotor balancing.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

MULTISTABILITY IN A BUTTERFLY FLOW

2013 ◽  
Vol 23 (12) ◽  
pp. 1350199 ◽  
Author(s):  
CHUNBIAO LI ◽  
J. C. SPROTT
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Fractal Structure ◽  
Equilibrium Points ◽  
Symmetric Pair ◽  
Coexisting Attractors ◽  
Quadratic Nonlinearities ◽  
Stable Equilibria ◽  
Basin Boundaries ◽  
Special Parameter

A dynamical system with four quadratic nonlinearities is found to display a butterfly strange attractor. In a relatively large region of parameter space the system has coexisting point attractors and limit cycles. At some special parameter combinations, there are five coexisting attractors, where a limit cycle coexists with two equilibrium points and two strange attractors in different attractor basins. The basin boundaries have a symmetric fractal structure. In addition, the system has other multistable regimes where a pair of point attractors coexist with a single limit cycle or a symmetric pair of limit cycles and where a symmetric pair of limit cycles coexist without any stable equilibria.

scholarly journals Limit Cycles in Planar Piecewise Linear Hamiltonian Systems with Three Zones Without Equilibrium Points

2020 ◽  
Vol 30 (11) ◽  
pp. 2050157
Author(s):  
Alexander Fernandes da Fonseca ◽  
Jaume Llibre ◽  
Luis Fernando Mello
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Linear Hamiltonian Systems

We study the existence of limit cycles in planar piecewise linear Hamiltonian systems with three zones without equilibrium points. In this scenario, we have shown that such systems have at most one crossing limit cycle.

Limit-Cycle Analysis of Planar Rotor/Autobalancer System Influenced by Alford's Force

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
DaeYi Jung ◽  
H. A. DeSmidt
Keyword(s):  
Steady State ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Journal Bearing ◽  
Dynamic Interactions ◽  
Rotor Imbalance ◽  
Synchronous Condition ◽  
Freely Moving ◽  
Automatic Balancing ◽  
Special Case

In recent years, there has been much interest in the use of so-called automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancers” consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steady-state. 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 and 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 steady-state positions resulting in increased rotor vibration. In this paper, an approximate analytical harmonic solution for the limit cycles is obtained for the special case of symmetric support stiffness together with the so-called Alford's force cross-coupling term. The limit-cycle stability is assessed via Floquet analysis with a perturbation. It is found that the stable balanced synchronous conditions coexist with undesirable nonsynchronous limit cycles. For certain combinations of bearing parameters and operating speeds, the nonsynchronous limit-cycle can be made unstable guaranteeing global asymptotic stability of the synchronous balanced condition. Additionally, the analytical bifurcation of the coexistence zone and the pure balanced synchronous condition is derived. Finally, the analysis is validated through numerical time- and frequency-domain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals ON PERIODIC SOLUTIONS OF LIÉNARD TYPE EQUATIONS

2013 ◽  
Vol 18 (5) ◽  
pp. 708-716 ◽  
Author(s):  
Svetlana Atslega ◽  
Felix Sadyrbaev
Keyword(s):  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Type Equation ◽  
Convex Shape ◽  
Period Annulus ◽  
Conservative Equation ◽  
Liénard Type Equation

The Liénard type equation x'' + f(x, x')x' + g(x) = 0 (i) is considered. We claim that if the associated conservative equation x'' + g(x) = 0 has period annuli then a dissipation f(x, x') exists such that a limit cycle of equation (i) exists in a selected period annulus. Moreover, it is possible to define f(x, x') so that limit cycles appear in all period annuli. Examples are given. A particular example presents two limit cycles of non-convex shape in two disjoint period annuli.

CHAOS AND TWO-TORI IN A NEW FAMILY OF 4-CNNS

2007 ◽  
Vol 17 (03) ◽  
pp. 953-963 ◽  
Author(s):  
XIAO-SONG YANG ◽  
YAN HUANG
Keyword(s):  
Neural Networks ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Cellular Neural Networks ◽  
Lyapunov Spectrum ◽  
Regular Array ◽  
Poincaré Maps ◽  
One Dimensional ◽  
Poincare Maps ◽  
New Family

In this paper we demonstrate chaos, two-tori and limit cycles in a new family of Cellular Neural Networks which is a one-dimensional regular array of four cells. The Lyapunov spectrum is calculated in a range of parameters, the bifurcation plots are presented as well. Furthermore, we confirm the nature of limit cycle, chaos and two-tori by studying Poincaré maps.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

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.

scholarly journals Dynamical Behaviour in Two Prey-Predator System with Persistence

2016 ◽  
Vol 16 ◽  
pp. 20-37
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Limit Cycle ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Behaviour ◽  
Positive Equilibrium ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
Necessary And Sufficient

In this work, the dynamical behavior of the system with two preys and one predator population is investigated. The predator exhibits a Holling type II response to one prey which is harvested and a Beddington-DeAngelis functional response to the other prey. The boundedness of the system is analyzed. We examine the occurrence of positive equilibrium points and stability of the system at those points. At trivial equilibrium E0and axial equilibrium (E1); the system is found to be unstable. Also we obtain the necessary and sufficient conditions for existence of interior equilibrium point (E6) and local and global stability of the system at the interior equilibrium (E6): Depending upon the existence of limit cycle, the persistence condition is established for the system. The numerical simulation infer that varying the parameters such as e and λ1it is possible to change the dynamical behavior of the system from limit cycle to stable spiral. It is also observed that the harvesting rate plays a crucial role in stabilizing the system.

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