ON STICK–SLIP HOMOCLINIC CHAOS AND BIFURCATIONS IN A MECHANICAL SYSTEM WITH DRY FRICTION

2001 ◽  
Vol 11 (07) ◽  
pp. 2019-2029 ◽  
Author(s):  
B. R. PONTES ◽  
V. A. OLIVEIRA ◽  
J. M. BALTHAZAR
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Chaotic Behavior ◽  
Prediction Method ◽  
Phase Portraits ◽  
Frequency Spectra ◽  
Stick Slip ◽  
Potential Wells ◽  
Homoclinic Chaos ◽  
Belt System

In this paper we consider a self-excited mechanical system by dry friction in order to study the bifurcational behavior of the arisen vibrations. The oscillating system consists of a mass block-belt-system which is self-excited by static and Coulomb friction. We analyze the system behavior numerically through bifurcation diagrams, phase portraits, frequency spectra and Poincaré maps, which show the existence of nonhomoclinic and homoclinic chaos and a route to homoclinic chaos. The homoclinic chaos is also analyzed analytically via the Melnikov prediction method. The system dynamic is characterized by the existence of two potential wells in the phase plane which exhibit rich bifurcational and chaotic behavior.

scholarly journals Study on Nonlinear Dynamics and Chaos Suppression of Active Magnetic Bearing Systems Based on Synchronization

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Frequency Spectra ◽  
Control Approach ◽  
Chaos Suppression

This study employed a variety of nonlinear dynamic analysis techniques to explore the complex phenomena associated with a nonlinear mathematical model of an active magnetic bearing (AMB) system. The aim was to develop a method with which to assume control over chaotic behavior. The bifurcation diagram comprehensively explicates rich nonlinear dynamics over a range of parameter values. In this study, we examined the complex nonlinear behaviors of AMB systems using phase portraits, Poincaré maps, and frequency spectra. Furthermore, estimates of the largest Lyapunov exponent based on the properties of synchronization confirmed the occurrence of chatter vibration indicative of chaotic motion. Thus, the proposed continuous feedback control approach based on synchronization characteristics eliminates chaotic oscillations. Finally, some simulation results demonstrated the feasibility and efficiency of the proposed control scheme.

A COMMENT ON A NONIDEAL CENTRIFUGAL VIBRATOR MACHINE BEHAVIOR WITH SOFT AND HARD SPRINGS

2006 ◽  
Vol 16 (04) ◽  
pp. 1083-1088 ◽  
Author(s):  
MÁRCIO JOSÉ HORTA DANTAS ◽  
JOSÉ MANOEL BALTHAZAR
Keyword(s):  
Mechanical System ◽  
Classical Result ◽  
Chaotic Behavior ◽  
Damping Coefficient ◽  
Plane Curves ◽  
Center Manifolds ◽  
Nonlinear Spring ◽  
Homoclinic Chaos ◽  
Rotating Machine ◽  
Bezout Theorem

This paper concerns a type of rotating machine (centrifugal vibrator), which is supported on a nonlinear spring. This is a nonideal kind of mechanical system. The goal of the present work is to show the striking differences between the cases where we take into account soft and hard spring types. For soft spring, we prove the existence of homoclinic chaos. By using the Melnikov's Method, we show the existence of an interval with the following property: if a certain parameter belongs to this interval, then we have chaotic behavior; otherwise, this does not happen. Furthermore, if we use an appropriate damping coefficient, the chaotic behavior can be avoided. For hard spring, we prove the existence of Hopf's Bifurcation, by using reduction to Center Manifolds and the Bezout Theorem (a classical result about algebraic plane curves).

scholarly journals Stability, Chaos Detection, and Quenching Chaos in the Swing Equation System

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Shun-Chang Chang
Keyword(s):  
Power Systems ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Equation System ◽  
State Feedback Control ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Frequency Spectra ◽  
Chaos Detection ◽  
The Rich

The main objective of this study is to explore the complex nonlinear dynamics and chaos control in power systems. The rich dynamics of power systems were observed over a range of parameter values in the bifurcation diagram. Also, a variety of periodic solutions and nonlinear phenomena could be expressed using various numerical skills, such as time responses, phase portraits, Poincaré maps, and frequency spectra. They have also shown that power systems can undergo a cascade of period-doubling bifurcations prior to the onset of chaos. In this study, the Lyapunov exponent and Lyapunov dimension were employed to identify the onset of chaotic motion. Also, state feedback control and dither signal control were applied to quench the chaotic behavior of power systems. Some simulation results were shown to demonstrate the effectiveness of these proposed control approaches.

LATERAL AND TORSIONAL VIBRATIONS OF A TWO-DISK ROTOR-STATOR SYSTEM WITH AXIAL CONTACT/RUBS

2009 ◽  
Vol 01 (02) ◽  
pp. 305-326 ◽  
Author(s):  
KUNPENG ZHANG ◽  
QIAN DING
Keyword(s):  
Numerical Simulations ◽  
Dry Friction ◽  
Contact Forces ◽  
Phase Portraits ◽  
Torsional Vibrations ◽  
Frequency Spectra ◽  
Friction Forces ◽  
Coupled Equations ◽  
Dynamical Behaviors ◽  
Disk Contact

The dynamics of a rotor system with axial contact/rub events between the disks and stator are investigated by numerical simulations. The formula for determining the contact/rub points, axial contact forces and dry friction forces are deduced. To account for their influence, the axial contact forces are substituted by equivalent forces acting at the disk centers, based on the equivalent moment rule. One-parametric model is used to estimate the contact-induced dry friction forces. The coupled equations of lateral and torsional motions of rotor and the lateral motion of disk are then established. Numerical simulations are carried out to reveal the lateral and torsional vibrations for both two-disk contact/rubs with different axial clearances, and one disk contact/rubs. Bifurcation diagrams, orbits, phase portraits, amplitude-frequency spectra and Poincaré maps are adopted to demonstrate the dynamical behaviors of the system. The results show that though both the lateral and torsional vibrations can reflect the influences of contact/rubs on rotor dynamics, the spectrum analyses of the torsional vibrations are more suitable to determine straight the extent of their effect.

scholarly journals Nonlinear Dynamics and Suppressing Chaos in Magnetic Bearing System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Magnetic Bearing ◽  
State Feedback Control ◽  
Phase Portraits ◽  
Largest Lyapunov Exponent ◽  
Frequency Spectra ◽  
Loop Control ◽  
Identification Technique ◽  
Bearing System

A nonlinear mathematical model of a magnetic bearing system has been obtained by applying a modified conventional identification technique based on the principle of harmonic balance. In this study, we examined the rich nonlinear dynamics of a magnetic bearing system with closed-loop control using phase portraits, Poincaré maps, and frequency spectra. The resulting bifurcation diagram can be used to evaluate the operational range of systems employing nonlinear actuators. Estimates of the largest Lyapunov exponent based on the properties of synchronization revealed the occurrence of chatter vibration indicative of chaotic motion. Various control methods, such as the state feedback control and the injection of dither signals, were then used to quench the chaotic behavior.

scholarly journals Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Power Systems ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Lyapunov Stability Theory ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Controlling Chaos ◽  
Period Doubling Bifurcations

This paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. The results of our simulation support the feasibility of using the proposed method. The robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.

Energy Balancing Method to Identify Nonlinear Damping of Bolted-Joint Interface

2016 ◽  
Vol 693 ◽  
pp. 318-323 ◽  
Author(s):  
Xin Liao ◽  
Jian Run Zhang
Keyword(s):  
Dry Friction ◽  
Harmonic Excitation ◽  
Joint Interface ◽  
Damping Factor ◽  
Viscous Damping ◽  
Bolted Joint ◽  
Energy Balancing ◽  
Stick Slip ◽  
Non Linear ◽  
Linear Damping

The interface of bolted joint commonly focuses on the research of non-linear damping and stiffness, which affect structural response. In the article, the non-linear damping model of bolted-joint interface is built, consisting of viscous damping and Coulomb friction. Energy balancing method is developed to identify the dry-friction parameter and viscous damping factor. The corresponding estimation equations are acquired when the input is harmonic excitation. Then, the vibration experiments with different bolted preloads are conducted, from which amplitudes in various input levels are used to work out the interface parameters. Also, the fitting curves of dry-friction parameters are also obtained. Finally, the results illustrate that the most interface of bolted joint in lower excitation levels occurs stick-slip motion, and the feasibility of the identification approach is demonstrated.

Vibration analysis of multiple degrees of freedom mechanical system with dry friction

2012 ◽  
Vol 227 (7) ◽  
pp. 1505-1514 ◽  
Author(s):  
SD Yu ◽  
BC Wen
Keyword(s):  
Mechanical System ◽  
Dry Friction ◽  
Degrees Of Freedom ◽  
Simple Procedure ◽  
Mathematical Problem ◽  
Equations Of Motion ◽  
Inequality Constraints ◽  
Integration Scheme ◽  
Time Step ◽  
Multiple Degrees Of Freedom

This article presents a simple procedure for predicting time-domain vibrational behaviors of a multiple degrees of freedom mechanical system with dry friction. The system equations of motion are discretized by means of the implicit Bozzak–Newmark integration scheme. At each time step, the discontinuous frictional force problem involving both the equality and inequality constraints is successfully reduced to a quadratic mathematical problem or the linear complementary problem with the introduction of non-negative and complementary variable pairs (supremum velocities and slack forces). The so-obtained complementary equations in the complementary pairs can be solved efficiently using the Lemke algorithm. Results for several single degree of freedom and multiple degrees of freedom problems with one-dimensional frictional constraints and the classical Coulomb frictional model are obtained using the proposed procedure and compared with those obtained using other approaches. The proposed procedure is found to be accurate, efficient, and robust in solving non-smooth vibration problems of multiple degrees of freedom systems with dry friction. The proposed procedure can also be applied to systems with two-dimensional frictional constraints and more sophisticated frictional models.

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