Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Hiroshi Yabuno ◽  
Yota Kunitho ◽  
Takuma Kashimura
Keyword(s):  
Steady State ◽  
Coulomb Friction ◽  
Multiple Scales ◽  
Time History ◽  
Steady States ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Trivial Equilibrium ◽  
The Difference ◽  
Unstable Focus

The effect of Coulomb friction on the nonlinear dynamics of a van der Pol oscillator is presented. A map from the magnitude of a peak to that of the succeeding valley in the time history is analytically described by considering both the exponential growth due to negative viscous damping and the switching condition due to Coulomb friction, which is a function of the sign of the velocity of the system. The steady states and their stability are clarified and the difference from those in the case without Coulomb friction is revealed. The addition of Coulomb friction makes the trivial equilibrium, which is an unstable focus in the system without friction, into a locally asymptotically stable equilibrium set. The branch of stable nontrivial steady states is not bifurcated from the trivial steady state by the effect of Coulomb friction and is different from the branch in the case without Coulomb friction, which is bifurcated from the trivial steady state through Hopf bifurcation. Furthermore, experiments are conducted and the theoretically predicted dynamics due to Coulomb friction is confirmed.

scholarly journals Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

2021 ◽  
pp. 146134842110264
Author(s):  
Chun-Hui He ◽  
Dan Tian ◽  
Galal M Moatimid ◽  
Hala F Salman ◽  
Marwa H Zekry
Keyword(s):  
Fixed Points ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Time History ◽  
Phase Portraits ◽  
Response Curves ◽  
Van Der Pol ◽  
Stability Performance ◽  
Linearized Stability

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

AMPLITUDE CONTROL OF LIMIT CYCLE IN VAN DER POL SYSTEM

2006 ◽  
Vol 16 (02) ◽  
pp. 487-495 ◽  
Author(s):  
JIASHI TANG ◽  
ZILI CHEN
Keyword(s):  
Limit Cycle ◽  
Multiple Scales ◽  
Numerical Study ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Amplitude Control ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Controlled Systems ◽  
Strongly Nonlinear System

The feedback controllers are designed to modify the amplitude of limit cycles in van der Pol oscillator and generalized van der Pol oscillator. Bifurcation control equations of weakly nonlinear systems are obtained by using the method of multiple scales. Gain-amplitude curves of controlled systems are drawn. Based on numerical study, the brief results of controlling amplitude of limit cycle are given for strongly nonlinear system.

scholarly journals Dynamics of Three Coupled Van der Pol Oscillators With Application to Circadian Rhythms

2005 ◽  
Author(s):  
Kevin Rompala ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Circadian Rhythms ◽  
Pineal Gland ◽  
Multiple Scales ◽  
Van Der Pol Oscillator ◽  
Slow Flow ◽  
Direct Connection ◽  
Van Der Pol ◽  
Software Products ◽  
Van Der Pol Oscillators ◽  
The Brain

In this work we study a system of three van der Pol oscillators, x, y and w, coupled as follows: x¨−ε(1−x2)x˙+x=εμ(w−x)y¨−ε(1−y2)y˙+y=εμ(w−y)w¨−ε(1−w2)w˙+p2w=εμ(x−w)+εμ(y−w) Here the x and y oscillators are identical, and are not directly coupled to each other, but rather are coupled via the w oscillator. We investigate the existence of the in-phase mode x = y for ε ≪ 1. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the software products MACSYMA and AUTO. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator (x and y). Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator (w).

scholarly journals Quasi-harmonic self-oscillations in discrete time: analysis and synthesis of dynamic systems

2022 ◽  
Vol 24 (4) ◽  
pp. 19-24
Author(s):  
Valery V. Zaitsev ◽  
Alexander V. Karlov
Keyword(s):  
Discrete Time ◽  
Oscillatory System ◽  
Discrete Models ◽  
Van Der Pol Oscillator ◽  
Time Analysis ◽  
Van Der Pol ◽  
Analog System ◽  
Analysis And Synthesis ◽  
The Difference ◽  
System Prototype

For sampling of time in a differential equation of movement of Thomson type oscillator (generator) it is offered to use a combination of the numerical method of finite differences and an asymptotic method of the slowl-changing amplitudes. The difference approximations of temporal derivatives are selected so that, first, to save conservatism and natural frequency of the linear circuit of self-oscillatory system in the discrete time. Secondly, coincidence of the difference shortened equation for the complex amplitude of self-oscillations in the discrete time with Eulers approximation of the shortened equation for amplitude of self-oscillations in analog system prototype is required. It is shown that realization of such approach allows to create discrete mapping of the van der Pol oscillator and a number of mappings of Thomson type oscillators. The adequacy of discrete models to analog prototypes is confirmed with also numerical experiment.

scholarly journals SYNCHRONIZATION OF FRACTIONAL VAN-DER-POL OSCILLATOR

2017 ◽  
Vol 18 (3.1) ◽  
pp. 116-122
Author(s):  
V.V. Zaitsev ◽  
A.V. Karlov ◽  
I.V. Stulov
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Fractional Order ◽  
Harmonic Signal ◽  
Equation Of Motion ◽  
Van Der Pol Oscillator ◽  
Feedback Circuit ◽  
Van Der Pol ◽  
Quasiharmonic Approximation ◽  
Synchronization Band

A model of self-oscillating system with a differential equation of motion of fractional order under the action of external harmonic signal is proposed. Solutions of equation of motion which correspond to the regime of steady-state synchronized oscillations and the regime of beats near the synchronization band are obtained in the quasiharmonic approximation. The amplitude frequency and phase-frequency characteristics of synchronization of fractional Van-der-Pol oscillator are analyzed. An analogy between the generator with a fractional feedback circuit and the generator with delayed feedback is established.

Periodic and Quasi-periodic Responses of Van der Pol–Mathieu System Subject to Various Excitations

2016 ◽  
Vol 17 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Q. Fan ◽  
A. Y. T Leung ◽  
Y. Y. Lee
Keyword(s):  
Steady State ◽  
Numerical Method ◽  
Parametric Study ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Perturbation Technique ◽  
Original System ◽  
External Excitation ◽  
Van Der Pol ◽  
Parametric And External Excitations

AbstractThis paper addresses the steady-state periodic and quasi-periodic responses of van der Pol–Mathieu system subject to three excitations (i.e., self, parametric and external excitations). Method of multiple scales and double perturbation technique are employed to study the original system. The cases of van der Pol–Mathieu oscillator with and without external excitation are considered, and periodic and quasi-periodic solutions are obtained and discussed. In the parametric study, the effects of various parameters and self, parametric and external excitations on the system behaviors are studied. Results from method of multiple scales well agree with those from numerical method.

Two-frequency parametric excitation and combination resonance of a nanocomposite laminated piezoelectric trapezoidal plate in subsonic airflow

2020 ◽  
pp. 095440622097543
Author(s):  
Marziye Noroozi ◽  
Firooz Bakhtiari-Nejad ◽  
Morteza Dardel
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Geometrical Nonlinearity ◽  
Time History ◽  
Dynamic Responses ◽  
Linear Potential ◽  
Response Curves ◽  
Aerodynamic Pressure ◽  
Averaged Equations ◽  
The Difference

In this study, an analytical approach is presented to analyze the bifurcations and nonlinear dynamics of a cantilevered piezoelectric nanocomposite trapezoidal actuator subjected to two-frequency parametric excitations in the presence of subsonic airflow. The assumption of uniformly distributed single-walled carbon nanotubes along the thickness is taken into the consideration. The governing equations are built by the von-Karman nonlinear strain-displacement relations to consider the geometrical nonlinearity and the linear potential flow theory. The present study focuses on a specific resonance case deals with the occurrence of simultaneous resonances in the principal parametric resonance of the first mode and combination of the parametric resonance of the difference type involving two modes. The multiple scales method is employed to obtain the four nonlinear averaged equations which are solved by using the Runge-Kutta method. Moreover, the frequency-response curves, bifurcation diagrams, time history responses, and phase portrait are obtained to find the nonlinear dynamic responses of the plate. The effects of the amplitude of piezoelectric excitation, piezoelectric detuning parameter, and aerodynamic pressure are also studied. The results indicate that, the chaotic, quasi-periodic and periodic motions of the plate exist under certain conditions and the variation of controlling parameters can change the form of motions of the nanocomposite piezoelectric trapezoidal thin plate.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol
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