Threshold of Multiple Stick-Slip Chaos for an Archetypal Self-Excited SD Oscillator Driven by Moving Belt Friction

2017 ◽  
Vol 27 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Z. X. Li ◽  
Q. J. Cao ◽  
A. Léger
Keyword(s):  
Multiple Solutions ◽  
Coulomb Friction ◽  
Homoclinic Orbits ◽  
The Self ◽  
External Excitation ◽  
Stick Slip ◽  
Sd Oscillator ◽  
Excited System ◽  
Analytical Expressions ◽  
Natural Characteristics

In this paper, we investigate the multiple stick-slip chaotic motion of an archetypal self-excited smooth and discontinuous (SD) oscillator driven by moving belt friction, which is constructed with the SD oscillator and the classical moving belt. The friction force between the mass and the belt is modeled as a Coulomb friction for this system. The energy introduction or dissipation during the slip and stick modes in the system is analyzed. The analytical expressions of homoclinic orbits of the unperturbed SD oscillator are derived by using a special coordinate transformation without any pronominal truncation to retain the natural characteristics, which allows us to utilize the Melnikov’s method to obtain the chaotic thresholds of the self-excited SD oscillator in the presence of the damping and external excitation. Numerical simulations are carried out to demonstrate the multiple stick-slip dynamics of the system, which show the efficiency of the prediction for stick-slip chaos of the perturbed self-excited system. The results presented herein this paper demonstrate the complicated dynamics of stick-slip periodic solutions, multiple stick-slip chaotic solutions and also coexistence of multiple solutions for the perturbed self-excited SD oscillator.

Analytical Approximations for Dry Friction-induced Stick-Slip and Pure-Slip Vibration Amplitudes of a Self-Excited SD Oscillator

2021 ◽  
Author(s):  
Junfeng Yan ◽  
Zehao Huang
Keyword(s):  
Perturbation Method ◽  
Friction Force ◽  
Dry Friction ◽  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Stick Slip ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Sd Oscillator ◽  
Analytical Expressions

Abstract An analytical and numerical investigation into pure-slip and stick-slip oscillations induced by dry friction between a rigid mass linked by an inclined spring, modeled by the archetypal self-excited smooth and discontinuous (SD) oscillator, and the classical moving rigid belt, is presented. The friction force between surface contacts is modeled in the sense of Stribeck effect to formulate the friction model that the friction force firstly decreases and then increases with increasing relative sliding speed. Some perturbation methods are considered into this system for establishing the approximate analytical expressions of the occurring conditions, vibration amplitudes, and base frequencies of dry friction-induced stick-slip and pure-slip oscillations. For pure-slip oscillations, two different approaches are applied to analyze this self-excited SD oscillator. One of them is the homotopy perturbation method by constructing the nonlinear amplitude and frequency. Based on the multiple-scales homotopy perturbation method, a nonlinear equation for amplitude of the analytical approximate solution is constructed, which containing all parameters of problem. For stick-slip oscillations, the analytical approximations for amplitude and frequency are obtained by perturbation methods for finite time intervals of the stick phase, which is linked to the subsequent slip phase under the conditions of continuity and periodicity. The accuracy of analytical approximations is verified by the comparison between analytical approximations and numerical simulations. These analytical expressions are needed for gaining a deeper understanding of dry friction-induced pure-slip and stick-slip oscillations for the friction system with geometric nonlinearity.

Stick-slip vibrations of a self-excited SD oscillator with Coulomb friction

2020 ◽  
Vol 102 (3) ◽  
pp. 1419-1435 ◽  
Author(s):  
Zhixin Li ◽  
Qingjie Cao ◽  
Zairan Nie
Keyword(s):  
Coulomb Friction ◽  
Stick Slip ◽  
Sd Oscillator

Asperity Spring Friction Model With Application to Belt-Drives

2003 ◽  
Author(s):  
Tamer M. Wasfy
Keyword(s):  
Finite Element ◽  
Coulomb Friction ◽  
Friction Model ◽  
Belt Drive ◽  
Finite Element Code ◽  
Time Step ◽  
Stick Slip ◽  
Normal Contact ◽  
Belt Drives ◽  
Coulomb Friction Model

An asperity spring friction model that uses a variable anchor point spring along with a velocity dependent force is presented. The model is incorporated in an explicit timeintegration finite element code. The friction model is used along with a penalty-based normal contact model to simulate the dynamic response of a two-pulley belt-drive system. It is shown that the present friction model accurately captures the stick-slip behavior between the belt and the pulleys using a much larger time-step than a pure velocity-dependent approximate Coulomb friction model.

Variational Formulation for Frictional Interface Dynamics

2012 ◽  
Author(s):  
Timothy Truster ◽  
Arif Masud ◽  
Lawrence A. Bergman
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Coulomb Friction ◽  
Bolted Joints ◽  
Lap Joint ◽  
Stick Slip ◽  
Element Simulation ◽  
Friction Models ◽  
Frictional Interface ◽  
Current Emphasis

The dynamic response of component bolted joints often plays a significant role in the overall behavior of a structural system. Accurate finite element simulation of these problems requires proper treatment of the interface conditions. We present a formulation carefully suited to these problems that incorporates discontinuous Galerkin (DG) treatment locally at the interface. The present work is an extension of our previous investigations of friction models within a finite element method for quasi-static problems. The current emphasis is on the treatment of the inertial term and ensuring that artificial resonance is not induced by the discrete interface. The weak imposition of continuity constraints allows the stick-slip behavior at the jointed surface to proceed smoothly, reducing the numerical instability compared to node-to-node contact techniques. As a model problem, we simulate the dynamic response of a lap joint subjected to an impulse axial force assuming Coulomb friction at the interface.

CHAOTIC ATTITUDE MOTION OF GYROSTAT SATELLITE VIA MELNIKOV METHOD

2001 ◽  
Vol 11 (05) ◽  
pp. 1233-1260 ◽  
Author(s):  
JINLU KUANG ◽  
SOONHIE TAN ◽  
KANDIAN ARICHANDRAN ◽  
A. Y. T. LEUNG
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Necessary Condition ◽  
Attitude Motion ◽  
Physical Parameters ◽  
Chaotic Oscillations ◽  
External Excitation ◽  
Integration Algorithm ◽  
Hamiltonian Equations

In this paper Deprit's variables are used to describe the Hamiltonian equations for attitude motions of a gyrostat satellite spinning about arbitrarily body-fixed axes. The Hamiltonian equations for the attitude motions of the gyrostat satellite in terms of the Deprit's variables and under small viscous damping and nonautonomous perturbations are suitable for the employment of the Melnikov's integral. The torque-free homoclinic orbits to the symmetric Kelvin gyrostat are derived by means of the elliptic function integral theory. With the help of residue theory of complex functions, the Melnikov's integral is utilized to analytically study the criterion for chaotic oscillations of the attitude motions of the symmetric Kelvin gyrostat under small, damping and periodic external disturbing torques. The Melnikov's integral yields an analytical criterion for the chaotic oscillations of the attitude motions in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the physical parameters. The dependence of the onset of homoclinic orbits on quantities such as body shapes, the initial conditions of the angular velocities or the two constants of motions of the torque-free gyrostat satellite is investigated in details. The dependence of the onset of chaos on quantities such as the amplitudes of the external excitation and the damping coefficients' matrix is discussed. The bifurcation curves based upon the Melnikov's integral are computed by using the combined parameters versus the frequency of the external excitation. The theoretical criterion agrees with the result of the numerical simulation of the gyrostat satellite by using the fourth-order Runge–Kutta integration algorithm. The numerical solutions show that the motions of the perturbed symmetric gyrostat satellite possess a lot of "random" characteristic associated with a nonperiodic solution.

scholarly journals Multiple Bifurcations of a Cylindrical Dynamical System

2016 ◽  
Vol 46 (1) ◽  
pp. 33-52
Author(s):  
Ning Han ◽  
Qingjie Cao
Keyword(s):  
Dynamical System ◽  
Limit Cycles ◽  
Homoclinic Orbits ◽  
Complex Dynamic ◽  
Pendulum System ◽  
Sd Oscillator ◽  
Bistable State ◽  
Different Types ◽  
Coupling Dynamics ◽  
Dynamic Bifurcation

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

Global convergence for two-pulse rest-to-rest learning for single-degree-of-freedom systems with stick-slip Coulomb friction

Robotica ◽  
2006 ◽  
Vol 25 (3) ◽  
pp. 307-313
Author(s):  
Brian J. Driessen ◽  
Nader Sadegh
Keyword(s):  
Global Convergence ◽  
Coefficient Of Friction ◽  
Coulomb Friction ◽  
Sliding Friction ◽  
Input Pulse ◽  
Iterative Learning ◽  
Degree Of Freedom ◽  
Stick Slip ◽  
Single Degree Of Freedom ◽  
Single Degree

SUMMARYIn this paper, we consider the problem of rest-to-rest maneu-ver learning, via iterative learning control (ILC), for single-degree-of-freedom systems with stick-slip Coulomb friction and input bounds. The static coefficient of friction is allowed to be as large as three times the kinetic coefficient of friction. The input is restricted to be a two-pulse one. The desired input's first pulse magnitude is required to be five times the largest possible kinetic (sliding) friction force. The theory therefore allows the stiction force to be as large as the desired second input pulse. Under these conditions, we prove global convergence of a simple iterative learning controller. To the best of our knowledge, such a global-convergence proof has not been presented previously in the literature for the rest-to-rest problem with stick-slip Coulomb friction.

Coulomb Friction Limit Cycles in Elastic Positioning Systems

1999 ◽  
Vol 121 (2) ◽  
pp. 298-301 ◽  
Author(s):  
A. Bonsignore ◽  
G. Ferretti ◽  
G. Magnani
Keyword(s):  
Coulomb Friction ◽  
Closed Loop ◽  
Tracking Error ◽  
Pole Placement ◽  
Stick Slip ◽  
Positioning Systems ◽  
Relay Control ◽  
Pure Coulomb ◽  
Pole Placement Controller ◽  
Motor Velocity

The state space control of a positioning system affected by torsional elasticity at the gearbox is considered, using a motor position transducer only. An output feedback, pole placement controller is used, with an additional integral action on the tracking error to cancel it at steady state. Both experiments and simulations point out that large oscillations may appear for some sets of closed-loop poles which yields, in contrast to stick-slip cycles, instantaneous motor velocity reversals. It is shown that such oscillations are induced by “pure” Coulomb friction. The period of the oscillations is predicted precisely following the Tsypkin’s relay control theory and also by the approximate describing function method. The latter also allows understanding of how oscillations depend on observer and feedback control design and on plant parameters; thus we are able to derive guidelines for the design of an oscillation free closed-loop system.

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