AN OSCILLATOR WITH CUBIC AND PIECEWISE-LINEAR SPRINGS

1991 ◽  
Vol 01 (02) ◽  
pp. 349-356 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Parameter Space ◽  
Periodic Motion ◽  
Chaotic Motion ◽  
Piecewise Linear ◽  
Cubic Nonlinearity ◽  
Nonlinear Spring ◽  
Resonant Orbits ◽  
Offset Distance ◽  
Linear Spring ◽  
Parameter Values

A system consisting of a mass attached to a nonlinear spring with negative linear stiffness and cubic nonlinearity, (i.e., a spring obeying Duffing's equation with negative linear stiffness) and a linear spring at a certain offset distance is studied. Melnikov's method is applied to determine the existence of homoclinic points for the Poincare map, and preserved resonant orbits and boundaries for these are given in the parameter space. This system is then compared to the system consisting of only the nonlinear spring with regard to the existence of parameter regimes where chaotic motion is possible. It is shown that if the linear spring is of appropriate stiffness the chaotic motion for a given set of parameter values occurring for the system consisting of only the nonlinear spring is replaced by periodic motion and the mechanism of this phenomenon is explained.

Experimental Study of a Symmetrical Piecewise Base-Excited Oscillator

1998 ◽  
Vol 65 (3) ◽  
pp. 657-663 ◽  
Author(s):  
M. Wiercigroch ◽  
V. W. T. Sin
Keyword(s):  
Experimental Study ◽  
Periodic Motion ◽  
Chaotic Motion ◽  
Piecewise Linear ◽  
Calibration Procedure ◽  
Experimental Setup ◽  
Stiffness Ratio ◽  
Bifurcation Diagrams ◽  
Parameter Values ◽  
Excited Oscillator

This paper presents an experimental study on a base-excited piecewise linear oscillator with symmetrical flexible constrains of high stiffness ratio (above 20). The details of the adopted design of the oscillator, the experimental setup, and calibration procedure are briefly discussed. The regions of chaotic motion predicted theoretically were confirmed by the experimental results arranged into bifurcation diagrams. Clearance, stiffness ratio, amplitude, and frequency of the external force were used as branching parameters. The discussion of the system dynamics is based on bifurcation diagrams and Lissajous curves. The investigated system tends to be periodic for large clearances and chaotic for small ones. This picture is reversed for the amplitude of the forcing changes, where periodic motion occurred for small values and chaos dominated for larger forcing. The same behavior is observed for increasing frequency ratio where, for values below the natural frequency, the most interesting dynamics occurs. For the investigated parameter values, the stiffness ratio variation produces only periodic motion.

Dynamics analysis of a gyrostat system with intermittent forcing

2021 ◽  
pp. 095965182110590
Author(s):  
Jianbin He ◽  
Jianping Cai
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Simulations ◽  
Periodic Motion ◽  
Chaotic Motion ◽  
Dynamics Analysis ◽  
Bifurcation Diagrams ◽  
Stable Point ◽  
Dynamical Characteristics ◽  
Main Work ◽  
Selection Of

The dynamical characteristics of a gyrostat system with intermittent forcing are investigated, the main work and contributions are given as follows: (1) The gyrostat system with an intermittent forcing is studied, and its dynamical characteristics are investigated by the corresponding Lyapunov exponent spectrums and bifurcation diagrams with respect to the amplitude of intermittent forcing. The modified gyrostat system exists chaotic motion when the amplitude of intermittent forcing belongs to a certain interval, and it can be at a state of stable point or periodic motion by the design of amplitude. (2) The gyrostat system with multiple intermittent forcings is also investigated through the combination of Lyapunov exponent spectrums and bifurcation diagrams, and it behaves periodic motion or chaotic motion when the amplitude or forcing width is different. (3) By the selection of parameters in intermittent forcings, the modified gyrostat system is at a state of stable point, periodic motion or chaotic motion. Numerical simulations verify the feasibility and effectiveness of the modified gyrostat system.

DYNAMICS AT INFINITY OF A CUBIC CHUA'S SYSTEM

2011 ◽  
Vol 21 (01) ◽  
pp. 333-340 ◽  
Author(s):  
MARCELO MESSIAS
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Numerical Study ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Cubic Nonlinearity ◽  
Poincaré Compactification ◽  
Global Study ◽  
Dimensional Dynamical System ◽  
Parameter Values

We use the Poincaré compactification for a polynomial vector field in ℝ3 to study the dynamics near and at infinity of the classical Chua's system with a cubic nonlinearity. We give a complete description of the phase portrait of this system at infinity, which is identified with the sphere 𝕊2 in ℝ3 after compactification, and perform a numerical study on how the solutions reach infinity, depending on the parameter values. With this global study we intend to give a contribution in the understanding of this well known and extensively studied complex three-dimensional dynamical system.

Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise Linear, Impacting Oscillator

2006 ◽  
Vol 129 (3) ◽  
pp. 276-284 ◽  
Author(s):  
Albert C. J. Luo ◽  
Lidi Chen
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Motion ◽  
Local Stability ◽  
Piecewise Linear ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Local Stability Analysis ◽  
Mapping Structures

The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings relative to the discontinuous boundaries of this piecewise system are introduced. Based on such mappings, the corresponding grazing conditions are obtained. The mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

scholarly journals Nonlinear Dynamic Analysis of Rotor Rub-Impact System

2019 ◽  
Vol 2019 ◽  
pp. 1-20
Author(s):  
Youfeng Zhu ◽  
Zibo Wang ◽  
Qiang Wang ◽  
Xinhua Liu ◽  
Hongyu Zang ◽  
...  
Keyword(s):  
Periodic Motion ◽  
Chaotic Motion ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Time History ◽  
Rotor System ◽  
Quasiperiodic Motion ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.

scholarly journals Study on bifurcation characteristics of multi-clearance bending torsional coupling gear transmission based on EHL

2020 ◽  
Vol 12 (6) ◽  
pp. 168781402093750
Author(s):  
Hao Dong ◽  
Jianwen Zhang ◽  
Libang Wang
Keyword(s):  
Friction Coefficient ◽  
Periodic Motion ◽  
Chaotic Motion ◽  
Hydrodynamic Lubrication ◽  
Surface Friction ◽  
Tooth Surface ◽  
Time Varying ◽  
Mesh Stiffness ◽  
Vibration Model ◽  
Non Linear

In order to study the influence of tooth surface friction on the non-linear bifurcation characteristics of multi-clearance gear drive system, a 6 degree-of-freedom bending torsional coupled vibration model was established. The time-varying mesh stiffness, backlash, support clearance and damping were considered comprehensively in this non-linear vibration model. Loaded tooth contact analysis was used to calculate the time-varying mesh stiffness. Based on the elasto-hydrodynamic lubrication, the time-varying friction coefficient was calculated. Runge–Kutta numerical method was used to solve the dimensionless dynamic differential equation. Using phase diagram, Poincaré diagram, time history diagram, and spectrum diagram, the influence of tooth surface friction on bifurcation characteristics was studied. The results show that the system undergoes a change from 1-periodic motion, multi-periodic motion, to chaotic motion through bifurcation and catastrophe when the speed changes independently. When the friction coefficient of tooth surface changes from 0, 0.05 to 0.09, the chaotic motion of the system is suppressed. Similarly, with the increase in tooth friction, the chaotic motion characteristics are suppressed. Tooth surface friction is the main factor affecting chaotic motion. With the increase in friction coefficient of tooth surface, the chaos characteristic does not change obviously and the vibration amplitude decreases slightly.

scholarly journals Fundamental study on countermeasures against subharmonic vibration of order 1/2 in automatic transmissions for cars

2018 ◽  
Vol 211 ◽  
pp. 13004
Author(s):  
Fumiya Takino ◽  
Takahiro Ryu ◽  
Takashi Nakae ◽  
Kenichiro Matsuzaki ◽  
Risa Ueno
Keyword(s):  
Piecewise Linear ◽  
Vibration Reduction ◽  
System Model ◽  
Torsional Vibrations ◽  
Fundamental Study ◽  
Automatic Transmissions ◽  
Linear Spring ◽  
Low Stiffness ◽  
Subharmonic Vibration ◽  
The One

In automatic transmissions for cars, a damper is installed in the lock-up clutch to absorb torsional vibrations caused by combustion in the engine. Although a damper with low stiffness reduces the torsional vibration, low-stiffness springs are difficult to use because of space limitations. To address this problem, dampers have been designed using a piecewise-linear spring having three different stages of stiffness. However, a nonlinear subharmonic vibration of order 1/2 occurs because of the nonlinearity of the piecewise-linear spring in the damper. In this study, we experimentally and analytically examined a countermeasure against the subharmonic vibration by increasing the stages of the piecewise-linear spring using the one-degree-of-freedom system model. We found that the gap between the switching points of the piecewise-linear spring was the key to vibration reduction. The experimental results agreed with results of the numerical analyses.

GENETIC SELECTION AND NEURAL MODELING OF PIECEWISE-LINEAR CLASSIFIERS

1996 ◽  
Vol 10 (05) ◽  
pp. 587-612 ◽  
Author(s):  
JACK SKLANSKY ◽  
MARK VRIESENGA
Keyword(s):  
Piecewise Linear ◽  
Genetic Selection ◽  
Aerial Images ◽  
Neural Modeling ◽  
Linear Classifiers ◽  
Mathematical Structures ◽  
Straight Lines ◽  
Parameter Values ◽  
Adaptive Pattern ◽  
Abnormal Tissue

Piecewise-linear mathematical structures form a convenient and important framework for implementing trainable and adaptive pattern classifiers. Neural networks and genetic algorithms offer additional approaches with important benefits for the design of such classifiers. In this paper we show how neural modeling and genetic selection can be applied to piecewise-linear structures to optimize both the topology and the parameter values of the network forming the classifier. Such a classifier will tend to have a low error rate and high robustness. We describe applications of these techniques to an adaptive detector of abnormal tissue in mammograms and a detector of straight lines and edges in noisy aerial images.

FLUCTUATIONS IN THE STATE OF CHAOS-CHAOS INTERMITTENCY OF CHUA’S CIRCUIT

1995 ◽  
Vol 05 (01) ◽  
pp. 271-273
Author(s):  
M. KOCH ◽  
R. TETZLAFF ◽  
D. WOLF
Keyword(s):  
Power Spectrum ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
The State ◽  
Circuit Parameter ◽  
Chua’S Circuit ◽  
External Excitation ◽  
Chua's Circuit ◽  
Parameter Values

We studied the power spectrum of the normalized voltage across the capacitor parallel to a piecewise-linear resistor of Chua’s circuit in the “chaos-chaos intermittency” state [Anishchenko et al., 1992]. The investigations included various initial conditions and circuit parameter values without and with external excitation. In all cases we found spectra showing a 1/ω2-decay over more than four decades.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

2015 ◽  
Vol 25 (03) ◽  
pp. 1530006 ◽  
Author(s):  
Anastasiia Panchuk ◽  
Iryna Sushko ◽  
Viktor Avrutin
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Tent Map ◽  
Linear Map ◽  
Piecewise Linear Map ◽  
Replacement Technique ◽  
Bifurcation Structures

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

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