scholarly journals Bifurcation and Chaos Prediction in Nonlinear Gear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Anooshirvan Farshidianfar ◽  
Amin Saghafi
Keyword(s):  
Numerical Simulation ◽  
Phase Plane ◽  
Homoclinic Bifurcation ◽  
Threshold Values ◽  
Bifurcation And Chaos ◽  
System Parameters ◽  
Transition To Chaos ◽  
Time Histories ◽  
Fourier Spectra ◽  
Gear Systems

The homoclinic bifurcation and transition to chaos in gear systems are studied both analytically and numerically. Applying Melnikov analytical method, the threshold values for the occurrence of chaotic motion are obtained. The influence of system parameters on the character of vibration is studied. The numerical simulation of the system including bifurcation diagram, phase plane portraits, Fourier spectra, and time histories is considered to confirm the analytical predictions for the occurrence of homoclinic bifurcation and chaos in nonlinear gear systems.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Vibrations of beams with a breathing crack and large amplitude displacements

2015 ◽  
Vol 230 (1) ◽  
pp. 34-54 ◽  
Author(s):  
Gonçalo Neves Carneiro ◽  
Pedro Ribeiro
Keyword(s):  
Time Domain ◽  
Equations Of Motion ◽  
Shape Functions ◽  
Breathing Crack ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Time Histories ◽  
Linear Effects ◽  
Fourier Spectra ◽  
The Time Domain

The vibrations of beams with a breathing crack are investigated taking into account geometrical non-linear effects. The crack is modeled via a function that reduces the stiffness, as proposed by Christides and Barr (One-dimensional theory of cracked Bernoulli–Euler beams. Int J Mech Sci 1984). The bilinear behavior due to the crack closing and opening is considered. The equations of motion are obtained via a p-version finite element method, with shape functions recently proposed, which are adequate for problems with abrupt localised variations. To analyse the dynamics of cracked beams, the equations of motion are solved in the time domain, via Newmark's method, and the ensuing displacements, velocities and accelerations are examined. For that purpose, time histories, projections of trajectories on phase planes, and Fourier spectra are obtained. It is verified that the breathing crack introduce asymmetries in the response, and that velocities and accelerations can be more affected than displacements by the breathing crack.

Observation and Evolution of Chaos for an Autonomous System

1984 ◽  
Vol 51 (3) ◽  
pp. 664-673 ◽  
Author(s):  
E. H. Dowell
Keyword(s):  
Elastic Plate ◽  
Physical Interpretation ◽  
Autonomous System ◽  
Phase Plane ◽  
Power Spectra ◽  
Poincaré Maps ◽  
Flowing Fluid ◽  
Poincare Maps ◽  
Time Histories

Time histories, phase plane portraits, power spectra, and Poincare maps are used as descriptors to observe the evolution of chaos in an autonomous system. Although the motions of such a system can be quite complex, these descriptors prove helpful in detecting the essential structure of the motion. Here the principal interest is in phase plane portraits and Poincare maps, their methods of construction, and physical interpretation. The system chosen for study has been previously discussed in the literature, i.e., the flutter of a buckled elastic plate in a flowing fluid.

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

Quasi-Periodicity, Chaos and Coexistence in the Time Delay Controlled Two-Cell DC–DC Buck Converter

2014 ◽  
Vol 24 (10) ◽  
pp. 1450124 ◽  
Author(s):  
Karama Koubaâ ◽  
Moez Feki
Keyword(s):  
Numerical Simulation ◽  
Time Delay ◽  
Chaotic Attractor ◽  
Chaotic Behavior ◽  
Buck Converter ◽  
Design Parameters ◽  
Bifurcation And Chaos ◽  
Different Types ◽  
2D Torus ◽  
Discrete Map

In addition to border collision bifurcation, the time delay controlled two-cell DC/DC buck converter is shown to exhibit a chaotic behavior as well. The time delay controller adds new design parameters to the system and therefore the variation of a parameter may lead to different types of bifurcation. In this work, we present a thorough analysis of different scenarios leading to bifurcation and chaos. We show that the time delay controlled two-cell DC/DC buck converter may also exhibit a Neimark–Sacker bifurcation which for some parameter set may lead to a 2D torus that may then break yielding a chaotic behavior. Besides, the saturation of the controller can also lead to the coexistence of a stable focus and a chaotic attractor. The results are presented using numerical simulation of a discrete map of the two-cell DC/DC buck converter obtained by expressing successive crossings of Poincaré section in terms of each other.

scholarly journals Hopf Bifurcation and Chaos of a Delayed Finance System

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Xuebing Zhang ◽  
Honglan Zhu
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Center Manifold ◽  
Chaotic State ◽  
Bifurcation And Chaos ◽  
Finance System ◽  
Center Manifold Theorem ◽  
Normal Form Method ◽  
Simulation Results ◽  
The Stability

In this paper, a finance system with delay is considered. By analyzing the corresponding characteristic equations, the local stability of equilibrium is established. The existence of Hopf bifurcations at the equilibrium is also discussed. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, numerical simulation results are presented to validate the theoretical analysis. Numerical simulation results show that delay can lead a stable system into a chaotic state.

Multifractality at Transition to Chaos Complex-Temperature Singularities

1997 ◽  
Vol 11 (21n22) ◽  
pp. 929-937
Author(s):  
A. Bershadskii
Keyword(s):  
Numerical Simulation ◽  
High Temperature ◽  
Finite Temperature ◽  
Multifractal Spectrum ◽  
Finite Radius ◽  
Transition To Chaos ◽  
Iterative System ◽  
Multifractal Spectra ◽  
Complex Temperature ◽  
Generalized Dimensions

It is shown, that multifractal complex-temperature singularities can play a significant role in the critical strange sets multifractality. These singularities lead to a finite radius of convergence of the real high-temperature expansions and, therefore, to necessity to use a finite-temperature expansions (an analytic continuation). It is shown, using analytic results on multifractality of strange attractors of the baker map and results of numerical computations of the multifractal spectra on all critical points of phase transitions from period-η-tupling to chaos in 1D iterative system (Chinese Phys. Lett.3, 285 (1986) and J. Phys.A25, 589 (1992)) as well as results of a recent numerical simulation of a quantum system with multifractal spectrum (J. Phys.A28, 2717 (1995)), that the finite-temperature expansions give good approximation for the generalized dimensions Dq in a representative interval of q.

Numerical Simulation and Experimental Validation on the Surf-Riding and Broaching of a Fishing Vessel

2016 ◽  
Author(s):  
Liwei Yu ◽  
Ning Ma ◽  
Sheming Fan ◽  
Peiyuan Feng ◽  
Xiechong Gu
Keyword(s):  
Numerical Simulation ◽  
Numerical Simulations ◽  
Model Test ◽  
Periodic Motion ◽  
Following Seas ◽  
Weakly Nonlinear ◽  
Model Experiments ◽  
Time Histories ◽  
Wave Heights ◽  
Surf Riding

Model experiments and numerical simulations on the surf-riding and broaching in following seas of a 42.5m long purse seiner are conducted. Firstly, the free running model experiments with various ship speeds and wave heights are performed in the towing tank to reproduce the phenomena of surf-riding and broaching. Then, the 6-DOF weakly nonlinear unified model is applied to simulate the motions of the purse seiner with the same cases as the model experiments. Through the comparison between results of model test and numerical simulation, the occurrence conditions of periodic motion, surf-riding and broaching are roughly determined. Finally, it is found that although it is difficult for the numerical simulations to get the same time histories as model tests, the modes of motion (periodic motion, surf-riding or broaching) obtained from the numerical simulations agree well qualitatively and quantitatively in part with the model test results.

HOMOCLINIC BIFURCATION AND CHAOS IN COUPLED SIMPLE PENDULUM AND HARMONIC OSCILLATOR UNDER BOUNDED NOISE EXCITATION

2005 ◽  
Vol 15 (01) ◽  
pp. 233-243 ◽  
Author(s):  
W. Q. ZHU ◽  
Z. H. LIU
Keyword(s):  
Harmonic Oscillator ◽  
Random Motion ◽  
Homoclinic Bifurcation ◽  
Mean Square ◽  
Bounded Noise ◽  
Bifurcation And Chaos ◽  
Simple Pendulum ◽  
Onset Of Chaos ◽  
Weakly Coupled ◽  
Threshold Amplitude

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

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