Rigid block dynamics confined between side-walls

1994 ◽  
Vol 347 (1683) ◽  
pp. 411-419 ◽  
Keyword(s):  
Nonlinear Equation ◽  
Symmetry Breaking ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Nuclear Industry ◽  
Rigid Block ◽  
Fuel Rods ◽  
Side Walls ◽  
The Stability ◽  
Period Doubling Bifurcations

A rigid block lies on a horizontal surface between two symmetrically placed- walls. Under sinusoidal horizontal excitation, the block can impact both the floor and both side-walls. Periodic orbits are obtained analytically by linking piecewise linear solutions which have as an unknown the time impact difference between the floor and the wall. One nonlinear equation is then obtained in this unknown and solved numerically. The stability of these solutions is also examined. Symmetry-breaking and period-doubling bifurcations are observed. These latter responses can occur with larger than expected impact velocities. This problem has application in the nuclear industry where fuel rods can impact the sides of their containers. The design implication of these results is considered.

On the motion of a rigid block, tethered at one corner, under harmonic forcing

1992 ◽  
Vol 439 (1905) ◽  
pp. 35-45 ◽  
Keyword(s):  
Period Doubling ◽  
Rigid Block ◽  
Harmonic Forcing ◽  
Wide Range ◽  
Side Walls ◽  
Multiple Impact ◽  
Digital Simulations ◽  
First Time ◽  
Subharmonic Orbits ◽  
Period Doubling Bifurcations

A rigid block, tethered at one corner, is subjected to harmonic forcing. The motion is shown to be equivalent to that of the inverted pendulum impacting one of a pair of asymmetrically placed side-walls. The dynamics of the problem contain subharmonic responses, multiple solutions, period-doubling bifurcations, etc. Stability boundaries are given for a wide range of parameters and orbits are shown to be possible for a large range of forcing amplitudes. Some orbits are possible at forcing amplitudes larger than those in the untethered case. A period- and impact-doubling sequence is shown explicitly for the first time, using digital simulations. Evidence is offered for the existence of more than one type of multiple-impact solution. Large amplitude subharmonic orbits are found.

Data Processing in Bifurcation Analysis of Maps with Time-Delays in the Frequency Domain

2014 ◽  
Vol 685 ◽  
pp. 634-637
Author(s):  
Li Zeng ◽  
Jun Wei Wang
Keyword(s):  
Data Processing ◽  
Frequency Domain ◽  
Time Delays ◽  
Period Doubling ◽  
Feedback Systems ◽  
Bifurcation Solution ◽  
Nonlinear Maps ◽  
The Stability ◽  
Frequency Domain Approach ◽  
Period Doubling Bifurcations

A unified frequency-domain approach to analyze the NS (Neimark-Sacker) bifurcations and the period-doubling bifurcations of nonlinear maps with time-delays in the linear feed-forward term is presented. The technique relies on the HBA (harmonic balance approximation, a very important method in data processing ) and feedback systems theory. The expressions of the bifurcation solution and the stability are derived.

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Transition to chaos in a two-sided collapsible channel flow

2021 ◽  
Vol 926 ◽  
Author(s):  
Qiuxiang Huang ◽  
Fang-Bao Tian ◽  
John Young ◽  
Joseph C.S. Lai
Keyword(s):  
Symmetry Breaking ◽  
Channel Flow ◽  
Period Doubling ◽  
External Pressure ◽  
Immersed Boundary ◽  
Limit Cycle Oscillation ◽  
External Pressures ◽  
Wide Range ◽  
Cycle Oscillation ◽  
The Stability

The nonlinear dynamics of a two-sided collapsible channel flow is investigated by using an immersed boundary-lattice Boltzmann method. The stability of the hydrodynamic flow and collapsible channel walls is examined over a wide range of Reynolds numbers $Re$ , structure-to-fluid mass ratios $M$ and external pressures $P_e$ . Based on extensive simulations, we first characterise the chaotic behaviours of the collapsible channel flow and explore possible routes to chaos. We then explore the physical mechanisms responsible for the onset of self-excited oscillations. Nonlinear and rich dynamic behaviours of the collapsible system are discovered. Specifically, the system experiences a supercritical Hopf bifurcation leading to a period-1 limit cycle oscillation. The existence of chaotic behaviours of the collapsible channel walls is confirmed by a positive dominant Lyapunov exponent and a chaotic attractor in the velocity-displacement phase portrait of the mid-point of the collapsible channel wall. Chaos in the system can be reached via period-doubling and quasi-periodic bifurcations. It is also found that symmetry breaking is not a prerequisite for the onset of self-excited oscillations. However, symmetry breaking induced by mass ratio and external pressure may lead to a chaotic state. Unbalanced transmural pressure, wall inertia and shear layer instabilities in the vorticity waves contribute to the onset of self-excited oscillations of the collapsible system. The period-doubling, quasi-periodic and chaotic oscillations are closely associated with vortex pairing and merging of adjacent vortices, and interactions between the vortices on the upper and lower walls downstream of the throat.

scholarly journals Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Eric A. Butcher ◽  
Oleg A. Bobrenkov ◽  
Ed Bueler ◽  
Praveen Nindujarla
Keyword(s):  
Cutting Force ◽  
Extreme Points ◽  
Period Doubling ◽  
Milling Process ◽  
Depth Of Cut ◽  
Approximation Technique ◽  
The Stability ◽  
Delay Differential ◽  
High Degree ◽  
Period Doubling Bifurcations

In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems such as milling are modeled by delay-differential equations with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up- and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found, and an in-depth investigation of the optimal stable immersion levels for down-milling in the vicinity of where the average cutting force changes sign is presented.

Milling Model With Variable Time Delay

2004 ◽  
Author(s):  
X.-H. Long ◽  
B. Balachandran
Keyword(s):  
Time Delay ◽  
Process Model ◽  
Period Doubling ◽  
Milling Process ◽  
Variable Time Delay ◽  
Variable Time ◽  
Stability Of Periodic Orbits ◽  
Loss Of Contact ◽  
The Stability ◽  
Period Doubling Bifurcations

Taking into account the effect of feed rate, a milling-process model with a variable time delay is presented in this article. Loss-of-contact effects are also considered. The development of this formulation is described and the efforts undertaken to examine the stability of periodic orbits of this system are discussed. A semi-discretization treatment is used for the stability analysis, and this analysis provides evidence for period-doubling bifurcations and secondary Hopf bifurcations. Good agreement is found between the numerical results obtained from this work and experimental results published in the literature.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

Chaos from a hysteresis and switched circuit

1995 ◽  
Vol 353 (1701) ◽  
pp. 47-57 ◽  
Keyword(s):  
Symmetry Breaking ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Analytic Formula ◽  
Period Doubling Bifurcation ◽  
Sufficient Condition ◽  
Laboratory Measurements ◽  
One Dimensional ◽  
Bifurcation Phenomena ◽  
Theoretical Results

This paper considers a simple piecewise linear hysteresis circuit. We define one-dimensional return map and derive its analytic formula. It enables us to give a sufficient condition for chaos generation and to analyse bifurcation phenomena rigorously. Especially, we have discovered period-doubling bifurcation with symmetry breaking. Some of theoretical results are verified by laboratory measurements.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

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