Dynamics and Stability of Partial Immersion Milling Operations

1999 ◽  
Author(s):  
M. X. Zhao ◽  
B. Balachandran ◽  
M. A. Davies ◽  
J. R. Pratt
Keyword(s):  
Hopf Bifurcation ◽  
Time Variation ◽  
Periodic Motion ◽  
Contact Time ◽  
Period Doubling ◽  
Experimental Investigations ◽  
Partial Immersion ◽  
Milling Operations ◽  
Wide Range ◽  
The Stability

Abstract In this paper, numerical and experimental investigations conducted into the dynamics and stability of partial immersion milling operations are presented. A mechanics based model is used for simulations of a wide range of milling operations and instabilities that arise due to regeneration and/or impact effects are studied. Poincaré sections are used to assess the stability of motions. The studies reveal that apart from Hopf bifurcation of a periodic motion, a period-doubling bifurcation of a periodic motion may also lead to chatter in partial immersion milling operations. Issues such as tooth contact time variation and structure of stability charts are also discussed.

Antimonotonicity, Bifurcation and Multistability in the Vallis Model for El Niño

2019 ◽  
Vol 29 (03) ◽  
pp. 1950032 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Zhouchao Wei ◽  
Durairaj Premraj ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
El Niño ◽  
El Nino ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
The Stability ◽  
First Time ◽  
Parameter Condition

In this paper, the well-known Vallis model for El Niño is analyzed for the parameter condition [Formula: see text]. The conditions for the stability of the equilibrium points are derived. The condition for Hopf bifurcation occurring in the system for [Formula: see text] and [Formula: see text] are investigated. The multistability feature of the Vallis model when [Formula: see text] is explained with forward and backward continuation bifurcation plots and with the coexisting attractors. The creation of period doubling followed by their annihilation via inverse period-doubling bifurcation known as antimonotonicity occurrence in the Vallis model for [Formula: see text] is presented for the first time in the literature.

ON THE NUMERICAL DETECTION OF CODIMENSION-3 BIFURCATIONS OF PERIODIC SOLUTIONS (WITH APPLICATION TO OPTICAL BISTABILITY)

1994 ◽  
Vol 04 (06) ◽  
pp. 1425-1446
Author(s):  
KLAUS-GEORG NOLTE ◽  
IVAN L’HEUREUX
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Optical Bistability ◽  
Continuation Method ◽  
Period Doubling ◽  
Bistable System ◽  
Dimensional System ◽  
Systems Of Equations ◽  
The Stability ◽  
Stationary Behavior

Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.

scholarly journals Nonlinear Vibration of a Continuum Rotor with Transverse Electromagnetic and Bearing Excitations

2012 ◽  
Vol 19 (6) ◽  
pp. 1297-1314 ◽  
Author(s):  
Haiyang Luo ◽  
Yuefang Wang
Keyword(s):  
Nonlinear Vibration ◽  
Periodic Motion ◽  
Primary Resonance ◽  
Period Doubling ◽  
Time History ◽  
Oil Film ◽  
Rotor Bearing ◽  
Transverse Electromagnetic ◽  
The Stability ◽  
Bearing System

The nonlinear vibration of a rotor excited by transverse electromagnetic and oil-film forces is presented in this paper. The rotor-bearing system is modeled as a continuum beam which is loaded by a distributed electromagnetic load and is supported by two oil-film bearings. The governing equation of motion is derived and discretized as a group of ordinary differential equations using the Galerkin's method. The stability of the equilibrium of the rotor is analyzed with the Routh-Hurwitz criterion and the occurrence of the Andronov-Hopf bifurcation is pointed out. The approximate solution of periodic motion is obtained using the averaging method. The stability of steady response is analyzed and the amplitude-frequency curve of primary resonance is illustrated. The Runge-Kutta method is adopted to numerically solve transient response of the rotor-bearing system. Comparisons are made to present influences of electromagnetic load, oil-film force and both of them on the nonlinear vibration response. Bifurcation diagrams of the transverse motion versus rotation speed, electromagnetic parameter and bearing parameters are provided to show periodic motion, quasi-periodic motion and period-doubling bifurcations. Diagrams of time history, shaft orbit, the Poincaré section and fast Fourier transformation of the transverse vibration are presented for further understanding of the rotor response.

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 Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Huayong Zhang ◽  
Ju Kang ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Shengnan Ma ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Food Web ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Food Web Structure ◽  
Center Manifold Theorem ◽  
Dynamical Complexity ◽  
Period 2 ◽  
The Stability

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

Bifurcation Analysis of a Belousov–Zhabotinsky Reaction Model

2015 ◽  
Vol 25 (06) ◽  
pp. 1550093 ◽  
Author(s):  
Xiaoli Wang ◽  
Yu Chang ◽  
Dashun Xu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Reaction Model ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
The Stability ◽  
Belousov Zhabotinsky Reaction

We investigate the bifurcation phenomena in a Belousov–Zhabotinsky reaction model by applying Hopf bifurcation theory in frequency domain and harmonic balance method. The high accurate predictions, i.e. fourth-order harmonic balance approximation, on frequencies, amplitudes, and approximation expressions for periodic solutions emerging from Hopf bifurcation are provided. We also detect the stability and location of these periodic solutions. Numerical simulations not only confirm the theoretical analysis results but also illustrate some complex oscillations such as a cascade of period-doubling bifurcation, quasi-periodic solution, and period-doubling route to chaos. All these results improve the understanding of the dynamics of the model.

scholarly journals Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
Yajuan Yu ◽  
Han Bao ◽  
Min Shi ◽  
Bocheng Bao ◽  
Yangquan Chen ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Hysteresis Loop ◽  
Fractional Order ◽  
Period Doubling ◽  
Order System ◽  
Periodic Signal ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Attraction Basins ◽  
The Stability

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

Countermeasure Against Regenerative Chatter in End Milling Operations With Vibration Absorbers

2010 ◽  
Author(s):  
Y. Nakano ◽  
H. Takahara
Keyword(s):  
End Milling ◽  
Spindle Speed ◽  
The Self ◽  
Depth Of Cut ◽  
Vibration Absorbers ◽  
Regenerative Chatter ◽  
Stability Lobe ◽  
Milling Operations ◽  
Wide Range ◽  
The Stability

Chatter can result in the poor machined surface, tool wear and reduced product quality. Chatter is classified into the forced vibration and the self-excited vibration in perspective of the generation mechanism. It often happens that the self-excited chatter becomes problem practically because this causes heavy vibration. Regenerative chatter due to regenerative effect is one of the self-excited chatter and generated in the most cutting operations. Therefore, it is very important to quench or avoid regenerative chatter (hereafter, simply called chatter). It is well known that chatter can be avoided by selecting the optimal cutting conditions which are determined by using the stability lobe of chatter. The stability lobe of chatter represents the boundary between stable and unstable cuts as a function of spindle speed and depth of cut. However, it is difficult to predict the stability lobe of chatter perfectly because the prediction accuracy of it depends on the tool geometry, the vibration characteristics of the tool system and the machine tool and the material behavior of the workpiece. In contrast, it is made clear that the stability lobe of chatter has been elevated in the wide range of spindle speed by the vibration absorber in the turning operations. However, it should be noted that none of the previous work has actually applied the vibration absorbers to the rotating tool system in the machining center and examined the effect of the vibration absorbers on chatter in the end milling operations to the best of authors’ knowledge. In this paper, the effect of the vibration absorbers on regenerative chatter generated in the end milling operations is qualitatively evaluated by the stability analysis and the cutting test. It is made clear the relationship between the suppression effect of the vibration absorbers and the tuning parameters of them. It is shown that the greater improvement in the critical axial depth of cut is observed in the wide range of spindle speed by the properly tuned vibration absorbers.

scholarly journals Another New Chaotic System: Bifurcation and Chaos Control

2020 ◽  
Vol 30 (11) ◽  
pp. 2050161
Author(s):  
Arnob Ray ◽  
Dibakar Ghosh
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Period Doubling ◽  
Slow Manifold ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Dimensional Parameter ◽  
Lyapunov Coefficient ◽  
The Stability

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

Periodic-Doubling and Hopf Bifurcations of a Two-Degree-of-Freedom System with Elastic Constraints and Clearances

2014 ◽  
Vol 1006-1007 ◽  
pp. 285-289
Author(s):  
Xi Feng Zhu ◽  
Quan Fu Gao
Keyword(s):  
Hopf Bifurcation ◽  
Period Doubling ◽  
Simulation Method ◽  
Degree Of Freedom ◽  
Local Bifurcations ◽  
Stiffness And Damping Coefficient ◽  
The Stability ◽  
Elastic Constraints ◽  
Stiffness And Damping ◽  
Two Degree Of Freedom

Based on the study of a dual component system with elastic constraints, the stability and local bifurcations of the soft-impacts system, such as piecewise property and singularity, was analyzed by using the Poincaré map and Runge-Kutta numerical simulation method. The routes from periodic motions to chaos, via Hopf bifurcation and period-doubling bifurcation, were investigated exactly. In the large constraint stiffness case, the period-doubling and Hopf bifurcation exist in the two-degree-of-freedom system with elastic constraints and clearances. The clearances of the system, stiffness and damping coefficient of the elastic constraints is the main reasons for influencing the chaotic motion. The steady 1-1-1 period orbits or 2-1-1 period orbits will exist within a wideband frequency range and the value of velocity will be higher when appropriate system parameters are chosen.

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