Active Chatter Suppression in Low Immersion Intermittent Milling Process

2020 ◽  
Vol 142 (10) ◽  
Author(s):  
Le Cao ◽  
Tao Huang ◽  
Da-Ming Shi ◽  
Xiao-Ming Zhang ◽  
Han Ding
Keyword(s):  
Period Doubling ◽  
Milling Process ◽  
Period Doubling Bifurcation ◽  
First Order ◽  
Chatter Suppression ◽  
Stability And Stabilization ◽  
Piecewise Model ◽  
Full Immersion ◽  
The Stability ◽  
Time Invariant

Abstract Chatter in low immersion milling behaves differently from that in full immersion milling, mainly because of the non-negligible time-variant dynamics and the occurrence of period doubling bifurcation. The intermittent and time-variant characteristics make the active chatter suppression based on Lyaponov theorem a non-trivial problem. The main challenges lie in how to deal with the time-variant directional coefficient and how to construct a suitable Lyaponov function so as to alleviate the conservation, as well as the saturation of the controller. Generally, the Lyaponov stability of time-invariant dynamics is more tractable. Hence, in our paper, a first-order piecewise model is proposed to approximate the low immersion milling system as two time-invariant sub-ones that are cyclically switched. To alleviate the conservation, a novel piecewise Lyaponov function is constructed to determine the stability of each subsystem independently. The inequality conditions for determining the stability and stabilization are derived. The validity of the proposed stabilization algorithm to suppress both the hopf and period doubling bifurcation, as well as to reduce the conservation of the controller parameters have been verified.

scholarly journals The Application of the Undetermined Fundamental Frequency Method on the Period-Doubling Bifurcation of the 3D Nonlinear System

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Gen Ge ◽  
Wei Wang
Keyword(s):  
Numerical Simulation ◽  
Nonlinear System ◽  
Fundamental Frequency ◽  
Three Dimensional ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Cubic Nonlinearity ◽  
Frequency Method ◽  
The Stability ◽  
Second Period

The analytical method to predict the period-doubling bifurcation of the three-dimensional (3D) system is improved by using the undetermined fundamental frequency method. We compute the stable response of the system subject to the quadratic and cubic nonlinearity by introducing the undetermined fundamental frequency. For the occurrence of the first and second period-doubling bifurcation, the new bifurcation criterion is accomplished. It depends on the stability of the limit cycle on the central manifold. The explicit applications show that the new results coincide with the results of the numerical simulation as compared with the initial methods.

Period Doubling Bifurcation Point Detection Strategy with Nested Layer Particle Swarm Optimization

2017 ◽  
Vol 27 (07) ◽  
pp. 1750101 ◽  
Author(s):  
Haruna Matsushita ◽  
Yusho Tomimura ◽  
Hiroaki Kurokawa ◽  
Takuji Kousaka
Keyword(s):  
Particle Swarm Optimization ◽  
Bifurcation Point ◽  
Particle Swarm ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Swarm Optimization ◽  
Detection Strategy ◽  
The Stability ◽  
Point Detection ◽  
Discrete Time Dynamical Systems

This paper proposes a bifurcation point detection strategy based on nested layer particle swarm optimization (NLPSO). The NLPSO is performed by two particle swarm optimization (PSO) algorithms with a nesting structure. The proposed method is tested in detection experiments of period doubling bifurcation points in discrete-time dynamical systems. The proposed method directly detects the parameters of period doubling bifurcation regardless of the stability of the periodic point, but require no careful initialization, exact calculation or Lyapunov exponents. Moreover, the proposed method is an effective detection technique in terms of accuracy, robustness usability, and convergence speed.

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.

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.

Active Control of an Active Magnetic Bearings Supported Spindle for Chatter Suppression in Milling Process

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
Tao Huang ◽  
Zhiyong Chen ◽  
Hai-Tao Zhang ◽  
Han Ding
Keyword(s):  
Integrated Control ◽  
Magnetic Bearing ◽  
Milling Process ◽  
Active Magnetic Bearing ◽  
Machining Process ◽  
Maintenance Cost ◽  
Machining Parameters ◽  
Operating Life ◽  
Chatter Suppression ◽  
The Stability

In machining process, chatter is an unstable dynamic phenomenon which causes overcut and quick tool wear, etc. To avoid chatter, traditional methods aim to optimize machining parameters. But they have inherent disadvantage in gaining highly efficient machining. Active magnetic bearing (AMB) is a promising technology for machining on account of low wear and friction, low maintenance cost, and long operating life. The control currents applied to AMBs allow not only to stabilize the supported spindle but also to actively suppress chatter in milling process. This paper, for the first time, studies an integrated control scheme for stability of milling process with a spindle supported by AMBs. First, to eliminate the vibration of an unloaded spindle rotor during acceleration/deceleration, we present an optimal controller with proper compensation for speed variation. Next, the controller is further enhanced by adding an adaptive algorithm based on Fourier series analysis to actively suppress chatter in milling process. Finally, numerical simulations show that the stability lobe diagram (SLD) boundary can be significantly expanded. Also, a practical issue of constraints on controller output is discussed.

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.

First-Order Recurrent Neural Networks and Deterministic Finite State Automata

1994 ◽  
Vol 6 (6) ◽  
pp. 1155-1173 ◽  
Author(s):  
Peter Manolios ◽  
Robert Fanelli
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Network Architecture ◽  
A Priori ◽  
Finite State Automata ◽  
First Order ◽  
Finite State ◽  
The Stability ◽  
Time Invariant ◽  
Training Sets

We examine the correspondence between first-order recurrent neural networks and deterministic finite state automata. We begin with the problem of inducing deterministic finite state automata from finite training sets, that include both positive and negative examples, an NP-hard problem (Angluin and Smith 1983). We use a neural network architecture with two recurrent layers, which we argue can approximate any discrete-time, time-invariant dynamic system, with computation of the full gradient during learning. The networks are trained to classify strings as belonging or not belonging to the grammar. The training sets used contain only short strings, and the sets are constructed in a way that does not require a priori knowledge of the grammar. After training, the networks are tested using various test sets with strings of length up to 1000, and are often able to correctly classify all the test strings. These results are comparable to those obtained with second-order networks (Giles et al. 1992; Watrous and Kuhn 1992a; Zeng et al. 1993). We observe that the networks emulate finite state automata, confirming the results of other authors, and we use a vector quantization algorithm to extract deterministic finite state automata after training and during testing of the networks, obtaining a table listing the start state, accept states, reject states, all transitions from the states, as well as some useful statistics. We examine the correspondence between finite state automata and neural networks in detail, showing two major stages in the learning process. To this end, we use a graphics module, which graphically depicts the states of the network during the learning and testing phases. We examine the networks' performance when tested on strings much longer than those in the training set, noting a measure based on clustering that is correlated to the stability of the networks. Finally, we observe that with sufficiently long training times, neural networks can become true finite state automata, due to the attractor structure of their dynamics.

Investigation of Period-Doubling Islands in Milling With Simultaneously Engaged Helical Flutes

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Firas A. Khasawneh ◽  
Oleg A. Bobrenkov ◽  
Brian P. Mann ◽  
Eric A. Butcher
Keyword(s):  
State Space ◽  
Period Doubling ◽  
Helix Angle ◽  
Milling Process ◽  
Collocation Methods ◽  
Depth Of Cut ◽  
Stability Behavior ◽  
The Stability ◽  
Step Over ◽  
High Depth

This paper investigates the stability of a milling process with simultaneously engaged flutes using the state-space TFEA and Chebyshev collocation methods. In contrast to prior works, multiple flute engagement due to both the high depth of cut and high step-over distance are considered. A particular outcome of this study is the demonstration of a different stability behavior in comparison to prior works. To elaborate, period-doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or nonzero helix angle are simultaneously cutting. We also demonstrate stability differences that arise due to the parity in the number of flutes, especially at full radial immersion. In addition, we study other features induced by helical tools such as the waviness of the Hopf lobes, the sensitivity of the period-doubling islands to the radial immersion, as along with the orientation of the islands with respect to the Hopf lobes.

Comparsion of Machining Stability of Up and Down Milling

2013 ◽  
Vol 680 ◽  
pp. 369-386
Author(s):  
Te Ching Hsiao ◽  
J.J. Junz Wang
Keyword(s):  
Milling Process ◽  
One Dimension ◽  
Depth Of Cut ◽  
Two Dimension ◽  
Feed Direction ◽  
Radial Depth ◽  
Dimension 2 ◽  
The Stability ◽  
Time Invariant ◽  
Function Matrix

The purpose of this research is to discuss the machining stability in of the up and down milling of three milling systems: 1) the feed-direction one-dimension, 2) the normal-to-feed-direction one-dimension and 3) the symmetric two-dimension milling systems. A simplified model with time-invariant parameters is adopted to simulate the milling process. In this model, the elementary cutting function, which represents the trajectory of cutting force for a certain local cutting edge, plays an important role in affecting machining stability of the different cutting configurations. This paper presents an in-depth discussion on the effects of cutting configurations and radial depth of cut on the elementary cutting function, and also on the stability lobes. It is found that the elementary cutting function of one-dimension milling system is a real number, as well as its positive and negative values will lead to totally different chatter features. Most technical literatures focus on the chatter features of positive elementary cutting function, while this research discuss that of both positive and negative ones. On the other hand, for the two-dimension milling system, its machining stability found to be dominated by the eigenvalues of the elementary cutting function matrix. The comparison of machining stability in up and down milling is then analyzed and three conclusions are drawn in this research. Firstly, the feed-direction one-dimension milling system has better machining stability in up milling. Secondly, the normal-to-feed-direction one-dimension milling system has better machining stability in down milling. Thirdly, up and down milling both show the same machining stability in the symmetric two-dimension milling system.

Sign in / Sign up

Export Citation Format

Share Document