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.

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.

scholarly journals On Adaptive Moving Average Algorithms for the Application of the Conservative Power Theory in Systems with Variable Frequency

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 1201
Author(s):  
Daniel dos Santos Mota ◽  
Elisabetta Tedeschi
Keyword(s):  
Time Delay ◽  
Moving Average ◽  
Voltage Source ◽  
Moving Averages ◽  
Nonlinear Load ◽  
Power Theory ◽  
Variable Time Delay ◽  
Variable Time ◽  
Conservative Power Theory ◽  
Conservative Power

The Conservative Power Theory (CPT) emerged in recent decades as a theoretical framework for coping with harmonically distorted and unbalanced electric networks of ac power systems with a high participation of converter interfaced loads and generation. The CPT measurements are intrinsically linked to moving averages (MA) over one period of the grid. If the CPT is to be used in a low-inertia isolated-grid scenario, which is subjected to frequency variations, adaptive moving averages (AMA) are necessary. This paper reviews an efficient way of computing MAs and turns it into an adaptive one. It shows that an easily available variable time delay block, from MATLAB, causes steady-state errors in the measurements when the grid frequency varies. A new variable time delay block is, thus, proposed. Nonetheless, natural pulsations in the instantaneous power slip through MAs when the discrete moving average window does not fit perfectly the continuously varying period of the grid. A method consisting of weighing two MAs is reviewed and a new and effective hybrid AMA is proposed. The CPT transducers with the different choices of AMAs are compared via computer simulations of a single-phase voltage source feeding either a linear or a nonlinear load.

scholarly journals Dynamics of a Duopoly Game with Two Different Delay Structures

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Shumin Jiang ◽  
Fei Xu ◽  
Zhanwen Ding ◽  
Chen Yang ◽  
Huanhuan Liu
Keyword(s):  
Time Delay ◽  
Market Price ◽  
Period Doubling ◽  
System Stability ◽  
Delayed Feedback Control ◽  
System Control ◽  
Heterogeneous Players ◽  
Cournot Game ◽  
The Stability ◽  
Time Delayed Feedback

Two different time delay structures for the dynamical Cournot game with two heterogeneous players are considered in this paper, in which a player is assumed to make decision via his marginal profit with time delay and another is assumed to adjust strategy according to the delayed price. The dynamics of both players output adjustments are analyzed and simulated. The time delay for the marginal profit has more influence on the dynamical behaviors of the system while the market price delay has less effect, and an intermediate level of the delay weight for the marginal profit can expand the stability region and thus promote the system stability. It is also shown that the system may lose stability due to either a period-doubling bifurcation or a Neimark-Sacker bifurcation. Numerical simulations show that the chaotic behaviors can be stabilized by the time-delayed feedback control, and the two different delays play different roles on the system controllability: the delay of the marginal profit has more influence on the system control than the delay of the market price.

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