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.

Stability of Up- and Down-Milling Using Chebyshev Collocation Method

2005 ◽  
Author(s):  
Eric A. Butcher ◽  
Praveen Nindujarla ◽  
Ed Bueler
Keyword(s):  
Cutting Force ◽  
Monodromy Matrix ◽  
Period Doubling ◽  
Milling Process ◽  
Depth Of Cut ◽  
Approximation Technique ◽  
Stability Properties ◽  
Chebyshev Collocation ◽  
Collocation Points ◽  
The Stability

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 like milling are modeled by delay-differential equations (DDEs) 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 representation of the solution at their extremum points, the Chebyshev collocation points. The stability properties are determined by the eigenvalues of the approximate monodromy matrix which maps function values at the collocation points from one interval to the next. We check the results for convergence by varying the number of Chebyshev collocation points and by simulation of the transient response via the DDE23 MATLAB routine. The milling model used here was derived by Insperger et al. [14]. Here, the specific cutting force profiles, stability charts, and chatter frequency diagrams are produced for up-milling and down-milling cases for one and four cutting teeth and 25 to 100 % immersion levels. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found which agree with the previous results found by other techniques. An in-depth investigation in the vicinity of the critical immersion ratio for down-milling (where the average cutting force changes sign) and its implication for stability is presented.

Self-Excited Vibrations in a Delay Oscillator: Application to Milling With Simultaneously Engaged Helical Flutes

2009 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Brian P. Mann ◽  
Oleg A. Bobrenkov ◽  
Eric A. Butcher
Keyword(s):  
Period Doubling ◽  
Helix Angle ◽  
Milling Process ◽  
Depth Of Cut ◽  
Frame Work ◽  
Continuous Delay ◽  
The Stability ◽  
Delay Differential ◽  
Step Over ◽  
High Depth

This paper investigates the stability of a milling process with simultaneously engaged flutes by extending the state-space temporal finite elements method. In contrast to prior works, multiple flute engagement due to both a high depth of cut and a high step-over distance are considered. A particular outcome of this study is the development of a frame work to determine the stability of periodic, piecewise continuous delay differential equations. Another major outcome is the demonstration of different stability behavior at the loss of stability in comparison to prior results. To elaborate more, period doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or non-zero helix angle are simultaneously cutting.

On the Investigation of Machine Tool Chatter in the Milling Process

2007 ◽  
Author(s):  
Mahsa Moghaddas ◽  
Mohammad H. Ghaffari Saadat
Keyword(s):  
Milling Process ◽  
Depth Of Cut ◽  
Non Linear Equation ◽  
Stability And Instability ◽  
Machine Tool Chatter ◽  
Stability Chart ◽  
Periodic Delay ◽  
The Stability ◽  
Two Parameters ◽  
Delay Differential

In this paper, the chatter phenomenon is investigated through a single degree of freedom model of the milling process. In this regard, the non-linear equation of motion obtained from modeling of the milling process, which is a time-periodic delay differential equation, is simulated, and by changing the parameters: spindle speed and depth of cut, and assuming constant quantities for other parameters of the system the stable and instable points for the system are gained according to these two parameters by numerical method. In the end, the stability chart for this system is plotted and the approximate boundaries between the stability and instability regions are obtained numerically.

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.

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.

Stability Analysis of a Variable Spindle Speed Milling Process

2005 ◽  
Author(s):  
X.-H. Long ◽  
B. Balachandran
Keyword(s):  
Milling Process ◽  
Spindle Speed ◽  
Depth Of Cut ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Milling Operations ◽  
Milling Operation ◽  
The Stability ◽  
Delay Differential

In this effort, a stability treatment is presented for a milling process with a variable spindle speed (VSS). This variation is caused by superimposing a sinusoidal modulation on a nominal spindle speed. The dynamics of the VSS milling process is described by a set of delay differential equations (DDEs) with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite dimensional transition matrix is converted to a finite dimensional matrix over this period. The eigenvalues of this finite dimensional matrix are used to determine the stability of the VSS milling operation with respect to selected control parameters, such as the axis depth of cut and the nominal spindle speed. The benefits of VSS milling operations are discussed by comparing the stability charts obtained for VSS milling operations with those obtained for constant spindle speed (CSS) milling operations.

scholarly journals Chatter identification in milling of Inconel 625 based on recurrence plot technique and Hilbert vibration decomposition

2018 ◽  
Vol 148 ◽  
pp. 09003 ◽  
Author(s):  
Paweł Lajmert ◽  
Rafał Rusinek ◽  
Bogdan Kruszyński
Keyword(s):  
Stability Limit ◽  
Recurrence Plot ◽  
Milling Process ◽  
Machining Process ◽  
Inconel 625 ◽  
Depth Of Cut ◽  
Force Measurements ◽  
Stability Lobe ◽  
Theoretical Finding ◽  
The Stability

In the paper a cutting stability in the milling process of nickel based alloy Inconel 625 is analysed. This problem is often considered theoretically, but the theoretical finding do not always agree with experimental results. For this reason, the paper presents different methods for instability identification during real machining process. A stability lobe diagram is created based on data obtained in impact test of an end mill. Next, the cutting tests were conducted in which the axial cutting depth of cut was gradually increased in order to find a stability limit. Finally, based on the cutting force measurements the stability estimation problem is investigated using the recurrence plot technique and Hilbert vibration decomposition method.

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.

Optimization of Process Parameters for Cutting Force for Aluminium 7010 in CNC End Milling Process

2018 ◽  
Vol 10 (6) ◽  
Author(s):  
M. Kishanth ◽  
P. Rajkamal ◽  
D. Karthikeyan ◽  
K. Anand
Keyword(s):  
Surface Roughness ◽  
Cutting Force ◽  
Process Parameters ◽  
End Milling ◽  
Milling Process ◽  
Machining Process ◽  
Depth Of Cut ◽  
Optimization Of Process Parameters ◽  
Cnc End Milling ◽  
End Milling Process

In this paper CNC end milling process have been optimized in cutting force and surface roughness based on the three process parameters (i.e.) speed, feed rate and depth of cut. Since the end milling process is used for abrading the wear caused is very high, in order to reduce the wear caused by high cutting force and to decrease the surface roughness, the optimization is much needed for this process. Especially for materials like aluminium 7010, this kind of study is important for further improvement in machining process and also it will improve the stability of the machine.

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.

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