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.

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.

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.

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.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

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.

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.

scholarly journals Stability Analysis in Positive Output and Negative Output DC to DC Converters Used in Renewable Energy Applications

2019 ◽  
Author(s):  
Maheswari Ellappan ◽  
Kavitha Anbukumar
Keyword(s):  
Renewable Energy ◽  
Stability Analysis ◽  
Output Voltage ◽  
Monodromy Matrix ◽  
Period Doubling ◽  
Current Mode ◽  
Power Converter ◽  
Control Technique ◽  
Step Down ◽  
The Stability

The renewable energy source plays a major role in the grid side power production. The stability analysis is very essential in the renewable energy converters. In this paper the bifurcation is analyzed in ZETA converter and Continuous input and output(CIO) power Buck Boost converter. The ZETA converter gives positive step down and step up output voltage and the CIO power converter gives the negative step up and step down output voltage. These converters are used in the DC micro grid with renewable energy as the source. The current mode control technique is applied to analyze the bifurcation behavior and the reference current is taken as the bifurcation parameter. When the reference current is varied, both the converters loses its stability and it enters into chaotic region through period doubling bifurcation. The simulation results are presented to study the performance behavior of both the converters. The stability region of both the converters are determined by deriving the Monodromy matrix approach.

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.

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.

Monitoring of Surface Roughness Based on Cutting Force and Cutting Temperature in Ball-End Milling Process by Utilizing Response Surface Analysis

2015 ◽  
Vol 799-800 ◽  
pp. 324-328
Author(s):  
Panrawee Yaisuk ◽  
Somkiat Tangjitsitcharoen
Keyword(s):  
Surface Roughness ◽  
Cutting Force ◽  
Response Surface ◽  
Feed Rate ◽  
End Milling ◽  
Cutting Temperature ◽  
Milling Process ◽  
Depth Of Cut ◽  
Tool Diameter ◽  
End Milling Process

The surface roughness is monitored using the cutting force and the cutting temperature in the ball-end milling process by utilizing the response surface analysis with the Box-Behnken design. The optimum cutting condition is obtained referring to the minimum surface roughness, which is the spindle speed, the feed rate, the depth of cut, and the tool diameter. The models of cutting force ratio and the cutting temperature are proposed and developed based on the experimental results. It is understood that the surface roughness is improved with an increase in spindle speed, feed rate and depth of cut. The cutting temperature decreases with an increase in tool diameter. The model verification has showed that the experimentally obtained surface roughness model is reliable and accurate to estimate the surface roughness.

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