Stability Boundaries of High-Speed Milling Corresponding to Period Doubling Are Essentially Closed Curves

Manufacturing ◽  
2003 ◽  
Author(s):  
Ro´bert Szalai ◽  
Ga´bor Ste´pa´n
Keyword(s):  
Characteristic Function ◽  
High Speed ◽  
Period Doubling ◽  
Stability Boundaries ◽  
Closed Curves ◽  
High Speed Milling ◽  
Complex Roots ◽  
Stability Chart ◽  
The Stability ◽  
Special Case

In this paper a new method for the stability analysis of high-speed milling processes is introduced. The approach is based on the construction of a characteristic function whose complex roots determine the stability of the system. By using the argument principle, the number of roots causing instability can be counted, and thus, an exact stability chart can be drawn. In the special case of period doubling bifurcation, the corresponding multiplier −1 is substituted into the characteristic function leading to an implicit formula of the stability boundaries. Further investigations show that all the period doubling boundaries are closed curves, except the first lobe at the highest cutting speeds. Together with the stability boundaries of Neimark-Sacker (or secondary Hopf) bifurcations, the unstable parameter domains are formed from the union of lobes and lenses.

Lobes and Lenses in the Stability Chart of Interrupted Turning

2003 ◽  
Vol 1 (3) ◽  
pp. 205-211 ◽  
Author(s):  
Róbert Szalai ◽  
Gábor Stépán
Keyword(s):  
Characteristic Function ◽  
Period Doubling ◽  
Stability Boundaries ◽  
Closed Curves ◽  
Complex Roots ◽  
Stability Chart ◽  
The Stability ◽  
Interrupted Turning ◽  
Special Case ◽  
Cutting Speeds

In this paper, a new method for the stability analysis of interrupted turning processes is introduced. The approach is based on the construction of a characteristic function whose complex roots determine the stability of the system. By using the argument principle, the number of roots causing instability can be counted, and thus, an exact stability chart can be drawn. In the special case of period doubling bifurcation, the corresponding multiplier −1 is substituted into the characteristic function, leading to an implicit formula for the stability boundaries. Further investigations show that all the period doubling boundaries are closed curves, except the first lobe at the highest cutting speeds. Together with the stability boundaries of Neimark-Sacker (or secondary Hopf) bifurcations, the unstable parameter domains are formed from the union of lobes and lenses.

Vibration Frequencies in High-Speed Milling Processes or a Positive Answer to Davies, Pratt, Dutterer and Burns

2004 ◽  
Vol 126 (3) ◽  
pp. 481-487 ◽  
Author(s):  
T. Insperger ◽  
G. Ste´pa´n
Keyword(s):  
High Speed ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Vibration Frequencies ◽  
High Speed Milling ◽  
Machine Tool Chatter ◽  
Special Cases ◽  
Universal Approach ◽  
The Stability ◽  
Excitation Model

The stability charts of high-speed milling are constructed. New unstable regions and vibration frequencies are identified. These are related to flip bifurcation, i.e. period doubling vibrations occur apart of the conventional self-excited vibrations well-known for turning or low-speed milling with multiple active teeth. The Semi-Discretization method is applied for the delayed parametric excitation model of milling providing the connection of the two existing and experimentally verified results of machine tool chatter research. The two extreme models in question, that is, the traditional autonomous delayed model of time-independent turning, and the recently introduced discrete map model of time-dependent highly interrupted machining, are both involved as special cases in the universal approach presented in this study.

Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments

2004 ◽  
Vol 127 (2) ◽  
pp. 197-203 ◽  
Author(s):  
Gabor Stepan ◽  
Robert Szalai ◽  
Brian P. Mann ◽  
Philip V. Bayly ◽  
Tamas Insperger ◽  
...  
Keyword(s):  
Small Parameter ◽  
High Speed ◽  
Period Doubling ◽  
Hopf Bifurcations ◽  
Cutting Condition ◽  
Flip Bifurcation ◽  
Machining Parameters ◽  
High Speed Milling ◽  
The Stability ◽  
Discrete Mathematical Model

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

scholarly journals Stability of High-Speed Milling

2000 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán
Keyword(s):  
Parametric Excitation ◽  
Stability Criterion ◽  
High Speed ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Vibration Frequencies ◽  
High Speed Milling ◽  
Low Speed ◽  
The Stability ◽  
Excitation Model

Abstract The stability charts of high-speed milling are constructed. Non-conventional unstable regions and vibration frequencies are identified. These are related to flip bifurcation, i.e. period doubling vibrations occur apart of the conventional self-excited vibrations typical for turning or low-speed milling with multiple active teeth. A new stability criterion is proposed and applied for the delayed parametric excitation model of milling.

Study on the Stability of High-Speed Milling of Thin-Walled Workpiece

2010 ◽  
Vol 97-101 ◽  
pp. 1849-1852
Author(s):  
Tong Yue Wang ◽  
Ning He ◽  
Liang Li
Keyword(s):  
Aluminum Alloy ◽  
Machine Tool ◽  
High Speed ◽  
Modal Parameters ◽  
Inertia Effect ◽  
Cutting Force Coefficients ◽  
High Speed Milling ◽  
Machining System ◽  
Thin Walled ◽  
The Stability

Thin-walled structure is easy to vibrate in machining. The dynamic milling model of thin-walled workpiece is analyzed based on the analysis of degrees in two perpendicular directions of machine tool-workpiece system. In high speed milling of 2A12 aluminum alloy, the compensation method based on the modification of inertia effect is proposed and accurate cutting force coefficients are obtained. The machining system is divided into “spindle-cutter” and “workpiece-fixture” two sub-systems and the modal parameters of two sub-systems are acquired via modal analysis experiments. Finally, the stability lobes for high speed milling of 2A12 thin-walled workpiece are obtained by the use of these parameters. The results are verified against cutting tests.

scholarly journals SUPPRESSION OF PERIOD DOUBLING CHATTER IN HIGH-SPEED MILLING BY SPINDLE SPEED VARIATION

2011 ◽  
Vol 15 (2) ◽  
pp. 153-171 ◽  
Author(s):  
Sébastien Seguy ◽  
Tamás Insperger ◽  
Lionel Arnaud ◽  
Gilles Dessein ◽  
Grégoire Peigné
Keyword(s):  
High Speed ◽  
Period Doubling ◽  
Spindle Speed ◽  
High Speed Milling ◽  
Speed Variation ◽  
Spindle Speed Variation

Global Attractors of High-Speed Milling: Analyses, Numerics and Experiments

2003 ◽  
Author(s):  
Gabor Stepan ◽  
Robert Szalai ◽  
Brian P. Mann ◽  
Philip V. Bayly ◽  
Tamas Insperger ◽  
...  
Keyword(s):  
Small Parameter ◽  
High Speed ◽  
Period Doubling ◽  
Global Attractors ◽  
Hopf Bifurcations ◽  
Cutting Condition ◽  
Flip Bifurcation ◽  
Machining Parameters ◽  
High Speed Milling ◽  
Discrete Mathematical Model

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

Avoiding Instability on the Milling of Parts with Thin Features

2006 ◽  
Vol 526 ◽  
pp. 37-42 ◽  
Author(s):  
Francisco Javier Campa ◽  
Luis Norberto López de Lacalle ◽  
S. Herranz ◽  
Aitzol Lamikiz ◽  
A. Rivero
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Spindle Speed ◽  
Depth Of Cut ◽  
Test Device ◽  
High Speed Milling ◽  
Thin Walls ◽  
Chatter Vibrations ◽  
The Stability ◽  
Selection Of

In this paper, a 3D dynamic model for the prediction of the stability lobes of high speed milling is presented, considering the combined flexibility of both tool and workpiece. The main aim is to avoid chatter vibrations on the finish milling of aeronautical parts, which include thin walls and thin floors. In this way the use of complex fixtures is eliminated. Hence, an accurate selection of both axial depth of cut and spindle speed can be accomplished. The model has been validated by means of a test device that simulates the behaviour of a thin floor.

Structure Dynamic Modification and Optimization for High-Speed Face Milling Cutter

2004 ◽  
Vol 471-472 ◽  
pp. 663-667 ◽  
Author(s):  
Wei Xiao Tang ◽  
Xing Ai ◽  
H.Y. Wu ◽  
Song Zhang ◽  
H. Jiang
Keyword(s):  
High Speed ◽  
Milling Process ◽  
Face Milling ◽  
Relative Speed ◽  
Structural Dynamic ◽  
High Speed Milling ◽  
Structural Dynamic Modification ◽  
The Face ◽  
Dynamic Modification ◽  
The Stability

Due to the complexity of high-speed milling process by high relative speed and interrupted cutting, the face milling cutters possess the multi-order modes and the vibrating displacements of the cutting edges under each modes affect adversely both the surface roughness and the life of machine/tool system worse than other structures. In order to improve the stability of milling process, this work focuses on the influence of the variables such as structure geometries and constraint conditions on the eigenfrequencies and modeshapes of cutter. As an example, the dynamic characteristics of several face cutters are analyzed and optimized by structural dynamic modification (SDM) techniques.

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