Research Papers

State Dependent Regenerative Effect in Milling Processes

[+] Author and Article Information
Dániel Bachrathy

Research Group on Dynamics of Vehicles and Machines, Hungarian Academy of Sciences, Budapest H-1111, Hungarybachrathy@mm.bme.hu

Gábor Stépán

Department of Applied Mechanics, Budapest University of Technology and Economics, Budapest H-1111, Hungarystepan@mm.bme.hu

János Turi

Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083-0688turi@utdallas.edu

J. Comput. Nonlinear Dynam 6(4), 041002 (Mar 22, 2011) (9 pages) doi:10.1115/1.4003624 History: Received December 31, 2009; Revised February 04, 2011; Published March 22, 2011; Online March 22, 2011

The governing equation of milling processes is generalized with the help of accurate chip thickness derivation resulting in a state dependent delay model. This model is valid for large amplitude machine tool vibrations and uses accurate nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool relative to the workpiece. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization around these periodic solutions by means of the semidiscretization method applied for the corresponding time-periodic delay system. Predictor-corrector method is developed to continue the periodic solutions as the bifurcation parameter, that is, the axial immersion is varied. It is observed that the influence of the state dependent delay on linear stability can be significant close to resonance where large amplitude forced vibrations occur. The existence of an unusual fold bifurcation is shown where a kind of hysteresis phenomenon appears between two different stable periodic motions.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Machined workpiece surfaces created during large amplitude near-resonant vibration

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Figure 2

Representation of the mechanical model

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Figure 3

Approximations of the cutting force function: (a) linear, (b) power, and (c) polynomial functions

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Figure 4

Accurate model of chip thickness calculation

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Figure 5

Values of the screen functions gr and gh

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Figure 6

Comparison of the stability charts of the standard (shaded region with continuous line) and the new state dependent delay model (dashed line with crosses); parameters: N=2, ymin=−∞, ymax=−1.2 mm, and v=1 μm/s

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Figure 7

Stability chart computed by the semidiscretization method (shaded region with continuous thin line) and the limit of first bifurcation points wb of the state dependent delay model (thick lines); parameters: N=2, ymin=−∞, and ymax=−1.2 mm

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Figure 8

The bifurcation diagram of the milling process and the path of the cutting edges. Black circles denote the stable process; white circles denote the unstable process. The straight line represents the system with constant time delay and its Hopf bifurcation. Continuous lines represent stable branches; dashed lines represent unstable branches. Dark clouds denote the hypothetic chaotic region. Parameters: N=2, Ω=2270 rad/s, ymin=−∞, ymax=−1.2 mm, v=50 mm/s, and τv=0.07 mm/tooth.

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Figure 9

Representation of the fly-over effect

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Figure 10

Computational error of the chip thickness in the case of dτj/dt>1

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Figure 11

Representation of the multiple solutions of the time delay τj in space and in time domain

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Figure 12

Limit points around the first resonant angular velocity (white circles: fly-over effect; black circles: multiple chip thickness), the first bifurcation point wb (dashed line with crosses), and the stability chart computed by the semidiscretization method (shaded region with continuous line); parameters: N=2, ymin=−∞, ymax=−1.2 mm, and v=50 mm/s




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