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Research Papers

# Estimation of the Bistable Zone for Machining Operations for the Case of a Distributed Cutting-Force Model

[+] Author and Article Information
Tamás G. Molnár

Department of Applied Mechanics,
Budapest University of
Technology and Economics,
Budapest 1111, Hungary
e-mail: molnar@mm.bme.hu

Tamás Insperger

Department of Applied Mechanics,
Budapest University of
Technology and Economics,
Budapest 1111, Hungary
e-mail: insperger@mm.bme.hu

S. John Hogan

Department of Engineering Mathematics,
University of Bristol,
Bristol BS8 1TH, UK
e-mail: s.j.hogan@bristol.ac.uk

Gábor Stépán

Department of Applied Mechanics,
Budapest University of
Technology and Economics,
Budapest 1111, Hungary
e-mail: stepan@mm.bme.hu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 19, 2015; final manuscript received December 30, 2015; published online February 3, 2016. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 11(5), 051008 (Feb 03, 2016) (10 pages) Paper No: CND-15-1222; doi: 10.1115/1.4032443 History: Received July 19, 2015; Revised December 30, 2015

## Abstract

Regenerative machine tool chatter is investigated for a single-degree-of-freedom model of turning processes. The cutting force is modeled as the resultant of a force system distributed along the rake face of the tool, whose magnitude is a nonlinear function of the chip thickness. Thus, the process is described by a nonlinear delay-differential equation, where a short distributed delay is superimposed on the regenerative point delay. The corresponding stability lobe diagrams are computed and are shown numerically that a subcritical Hopf bifurcation occurs along the stability boundaries for realistic cutting-force distributions. Therefore, a bistable region exists near the stability boundaries, where large-amplitude vibrations (chatter) may arise for large perturbations. Analytical formulas are obtained to estimate the size of the bistable region based on center manifold reduction and normal form calculations for the governing distributed-delay equation. The locally and globally stable parameter regions are computed numerically as well using the continuation algorithm implemented in dde-biftool. The results can be considered as an extension of the bifurcation analysis of machining operations with point delay.

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## References

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## Figures

Fig. 1

Single-degree-of-freedom model of turning operations with distributed cutting force

Fig. 2

Force characteristics of two different cutting-force models: Taylor force (a) and Tobias force (b)

Fig. 3

Distribution of the shear stress along the rake face of the tool

Fig. 4

Stability charts of the nonlinear turning model with cutting-force distribution (8) showing the linear stability boundaries (solid line), and the analytically estimated (dashed line) and numerically determined (dashed-dotted line) boundaries of the bistable region

Fig. 5

Ratio of the size of the bistable region and the linearly stable region assuming Tobias force (dashed line: analytical estimate, dots: numerical results)

Fig. 6

Bifurcation diagram showing the amplitude of periodic orbits in the vicinity of the parameter points A, B, and C in Fig. 4 (dashed line: analytical estimate, dashed-dotted line: numerical result)

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