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Research Papers

# Increased Stability of Low-Speed Turning Through a Distributed Force and Continuous Delay Model

[+] Author and Article Information
Firas A. Khasawneh1

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708firas.khasawneh@duke.edu

Brian P. Mann

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708

Tamás Insperger, Gabor Stépán

Department of Applied Mechanics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary

1

Corresponding author.

J. Comput. Nonlinear Dynam 4(4), 041003 (Aug 24, 2009) (12 pages) doi:10.1115/1.3187153 History: Received December 31, 2007; Revised September 26, 2008; Published August 24, 2009

## Abstract

This paper investigates the increased stability behavior commonly observed in low-speed machining. In the past, this improved stability has been attributed to the energy dissipated by the interference between the workpiece and the tool relief face. In this study, an alternative physical explanation is described. In contrast to the conventional approach, which uses a point force acting at the tool tip, the cutting forces are distributed over the tool-chip interface. This approximation results in a second-order delayed integrodifferential equation for the system that involves a short and a discrete delay. A method for determining the stability of the system for an exponential shape function is described, and temporal finite element analysis is used to chart the stability regions. Comparisons are then made between the stability charts of the point force and the distributed force models for continuous and interrupted turning.

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## Figures

Figure 4

Schematic diagram of the interrupted cutting process. The force is proportional to the uncut chip area when the tool is in contact with the work piece but drops to zero when the tool vibrates freely (31).

Figure 5

Short time delay embedding into the force model. The area of a certain segment of the chip flowing over the rake face in (b) can be described by the area of the same segment as it was being cut at the tool tip in (a) at time t−t̂.

Figure 6

Stability charts for the point force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratios used are (a) ζ=0.0038 and (b) ζ=0.02 (unstable regions shaded).

Figure 9

Stability charts for the point force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting are (a) ρ=0.05, (b) ρ=0.10, and (c) ρ=0.20 (unstable regions shaded).

Figure 1

Qualitative stability charts shown to illustrate the phenomenological increase in stability at relatively low cutting speeds. Graph (a) shows the stability boundaries for a typical point force model and graph (b) provides a qualitative representation of the commonly observed stability increase at low speeds.

Figure 2

Schematic diagram of a: (a) distributed force model and (b) conventional point force model. Case (a) uses a stress distribution over the tool rake face and applies a finite time for the chip to travel along the tool-chip interface. Case (b) is the conventional modeling approach of using a point force and a discrete delay model.

Figure 3

Cutting force and chip area relation

Figure 7

Stability charts for the distributed force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The delay ratio used is r=0.001 and the damping ratios used are (a) ζ=0.0038 and (b) ζ=0.02 (unstable regions shaded).

Figure 8

Stability charts for the distributed force model of continuous turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the delay ratios used are (a) r=0.03, (b) r=0.05, and (c) r=0.10 (unstable regions shaded).

Figure 10

Stability charts for the distributed force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting and different delay ratios are (a) ρ=0.05 and r=0.001, (b) ρ=0.10 and r=0.001, (c) ρ=0.20 and r=0.001, (d) ρ=0.05 and r=0.03, (e) ρ=0.10 and r=0.03, and (f) ρ=0.20 and r=0.03 (unstable regions shaded).

Figure 11

Stability charts for the distributed force model of interrupted turning plotted as a function of the nondimensionalized cutting speed and depth of cut. The damping ratio used is ζ=0.0038 and the cases for different fractions of the workpiece revolution that the tool is cutting and different delay ratios are (a) ρ=0.05 and r=0.05, (b) ρ=0.10 and r=0.05, (c) ρ=0.20 and r=0.05, (d) ρ=0.05 and r=0.10, (e) ρ=0.10 and r=0.10, and (f) ρ=0.20 and r=0.10 (unstable regions shaded).

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