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RESEARCH PAPERS

# Analysis of the Chatter Instability in a Nonlinear Model for Drilling

[+] Author and Article Information
Sue Ann Campbell

Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 Canadasacampbell@uwaterloo.ca

Emily Stone

Department of Mathematical Sciences, The University of Montana, Missoula, MT 59812

J. Comput. Nonlinear Dynam 1(4), 294-306 (Apr 20, 2006) (13 pages) doi:10.1115/1.2338648 History: Received November 15, 2005; Revised April 20, 2006

## Abstract

In this paper we present stability analysis of a non-linear model for chatter vibration in a drilling operation. The results build our previous work [Stone, E., and Askari, A., 2002, “Nonlinear Models of Chatter in Drilling Processes,” Dyn. Syst., 17(1), pp. 65–85 and Stone, E., and Campbell, S. A., 2004, “Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling,” J. Nonlinear Sci., 14(1), pp. 27–57], where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we analyze the effect of varying cutting depth. We show that qualitatively different stability lobes are produced in this case. We analyze the criticality of the Hopf bifurcation associated with loss of stability and show that changes in criticality can occur along the stability boundary, resulting in extra periodic solutions.

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## Figures

Figure 1

Stability diagram for regenerative chatter. Steady cutting solution (the trivial solution) is stable in the region beneath the lobes.

Figure 2

Diagram of the drill geometry, showing the chip width, w, and the chip thickness (nominal cutting depth), t1. Ω is the angular speed of the drill.

Figure 3

Diagram of angles and forces for orthogonal cutting. The angle α is referred to as the rake angle, λ is the friction angle, ϕ is the shear plane angle. The cutting thickness is t1, V is the cutting speed, R is the cutting force, and Fs is the cutting force resolved along the shear plane.

Figure 4

Stability diagram for the traditional case: γ=0.5, p0cosθ=0.8, p̃1=0.2. Moving from top to bottom zooms out from the area near the lower part of the boundary for lower cutting speeds. The trivial solution is unstable in the region above the boundaries, and stable below.

Figure 5

Stability diagram for axial-torsional vibration: γ=0.5, p0cosθ=−0.8, p̃1=−0.4. Moving from bottom to top zooms in on the area near the lower part of the boundary for lower cutting speeds. The trivial solution is unstable in the region above the boundaries, and stable below.

Figure 6

Stability boundary showing criticality of Hopf bifurcation. The top diagram is a blow-up of the bottom diagram, and hatched lines indicate a subcritical Hopf bifurcation, solid lines a supercritical Hopf. Traditional case with p0cos(θ)=0.8, p̃1=0.2, p̃2=0.1, γ=0.5.

Figure 7

Stability boundary showing criticality of Hopf bifurcation. The top diagram is a blow-up of the bottom diagram, and hatched lines indicate a subcritical Hopf bifurcation, solid lines a supercritical Hopf. Traditional case with p0cos(θ)=0.8, p̃1=0.2, p̃2=0.1, γ=2.0.

Figure 8

Linear stability boundaries varying β, t1 versus 1T, traditional vibration parameters

Figure 9

Linear stability boundaries varying β, t1 versus 1∕T, axial-torsional vibration parameters

Figure 10

Extrema envelope of t1 boundary as function of β, traditional vibration parameters

Figure 11

Extrema envelope of t1 boundary as function of β, axial-torsional vibration parameters

Figure 12

Stability boundary showing criticality of Hopf bifurcation in the traditional case for (a)β=0.1 and (b)β=0.4. Other parameter values are: p0cos(θ)=0.8, p̃1=0.2, p̃2=0.1, γ=0.1. Thin lines indicate a subcritical Hopf bifurcation, thick lines a supercritical Hopf.

Figure 13

Stability boundary showing criticality of Hopf bifurcation in the axial torsional case for (a)β=0.1 and (b)β=0.5. Other parameter values are p0cos(θ)=−0.8, p̃1=−0.2, p̃2=−0.1, γ=0.1. Thin lines indicate a subcritical Hopf bifurcation, thick lines a supercritical Hopf.

Figure 14

Numerical simulations of Eq. 12 illustrating a supercritical Hopf bifurcation for the traditional vibration mode, close to the point of change in criticality. Shown is η̇ versus η for two different initial conditions. Parameter values: β=0.1,γ=0.1,1∕T=0.4,p0=0.8,p1=0.2,p2=0.1,θ=0. (a) Before Hopf bifurcation, t1=5. (b) After Hopf bifurcation, t1=7.

Figure 15

Numerical continuation of branches of periodic solutions from the Hopf bifurcation when β=0.1. Other parameter values are (a)p0cos(θ)=−0.8, p̃1=−0.2, p̃2=−0.1, γ=0.1; (b)p0cos(θ)=−0.8, p̃1=−0.2, p̃2=−0.1, γ=0.1. The curves show the maximum amplitude of η on the periodic solution for the corresponding value of 1∕T. (a) Traditional case. (b) Axial-torsional case.

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