State Dependent Regenerative Delay in Milling Processes

Insperger, Tamás; ́n, Ga ́bor Ste ́pa; Hartung, Ferenc; Turi, János

doi:10.1115/detc2005-85282

Cited by 22 publications

(29 citation statements)

References 22 publications

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“…Thus, if the delay is constant and equal to its steady-state value τ n0 , i.e., the angular velocity of the bit is constant, the angular perturbation is constant, ϕ =φ, and the term in α 0 disappears from the evolution equations (15)- (16).…”

Section: Scalingmentioning

confidence: 99%

“…In this respect, we follow the linearization approach presented in [16,17], which is itself based on the proof proposed in [24]. It relies on the stability assessment by localization in the complex plane of the roots of the characteristic equation associated with an equivalent linear system, i.e., a system with constant delay that has stability properties identical to those of the original system.…”

Section: Preamblementioning

confidence: 99%

“…Models built on the same rate-independent bit/rock interface law, but that include additional degrees of freedom and viscous dissipation in the representation of the drillstring and/or consider alternate boundary conditions at the rig, have recently been studied [12][13][14][15]; they yield similar although richer responses than the basic RGD model. Also, comparable state-dependent delay models to analyze regenerative chatter in metal turning processes have been independently developed by Insperger et al [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Instability regimes and self-excited vibrations in deep drilling systems

Depouhon

Detournay

2014

Journal of Sound and Vibration

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This paper analyzes the stability of the discrete model proposed by Richard et al. [1,2] to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω 0 of the drillstring imposed at the rig. On the one hand, if Ω 0 is larger than a critical velocity Ω c , the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω 0 is smaller than Ω c , the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.

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Section: Scalingmentioning

confidence: 99%

Section: Preamblementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Instability regimes and self-excited vibrations in deep drilling systems

Depouhon

Detournay

2014

Journal of Sound and Vibration

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“…This consideration results in a state-dependent delay differential equation, with the delay being the solution of an implicit equation that relies on the history of the bit angular motion. Thus, in contrast to the work on the self-excited vibrations of machine tools, where with few exceptions [20,21] the delay is considered constant, the variation of the angular velocity affecting the bit due to the large torsional compliance of the drilling structure needs to be accounted for. Combination of the depth of cut, the difference between the current and a delayed axial position of the bit, as the kinematical state variable of the interface laws with a variable delay conspires to create a two-way coupling between the axial and torsional modes of vibrations at the bit-rock interface.…”

Section: Introductionmentioning

confidence: 99%

Multiple mode analysis of the self-excited vibrations of rotary drilling systems

Germay¹,

Denoël

Detournay

2009

Journal of Sound and Vibration

133

101

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This paper extends the approach proposed by to analyze the axial and torsional vibrations of drilling systems that are excited by the particular boundary conditions at the drag bit-rock interface, by basing the formulation of the model on a continuum representation of the drillstring rather than on a characterization of the drilling structure by a two degree of freedom system. These boundary conditions account for both cutting and frictional contact at the interface. The cutting process combined with the quasi-helicoidal motion of the bit leads to a regenerative effect that introduces a coupling between the axial and torsional modes of vibrations and a state-dependent delay in the governing equations, while the frictional contact process is associated with discontinuities in the boundary conditions when the bit sticks in its axial and angular motion. The dynamic response of the drilling structure is computed using the finite element method. While the general tendencies of the system response predicted by the discrete model are confirmed by this computational model (for example that the occurence of stick-slip vibrations as well as the risk of bit bouncing are enhanced with an increase of the weight-on-bit or a decrease of the rotational speed), new features in the self-excited response of the drillstring can be detected. In particular, stickslip vibrations are predicted to occur at natural frequencies of the drillstring different from the fundamental one (as sometimes observed in field operations), depending on the operating parameters.

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“…Isolated islands of instability due to the periodicity of the process [22] or nonzero helix angles [23,24] have recently been discovered. Most recently, the milling process has been investigated for the special cases of variable timedelays [25], variable pitch [26], and state-dependent regenerative delay [27].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

Butcher

Bobrenkov

Bueler

et al. 2009

Journal of Computational and Nonlinear Dynamics

112

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In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems like milling are modeled by delay-differential equations (DDEs) with timeperiodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up-and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces.The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations CND-08-1016, Butcher 2 are found, and an in-depth investigation of the optimal stable immersion levels for downmilling in the vicinity of where the average cutting force changes sign is presented.

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State Dependent Regenerative Delay in Milling Processes

Cited by 22 publications

References 22 publications

Instability regimes and self-excited vibrations in deep drilling systems

Instability regimes and self-excited vibrations in deep drilling systems

Multiple mode analysis of the self-excited vibrations of rotary drilling systems

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

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