Chaotic Vibration of a Simple Model of the Machine Tool-Cutting Process System

Wiercigroch, Marian

doi:10.1115/1.2889747

Cited by 86 publications

(51 citation statements)

References 11 publications

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“…Bespoke computer algebra software is used to generate the perturbation equations (see references [33,34]) and to provide the required analytical solutions to a pre-specified level of approximation accuracy. It is also shown that the analytically complicated equations of motion due to Wiercigroch [23] can be simplified somewhat without too much loss of generality or accuracy. This simplification applies specifically to the orthogonal cutting forces f x and f y and is discussed further within the following analysis.…”

Section: Introductionmentioning

confidence: 99%

“…Chaotic motions have been observed in the responses of chattering machine-tool systems by Grabec [4,5] and Wiercigroch [22,23], Wiercigroch and Cheng [24], Warmi ! n nski et al [25], Kalmar-Nagy and Pratt [26], and Wiercigroch and Krivtsov [27].…”

Section: Introductionmentioning

confidence: 99%

“…This paper looks, for the first time, at the stability regions in the context of the depth of cut and the cutting speed co-ordinates, and in order to achieve this the principal foundation of the work has been based on an investigation of the original dynamical model proposed by Wiercigroch [23], noting that this model has been derived from that of Grabec [4,5]. In this paper, Wiercigroch's model has been developed to include the effects of Coulomb friction, but in the context of analytical rather than numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

Warmiński

Litak

Cartmell³

et al. 2003

Journal of Sound and Vibration

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This paper considers an accepted model of the metal cutting process dynamics in the context of an approximate analysis of the resulting non-linear differential equations of motion. The process model is based upon the established mechanics of orthogonal cutting and results in a pair of non-linear ordinary differential equations which are then restated in a form suitable for approximate analytical solution. The chosen solution technique is the perturbation method of multiple time scales and approximate closed-form solutions are generated for the most important non-resonant case. Numerical data are then substituted into the analytical solutions and key results are obtained and presented. Some comparisons between the exact numerical calculations for the forces involved and their reduced and simplified analytical counterparts are given. It is shown that there is almost no discernible difference between the two thus confirming the validity of the excitation functions adopted in the analysis for the data sets used, these being chosen to represent a real orthogonal cutting process. In an attempt to provide guidance for the selection of technological parameters for the avoidance of primary chatter, this paper determines for the first time the stability regions in terms of the depth of cut and the cutting speed co-ordinates. #

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

Warmiński

Litak

Cartmell³

et al. 2003

Journal of Sound and Vibration

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“…Recently many papers tackled that problem focusing on the conditions of appearing such vibrations. Apart from regular (periodic or quasi periodic) vibrations, possibilities of chaotic ones were also investigated theoretically [1][2][3][4][5][6][7][8][9][10] as well as experimentally [5,11,12]. The aim of this paper is to explore the mechanism of the chatter vibration generation using a simple model of a cutting process with one degree of freedom.…”

Section: Introductionmentioning

confidence: 99%

Chaotic vibrations in a regenerative cutting process

Litak

2002

Chaos, Solitons & Fractals

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We have analyzed vibrations generated in an orthogonal cutting process. Using a simple one degree of freedom model of the regenerative cutting we have observed the complex behaviour of the system. In presence of a shaped cutting surface, the nonlinear interaction between the tool and a workpiece leads the to chatter vibrations of periodic, quasi-periodic or chaotic type depending on system parameters. To describe the profile of the surface machined by the first pass we used a harmonic function. We analyzed the impact phenomenon between the tool and a workpiece after their contact loss.

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“…Specific examples include atomic force microscopy [5,6], gear assemblies [7,8], metal cutters [9,10], and vibrating heat-exchanger tubes [11,12]. Piecewisesmooth systems are characterized by the presence of codimension-one regions of phase space, termed switching manifolds, across which the functional form of the system changes.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Regular Grazing Bifurcations

Simpson¹,

Hogan²,

Kuske³

2013

SIAM J. Appl. Dyn. Syst.

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A grazing bifurcation corresponds to the collision of a periodic orbit with a switching manifold in a piecewise-smooth ODE system and often generates complicated dynamics. The lowest order terms of the induced Poincaré map expanded about a regular grazing bifurcation constitute a Nordmark map. In this paper we study a normal form of the Nordmark map in two dimensions with additive Gaussian noise of amplitude, ε. We show that this particular noise formulation arises in a general setting and consider a harmonically forced linear oscillator subject to compliant impacts to illustrate the accuracy of the map. Numerically computed invariant densities of the stochastic Nordmark map can take highly irregular forms, or, if there exists an attracting period-n solution when ε = 0, be well approximated by the sum of n Gaussian densities centred about each point of the deterministic solution, and scaled by 1 n , for sufficiently small ε > 0. We explain the irregular forms and calculate the covariance matrices associated with the Gaussian approximations in terms of the parameters of the map. Close to the grazing bifurcation the size of the invariant density may be proportional to √ ε, as a consequence of a square-root singularity in the map. Sequences of transitions between different dynamical regimes that occur as the primary bifurcation parameter is varied have not been described previously.

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Chaotic Vibration of a Simple Model of the Machine Tool-Cutting Process System

Cited by 86 publications

References 11 publications

Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

Approximate Analytical Solutions for Primary Chatter in the Non-Linear Metal Cutting Model

Chaotic vibrations in a regenerative cutting process

Stochastic Regular Grazing Bifurcations

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