Nonlinear Regenerative Machine Tool Vibrations

1997 ◽  
Author(s):  
Gábor Stépán ◽  
Tamas Kalmar-Nagy
Keyword(s):  
Machine Tool ◽  
Analytical Formula ◽  
Chip Thickness ◽  
Domain Of Attraction ◽  
Equation Model ◽  
Center Manifold Reduction ◽  
Nonlinear Part ◽  
Dimensional Phase Space ◽  
Lengthy Calculation ◽  
Delay Differential

Abstract The existence and the nature of the Hopf bifurcation is presented in the delay-differential equation model of the so-called regenerative machine tool vibration. The relevant nonlinearity is considered at the cutting force dependence on the chip thickness. The delayed terms show a special algebraic structure in the nonlinear part of the equation of motion. This results in a surprisingly simple and useful analytical formula in the end of the lengthy calculation based on center manifold reduction in the corresponding infinite dimensional phase space. The result gives a simple way to estimate the domain of attraction of the stable stationary cutting as well as an estimation of that technological parameter domain, where the cutting is globally stable.

scholarly journals On the bistable zone of milling processes

2015 ◽  
Vol 373 (2051) ◽  
pp. 20140409 ◽  
Author(s):  
Zoltan Dombovari ◽  
Gabor Stepan
Keyword(s):  
Periodic Solution ◽  
Domain Of Attraction ◽  
Cutting Parameters ◽  
Modal Testing ◽  
Structural Behaviour ◽  
Infinite Dimensional ◽  
Dimensional Phase Space ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasi-periodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasi-periodic solutions and the checking of their grazing to a time-periodic switching surface in the infinite-dimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains.

Experimental and Analytical Investigation of the Subcritical Instability in Metal Cutting

1999 ◽  
Author(s):  
Tamás Kalmár-Nagy ◽  
Jon R. Pratt ◽  
Matthew A. Davies ◽  
Michael D. Kennedy
Keyword(s):  
High Speed ◽  
Metal Cutting ◽  
Stability Boundary ◽  
Analytical Investigation ◽  
Equation Model ◽  
Center Manifold Reduction ◽  
Single Degree Of Freedom ◽  
Subcritical Hopf Bifurcation ◽  
Delay Differential ◽  
Manifold Reduction

Abstract A single-degree-of-freedom dynamic cutting fixture is used to map out a part of the lobed stability boundary in a simple high-speed machining experiment. The experiment reveals the hysteretic nature of the instability. A 1 DOF mechanical model is derived using parameters identified from the experiment. We then show the existence of a subcritical Hopf bifurcation in this delay-differential equation model which corresponds to the observed experimental instability. The calculation is based on center manifold reduction. Then time domain simulation is used to solve the full nonlinear equation of motion that allows for the tool to leave the workpiece giving excellent agreement with the experiment.

scholarly journals A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

2014 ◽  
Vol 9 ◽  
pp. 53-68 ◽  
Author(s):  
B. El Boukari ◽  
N. Yousfi
Keyword(s):  
Reproduction Number ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Differential Equation Model ◽  
Globally Asymptotically Stable ◽  
Intracellular Delay ◽  
Immunodeficiency Virus ◽  
Delay Differential ◽  
Theoretical Results ◽  
Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeficiency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely defined by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay affects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

scholarly journals Non-Spiking Laser Controlled by a Delayed Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2069
Author(s):  
Anton V. Kovalev ◽  
Evgeny A. Viktorov ◽  
Thomas Erneux
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Optical Feedback ◽  
Laser Pulses ◽  
Equation Model ◽  
Rate Equations ◽  
Differential Equation Model ◽  
The Stability ◽  
Delay Differential ◽  
Intensity Laser

In 1965, Statz et al. (J. Appl. Phys. 30, 1510 (1965)) investigated theoretically and experimentally the conditions under which spiking in the laser output can be completely suppressed by using a delayed optical feedback. In order to explore its effects, they formulate a delay differential equation model within the framework of laser rate equations. From their numerical simulations, they concluded that the feedback is effective in controlling the intensity laser pulses provided the delay is short enough. Ten years later, Krivoshchekov et al. (Sov. J. Quant. Electron. 5394 (1975)) reconsidered the Statz et al. delay differential equation and analyzed the limit of small delays. The stability conditions for arbitrary delays, however, were not determined. In this paper, we revisit Statz et al.’s delay differential equation model by using modern mathematical tools. We determine an asymptotic approximation of both the domains of stable steady states as well as a sub-domain of purely exponential transients.

Cutting Dynamics in Machine Tool Chatter: Contribution to Machine-Tool Chatter Research—3

1965 ◽  
Vol 87 (4) ◽  
pp. 464-470 ◽  
Author(s):  
R. L. Kegg
Keyword(s):  
Machine Tool ◽  
Metal Cutting ◽  
Chip Thickness ◽  
Cutting Process ◽  
Dynamic Force ◽  
General Technique ◽  
Cross Sensitivity ◽  
Machine Tool Chatter ◽  
Measuring Techniques ◽  
Tool Chatter

This is one of four papers presented simultaneously on the general subject of chatter. This work is concerned with finding a representation of the dynamic metal-cutting process which is suitable for use in a linear closed-loop theory of stability of the system composed of the machine tool structure, the cutting process, and their means of combining. Measuring techniques for experimentally determining this behavior are discussed and some problems in the dynamic measurement of forces are explored. It is found that it is not at all sufficient to simply build a dynamometer whose lowest natural frequency is well beyond the range of interest. It is also shown that dynamic cross sensitivity can far exceed static cross sensitivity so that a more general technique for data correction developed in the present work must be used to calibrate dynamic force data. Results obtained to date with an oscillating tool and a flat uncut surface show that some phase, increasing with frequency, is always present between the dynamic cutting forces and the oscillatory uncut chip thickness. This phase is different for the two components of the resultant cutting force. It is felt that two mechanisms, both associated with the tool clearance flank, can explain most of the dynamic cutting effects found in testing.

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