### Abstract

In this paper the nonlinear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.

Original language | English |
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Publication status | Published - 17 Jul 2007 |

### Bibliographical note

Additional information: Preprint submitted to the International Journal of Non-Linear MechanicsSponsorship: T.I. is supported by a János Bolyai research scholarship from the Hungarian Academy of Sciences and by the Hungarian National Science Foundation under grant no. OTKA F047318. D.A.W.B. is supported by a fellowship from the Lloyds Tercentenary Foundation. G.S. is supported by the Hungarian National Science Foundation under grant no. OTKA T043368.

### Keywords

- machine tool chatter
- Hopf bifurcation
- state-dependent delay

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## Cite this

Insperger, T., Barton, DAW., & Stepan, G. (2007).

*Criticality of Hopf bifurcation in state-dependent delay model of turning processes*. http://hdl.handle.net/1983/947