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Delay induced canards in a model of high speed machining. (English) Zbl 1185.37135

The authors formulate the evolution equation for a machine-tool system subjected to a delayed feedback and investigate the small delay limit corresponding to the high speed revolution of the spindle of the cutting machine. The equation of motion for zero delay is reformulated as a weakly perturbed conservative problem with a non-trivial effect of the delayed feedback. It is constructed the limit-cycle solution which emerges from the Hopf bifurcation at the usage of averaging technique for weakly perturbed but strongly nonlinear conservative oscillators. The critical value of the control parameter is determined, where the amplitude of the oscillations is suddenly increased (so named canard explosion). The validity of the suggested asymptotic analysis is evaluated by comparisons between the numerical bifurcation diagram for the original delayed differential equation, its approximation for small delay and the analytical predictions.

MSC:

37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
70K20 Stability for nonlinear problems in mechanics
34K19 Invariant manifolds of functional-differential equations
70F40 Problems involving a system of particles with friction
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References:

[1] Callot J, C.R. Acad. Sc. Paris Série A 286 pp 1059– (1978)
[2] Benoît E, Collectanea Mathematica 31 pp 37– (1981)
[3] DOI: 10.1007/BFb0062381 · Zbl 0509.34037 · doi:10.1007/BFb0062381
[4] DOI: 10.1002/9783527622313 · Zbl 1130.93001 · doi:10.1002/9783527622313
[5] DOI: 10.1016/j.physd.2004.01.038 · Zbl 1099.34065 · doi:10.1016/j.physd.2004.01.038
[6] Erneux T, Applied Delay Differential Equations, Series: Surveys and Tutorials in the Applied Mathematical Sciences 3 (2009)
[7] DOI: 10.1080/14689360110105788 · Zbl 1095.70011 · doi:10.1080/14689360110105788
[8] DOI: 10.1063/1.1707586 · doi:10.1063/1.1707586
[9] Oxley P, The mechanics of machining (1989)
[10] DOI: 10.1007/s00332-003-0553-1 · Zbl 1056.37098 · doi:10.1007/s00332-003-0553-1
[11] DOI: 10.1137/S0036139993248853 · Zbl 0809.34077 · doi:10.1137/S0036139993248853
[12] DOI: 10.1006/jdeq.1995.1144 · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[13] DOI: 10.1103/PhysRevE.49.203 · doi:10.1103/PhysRevE.49.203
[14] DOI: 10.1137/0146047 · Zbl 0614.92008 · doi:10.1137/0146047
[15] DOI: 10.1137/0152095 · Zbl 0782.34040 · doi:10.1137/0152095
[16] DOI: 10.1016/S0022-0396(02)00148-1 · Zbl 1044.34026 · doi:10.1016/S0022-0396(02)00148-1
[17] DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[18] Engelborghs K, Technical Report TW-330, in: DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations (2001)
[19] DOI: 10.1137/1.9780898718195 · Zbl 1003.68738 · doi:10.1137/1.9780898718195
[20] DOI: 10.1142/S0218127400001742 · Zbl 0978.34038 · doi:10.1142/S0218127400001742
[21] DOI: 10.1142/S0218127401003486 · Zbl 1091.65510 · doi:10.1142/S0218127401003486
[22] DOI: 10.1006/jdeq.2000.3929 · Zbl 0994.34032 · doi:10.1006/jdeq.2000.3929
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