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Identification of nonlinear characteristics based on bistability in delayed model of cutting

Identification of nonlinear characteristics based on bistability in delayed model of cutting. G Stepan , Z Dombovari Department of Applied Mechanics Budapest University of Technology and Economics J Munoa Ideko Research Alliance IK4, Danobat Group. Introduction to cutting.

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Identification of nonlinear characteristics based on bistability in delayed model of cutting

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  1. Identification of nonlinear characteristics based on bistability in delayed model of cutting G Stepan, Z Dombovari Department of Applied Mechanics Budapest University of Technology and Economics J Munoa Ideko Research Alliance IK4, Danobat Group

  2. Introduction to cutting Specific amount of material cut within a certain time where w – chip width h – chip thickness v – cutting speed Ω~cutting speed Cutting force

  3. Introduction to milling Number of cutting edgesin contact varies periodically with periodequal to the delay between two subsequent cutting edges. Thus, the resultant cutting force also varies with the same period.

  4. The goal – cutting force characteristics “high performance”

  5. Cutting force characteristics How to measure/identify? { } nonlinearities? Linear (Taylor): } Shifted lin. (Altintas): uniqueness? Power (Kienzle): Exponential (Endres): Cubic pol. (Tobias):

  6. Preliminaries • Classical experiment (Tobias, Shi, 1984) • cutting process is sensitive tolarge perturbations • self excited vibrations (chatter) “around” stable cutting • important effect of chip thicknesson size of unsafe zone 2/17

  7. Mechanical model of turning τ – time period of revolution

  8. A pair of complex conjugate roots at stability limit Transversality condition Linear stability & Hopf Bifurcation 18/27

  9. Centre manifold reduction, and calculation of Poincare-Ljapunov constant (PLC) since and Subcritical Hopf bifurcation 19/27

  10. Unstable limit cycle and bi-stable zone  20/27

  11. Fly-over • Dombovari, Barton, Wilson • Stepan, 2010

  12. Variation of the bi-stable zone Tobias, Shi  9/10

  13. Model of milling Mechanical model: - number of cutting edgesin contact varies periodically with periodequal to the delay

  14. High-speedmilling Theory &experiments: stability chart (Insperger,Mann, Stepan,Bayly, 2004, also groupsin Dortmund,Ljubljana,…)

  15. Turning (Tobias, Tlusty, 1960)

  16. Newtonian impact theory and regenerative effect (Davies, Burns, Dutterer, Pratt,…Insperger, Stépán, 2001 Szalay, Stépán, 2002 – subcr, flip)

  17. Semi-discretization method – Insperger, Stépán Multi-frequency method – Merdol, Altintas Time Finite Element method – Bayly, Mann,… Full discretization – Altintas, Balachandran,… Period-doubling(Corpus, Endres)

  18. = 0.05… 0.1 … 0.2 Characteristic matrices(Szalai, 2006) Experiments on lenses/islands (Zatarian, Mann, 2008)

  19. Time averaging (basic Fourier component) provides satisfactory stability limits, bifurcations (Tobias, Tlusty, Minis,… 1965…1995, Altintas, Budak – multi DoF, single frequency… 1998), but the frequency content is rich (Insperger,... 2003)

  20. Dynamic experiment for cutting force

  21. Unsafe/bistable zone identification

  22. Checking the hysteresis loop

  23. Differential equation of cutting force characteristics + 2= = From the Hopf calculation: where we can measure: 

  24. Example: size w of bistable zone does not depend on chip thickness h Eulerian-type diff. equ, , , With the boundary conditions , softening  With a typical measured value of Typical power law

  25. The experiment

  26. Evaluation of the results

  27. Force characteristics reconstruction

  28. Conclusion The invers application of the results of the Hopf bifurcation calculation in case of regenerative machine tool vibrations makes it possible to measure the nonlinear cutting force characteristics with cheap accelerometers only in a fast and accurate way. Thank you for your attention!

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