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

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

The goal – cutting force characteristics “high performance”

Cutting force characteristics nonlinearities? uniqueness? } } { How to measure/identify?

Preliminaries Classical experiment (Tobias, Shi, 1984) cutting process is sensitive to large perturbations self excited vibrations (chatter) “around” stable cutting important effect of chip thickness on size of unsafe zone 2/17

Mechanical model of turning τ – time period of revolution

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

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

Unstable limit cycle and bi-stable zone 20/27

Fly-over Dombovari, Barton, Wilson Stepan, 2010

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

Model of milling Mechanical model: - number of cutting edges in contact varies periodically with period equal to the delay

High-speed milling Theory & experiments: stability chart (Insperger, Mann, Stepan, Bayly, 2004, also groups in Dortmund, Ljubljana,…)

Turning (Tobias, Tlusty, 1960)

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

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

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

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, )

Dynamic experiment for cutting force

Unsafe/bistable zone identification

Checking the hysteresis loop

Differential equation of cutting force characteristics From the Hopf calculation: where we can measure:

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

The experiment

Evaluation of the results

Force characteristics reconstruction

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!