Can nonlinear dynamics contribute to chatter suppression? Gábor Stépán

1. Can nonlinear dynamics contribute to chatter suppression?Gábor Stépán Department of Applied MechanicsBudapest University of Technology and Economics

2. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • State dependent regenerative effect

3. Motivation: Chatter ~ (high frequency)machine tool vibration “… Chatter is the most obscure and delicate of all problems facing the machinist – probably no rules or formulae can be devised which will accurately guide the machinist in taking maximum cuts and speeds possible without producing chatter.” (Taylor, 1907). (Moon, Johnson, 1996)

4. Efficiency of cutting Specific amount of material cut within a certain time where w – chip width h – chip thickness Ω~cutting speed

6. Modelling – regenerative effect Mechanical model τ – time period of revolution Mathematical model

7. Milling Mechanical model: - number of cutting edgesin contact varies periodically with periodequal to the delay

8. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • State dependent regenerative effect

9. Stabilizing inverted pendula Stephenson (1908): periodically forced pendulum Mathematical background: Mathieu equation (1868) x = 0 can be stable inLjapunov sense for  < 0 .

10. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • State dependent regenerative effect

11. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swingDelayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • State dependent regenerative effect

12. Balancing with reflex delay  instability

13. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulumUnstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • Outlook: Act & wait control, periodic flow control

14. Stick&slip – unstable periodic motion Experiments with brakepad-like arrangements(R Horváth, Budapest / Auburn)

15. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ • Nonlinear vibrations of cutting processes • State dependent regenerative effect

16. The delayed Mathieu equation Analytically constructed stability chart for testing numerical methods and algorithms Time delay and time periodicity are equal: Damped oscillator Mathieu equation (1868) Delayed oscillator (1941– shimmy)

17. The damped oscillator stable Maxwell(1865) Routh (1877) Hurwitz (1895) Lienard & Chipard (1917)

18. Stability chart – Mathieu equation Floquet (1883) Hill (1886) Rayleigh(1887) van der Pol & Strutt (1928) Sinha (1992) Strutt – Ince (1956) diagramswing(2000BC) Stephenson’s inverted pendulum (1908)

19. The damped Mathieu equation

20. The delayed oscillator Hsu & Bhatt (1966) Stepan, Retarded Dynamical Systems (1989)

21. Delayed oscillator with damping

22. The delayed Mathieu – stability charts b=0 ε=1 ε=0

23. Stability chart of delayed Mathieu Insperger, Stepan (2002)

24. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ • Nonlinear vibrations of cutting processes • State dependent regenerative effect

25. Modelling – regenerative effect Mechanical model τ – time period of revolution Mathematical model

26. Cutting force ¾ rule for nonlinear cutting force Cutting coefficient

27. Linear analysis – stability Dimensionless time Dimensionless chip width Dimensionless cutting speed TobiasTlusty, Altintas, BudakGradisek, Kalveram, Insperger

28. Stability and bifurcations of turning Subcritical Hopfbifurcation:unstable vibrations around stable cutting

29. The unstable periodic motion Shi, Tobias (1984) – impactexperiment

30. Case study – thread cutting m= 346 [kg] k=97 [N/μm] fn=84.1 [Hz] ξ=0.025 gge=3.175[mm]

31. Stability of thread cutting – theory&exp. Ω=344 f/p Quasi-periodic vibrations: f1=84.5 [Hz] f2=90.8 [Hz]

32. Machined surface D=176 [mm], τ =0.175 [s]

33. Self-interrupted cutting

34. High-speed milling Parametrically interrupted cutting Low number of edges Low immersion Highly interrupted

35. Highly interruptedcutting Two dynamics: • free-flight • cutting with regenerative effect– like an impact

36. Stability chart of H-S milling Sense of the period doubling (or flip) bifurcation? Linear model (Davies, Burns, Pratt, 2000)Simulation (Balachandran, 2000)

37. Subcritical flip bifurcation

38. Bifurcation diagram – chaos

39. The fly-over effect

40. Both period-2s unstable at b)

41. Milling Mechanical model: - number of cutting edgesin contact varies periodically with periodequal to the delay

43. Phase space reconstruction A – secondary B – stable cutting C – period-2 osc. Hopf (tooth pass exc.) (no fly-over!!!) noisy trajectory from measurement noise-free reconstructed trajectory cutting contact(Gradisek,Kalveram)

44. The stable period-2 motion

45. Lobes & lenses with =0.02 (Szalai, Stepan, 2006)

46. with =0.0038 (Insperger, Mann, Bayly, Stepan, 2002)

47. Phase space reconstruction at A Stable milling Unstable milling with (Gradisek et al.) stable period-2(?) or quasi-periodic(?) oscillation

48. Bifurcation diagram (Szalai, Stepan, 2005)

49. Stability of up- and down-milling Stabilization by time-periodic parameters! Insperger, Mann, S, Bayly (2002)

50. Contents • Motivation – high-speed milling • Physical background Periodically constrained inverted pendulum, and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip • Periodic delayed oscillators: delayed Mathieu equ. • Nonlinear vibrations of cutting processes • State dependent regenerative effect