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The evolution of time- delay models for high-performance manufacturing G ábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics. Contents. (1900…) 1950… turning - single discrete delay (RDDE) process damping - distributed delay (RFDE)

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  1. The evolution oftime-delay models for high-performance manufacturingGábor StépánDepartment of Applied MechanicsBudapest University of Technology and Economics

  2. Contents (1900…) 1950… • turning - single discrete delay (RDDE) • process damping - distributed delay (RFDE) • nonlinearities - bifurcations in RFDE • milling - non-autonomous RDDE • varying spindle speed - time-periodic delay • high-performance - state-dependent delay • forging - neutral DDE …2006

  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).

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

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

  6. Time delay models Delay differential equations (DDE): - simplest (populations) Volterra (1923) - single delay (production based on past prices) - average past values (production based on statisticsof past/averaged prices) • weighted w.r.t. the past(Roman law)

  7. Modelling – regenerative effect Mechanical model (Tlusty 1960, Tobias 1960) τ – time period of revolution Mathematical model

  8. Linear analysis – stability Dimensionless time Dimensionless chip width Dimensionless cutting speed

  9. Delay Diff Equ (DDE) – Functional DE Time delay & infinite dimensional phase space: Myshkis (1951) Halanay (1963) Hale (1977) Riesz Representation Theorem

  10. The delayed oscillator Pontryagin (1942) Nyquist (1949) Bellman & Cooke (1963) Olgac, Sipahi Hsu & Bhatt (1966) (Stepan: Retarded Dynamical Systems, 1989)

  11. Stability chart of turning But: better stability properties experienced at low and high cutting speeds!

  12. Short regenerative effect Stepan (1986)

  13. Weight functions

  14. Weight functions Experiments Usui (1978) Bayly (2000) Finite Elements Ortiz (1995) Analitical Davies (1998)

  15. Nonlinear cutting force ¾ rule for nonlinear cutting force Cutting coefficient

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

  17. Case study – thread cutting(1983) m= 346 [kg] k=97 [N/μm] fn=84.1 [Hz] ξ=0.025 gge=3.175[mm]

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

  19. Stability and bifurcations of turning Hale (1977) Hassard (1981) Subcritical Hopfbifurcation (S, 1997):unstable vibrations around stable cutting

  20. Bifurcation diagram

  21. Phase space structure

  22. Milling (1995 - ) Mechanical model: - number of cutting edgesin contact varies periodically with periodequal to the delay

  23. The delayed Mathieu – stability charts b=0 (Strutt, 1928) ε=1 ε=0 (Hsu, Bhatt, 1966)

  24. Stability chart of delayed Mathieu Insperger, Stépán (2002)

  25. Test of damped delayed Mathieu equ.

  26. Measured and processed signals A B C

  27. 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)

  28. Animation of stable period doubling

  29. Lenses

  30. Stability chart  = 0.05 … 0.1 … 0.2

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

  32. Stabilization by time-periodic time delay Chatter suppression by spindle speed modulation:

  33. Improved stability properties (Hard to realize…)

  34. State dependent regenerative effect

  35. State dependent regenerative effect State dependent time delay  (xt): Without state dependence (at fixed point): Trivial solution: With state dependence, the chip thickness is , fz – feed rate, Krisztin, Hartung (2005), Insperger, S, Turi (2006)

  36. 2 DoF mathematical model Linearisation at stationary cutting (Insperger, 2006) Realistic range of parameters: Characteristic function

  37. Stability charts – comparison

  38. Forging Lower tup: 105 [t]  (Upper tup: 21 [t]“hammer”)

  39. with boundary conditions  Initial conditions: Traveling wave solution

  40. Neutral DDE  With initial function

  41. Impact – elastic & plastic traveling waves

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