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The evolution of time-delay models for high-performance manufacturing The evolution of time-delay models for high-performance manufacturing Gábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics

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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

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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).

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

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

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

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

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

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

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

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

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Short regenerative effect Stepan (1986)

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Weight functions

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Experiments Usui (1978) Bayly (2000) Finite Elements Ortiz (1995) Analitical Davies (1998)

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Nonlinear cutting force ¾ rule for nonlinear cutting force Cutting coefficient

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The unstable periodic motion Shi, Tobias (1984) – impact experiment

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

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

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

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Bifurcation diagram

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Phase space structure

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Milling ( ) Mechanical model: - number of cutting edges in contact varies periodically with period equal to the delay

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

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Stability chart of delayed Mathieu Insperger, Stépán (2002)

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Test of damped delayed Mathieu equ.

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Measured and processed signals ABCABC

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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)

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Animation of stable period doubling

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Lenses

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Stability chart = 0.05 … 0.1 … 0.2

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

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

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Improved stability properties (Hard to realize…)

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State dependent regenerative effect

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State dependent time delay (x t ): Without state dependence (at fixed point): Trivial solution: With state dependence, the chip thickness is, f z – feed rate, Krisztin, Hartung (2005), Insperger, S, Turi (2006)

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

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Stability charts – comparison

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Forging Lower tup: 105 [t] (Upper tup: 21 [t] hammer)

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with boundary conditions Initial conditions: Traveling wave solution

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Neutral DDE With initial function

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Impact – elastic & plastic traveling waves

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