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

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

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).

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

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

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)

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

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

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

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

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

Short regenerative effect Stepan (1986)

Weight functions

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

Nonlinear cutting force ¾ rule for nonlinear cutting force Cutting coefficient

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

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

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

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

Bifurcation diagram

Phase space structure

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

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

Stability chart of delayed Mathieu Insperger, Stépán (2002)

Test of damped delayed Mathieu equ.

Measured and processed signals ABCABC

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)

Animation of stable period doubling

Lenses

Stability chart = 0.05 … 0.1 … 0.2

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

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

Improved stability properties (Hard to realize…)

State dependent regenerative effect

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)

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

Stability charts – comparison

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

with boundary conditions Initial conditions: Traveling wave solution

Neutral DDE With initial function

Impact – elastic & plastic traveling waves