The evolution of time-delay models for high-performance manufacturing G

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The evolution of time-delay models for
high-performance manufacturingGábor
StépánDepartment of Applied MechanicsBudapest
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
• O 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
• O 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 statisticsof
  past/averaged prices)
• weighted w.r.t. the past(Roman law)

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Modelling regenerative effect

• Mechanical model (Tlusty 1960, Tobias 1960)
• t 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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Weight functions

• 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)
• impactexperiment

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Case study thread cutting (1983)

• 
  m 346 kg
• 
  k97 N/µm
• 
  fn84.1 Hz
• 
  ?0.025
• 
  gge3.175mm

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Machined surface

• D176 mm, t 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

• (1995 - )
• Mechanical model
• - number of cutting edgesin contact varies
  periodically with periodequal to the delay

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The delayed Mathieu stability charts

• 
  b0 (Strutt, 1928)
• 
  e1
  e0 (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

• A
• B
• C

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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 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)

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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 thammer)

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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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(No Transcript)
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Impact elastic plastic traveling waves