the evolution of time-delay models for high-performance manufacturing the evolution of time-delay...
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The evolution ofThe evolution of time-time-delay delay models for high-performance models for high-performance
manufacturingmanufacturingGábor Stépán
Department 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
Ω ~ cutting speed
2D
whV .
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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
2D
whV .
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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)
)1()( txtx
)()( tcxtx
d)()(0
txtx
d)()()(0
txwtx
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Modelling – regenerative effect
Mechanical model (Tlusty 1960, Tobias 1960)
τ – time period of revolution
Mathematical model)(
12 2 hF
mxxx xnn
)()()( 0 txtxhth
)()()()( 0 txtxhthth
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Linear analysis – stability
Dimensionless time
Dimensionless chip width
Dimensionless cutting speed
tt n~
)~(~)~()~1()~(2)~( ntxwtxwtxtx
kk
mk
wn
12
1~
nn
n
2
22~
2
)()()()(2)( 112 txm
ktx
m
ktxtx nn
0e~)~1(22 nww
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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
)()( txLtx
)()( txxt
]0,[
0
)(d)()(
txtx
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The delayed oscillator
Pontryagin (1942)
Nyquist (1949)
Bellman &
Cooke (1963)
Olgac, Sipahi
Hsu & Bhatt (1966)
(Stepan: Retarded Dynamical Systems, 1989)
)2()()( txbtxtx
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Stability chart of turning
But: better stability properties experienced at low and high cutting speeds!
211
atn1
21
jn
j
)1(2~ crw 21 ncr
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Short regenerative effect
Stepan (1986)
d)()()(0
1
txtxpmk
)()(2)( 2 txtxtx nn
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Weight functions
0
1d)( p
/e
1)( p
q
05.0
01.02
D
lq
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Weight functions
Experiments
Usui (1978)
Bayly (2000)
Finite Elements
Ortiz (1995)
Analitical
Davies (1998)
)cos(1
1)(
p
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Nonlinear cutting force
¾ rule for nonlinear
cutting force
Cutting coefficient
4/31),( whchwFx
4/10101 4
3),(),(
0
whc
h
hwFhwk
h
x
...)()( 33
2210, hkhkhkFF xx
0
12 8
1hk
k
20
13 96
5hk
k
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The unstable periodic motion
Shi, Tobias
(1984) –
impactexperiment
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Case study – thread cutting (1983)
m= 346 [kg]
k=97 [N/μm]
fn=84.1 [Hz]
ξ=0.025
gge=3.175[mm]
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Machined surface
D=176 [mm], τ =0.175 [s]
]Hz[0.883.15
221
ff
]Hz[5.3)5.122(
3.152
21
ff
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Stability and bifurcations of turning
Hale (1977)
Hassard (1981)
Subcritical Hopf bifurcation (S, 1997): unstable vibrations around stable cutting
211
atn1
21
jn
j
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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
)()(
)())(
()(2)( 112 txm
tktx
m
tktxtx nn
)()( 11 tktk
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The delayed Mathieu – stability charts
b=0 (Strutt, 1928)
ε=1 ε=0 (Hsu, Bhatt, 1966)
)2()()cos()( txbtxttx
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Stability chart of delayed Mathieu
Insperger,
Stépán (2002)
)2()()cos()( txbtxttx
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Test of damped delayed Mathieu equ.)2()()cos()()( txbtxttxtx
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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:
))~(~~(~)~()~1()~(2)~( ttxwtxwtxtx
)~~cos(~~)~(~10 tt m
)~~/(2/ 00 mmP TR
)~/~/ 0101 AR
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Improved stability properties
(Hard to realize…)
2PR
1.0AR
0AR
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State dependent regenerative effect
/zfv
3.0x
yr K
Kk
R
f
R
v z
2
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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)
]0,[),()(),()()(2 rtxxxtxtxRR tt
2
))(()()(d)()()(
tt
t
xt
xtytyxvyvtht
/zfv
R
f
Rv z
2
y
qy
x
qx
k
vwKy
k
vwKx
)(,
)(
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2 DoF mathematical model
Linearisation at stationary cutting (Insperger, 2006)
Realistic range of parameters:
Characteristic function
q
ttyyy
qttxxx
xtytyxvwKtyktyctym
xtytyxvwKtxktxctxm
))(()()()()()(
))(()()()()()(
)()()()()()()()(
)()()()()()()()(1
1
ttttvwKtktctm
ttttvwKtktctmq
yyy
qxxx
01.0001.0
R
v
0e111212 11
22
nq
r
Kk
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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
)(
)(
)(
)(
ctf
tx
tx
t w
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Impact – elastic & plastic traveling waves