Date post: | 29-Mar-2015 |
Category: |
Documents |
Upload: | aryanna-hammill |
View: | 214 times |
Download: | 1 times |
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
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
2D
whV .
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 .
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
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
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
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
The delayed oscillator
Pontryagin (1942)
Nyquist (1949)
Bellman &
Cooke (1963)
Olgac, Sipahi
Hsu & Bhatt (1966)
(Stepan: Retarded Dynamical Systems, 1989)
)2()()( txbtxtx
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
Short regenerative effect
Stepan (1986)
d)()()(0
1
txtxpmk
)()(2)( 2 txtxtx nn
Weight functions
0
1d)( p
/e
1)( p
q
05.0
01.02
D
lq
Weight functions
Experiments
Usui (1978)
Bayly (2000)
Finite Elements
Ortiz (1995)
Analitical
Davies (1998)
)cos(1
1)(
p
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
The unstable periodic motion
Shi, Tobias
(1984) –
impactexperiment
Case study – thread cutting (1983)
m= 346 [kg]
k=97 [N/μm]
fn=84.1 [Hz]
ξ=0.025
gge=3.175[mm]
Machined surface
D=176 [mm], τ =0.175 [s]
]Hz[0.883.15
221
ff
]Hz[5.3)5.122(
3.152
21
ff
Stability and bifurcations of turning
Hale (1977)
Hassard (1981)
Subcritical Hopf bifurcation (S, 1997): unstable vibrations around stable cutting
211
atn1
21
jn
j
Bifurcation diagram
Phase space structure
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
The delayed Mathieu – stability charts
b=0 (Strutt, 1928)
ε=1 ε=0 (Hsu, Bhatt, 1966)
)2()()cos()( txbtxttx
Stability chart of delayed Mathieu
Insperger,
Stépán (2002)
)2()()cos()( txbtxttx
Test of damped delayed Mathieu equ.)2()()cos()()( txbtxttxtx
Measured and processed signals
A
B
C
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:
))~(~~(~)~()~1()~(2)~( ttxwtxwtxtx
)~~cos(~~)~(~10 tt m
)~~/(2/ 00 mmP TR
)~/~/ 0101 AR
Improved stability properties
(Hard to realize…)
2PR
1.0AR
0AR
State dependent regenerative effect
/zfv
3.0x
yr K
Kk
R
f
R
v z
2
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
)(,
)(
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
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
)(
)(
)(
)(
ctf
tx
tx
t w
Impact – elastic & plastic traveling waves