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