Can nonlinear dynamics Can nonlinear dynamics contribute to chatter contribute to chatter

suppression?suppression?

Gábor StépánGábor Stépán

Department of Applied MechanicsBudapest University of Technology

and Economics

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

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).

(Moon, Johnson, 1996)

Efficiency of cutting

Specific amount of material cut within a certain time

where

w – chip width

h – chip thickness

Ω ~ cutting speed

2D

whV .

Modelling – regenerative effect

Mechanical model

τ – time period of revolution

Mathematical model

)(1

2 2 hFm

xxx xnn .. .

)()()( 0 txtxhth )()()( 0 txtxhthh

Milling

Mechanical model:

- number of cutting edgesin contact varies periodically with periodequal to the delay

)()(

)())(

()(2)( 112 txm

tktx

m

tktxtx nn

)()( 11 tktk

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

Stabilizing inverted pendula

Stephenson (1908): periodically forced pendulum

Mathematical background:

Mathieu equation (1868)

x = 0 can be stable inLjapunov sense for < 0

.

0))cos(()( 22 tlrmlgmlm SSSS

0)()cos()( txttx

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

Balancing with reflex delay

instabilityg

lcr 3

]s[1.0

]m[3.0

l

]Hz[5.24

10

f

)()()( tDtPtQ

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- Outlook: Act & wait control, periodic flow control

Stick&slip – unstable periodic motion

Experiments with brakepad-like arrangements(R Horváth, Budapest / Auburn)

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

The delayed Mathieu equation

Analytically constructed stability chart for testing numerical methods and algorithms

Time delay and time periodicity are equal:

Damped oscillator

Mathieu equation (1868)

Delayed oscillator (1941 – shimmy)

)2()()cos()()( txbtxttxtx

2T

0b0

0b

The damped oscillator

stable

Maxwell(1865)

Routh (1877)

Hurwitz (1895)

Lienard & Chipard (1917)

0)()()( txtxtx

Stability chart – Mathieu equation

Floquet (1883)

Hill (1886)

Rayleigh(1887)

van der Pol &

Strutt (1928)

Sinha (1992)

Strutt – Ince (1956) diagram swing(2000BC)

Stephenson’s inverted pendulum (1908)

0)()cos()( txttx

The damped Mathieu equation0)()cos()()( txttxtx

The delayed oscillator

Hsu & Bhatt (1966)

Stepan, Retarded Dynamical Systems (1989)

)2()()( txbtxtx

Delayed oscillator with damping

)2()()()( txbtxtxtx

The delayed Mathieu – stability charts

b=0

ε=1 ε=0

)2()()cos()( txbtxttx

Stability chart of delayed Mathieu

Insperger, Stepan (2002)

)2()()cos()( txbtxttx

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

Modelling – regenerative effect

Mechanical model

τ – time period of revolution

Mathematical model)(

12 2 hF

mxxx xnn

)()()( 0 txtxhth

)()()( 0 txtxhthh

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

Linear analysis – stability

Dimensionless time

Dimensionless chip width

Dimensionless cutting speed

TobiasTlusty, Altintas, BudakGradisek, Kalveram, Insperger

tt n~

)~(~)~()~1()~(2)~( ntxwtxwtxtx

kk

mk

wn

12

1~

nn

n

2

22~

2

)()()()(2)( 112 txmk

txmk

txtx nn.. .

Stability and bifurcations of turning

Subcritical Hopf bifurcation: unstable vibrations around stable cutting

211

atn1

21

jn

j

)1(2~ crw 21 ncr

The unstable periodic motion

Shi, Tobias

(1984) –

impactexperiment

Case study – thread cutting

m= 346 [kg]

k=97 [N/μm]

fn=84.1 [Hz]

ξ=0.025

gge=3.175[mm]

Stability of thread cutting – theory&exp.

Ω=344 f/p

Quasi-periodic

vibrations:

f1=84.5 [Hz]

f2=90.8 [Hz]

Machined surface

D=176 [mm], τ =0.175 [s]

]Hz[0.883.15

221

ff

]Hz[5.3)5.122(

3.152

21

ff

Self-interrupted cutting

High-speed milling

Parametrically interrupted cutting

Low number of edges

Low immersion

Highly interrupted

Highly interrupted cutting

Two dynamics:

- free-flight

- cutting with regenerative effect– like an impact

00,;3,2

111

100

Fm

vxbv

x

v

x

khkh

kj

hjhk

j

j

j

j A

),[ jj ttt

),[ jj ttt

Stability chart of H-S milling

Sense of the

period

doubling

(or flip)

bifurcation?

Linear model (Davies, Burns, Pratt, 2000)Simulation (Balachandran, 2000)

Subcritical flip bifurcation

Bifurcation diagram – chaos

The fly-over effect

00,;3,2

111

100

Fm

vxbv

x

v

x

khkh

kj

hjhk

j

j

j

j A

Both period-2s unstable at b)

Milling

Mechanical model:

- number of cutting edgesin contact varies periodically with periodequal to the delay

)()(

)())(

()(2)( 112 txm

tktx

mtk

txtx nn.. .

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)

The stable period-2 motion

Lobes & lenses with =0.02

(Szalai, Stepan, 2006)

with =0.0038

(Insperger,

Mann, Bayly,

Stepan, 2002)

Phase space reconstruction at A

Stable milling Unstable milling with (Gradisek et al.) stable period-2(?) or quasi-periodic(?) oscillation

Bifurcation diagram

(Szalai, Stepan, 2005)

Stability of up- and down-milling

Stabilization by time-periodic parameters!Insperger, Mann, S, Bayly (2002)

Contents- Motivation – high-speed milling

- Physical background Periodically constrained inverted pendulum,

and the swing Delayed PD control of the inverted pendulum Unstable periodic motion in stick-slip

- Periodic delayed oscillators: delayed Mathieu equ.

- Nonlinear vibrations of cutting processes

- State dependent regenerative effect

State dependent regenerative effect

3.0x

yr K

Kk

State dependent regenerative effect

State dependent time delay (x):

Without state dependence:

With state dependence, the chip thickness is

, fz – feed rate,

]0,[),()(

)()()(2

rsstxsx

xtxtxRR

t

t

2

))(()()(d)()()(

tt

t

xt

xtytyxvssyvtht

/zfv

R

f

Rv z

2

2 DoF mathematical model

Linearisation at stationary cutting (Insperger, 2006)

Realistic range of parameters:

Characteristic function

q

ttyyy

qttxxx

xtytyxvwKtyktyctym

xtytyxvwKtxktxctxm

))(()()()()()(

))(()()()()()(

)()()()()()()()(

)()()()()()()()(

1

1

ttRv

ttvwKtktctm

ttRv

ttvwKtktctm

qyyy

qxxx

01.0001.0 Rv

0e111212 11

22

nq

r

Kk

Stability chart – comparison

Conclusion

- Periodic modulation of cutting coefficient may result improvements in the stability, e.g., for high-speed milling, but

- It may also cause loss of stability via period-2 oscillations, leading to lenses (& lobes), too.

- Subcriticality results reduction in safe chatter-free parameter domain for turning, milling,…

- There is no nonlinear theory for state-dependent regenerative effect.

Thank you for your attention!