Department of Applied Mechanics
Budapest 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
V
.
 wh

D
2 where w
– chip width h
– chip thickness
Ω
~ cutting speed

Modelling – regenerative effect
Mechanical model h ( t )
 h

 h
0
 x ( t h ( t )
 h
0
 
)

 x ( t x ( t )
 
)
 x ( t )
τ – time period of revolution
Mathematical model
..
x

2
 n x
.
  n
2 x

1 m

F x
(
 h )

Milling
Mechanical model:
- number of cutting edges in contact varies periodically with period equal to the delay
 x  ( t )

2
 n x  ( t )

(
 n
2  k
1
( t )
) x ( t ) m
 k
1
( t ) x ( t m k
1
( t
 
)
 
)
 k
1
( t )

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
( m l
S
2  
S
)
  
(
 m g l
S
 m r

2 l
S cos(
 t ))
 
0
.
Mathematical background:
Mathieu equation (1868)
 x  ( t )

(
   cos t ) x ( t )

0 x = 0 can be stable in
Ljapunov sense for

< 0

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
   cr
 l
 instability
3 g l


0 .
3 [ m ]


0 .
1 [ s ]
Q ( t )

P

( t
 
)

D
  ( t
 
)
0
 f

1
4


2 .
5 [ Hz ]

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
 x  ( t )
  x  ( t )

(
   cos t ) x ( t )
 b x ( t

2

) b
Time delay and time periodicity are equal:
T
  
2

  
0 Damped oscillator b



0
0
Mathieu equation (1868)
Delayed oscillator (1941 – shimmy)

The damped oscillator
 x  ( t )
   ( t )
  x ( t )

0 stable
Maxwell(1865)
Routh (1877)
Hurwitz (1895)
Lienard &
Chipard (1917)

Stability chart – Mathieu equation
 x  ( t )

(
   cos t ) x ( t )

0
Floquet (1883)
Hill (1886)
Rayleigh(1887) van der Pol &
Strutt (1928)
Sinha (1992)
Strutt – Ince (1956) diagram swing(2000BC)
Stephenson’s inverted pendulum (1908)

The damped Mathieu equation
 x  ( t )
   ( t )

(
   cos t ) x ( t )

0

The delayed oscillator
 x  ( t )
  x ( t )
 b x ( t

2

)
Hsu & Bhatt (1966)
Stepan, Retarded Dynamical Systems (1989)

Delayed oscillator with damping
 x  ( t )
  x  ( t )
  x ( t )
 b x ( t

2

)

The delayed Mathieu – stability charts
 x  ( t )

(
   cos t ) x ( t )
 b x ( t

2

) b =0
ε
=0
ε =1

Stability chart of delayed Mathieu
 x  ( t )

(
   cos t ) x ( t )
 b x ( t

2

)
Insperger,
Stepan (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

Modelling – regenerative effect
Mechanical model h ( t )
 h
0
 x ( t
 
)
 x ( t )
 h
 h ( t )
 h
0
 x ( t
 
)
 x ( t )
τ – time period of revolution
Mathematical model
 x  
2
 n x    2 n x

1 m

F x
(
 h )

Cutting force
¾ rule for nonlinear cutting force
F x
( w , h )
 c
1 wh
3 / 4
F x

F x , 0
 k
1
 h
 k
2
(
 h )
2
Cutting coefficient k
1
( w , h
0
)


F x
( w
 h
, h ) h
0

 k
3
(
 h )
3 
...
3
4 c
1 wh
0

1 / 4 k
2
 
1
8 k h
0
1 k
3

5
96 k
1 h
0
2

Linear analysis – stability
..
x ( t )

2
 n
.
x ( t )

(
 n
2  k m
1 ) x ( t )
 k m
1 x ( t
 
)
Dimensionless time t
~   n t x

( t
~
)

2
 x

( t
~
)

( 1
 ~
) x ( t
~
)

Dimensionless chip width
Dimensionless cutting speed
Tobias
Tlusty, Altintas, Budak
2

 ~
Gradisek, Kalveram, Insperger x ( t
~   n

)
 k
1 m
 n
2
 k
1 k

2

 n



2

2
 n 


 n

Stability and bifurcations of turning
Subcritical Hopf bifurcation: unstable vibrations around stable cutting

 n j w cr

1
 j

1
 atn

2

( 1

 cr


 n
)
2

1
1

2

1

2


The unstable periodic motion
Shi, Tobias
(1984) – impact experiment

Case study – thread cutting m= 346 [kg] k= 97 [N/μm] f n
=84.1 [Hz]
ξ=0.025
gge=3.175[mm]

Stability of thread cutting – theory&exp.
Ω=344 f/p
Quasi-periodic vibrations: f
1
= 84.5 [Hz] f
2
= 90.8 [Hz]

Machined surface f
1 f
1
D =176 [mm],
τ
=0.175 [s]


2
2 f f
2
2


(
15 .
3
2



15 .
3
88 .
0
12 .
5 )

[

Hz
3 .
5
]
[ Hz ]

Self-interrupted cutting

High-speed milling
Parametrically interrupted cutting
Low number of edges
Low immersion
Highly interrupted

Highly interrupted cutting
Two dynamics:
- free-flight t

[ t j
 
, t j
 
)
- cutting with regenerative effect t
– like an impact

[ t j
 
, t j
)


 x v j j




A


 x v j j

1

1






 h
 k

2 ,

3 ; h , k
0

0 b hk x j h

1 v k j

1








0
F
0 m




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


 x v j j




A


 x v j

1 j

1






 h
 k
0


2 , 3 ; h , k

0 b hk x j h

1 v k j

1








0
F
0 m




Both period-2s unstable at b)

Milling
Mechanical model:
- number of cutting edges in contact varies periodically with period equal to the delay
..
x ( t )

2
 n
.
x ( t )

(
 n
2  k
1
( t )
) x ( t ) m
 k
1
( t ) x ( t m
 
)

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 k r

K y
K x

0 .
3

State dependent time delay

( x ):
R
  
2 R
  x ( t )
 x ( t
 
)

Without state dependence:
  
( x t
)
 x t

( s )
 
 x ( t
2


 s ), s

[
 r , 0 ]
With state dependence, the chip thickness is h ( t )
 t
 
 t
( x t
 v
)
 v
 y  f z
( s )
 d
/
 s

, f z v

( x t
)
 y ( t
– feed rate,
)

 y ( t

 v

R

( x t

))
2 f z
R


2 DoF mathematical model m  x  ( t )
 m  y  ( t )
 c x x  ( t ) c y
( t )

 k x x ( t ) k y y ( t )


K x

K w
 v

( x t y w
 v

)
( x t

) y ( t )

 y ( t ) y ( t

 
( x t y ( t
))
 q
 
( x t
))
 q
Linearisation at stationary cutting (Insperger, 2006) m
  
( t )
 c x
 
( t )
 k x

( t )

K x w ( v

) q

1

( t )
 
( t
 
)
 v
R

 
( t )
 
( t
 
)
 m

 ( t )
 c y

 ( t )
 k y

( t )
 
K y w ( v

) q

1

( t )
 
( t
 
)

Realistic range of parameters: v
R
 v
R

 
( t )
 
( t
 

0 .
001

0 .
01
)




Characteristic function


2 
2
 
1


2 
2
 
1



1

 k r

 K
1
 q

1

1
 e
  n
 


0

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 chatterfree parameter domain for turning, milling,…
- There is no nonlinear theory for state-dependent regenerative effect.
Thank you for your attention!