### periodically

```Can nonlinear dynamics
contribute to chatter
suppression?
Gábor Stépán
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
.
D
V  wh 
2
where
w – chip width
h – chip thickness
Ω ~ cutting speed
Modelling – regenerative effect
Mechanical model
h(t )  h0  x(t   )  x(t )
h  h(t )  h0  x(t   )  x(t )
τ – time period of
revolution
Mathematical model
1
..
.
2
x  2 n x   n x  Fx (h)
m
Milling
Mechanical model:
- number of cutting edges
in contact varies
periodically with period
equal to the delay
k1 (t )
k1 (t )
x(t )  2 n x (t )  ( 
) x(t ) 
x(t   )
m
m
k1 (t   )  k1 (t )
2
n
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
2
(m lS  S )   (m g lS  m r  2 lS 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
3g
l  0.3 [m] 
  0.1 [s]
Q(t )  P (t   )  D (t   )
1
0 f 
 2.5[Hz]
4
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
(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 )  (   cost ) x(t )  b x(t  2 )
Time delay and time periodicity are equal:
T    2
b    0 Damped oscillator
b  0 Mathieu equation (1868)
  0 Delayed oscillator (1941 – shimmy)
The damped oscillator
x(t )   x (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 )   x (t )  (   cost ) 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 )  h0  x(t   )  x(t )
h  h(t )  h0  x(t   )  x(t )
τ – time period of
revolution
Mathematical model
1
2
x  2 n x   n x  Fx (h)
m
Cutting force
¾ rule for nonlinear
cutting force
Fx ( w, h)  c1wh 3 / 4
Fx  Fx,0  k1h  k2 (h)  k3 (h)  ...
1 k1
Cutting coefficient
k2  
8
h
0
Fx ( w, h)
3
1/ 4
k1 ( w, h0 ) 
 c1wh0
5 k1
h
4
k3 
h
2
96 h0
2
0
3
Linear analysis – stability
k1
k1
..
.
2
x(t )  2 n x(t )  ( n  ) x(t )  x(t   )
m
m
~
Dimensionless time t   t
n
~ ) x( ~
~ x( ~
x(~
t )  2 x(~
t )  (1  w
t)w
t   n )
k1
k1
~
w

Dimensionless chip width
2
m n k
Dimensionless cutting speed
2

2

2


Tobias



~ 
2



n
n
Tlusty, Altintas, Budak
n

Stability and bifurcations of turning
Subcritical Hopf
bifurcation:
unstable vibrations
around stable cutting
j
n

1  2
1
1
j  atn

1  2
~  2 (1   )
w
cr
 cr   n 1 2
The unstable periodic motion
Shi, Tobias
(1984) –
impact
experiment
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]
f1  f 2 15.3

 88.0[Hz ]
2

f1  f 2
15.3

 3.5[Hz ]
2
(2 12.5)
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
– like an impact
t  [ t j   , t j )
0
  0 
x j 
 x j 1  
h
k    

v   A v   
b
x
v
F0 

hk j 1 j 1 


j
j

1
 

 h  k  2,3; h, k 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
0
  0 
x j 
 x j 1  
h
k    

v   A v   
b
x
v
F0 

hk j 1 j 1 


j
j

1
 

 h  k  2,3; h, k 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
k1 (t )
k1 (t )
..
.
2
x(t )  2 n x(t )  ( n 
) x(t ) 
x(t   )
m
m
Phase space reconstruction
A – secondary
Hopf
B – stable cutting C – period-2 osc.
(tooth pass exc.) (no fly-over!!!)
noisy trajectory from measurement
noise-free reconstructed trajectory
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
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
kr 
Ky
Kx
 0 .3
State dependent regenerative effect
State dependent time delay  (x):
R  2R  x(t )  x(t   )     ( xt )
xt (s)  x(t  s), s [r ,0]
Without state dependence:
2
  

With state dependence, the chip thickness is
t
h (t ) 
 v  y ( s) d s  v ( x )  y (t )  y (t   ( x ))
t
t  ( xt )
t
fz
v

v  f z /  , fz – feed rate,  
R 2 R
2 DoF mathematical model
mx(t )  c x x (t )  k x x(t )  K x wv ( xt )  y (t )  y (t   ( xt )) 
q
my(t )  c y y (t )  k y y (t )   K y wv ( xt )  y (t )  y (t   ( xt )) 
q
Linearisation at stationary cutting (Insperger, 2006)
v
q 1 



 (t )   (t   ) 
m (t )  c x (t )  k x (t )  K x w(v )  (t )   (t   ) 
R


v
 (t )   (t   ) 
m(t )  c y (t )  k y (t )   K y w(v ) q 1  (t )   (t   ) 
R


Realistic range of parameters:
Characteristic function
v
 0.001  0.01
R
 2



  2  1    2  1  1 
 kr


2


 K1  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.