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Transformation of variable delays to constant delays with application to machine tool chatter Andreas Otto*, David Müller and Günter Radons
Institute of Physics, Chemnitz University of Technology, Germany*E-Mail: [email protected]
Pipe delays(variable transport delays)
● Delays often occur due to transport mechanisms ● Flow of an incompressible fluid through a pipe of length c with strictly
positive velocity v(t) → time delay τ(t) between inlet and outlet● e.g. material flow in reactors, conveyor belts
FIFO buffers (first in – first out) regenerative effect in machining
● Bijective mapping between spatial position φ of fluid elements and time t
Dynamics can be described in three different ways:1) Partial differential equation (PDE) with non-local boundary conditions
1) Delay differential equation (DDE)
1) DDE with constant delay
(PDE)
(Boundary condition)
(Initial condition)
(DDE)
(Retarded argument)
(Initial condition)
(DDE)
(Initial condition)
t
ϕ
Φ ( t )
Transformation from variable to constant delay
Identification of pipe delays(only time-varying delay τ(t) is known)
● Retarded argument r(t)=t-τ(t) defines an iterated map● Constant delay c is determined by rotation number ● Rotation number can be rational or irrational● Necessary and sufficient condition for pipe delays:
Pipe delay Non-pipe delay(rational c=3/5) (irrational c≈17/28)
Calculation of time scale transformation possible:
Example: sinusoidally varying delay
White areas: → pipe delay / transformation to constant delay c
Colored regions: → non-pipe delay / no transformation to constant delay→ phase locking / Arnold tongues of circle map
Machine tool chatter(e.g. regenerative effect in turning)
● Undesired vibrations v(t) in metal cutting processes between tool and workpiece
● Normal vibrations at the present cut y(t) and the previous cut y(t-τ) affect chip thickness and cutting force F→ self-excited vibrations (regenerative effect)
● Time delay τ between present and previous cut● Tangential vibrations at the current cut x(t) and
at the previous cut x(t-τ) affect the time delay τ→ tangential regenerative effect→ state-dependent delay τ=τ(x)
Simulation of regenerative effect with delay differential equations with time- and state-dependent delay
R
Pipe delay for
Example: Spindle speed variation (SSV)Calculation of the stability possible with a delay differential equation with
● time-dependent delay (time domain)● constant delay 2π and an additional
time-dependent coefficient (angular domain)
Conclusion● Variable transport in pipes is often the mechanism behind variable delays ● Variable pipe delays in time domain can be transformed to constant delays via a
nonlinear time scale transformation● Identification of pipe delays and the transformation to constant delay is possible
with the iterated map of the retarded argument and the equivalence criterion
● Time delay in the regenerative effect in machining is a pipe delay● Description of machine tool chatter in the angular domain with constant
delay is possible (and often more suitable!)● Effect of state-dependent delays due to vibrations in tangential direction can
be also modeled with a pipe delay
for time delay systems in general: for vibrations at machine tools:
ϕ=Φ(t ) ⇔ t=Φ−1(t) , Φ(t )=v (t)
ut (x , t )+v (t ) ux(x , t )=0
−v (t ) ux(0, t )= f (t , u (0, t ) , u(−c ,t ))
u(x ,0)=u0(x)
u( x , t )= y (Φ−1(Φ(t )+ x))˙y (t )= f (t , y (t ) , y (r (t )))
y (θ)=u0(Φ(0)−Φ(θ)) , θ<0
r (t )=t−τ(t )=Φ−1(Φ(t )−c)
u( x , t )= z (Φ(t )+x )
z ' (ϕ)=(Φ−1) ' (ϕ) f (Φ−1(ϕ) , z (ϕ) , z (ϕ−c))
z (φ)=u0(Φ(0)−φ) , −c<φ<0
limk→∞
1k(r k (t)−t )=−c
t k=r k (t0)
c=limn→∞
pn /qnc= p /q
rq(t )+ p=t limn→∞
rqn(t )+ pn=t
Φ(t )=1q ∑
k=0
q−1
r k(t ) Φ(t )=lim
n→∞
1qn
∑k=0
qn−1
r k (t )
r (t )=t−τ0−A
2πsin (2π t )
Φ(t , x)−2 π=Φ(t−τ , x)
ϕ=Φ(t , x)=Φ0+∫0
t
Ω(t ' )dt '−x (t )
RΩ(t )>
x (t )R