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: andreas.otto@physik.tu-chemnitz.de

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