ANALYSIS OF SYSTEMS WITH STATE-DEPENDENT DELAYS AND APPLICATIONS IN METAL CUTTING

Abstract

The equivalence between systems with state-dependent delays and systems with constant delays is studied. Systems with state-dependent delays appear when the dynamics of the system does not only depend on the instantaneous configuration but also on a delayed configuration of the system, and the size of the delay depends on the state of the system. It is shown that, indeed, the condition for a transformation to a constant delay is related to the mechanism that generates the state-dependent delay. With an example for the cutting force in turning it is demonstrated that variable and state-dependent delays in metal cutting vibrations can be transformed to constant delays.

Full text

XXIV ICTAM, 21-26 August 2016, Montreal, Canada
ANALYSIS OF SYSTEMS WITH STATE-DEPENDENT DELAYS AND APPLICATIONS IN
METAL CUTTING
Andreas Otto ∗1and G¨unter Radons1
1Institute of Physics, Chemnitz University of Technology, Chemnitz, Germany
Summary The equivalence between systems with state-dependent delays and systems with constant delays is studied. Systems with state-
dependent delays appear when the dynamics of the system does not only depend on the instantaneous configuration but also on a delayed
configuration of the system, and the size of the delay depends on the state of the system. It is shown that, indeed, the condition for a
transformation to a constant delay is related to the mechanism that generates the state-dependent delay. With an example for the cutting
force in turning it is demonstrated that variable and state-dependent delays in metal cutting vibrations can be transformed to constant delays.
INTRODUCTION
Time delay systems can be found in many engineering applications. They range from metal cutting vibrations and laser
dynamics to traffic models and teleoperation of mobile robots (see [1] and Refs. therein). In some systems, as for example
in laser dynamics, the assumption of time-invariant delays is quite reasonable. However,in many applications, the delays are
actually time-varying or state-dependent. For example, in metal cutting the vibrations of the tool directly affect the time delay
and lead to a state-dependent delay [2]. Moreover, variable delays are used for a better control of the system. Indeed, metal
cutting vibrations can be suppressed by an active variation of the spindle speed, which is equivalent to a time-varying delay
[3]. Thus, the analysis of systems with time- or state-dependent delays is relevant in many research areas.
Here, we focus on the analysis of systems with state-dependent delays because time-varying delays can be put in the
framework of state-dependent delays. However, in contrast to the well-established theory for systems with constant delays
there are many open problems in systems with state-dependent delays [4]. In this contribution we shown how a system with
state-dependent delay can be transformed to a system with constant delay. It turns out that, indeed, the mathematical condition
for the existence of such a transformation is fulfilled in many real word examples.
TRANSFORMATION FROM STATE-DEPENDENT DELAYS TO CONSTANT DELAYS
In general, a system with state-dependent delay τ(xt)can be described by the delay differential equation (DDE)
˙
x(t) = f(x(t),x(t−τ(xt))) ,(1)
where x(t)is the configuration of the system at time t. The state of the DDE can be specified by the vector function xt=
x(t−θ),θ∈[0, τmax], where the constant τmax is the maximum delay [4]. A nonlinear time scale transformation ϕ= Φ(t)
can be used to change the delayed argument t−τ(xt)of the DDE Eq. (1). The function Φ(t)is assumed to be bijective,
which means that there is a one-to-one mapping between the time tand the new independent variable ϕ. The configuration
in terms of the new variable ϕis given by y(Φ(t)) = x(t). We demand for a constant delay δin the new representation,
x(t−τ(xt)) = y(ϕ−δ), which is equivalent to the condition
Φ(t)−δ= Φ(t−τ(xt)) ↔δ=
t
Z
t−τ(xt)
˙
Φ(t′)dt′.(2)
Eq. (2) can be interpreted as follows. The state-dependent delay τ(xt)is the traveling time for a transport of a particle over
the constant distance δwith the velocity ˙
Φ(t). The absolute distance covered by the particle at time tis given by the function
Φ(t). A graphical illustration of the condition is presented in Fig. 1 a). The function Φ(t)defines a mapping between time t
(horizontal) and space ϕ(vertical). A vertical shift of the function Φ(t)by the constant distance δleads to a variable horizontal
displacement (red lines), which is equal to the time delay τ(xt). In particular, a state-dependent delay occurs if the absolute
distance Φ(t)covered by the particles is a component of the configuration x(t)or affected by the configuration x(t)of the
system. In the spatial domain the system can be described by the DDE
y′(φ) = Φ−1′(ϕ)f(y(ϕ),y(ϕ−δ)) .(3)
Thus, a DDE with state-dependent delay τ(xt)Eq. (1), whose time delay is specified by the condition Eq. (2), is equivalent
to the DDE Eq. (3) with constant delay δ. In other words, every delay that can be interpreted as a variable transport over a
constant distance can be transformed to a constant delay. These delays are sometimes called variable transport delays and
occur in many applications [5]. On the other hand, it is also possible that variable delays are generated by a transport over a
variable distance. In this case it is not clear a priori if the delay is a variable transport delay or not.
∗Corresponding author. Email: otto.a@mail.de

F
Ω
x(t)
x(t-τ)
vf
Φ
Φ(t)
t-τ(xt )t
φ
φ-δΦ(t)-δ
τ(xt)
space
time
a) b)
x1
x2
R
tool
workpiece
Figure 1: a) Illustration of the condition Eq. (2) between the constant delay δ(vertical shift) and the state-dependent delay
τ(xt)(red horizontal displacement). b) State-dependent delay in turning (see text for explanations).
STATE-DEPENDENT DELAYS IN METAL CUTTING APPLICATIONS
The occurrence of undesired large vibrations at machine tools is still a significant problem in metal cutting applications.
These so-called chatter vibrations are not acceptable in manufacturing industry because they cause noise, bad surface finish
and increased tool wear. As a consequence the prediction of machine tool vibrations and the stability of metal cutting processes
is an important field in research and industry.
An example of a turning process is shown in Fig. 1 b). Small fluctuations of the cutting force F(t)lead to dynamic
displacements x(t) = col(x1(t), x2(t)) of the turning tool and a wavy surface of the workpiece. The instantaneous and time-
delayed vertical displacements x2(t)and x2(t−τ(xt)) determine the inner and the outer surface of the chip, respectively,
and lead to dynamic variations of the chip thickness and the cutting force. In particular, the time delay τ(xt)between the
present and the previous cut at the same location of the workpiece is equivalent to the time for one revolution of the workpiece.
Hence, the delay in turning is defined by a variable transport delay similar to Eq. (2) with the constant distance δ= 2πR,
where Ris the radius of the workpiece. In this case, the spatial variable Φ(t)can be interpreted as the absolute machined
distance given by Φ(t) = ΩtR −x1(t), where Ωis the angular velocity of the workpiece and x1(t)are the horizontal dynamic
displacements of the tool. This mechanism of self-excitation of the vibrations in machining is called regenerative effect (see
[2], [3]). If the closed-loop of the regeneration is unstable, chatter vibrations occur. In many other cutting processes the time
delay is similarly defined by a constant distance between two subsequent cuts.
Since the structural dynamics can be described by ordinary differential equations, metal cutting processes, in general, can
be described either in time domain by Eq. (1) with state-dependent delay or by a system similar to Eq. (3) with constant
delay. For Eq. (3) the linearization of the system is much simpler, whereas for systems with state-dependent delay similar to
Eq. (1) the linearization is not straight forward [2], [3]. In addition, many methods for the stability analysis of linear non-
autonomous DDEs can be only applied to systems with constant delay, which followfrom the linearization of Eq. (3), whereas
the equivalent linear system of Eq. (1) is a DDE with time-varying delay.
CONCLUSIONS
The transformation of a system with state-dependent delay to a system with constant delay is studied, which can be used
to simplify the analysis in many applications. It is shown that the transformation is possible if the state-dependent delay is
generated by a variable transport over a constant distance. Using the example of a turning process it is demonstrated that the
time-varying and state-dependent delays in metal cutting applications belong to the class of variable transport delays and can
be transformed to constant delays.
References
[1] Kyrychko Y.N., Hogan S.J.: On the Use of Delay Equations in Engineering Applications. J. Vibr. Control 16:943960, 2010.
[2] Insperger T., Stepan G., Turi J.: State-dependent delay in regenerative turning processes. Nonlin. Dyn. 47:275-283, 2007.
[3] Otto A., Radons G.: Application of spindle speed variation for chatter suppression in turning. CIRP J. Manuf. Sci. Technol. 6:102-109, 2013.
[4] Hartung F., Krisztin T., Walther H.-O., Wu J.: Functional Differential Equations with State-Dependent Delays: Theory and Applications. In: Canada
A., Drabek P., Fonda A. (editors): Handbook of Differential Equations: Ordinary Differential Equations 3:435-545, North-Holland, 2006.
[5] Zhang F., Yeddanapudi M.: Modeling and simulation of time-varying delays. Proc. TMS/DEVS 34:1-8, San Diego, CA, USA, 2012.