Analysis of stochastic processes governed by the Langevin equation is discussed. The analysis is based on a general method for non-parametric estimation of deterministic and random terms of the Langevin equation directly from given data. Separate estimation of the terms corresponds to the decomposition of process dynamics into deterministic and random components. Part I of the paper presented several possibilities for qualitative and quantitative analysis of process dynamics based on such decomposition. In Part II, some of these analysis possibilities are applied to experimental datasets from metal cutting and laser-beam welding.
Most experimental data are to some extent noisy. Data can be noisy due to either the
measurement procedure or the process generating the data. In the former case, the noise is
superimposed on the measured data and uncorrelated to the process dynamics, while in the
latter case the noise represents a constitutive part of the process dynamics, and the process is
therefore stochastic. In Part I of this paper [1], analysis of stochastic processes with
uncorrelated Gaussian noise was discussed. Such processes can be modelled by the Langevin
equation, in which the temporal evolution of a process is determined by a sum of deterministic
and random terms. The deterministic term usually describes the global dynamics of the
process, whereas the random term describes some kind of environmental noise or noisy input
which a
!
ects the process state but does not a
!
ect the process parameters. It was shown in
Part I [1] as to how both the deterministicand random terms can be estimated from data and
analyzed. The aim here is to apply these analysis methods to analyze experimental data from
metal cutting and laser-beam welding. Metal cutting is an example in which the deterministic
and random terms of the Langevin equation can be related quite reasonably to the actual
physical phenomena involved in the process. For laser-beam welding, such relations are not
easy to establish. However, the analysis methods presented in reference [1] can nevertheless
be used to extract relevant information about the process dynamics from stochastic data
This paper discusses the possibilities for qualitative and quantitative analysis of
stochastic processes based on measured data. In the
"
rst part of the paper, the analysis
possibilities were presented and illustrated using syntheticdatasets. In this part of the paper,
the methods were applied to experimental datasets from two regimes of turning and
CO
-laser-beam welding. In the turning example, the chatter-free and the chatter-cutting
regime were analyzed based on the recorded
#
uctuations of the cutting force. It was found
that the dynamics of the two regimes can be described as random
#
uctuations around
a stable
"
xed point and a stable limit cycle respectively. These results support the Figure 10. Dependence of estimated terms in deep-penetration welding on the time step
. A cross-section of the
terms at
x
+
1 is shown. (a) Component of the deterministicterm
h
; (b) component of the random term
G
.
x
(
t
)
"
(
I
(
t
!
0
)
16 ms),
I
(
t
)).
description of chatter onset as a Hopf bifurcation. In the CO
-laser-beam welding example,
deep- and shallow-penetration welding regimes were analyzed based on the recordings of
the light intensity
#
uctuations emitted by the welding process. The dynamics of the two
regimes were described as random
#
uctuations around a stable
"
xed point. In
deep-penetration welding the
"
xed point was a node, whereas in shallow-penetration
welding it was a focus. It was suggested that such a distinction between the dynamics of the
two regimes could be exploited to detect a transition between the two welding regimes.
Both processes were analyzed in a two-dimensional phase space reconstructed from
a measured scalar variable. The reasons for restricting the analyses to two dimensions are
the following. Various non-linear deterministic models of metal cutting on a macroscopic
scale suggest that cutting dynamics evolve on a low-dimensional attractor [2
}
4, 13],
although phase spaces of these models have four, or even in
"
nitely many, dimensions. The
experimental data have also been analyzed in a three-dimensional reconstructed space [5],
and the results obtained are equivalent to those quoted above. The situation is di
!
erent in
the case of laser-beam welding, where very little information is available about the original
phase space, the underlying deterministic attractor, their properties and dimensions. The
relation between the measured physical quantity and the process dynamics is also not clear.
It, therefore, seems reasonable to analyze data in a low-dimensional space before extending
the analysis to higher dimensions. In the present case, analysis of data in two- and
three-dimensional reconstructed spaces yielded similar results.
When there are indications that the process under inspection is chaotic, the reconstructed
phase space should certainly span more than two dimensions. In order to verify whether the
phase space dimension is su
$
ciently large, one should compare the reconstructed
deterministic trajectories rather than the vector
"
elds or their cross-sections. Once the
su
$
cient phase space dimension has been chosen, the trajectories will not change
signi
"
cantly as the dimension is further increased. In the case of the stochastic Lorenz
system in a chaotic regime analyzed in Part I of this paper [1], the reconstructed
deterministic trajectories visit both lobes of the attractor interchangeably only if the
reconstructed phase space is at least three-dimensional.
The analysis methods presented in this paper are applicable in principle only to stochastic
processes which can be described by the Langevin equation. These processes contain
uncorrelated dynamic noise which does not a
!
ect the process parameters. The
metal-cutting process, for example, can be modelled as a mechanical oscillatory system
in
#
uenced by the
#
ow of the non-homogeneous cut material. Other mechanical systems
which can be modelled analogously include: an airplane wing in
#
uenced by the air
#
ow
[14, 15], a vehicle system in
#
uenced by road conditions, etc. However, analysis of processes for which the relations between the physical phenomena and the terms of the Langevin
equation cannot be reliably established, may also yield information relevant for both
modelling and monitoring purposes. This shows that the methods are applicable to analysis
of stochastic data, especially when noisiness of the data cannot be neglected.
