Noise-sensitivity in machine tool vibrations
E. Buckwar, 1 R. Kuske 2 B. L’Esperance, 3 T. Soo
4
Abstract
We consider the effect of random variation in the material parameters in a model for machine
tool vibrations, specifically regenerative chatter. We show that fluctuations in these parameters
appear as both multiplicative and additive noise in the model. We focus on the effect of additive
noise in amplifying small vibrations which appear in subcritical regimes. Coherence resonance
is demonstrated through computations, and is proposed as a route for transitions to larger
vibrations. The dynamics also exhibit scaling laws observed in the analysis of general stochastic
delay differential models.
1
Introduction
Machine tool vibrations, commonly known as chatter, are essentially self-induced oscillations in
a machining process, such as metal cutting, milling, or drilling. Chatter can be responsible for
machine or workpiece damage, and it can lower productivity or precision in the process. Factors
contributing to chatter are varied and research into the physical sources has been ongoing for
decades. Recent improvements in technology, particularly related to high-speed machining, have
contributed to the sources for vibrations, so that the understanding of this phenomenon is far from
complete. Appropriate models for machining are necessary, particularly in predicting the behavior
and thus improving efficiency and prolonging machine life. New directions in virtual machine tools
rely on modeling and computational power, with chatter prediction as a crucial building block in
the design of a complete machining process [1] [2].
Early efforts by Tlusty [3] and Tobias [4] have provided a basis for modeling machine tool
vibrations, and illustrate why oscillations are inherent in the system. A crucial aspect in these
models is that the force on the tool depends on the previous cut. In Figure 1 we show the case
where the cutting process takes place as the workpiece is rotated. Then variations in the cut depth
from the previous rotation can feedback into the system to excite further vibrations, thus the name
regenerative chatter. Consequently mathematical models for machine tool vibrations must involve
a delay, representing the time of one revolution of the workpiece. It is well known in the modeling
literature that delays naturally give rise to oscillatory and even chaotic behavior, depending on the
model [5]-[11], and machining processes are no exception.
The dynamics of machine tool vibrations have been studied from a variety of perspectives,
including analysis, computations, and experiments (see [12]-[19] and references therein). In all
examples the goal is to understand the source of chatter in order to avoid it. Certain models have
considered the effect of variable speed on chatter [20, 21], as well as understanding the nonlinear
behavior with a frequency different from the natural frequency of the system [19][22]. Other issues
include the geometry of the cutting which can affect the bifurcation structure of the nonlinear
behavior [23]. Most of the analytical and computational studies have concentrated on the deterministic dynamics. Since these models naturally include time delays, they are already difficult to
analyze in the deterministic setting. When noise is included, the collection of methods for studying
these models in limited, both computationally and analytically.
In order to focus the discussion, we consider a particular aspect of the effect of noise on regenerative chatter. We consider the system in a parameter regime where, without noise, vibrations
1
Institut für Mathematik Humboldt-Universität Berlin, DE,email:buckwar@mathematik.hu-berlin.de
Department of Mathematics, University of British Columbia, Vancouver, BC Canada , email: rachel@math.ubc.ca
3
University of Montreal, Quebec, Canada email: bruno.lesperance@UMontreal.CA
4
Department of Mathematics, University of British Columbia, Vancouver, BC Canada , email:tsoo@math.ubc.ca
2
1
chip
tool
previous cut
present cut
Figure 1: Schematic illustrating the influence of the delay on the cutting process. The inset shows the
setting when the workpiece is rotated while the tool is held fixed. Ideally, chip thickness and the force on
the tool are in equilibrium (2.3). In general, the force depends on chip thickness which in turn depends on
both the previous and present cut. In (2.8) variations in chip thickness are related to the difference in tool
location over one revolution z(t − τ ) − z(t) with τ the time of one revolution.
would decay and the equilibrium position would be stable. The choice of the parameter regimes
reflects the balance of two goals: choosing operating regimes where the oscillations are damped
while maximizing the range of parameter values for operation. Given the complex stability regions
found for machine tools, shown in Figure 3 in Section 3, the balance of these two goals can result in
operating near the stability boundaries for the deterministic steady, non-oscillatory behavior. As
shown in a variety of other contexts, even when the system is tuned to a regime where oscillations
would decay in a noise-free setting, the presence of noise can cause a resonance which sustains the
oscillations. This phenomenon is known as autonomous stochastic resonance or coherence resonance, and has been studied in a number of contexts, with and without delays [5]-[9], [24]-[27]. In
these examples the resonance effect can be amplified in parameter regimes near stability boundaries for the equilibrium state. Here we investigate this phenomenon in the context of regenerative
chatter.
To illustrate the noise sensitivity in the context of machine tool vibrations, we begin with a
simulation of the simple model,
dx = y ds
dy = (−2κy − x + c1 (x(s − τ ) − x(s)))ds + δdW (s).
(1.1)
In the following section we show that this is the linearized equation for variations from a desired chip
thickness, due to vibrations. Here W is a standard Brownian motion, κ is a damping coefficient,
c1 is a non-dimensionalized material parameter, and τ is a delay due to rotation in the machining
process. Temporarily we have dropped the terms with multiplicative noise which are present in
the full model (2.9) but do not play a significant role in the parameter regime considered in Figure
2. In Section 3.3 we show the significance of both types of noise when we discuss transitions from
small oscillations to nonlinear behavior in the context of the full model (2.9).
In Figure 2 we show that, keeping the noise at a low level, the vibrations are amplified as the
other parameters vary, even though these parameters remain in the range where the zero state is
stable in the absence of noise. This phenomenon is due to two factors: the resonance of the noise
with the primary oscillation mode and the factor of the amplification of noise, which can be shown
2
to be inversely proportional to the square root of the distance in parameter space from the stability
boundary for the equilibrium state [27]. The same phenomenon is observed for the complete model
(2.9) which is studied in Section 3.3. We note a few key features of the behavior shown in Figure
0.8
0.6
0.4
0.2
0
x
−0.2
−0.4
−0.6
−0.8
700
750
800
850
900
s
Figure 2: The numerical simulation of (1.1) for κ = .05, δ = .05, τ = 66 for large s with a small initial
condition, below the noise level δ. The solid line corresponds to c1 = .07, and the dash-dotted line is
c1 = .11. Note that the noise level and the delay is the same for both realizations, but for the dash-dotted
line we have chosen a value of c1 which is closer to the stability boundary (shown in Figure 3). Even though
the parameters chosen are below the deterministic threshold for growth of oscillations, the variance of the
amplitude of the oscillation does not decrease to zero.
2. Firstly, the oscillations are a noise induced phenomenon; without noise the oscillations would
decay to zero, since the parameters are chosen in the region where the zero solution is stable in
the deterministic case. Secondly, even though they are caused by noise, the oscillations appear
to have a regular frequency in the oscillations, with a variation in the amplitude or envelope of
the oscillations. Indeed, an analysis of the power spectrum shows a strong peak near the natural
frequency of the deterministic system. Thirdly, the amplitude of the oscillations is an order of
magnitude larger than the actual noise level. The noise coefficient for both simulations is δ = .05,
while the maximum amplitude regularly observed in the oscillations increases from approximately
|x| ∼ .2 for c1 = .07 to |x| ∼ .6 for c1 = .11.
In the following section we derive the model, showing how both additive and multiplicative noise
play a role in the dynamics. In Section 3 we give simulations of the model to demonstrate the effect
of the noise. There we show an amplification of the noise which is consistent with analytical results
predicted for general linear and nonlinear stochastic differential delay models [27]. These results,
based on the method of multiple scales, predict that when the system in operated in a regime near
the stability boundary of the equilibrium for the deterministic system, the noise is amplified by an
order of magnitude, related to the distance to the stability boundary for the steady equilibrium
state.
3
2
The model
We begin with the one degree of freedom model for machine tool dynamics [12]
d2 z
dz
F (f )
+ 2ακ + α2 z =
.
dt2
dt
m
(2.1)
Here z is the machine tool position, κ is the damping factor, and α 2 = ks /m is the ratio of stiffness,
ks , to the mass of the machine tool, m, so that α 2 gives the natural frequency of the undamped
system. The cutting force is given by F which depends on the chip thickness f .
In general, the force is adjusted so that there is an equilibrium position z 0 corresponding to a
desired equilibrium chip thickness f 0
α2 z0 =
F (f0 )
.
m
(2.2)
The functional form of the force is determined experimentally, viewing the cutting force as an
empirical function of the physical parameters as discussed in [3] [4] [12],
3/4
F (f0 )
K0 wf0
=
m
m
.
(2.3)
Under ideal circumstances, the machine tool stays at this equilibrium position z 0 , yielding a uniform
desired chip thickness. Here K0 is a constant that depends on material properties and w is the
chip width.
We consider the case where variations in the system enter through variations in the material
properties, so we write K = K0 (1 + η), with η viewed as a percentage of K 0 . In particular, we are
interested in how these variations translate into vibrations in the machine tool position. In order
to determine circumstances in which there is a significant deviation from z 0 , we consider variations
about z = z0 . We also non-dimensionalize the equations by using the change of variables,
z = z0 (1 + x),
s = αt,
(2.4)
to get
dx
Kw 3/4
K0 w
d2 x
+ 2κ
(1 + η)(f0 + (f − f0 ))3/4 −
f ,
+x=
ds2
ds
mα2 z0
mα2 z0 0
(2.5)
3/4
where we have added and subtracted K 0 wf0 /m to the right hand side of (2.1) before dividing
by z0 α2 and using (2.2)-(2.3) A reasonable approximation can be obtained by using a Taylor series
expansion about f − f0 = 0, keeping terms up to (f − f0 )3 [12] [23]. Then on the right hand side
of (2.5) we have
3
X
K0 w 3/4
K0 w
3/4
(1
+
η)(f
+
(f
−
f
))
−
f
=
η
+
cj (1 + η)(f − f0 )j ,
0
0
mα2 z0
mα2 z0 0
j=1
(2.6)
where
c1 =
3 K0 w
,
4 (mα2 f 1/4 )
0
c2 = −
1 c1 z0
,
8 f0
4
c3 =
5 c1 z02
.
96 f02
(2.7)
The chip thickness variation f − f0 can be expressed simply as the difference between the tool
edge position delayed by one rotation z(t−2π/Ω) and the present position z(t), so that the equation
for x is
3
X
d2 x
dx
+
2κ
+
x
=
η
+
cj (1 + η) [x(s − τ ) − x(s)]j ,
ds2
ds
j=1
with τ =
2πα
Ω
(2.8)
and Ω the angular velocity of the rotation in the machining process. In our model we take the
variation η in K to be modeled by white noise δdW as a simple model for variations in the
material properties encountered in the cutting process. We take δ in the range .01 < δ < .15, as
representative of the percentage variation in K. It is also convenient to write the second order
equation as a first order system:
dx = y ds
dy = (−2κy − x +
3
X
cj [x(s − τ ) − x(s)]j )ds + δdW +
j=1
3
X
cj [x(s − τ ) − x(s)]j δdW. (2.9)
j=1
The choice for η is convenient since it allows us to compare the behavior of the oscillations to
that found in previous analyses of similar stochastic delay-differential models [27]. If we replace
the white noise with colored noise in (2.9), the oscillations with the natural frequency are again
amplified as seen in Figure 2. As suggested by the simulations in this paper and confirmed by the
analysis in [27], the important aspect is the resonance of the noise with the natural frequency, so
that a similar coherence resonance is observed as long as the noise contains this frequency.
Then (2.9) can be viewed as the equation for the variation from the desired chip thickness
z0 , with its magnitude |x| given as the percentage of z 0 . The key parameters are the delay τ ,
proportional to the inverse rotational velocity, and c 1 , which is a non-dimensionalized material
parameter. The noise coefficient δ can be viewed as the percentage of variation in the material
properties. In the following we give numerical simulations for fixed damping κ, considering a range
of c1 and δ and a few different values for τ .
3
Numerical simulations and resonant behavior
In this section we demonstrate numerically the effect of the noise in combination with the other
parameters in (2.9). In particular, we consider the stochastic resonance as we vary c 1 and τ , which
is equivalent to considering different combinations of K 0 and Ω. We fix the remaining parameters
at α = 775 rad/sec , m = 50 kg, κ = .05, f0 = .001 m. The ranges of parameter values are chosen
to be comparable to parameters used in [12, 17, 18]. First we review the linear stability analysis
for steady state solution of the deterministic system (2.9) with δ = 0. Then we show simulations of
the linear system, to demonstrate that the resonance of the primary mode with the noise sustains
significant oscillations, even for small noise. Finally we show the implications of this resonance
on the fully nonlinear system. The numerical simulations have used both the Euler-Maruyama
method [28] and the multi-step method for stochastic delay differential equations [29]. By using a
higher order method, we verify that the oscillations observed in the simulation are not due to the
numerical method.
3.1
Linear stability
We review the linear stability analysis of the steady state x = 0 for the deterministic system, which
has been given previously in [30], and also discussed in [13]. We consider system (2.8) with δ = 0
5
and linearized about x = 0, which yields
dx
d2 x
− 2κ
+ x = c1 (x(s − τ ) − x(s)).
2
ds
ds
(3.10)
Substituting x = eλt yields the characteristic equation
λ2 − 2κλ + 1 = c1 (e−λτ − 1).
(3.11)
The stability boundary is determined by setting λ = iω in (3.11), dividing this equation into its
real and imaginary parts, and solving these two equations for c 1 and τ . These curves are shown
in Figure 3, in terms of the non-dimensional parameters and the dimensional parameters. The
c1
0.115
0.11
0.105
0.1
40
45
50
55
60
65
70
75
τ
6
3.7
x 10
3.6
3.5
3.4
K0
3.3
3.2
3.1
3
700
800
900
1000
1100
1200
Ωrpm
Figure 3: The stability boundary for the zero solution of the linearized deterministic equation (3.12) with
δ = 0; that is, there is a stable equilibrium at x = 0 for parameter values below these curves. There is
a series of curves, due to the periodicity of the functions in (3.11) with λ = iω. The top figure gives the
stability boundary in terms of the non-dimensional parameters c1 and τ . The bottom figure shows the same
boundary for the dimensional parameters K0 (N/m1.75 ) and Ωrpm (rpm). Here Ωrpm = 60 · Ω/(2π) is used
for consistency with graphs given in previous references, for example [12].
shape of the stability boundary implies that for certain values of τ there is a larger range of c 1 for
which the steady state x = 0 is linearly stable. Since the delay τ is inversely proportional to Ω,
the angular velocity of the rotating workpiece, then there may be certain preferred values of Ω for
which the machining process is expected to be stable for a larger range of the material parameter
K0 .
3.2
Resonance in the noisy linearized system
Now we consider the effect of noise in (2.9), linearized about x = 0,
dx = y ds
dy = (−2κy − x + c1 [x(s − τ ) − x(s)])ds + δdW + c1 [x(s − τ ) − x(s)] δdW
6
(3.12)
8
7
6
5
4
p(x)
3
2
1
0
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
Figure 4: The probability density p(x) for τ = 39 and c1 = .09 (dashed line), c1 = .11 (dashed-dotted
line) and c1 = .115 (diamonds) for δ = .02. All values are in the stability regime for the steady state of
the deterministic equation. Nevertheless, oscillations are sustained, with larger oscillations occurring with
increased probability for parameter values closer to the stability boundary.
We focus on this system in order to demonstrate the impact of the resonance with the primary
mode of the linearized system (3.12) with frequency ω = 1 + O(κ 2 ) for κ 1. The linearization
also allows us to focus on this resonance, while temporarily neglecting other large amplitude states
that exist in the fully nonlinear system. In Section 3.3 we discuss the implications of this resonance
in the full system.
Results from simulations of the linearized system (3.12) for a range of values of c 1 and fixed
τ are shown in Figures 4 and 5 via the stationary probability density function p(x). This density
is obtained by running simulations over a sufficiently long time (in this case s = O(α) which
is t = O(1) in the original time scale), for 5000 realizations. From these realizations we get a
reasonable approximation for the stationary probability density for x, the deviation of tool location
as a percentage of the desired position z 0 . That is, the simulations are run over a sufficiently long
time so that the density does not change if the simulations are run for larger values of the nondimensionalized time s. The purpose of showing first the stationary density for the linear system is
to illustrate how the oscillations are amplified via a resonance, particularly with the additive noise.
Later we discuss how this resonance plays a role in the nonlinear system.
In Figure 4, we show simulations for δ = .02, τ = 39 (Ω rpm ≈ 1192) with c1 = .09, .110, .115.
Referring to Figure 3, we see that these values of c 1 are in the stability region for the zero solution
of the deterministic system. However, the oscillations do not decay over time, as shown by the
graphs
of the probability density. Rather, the density shows a standard deviation of approximately,
p
Var(x) =.055, .095, .13, respectively, for the values of c 1 given above, with amplitudes typically
observed over the range of .1 < |x| < .35, that is, 10−35% variation of the equilibrium tool position
z0 . We show the stationary probability density, which means that these oscillations are sustained
over time and do not damp out, even though the parameter values are chosen in the stability region
for the deterministic system. Note that the larger oscillations are observed for values of c 1 closer
7
to the stability boundary for x = 0. From Figure 4 we see that even though the noise is small
δ = .02, there is a significant probability at any time that the amplitude of oscillations is an order
of magnitude larger than δ.
In Figure 5, we show simulations for δ = .07, τ = 39 and τ = 66 (Ω rpm ≈ 1192 and Ωrpm ≈
704.5), and varying values of c1 , again chosen in the stability region for the zero solution of the
linear deterministic system. The noise is slightly larger (δ = .07) than in Figure 4 but it is still
small. Again the oscillations are amplified by an order of magnitude and indeed oscillations with
an O(1) amplitude can be observed.
2.5
1.8
1.6
2
1.4
1.2
1.5
p(x)
1
0.8
p(x)
1
0.6
0.4
0.5
0.2
0
−1.5
−1
−0.5
0
x
0.5
1
0
−1.5
1.5
−1
−0.5
0
0.5
1
1.5
x
Figure 5: On the left, we show the probability density p(x) with τ = 39 and c1 = .09 (dash-dotted lined)
and c1 = .11 for δ = .07 (diamonds). On the right, we show the probability density p(x) with τ = 66 and
c1 = .09 (dash-dotted line) and c1 = .105 for δ = .07 (diamonds). All parameter values are in the stability
regime for the deterministic equation. However, substantial oscillations are sustained, with larger oscillations
occurring with increased probability for values closer to the stability boundary for x = 0.
The amplification of the noise can be calculated analytically using a multi-scale analysis as
in [27], for general linear and non-linear stochastic differential delay equations with additive and
multiplicative noise. In that previous study, equations for the amplitude of the oscillations are
obtained for long time scales. These equations include an effective noise term, which can be derived
as a result of the resonance with the primary mode of the deterministic system. The magnitude
of the effective noise given in [27] is δ/, where δ is the noise level of the original system, and 2
is the distance of the control parameter(s) from the stability boundary for the steady state. For
the present system, δ is also the noise level, and the parameter c 1 or τ plays the role of the control
parameter. For example, if (c∗1 , τ ∗ ) corresponds to a point on the stability boundary in the top of
Figure 3, then we could define 2 = c∗1 − c1 with τ = τ ∗ . In the simulations shown above, ranges
from approximately .1 < < .3, which suggests an amplification factor between 3 and 10 for the
noise level in the dynamics. In fact, this is what is observed in Figure 2, and in the simulations
for the densities shown in Figures 4 -5. Calculations similar to those in [27] can be used to derive
stochastic amplitude equations and the amplification factor of the noise for the present system (2.9)
and (3.12), and we leave that to future work.
8
3.3
Implications for the oscillations in the nonlinear system
In Figure 6 we show the probability P (x > x c ), that is the probability that x exceeds a threshold
value xc . Over long times, this probability can be viewed as the percentage of time that the system
(3.12) has oscillations with amplitudes that exceed 100x c % of the equilibrium value z0 . P (x > xc )
can be calculated by integrating the approximation to the probability density computed from the
simulations as shown in Figures 4 and 5,
P (x > xc ) =
Z
∞
p(x)dx .
(3.13)
xc
Clearly this probability increases with increased amplitude of the oscillations. From Figure 6 we
see that this probability increases either as the noise level increases or as the material parameter c 1
and/or the delay parameter τ values approach the stability boundary. Then P (x > x c ) can be used
as a measure of the variation of the oscillations in the tool position, illustrating the information
contained in p(x).
0.35
0.3
0.25
P (x > xc )
0.2
0.15
0.1
0.05
0
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
c1
Figure 6: The graph of P (x > xc ), the probability that the linearized system exceeds xc for varying values
of c1 and τ = 39. The ∗’s correspond to a noise level of δ = .02 with a threshold of xc = .15. The o’s
correspond to noise level δ = .07 and a threshold of xc = .35. Notice the increase of P (x > xc ) as the values
of c1 approach the stability boundary for x = 0.
Above we have demonstrated the behavior of the probability density for the linear system. We
outline the implications of this behavior for the nonlinear system in the setting of a subcritical
bifurcation, as shown in Figure 7 and observed in a number of studies of deterministic models
for machine tool vibrations [12, 22, 23]. In Figure 7a, we give a sketch of the typical bifurcation
scenario for the deterministic system, similar to that shown in [12]. The sketch is shown in terms
of the non-dimensional material parameter c 1 as the bifurcation parameter. The branch shown
bifurcates from the equilibrium state at the Hopf bifurcation point c ∗ . We consider the behavior
for c < c∗ , as used in the results shown in Figures 4-6. In the absence of noise, small perturbations
to the zero state are damped, since small oscillations are unstable. However, for large enough
perturbations, the system can move to large amplitude nonlinear oscillations which are stable. In
the presence of noise we have shown that small oscillations can be amplified through coherence
resonance, so that once these oscillations are large enough, the nonlinear system may make the
9
transition to the branch of large amplitude behavior, as discussed in [12]. Then the system may
continue in a non-linear mode, and for sufficiently large x may move into a mode where the tool
loses contact with the workpiece. Then quantities such as P (x > x c ) indicate the probability that
the system makes a transition from small, nearly linear oscillations to large nonlinear oscillations.
In particular, we can compute the probability that x exceeds a critical value, which can be set as
the threshold for such a transition.
The transition to a nonlinear mode with large oscillations is demonstrated in Figure 7b for the
fully nonlinear system (2.9), comparing realizations of the system for parameters in the stability
range for the zero solution, with and without noise. That is, in the absence of noise, the oscillations
in the system damp out to zero, so that the system goes to steady state. However, the small
oscillations are amplified even with small noise (in this case δ = .05), so that eventually the system
makes the transition to oscillations with O(1) amplitude. As can be seen from (2.9), once the system
makes a transition to large oscillations, the multiplicative noise then plays a role in contributing
to the irregularity, and the additive noise does not influence the dynamics as much, since it is
relatively small. We leave a complete exploration of the fully nonlinear noisy system for future
work.
maximum
amplitude
1.5
1
large amplitude
nonlinear mode
0.5
x
0
−0.5
small amplitude
nonlinear mode
zero solution
−1
−1.5
2200
control parameter (c1 )
2250
2300
2350
2400
2450
2500
s
Figure 7: Left: Bifurcation branch for a subcritical bifurcation from the zero state. In the deterministic case
small amplitude oscillations are unstable, while large amplitude fully nonlinear solutions may be stable. For
subcritical values of the control parameter, such as the material parameter c 1 , noise can cause a transition
to large amplitude oscillations, through a resonance response which allows significant variation beyond the
unstable small oscillations. Right: Realization for the nonlinear system with noise, δ = .05 (dotted line)
and without noise (solid line), with small initial condition. Here τ = 65 and c1 = .104. Without noise, the
solution decays slowly to zero, which is stable for small initial conditions. With noise the small amplitude
oscillations are amplified through a resonance with the noise, and the system makes a transition to a large
amplitude oscillation, corresponding to a nonlinear mode.
4
Summary
We have derived a model for machine tool vibrations when the material properties of the workpiece
fluctuate randomly. The randomness appears as both additive and multiplicative noise in the nondimensionalized equation for the percent variation from the machine tool equilibrium position. We
10
show that the resonance of the additive noise with the natural mode of the linearized system plays a
dominant role in the dynamics, particularly in the subcritical regime where small vibrations would
be otherwise damped. Then oscillations can be sustained in subcritical regimes through coherence
resonance. The effective amplification factor for the noise is −1 where is a measure of proximity
to the linear stability threshold of the equilibrium state in the deterministic system. That is, the
amplification factor increases both with the size of the noise and as the model parameters approach
critical values corresponding to a Hopf bifurcation in the absence of noise.
One important aspect in this setting is the concept of stability. Clearly in the deterministic case
stability is defined via boundaries in parameter space between growth and decay of oscillations. In
the stochastic setting the stability is not clearly defined. Typically Lyapunov exponents and structure of the invariant densities have been used to classify stability in stochastic dynamical systems
[26] [31] [32]. Here we have shown that certain critical combinations of noise levels and parameter
regimes near stability boundaries can allow and even promote transitions between different stable
states, particularly in the context of subcritical bifurcations. The results shown above suggest that
in practical settings factors such as the amplification of the oscillations is of central importance.
Even though the amplitude of the oscillations is not growing in this setting, it is not decaying, so
variance of the amplitude is a contributing factor towards transitions to nonlinear states.
Figure 6 suggests a second quantity based on the probability density function p(x) which provides an indicator for transitions in finite time. There the quantity P (x > x c ) is defined as the
probability that deviations x from the equilibrium z 0 is above a threshold xc for a certain percentage of the time. In the context of a system with a supercritical bifurcation, or one where subcritical
oscillations are unstable, this probability defines the percentage of time in which chatter is above
a given threshold for subcritical parameter regimes. In the context of subcritical bifurcations with
unstable small amplitude oscillations and stable large amplitude behavior, there is a threshold
above which nonlinear behavior can be observed. When there is a finite probability that the oscillations exceed a threshold xc , then there is a finite probability that a small perturbation can result
in a transition to the nonlinear state. For machine tool dynamics, calculating the probability that
a transition can occur on an O(1) time scale gives significant insight into the incidence of chatter.
Quantities of this type are of greater interest than stability measures obtained over long time scales,
particularly when transitions on O(1) time scales can be catastrophic.
References
[1] Manufacturing Automation Laboratories,http://www.malinc.com/.
[2] P. Zelinski,“The Dynamics of Better Drilling”, Modern Machine Shop,
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