NOISE AND SIGNAL
INTERACTION
Multiple-Scales Approximation
of a Coherence Resonance
Route to Chatter
This article considers the effect of random variation in the material parameters of a model
of machine tool vibrations—specifically, it examines a model of regenerative chatter.
I
n contrast to deterministic models, which
predict certain regimes for a machine tool’s
stable equilibrium position, a stochastic
model shows that noise can cause significant
stochastic variation in the tool’s position, leading to
transitions from an equilibrium state to one of
chatter. A reduced model—obtained by using a
multiple-scales method adapted for stochastic dynamics—can capture the mechanism for these amplified oscillations via coherence resonance and
provide an efficient computational method for the
probability density of the machine tool’s position.
This article gives the full model and shows how a
multiple-scales approach leads to a reduced system
that can describe the chatter’s sustained amplification. A reduction provides an efficient semi-analytical approach for computing the dynamics:
simulations of the behavior via the reduced system
are a factor of ⑀–2 faster than simulations of the original system, where ⑀–1 is the oscillations’ amplification factor. We’ll also compare these results via
computations of the probability densities and see
how noise plays a role in the global dynamics.
1521-9615/06/$20.00 © 2006 IEEE
Copublished by the IEEE CS and the AIP
RACHEL KUSKE
University of British Columbia, Vancouver
50
Vibrations
Machine tool vibrations, commonly known as chatter, are essentially self-induced oscillations in a machining process, such as metal cutting, milling, or
drilling. In addition to causing damage, chatter can
also lower productivity and precision. Recent technological improvements, particularly those related
to high-speed machining, have contributed to the
varied sources of chatter. New directions in virtual
machine tools rely on modeling and computational
power, with chatter prediction serving as a crucial
building block in the design of an efficient machining process.1
Early efforts by J. Tlusty2 and S.A. Tobias3 illustrate why these vibrations are inherent in the system: the force on the tool depends on the previous
cut, as illustrated in Figure 1. Variations in the cut’s
thickness from the previous rotation can feed back
into the system and excite further vibrations, thus
the term regenerative chatter. Consequently, mathematical models for machine tool vibrations must
involve a delay, representing the time of one revolution of the work piece. Delays naturally give rise
to oscillatory and even chaotic behavior, depending on the model.4–10
The dynamics of machine tool vibrations have
been studied from a variety of perspectives, including analysis, computation, and experiment.11–19
Additional studies have considered variable
speed,20,21 nonlinear effects on frequency,19,22 and
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the geometry of the cutting, which can change the
bifurcation structure.23 Most analytical and computational studies focus on deterministic dynamics,
but in contrast to deterministic models (which predict stability for the equilibrium position in certain
parameter regimes), stochastic models capture a
significantly different behavior, in which the interaction of noise and intrinsic oscillations can amplify
otherwise-damped vibrations. It’s thus crucial to
understand the variation in the tool’s position,
which can be done via its probability density. In a
stochastic setting, the collection of both computational and analytical methods for models with
memory is limited. However, a multiple-scales
method provides a reduced stochastic model that
can capture the mechanism for the noise-induced
amplification of the vibrations and provide an efficient computational approach for the probability
density of the machine tool’s position.
A simple model demonstrates this phenomenon:
Inherent damped
vibrations of
the machine tool
Chip thickness
Tool
Tool location
at previous cut
Tool location at present cut
Figure 1. Delay and the cutting process. The inset shows the setting
in which the work piece is rotated with the tool held fixed. The
force on the tool depends on the work piece’s chip thickness and
the difference between previous and present tool positions.
1.0
0.8
dx = y dt
0.6
dy = (–2␬y – x + c1(x(t – ␶) – x(t)))dt + ␦dw(t), (1)
MAY/JUNE 2006
0.2
x
which is the linearized equation for variations x
from a desired chip thickness due to vibrations.
Note that this equation describes a spring with
damping coefficient ␬, subject to both a force proportional to chip thickness x(t – ␶) – x(t) with coefficient c1, a non-dimensionalized material
parameter, and noise, where w is a standard Brownian motion. Here, ␶ is a delay due to the cutting
tool’s rotation, as illustrated in Figure 1. Later, we’ll
see that this linearized model captures the amplification of the intrinsic oscillations, which play a
dominant role in transitions from equilibrium
states to larger vibrations.
Figure 2 shows simulation results for Equation 1
in a setting in which the oscillations would decay
to zero in the absence of noise—we choose the parameters in the region in which the equilibrium (x
= 0) solution is stable in the deterministic case. This
phenomenon is known as autonomous stochastic resonance or coherence resonance: the presence of noise
causes a resonance that sustains otherwise-damped
oscillations. Although the sustained oscillations are
noise-induced, they have a dominant regular frequency that can be verified by a standard computation of the power spectral density (not shown),
which is strongly peaked at a particular frequency.
The amplitude or envelope of these oscillations has
significant variation—in fact, an order of magnitude larger than the actual noise level. The noise
coefficient for both simulations is ␦ = .05, whereas
0.4
0.0
–0.2
–0.4
–0.6
–0.8
–1.0
175
180
185
190
195
200
t
Figure 2. Numerical simulation of Equation 1 for ␬ =.05, ␦ =.05, and
␶ = 66 for large time with a small initial condition. The blue line
corresponds to c1 = .07, and the pink line is c1 = .11, which is closer
to the deterministic stability boundary of the equilibrium x = 0.
the maximum amplitude regularly observed in the
oscillations increases from approximately |x| ~ .2
for c1 = .07 to |x| ~ .7 for c1 = .11.
Researchers have studied coherence resonance
in several contexts, with and without delays.4–8
However, few analytical methods have been developed for problems with delays and noise: computational methods are typically slow, and the
computational error in the noise-sensitive regime
can induce additional oscillations that interfere
with the coherence resonance phenomenon. A
51
multiple-scales analysis can uncover the mechanism for noise-amplified oscillations by showing
that the main factors are the noise’s resonance with
the primary oscillation mode and its close proximity to the steady state’s stability boundary. In the
context of regenerative chatter, the complex stability regions found for uniform cutting suggest that
it can be advantageous to operate near these stability boundaries for steady, nonoscillatory behavior.
However, analysis of the resonance effect also
shows that vibrations can be amplified in parameter regimes near these stability boundaries, capturing the variability with an explicit noise
amplification factor inversely proportional to the
square root of the distance in parameter space from
the stability boundary.24
The Model
Let’s begin with the one-degree-of-freedom model
for machine tool dynamics, which is discussed in
detail elsewhere:11
d 2z
ds 2
+ 2ακ
F( f )
dz
+ α 2z =
.
ds
m
ds
2
+ 2ακ
F( f )
dz
+ α 2z =
.
ds
m
(2)
(3)
The functional form of the force F is determined
experimentally, in which the cutting force is viewed
as an empirical function of the physical parameters.2,3,11 Under ideal circumstances, the machine
tool will stay uniformly at this equilibrium position
z0. To consider variations about the equilibrium position z = z0, and how they translate into vibrations
in the machine tool position, we must introduce
the nondimensional variables
z = z0(1 + x), t = ␣s.
(4)
We write the force F in terms of actual chip thickness f = f0 + (f – f0), expressing the chip thickness
variation f – f0 as the difference of the tool-edge position delayed by one rotation z(t – ␶) and the present position z(t). Substituting this into the
expression for force (Equation 3), we can obtain a
52
dx = y dt
3
⎛
⎞
dy = ⎜ −2κ y − x + ∑ c j [ x ( t − τ ) − x ( t )] j ⎟ dt
⎝
⎠
1
3
+δ dw + ∑ c j [ x ( t − τ ) − x ( t )] j δ dw,
(5)
1
Here, z is the machine tool position, ␬ is the damping factor, and ␣2 = ks/m is the ratio of stiffness ks to
m mass of the machine tool, so ␣2 gives the natural
frequency of the undamped system’s oscillations—
that is, the machine tool’s natural vibrations. In
general, the cutting force F is adjusted so that an
equilibrium position z0 corresponds to a desired
equilibrium chip thickness f0:
d 2z
reasonable approximation by using a Taylor series
of F(f0 + [z(t – ␶) – z(t)]) about z(t – ␶) – z(t) = 0,
keeping terms up to [z(t – ␶) – z(t)]3.11,23 To model
variations in the material properties, we write K =
K0(1 + ␩), with ␩ viewed as a percentage of K0. We
model ␩ with white noise that has a coefficient ␦ <<
1 as a simple model for variations in the material
properties encountered in the cutting process. The
choice for ␩ is convenient for analyzing delay-differential models,24 but we see a similar phenomena
if we use colored noise. Substituting these expressions for the force and variation in Equation 2, and
writing the second-order equation as a first-order
system, we get
where
c1 =
K 0w
3
1 c
, c2 = − 1 ,
2
1
4
/
4 ( mα f 0 )
8 f0
c3 =
5 c1
2πα
,τ =
,
96 f 02
Ω
(6)
with representing the rotating work piece’s angular velocity. Complete details of the derivation appear elsewhere.25 We take ␦ in the range .01 < ␦ < .15
as representative of typical variations in material
properties. Equation 5 describes the variation |x| as
a percentage of the desired tool position z0. The key
parameters are the delay ␶, proportional to the inverse rotational frequency, c1, the non-dimensionalized material parameter, and ␦, the percentage of
variation in material properties. Next, we’ll examine
a reduced model and compare it with numerical simulations for fixed damping ␬ = .05, considering a
range of c1 and ␦ and a few different values for ␶.25
Multiscale and Coherence Resonance
The multiscale analysis described in this section
demonstrates how the stochastic amplitude of the
nearly regular oscillations is amplified. One important ingredient in this phenomenon is the proximity of the parameters to the linear stability boundary
of the equilibrium x = 0. As Figure 2 shows, for the
same noise level, the oscillations’ amplitude increases as c1 approaches this stability boundary (see
Figure 3), even though the noise level remains the
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0.115
c1
0.110
0.105
0.100
35
40
45
50
55
τ
60
65
70
75
Figure 3. Stability boundary for the zero solution in Equation 7. The equilibrium at x = 0 is stable for
parameter values below these curves. The series of curves is a result of the periodicity e␭t in Equation 8
for ␭ = i␸.
same. Another important ingredient is the noise’s
resonance with the tool position’s intrinsic oscillations. We can see this in the derivation of the amplitude equation, which also provides a reduced
model for efficient computations of the dynamics.
the dimensional parameters K0 and in terms of c1
and ␶, respectively; the stability boundary then
shows how advantageous it is to operate at certain
values of , which should be stable for a larger
range of the material parameter K0.
Linear Stability
Multiscale Analysis and Resonance
The stability boundary for the deterministic system
has appeared in previous work.11,26 Here, we consider the system in Equation 5, linearized about the
steady state equilibrium x = 0 (corresponding to no
chatter, with ␦ = 0). This linearization yields
The linear stability analysis given earlier illustrates
that for values near the neutral stability curve—for
example, for c1 = c1c + ⑀ 2c12 for ⑀ << 1—the eigenvalue in Equation 8 has the form ␭ ~ ⑀2r + i(␸ +
⑀ 2␸2). Here, we can obtain the real part ⑀2r and correction to the frequency ⑀ 2␸2 via a straightforward
perturbation expansion of Equation 8 for c1 = c1c +
⑀ 2c12. The real part of ␭ is small (O(⑀ 2)), and r < 0 for
c12 < 0—that is, for values c1 < c1c below the stability
boundary. Without noise (␦ = 0), we have a slow
time scale T = ⑀2t for the decay of perturbations with
critical frequency ␸. As we’ll see later, the introduction of a slow time scale is indeed critical for obtaining a reduced system that can describe the
noise’s coherence resonance with oscillations.
An approximation of the solution of Equation 5
via a stochastic multiscale analysis captures the
main features of the dynamics. Particularly in the
region below the stability boundary for the steady
state, we observe the following noise-induced oscillations by using the linearization of Equation 5,
d 2x
dt 2
+ 2κ
dx
+ x = c1 ( x ( t − τ ) − x ( t )) .
dt
(7)
Substituting x = e␭t, we get the characteristic equation
␭2 + 2␬␭ + 1 = c1(e–␭␶ – 1).
(8)
The stability boundary is determined by finding
the curves satisfied by the equations for the real and
imaginary parts of Equation 8 for ␭ = i␸. Figure 3
shows these curves in terms of the nondimensional
parameters c1 and ␶. The linear stability criteria obtained from Equation 8 indicates that for a given
value of ␶, a critical value c1c exists on the stability
curve in Figure 3, as does a corresponding frequency ␸, both obtained from Equation 8 with ␭ =
i␸. This point corresponds to a Hopf bifurcation.12
For values of the material parameter c1 < c1c, the
equilibrium solution x = 0 is stable, and oscillations
decay; for c1 > c1c, the equilibrium is unstable, and
oscillations with frequency ␸ are sustained. Notice
that in certain values of ␶, the steady equilibrium
state is stable over a larger range of c1. We can write
MAY/JUNE 2006
dx = y dt
dy = (–2␬y – x + c1[x(t – ␶) – x(t)])dt + ␦dw
+ c1[x(t – ␶) – x(t)]␦dw.
(9)
Here, we’ve dropped the nonlinear terms—that is,
we take c2 = c3 = 0, which is an appropriate approxi-
53
slow time scale. We obtain a reduced system for A
and B by using the multiple-scales method, which
is based on the approximation that treats the fast
and slow time scales, t and T, as independent.27 This
assumption is valid when the time scales are “separate”—that is, ⑀ << 1 so that T is indeed slow compared to t. Equation 10 describes the amplification
of the noise in the oscillation with a nearly deterministic frequency ␸, and the resulting reduced system gives an efficient means for computing an
approximate probability density function over the
long time scale T. The ansatz, or proposed, form for
the equations for A and B is
5.0
4.5
4.0
3.5
p (x)
3.0
2.5
2.0
1.5
1.0
⎛ dA⎞ ⎛ ψ A ⎞
⎛ σ11 σ12 ⎞ ⎛ dξ1 (T ) ⎞
⎜⎝ dB ⎟⎠ = ⎜⎝ ψ ⎟⎠ dT + ⎜⎝ σ
⎟⎜
⎟
B
21 σ 22 ⎠ ⎝ d ξ2 (T )⎠
0.5
0.0
–0.5
–0.4
–0.3
–0.2
–0.1
0.0
0.1
0.2
0.3
0.4
3
⎛ A⎞ ⎛ d β j1 (T ) ⎞
+∑ Σ j ⎜ ⎟ ⎜
⎟,
⎝ B ⎠ ⎝ d β j 2 (T )⎠
j =1
0.5
x
Figure 4. Comparison of the stationary probability density p(x) for ␦
= .02 computed from Equation 9. This linear system includes
multiplicative noise (blue line) and uses the multiple-scales
approximation, computed via Equation 11 (pink diamonds).
mation in this region for small deviations from x =
0. The analysis in Equation 9 leads to a reduced system and demonstrates the impact of the noise’s resonance with the primary mode, which has frequency
␸ (obtained in the previous section). This lets us focus on resonance for small perturbations around the
equilibrium steady state while temporarily neglecting other large-amplitude states that exist in the fully
nonlinear system. Later, we’ll discuss the implications of this resonance in the full system.
We use the approximation
where ␹i and ␤ji are independent standard Brownian
motions. The multiscale analysis’ goal is to derive
the equations and Equation 11’s drift coefficients ␷A
and ␷B and diffusion coefficients, the matrices j and
constants ␴ij. The analysis also demonstrates that the
ansatz for the form of the amplitude equations in
Equation 11 is indeed consistent.
For clarity, let’s review the results here (details of
the analysis appear in the “Appendix” sidebar).
(f
2
1
)
+ ( 2κω cos ωτ − ω sin ωτ )( c1c ∆B + c12B
− c12 (ω A + 2κ B ))]
+ f1[( c1c ∆A + c12 A)( 2κω cos ωτ − ω sin ωτ
− ( c1c ∆B + c12B )(ω cos ωτ + 2κ sin ωτ )
− c12 ( −ω B + 2κ A)]
(10)
where T = ⑀ 2t. As described earlier, c1c and ␸ correspond to a fixed delay ␶, and c12 < 0 (for the remainder of this article, c12 = –1 without loss of
generality). The multiscale form for x and y in
Equation 10 is identical to that used for the leading
order in a deterministic system near a Hopf point,27
but the difference here is that the slowly varying
amplitudes A and B must capture stochastic behavior. Equation 10 also reflects the stochastically modulated oscillation with fixed frequency ␸, which is
consistent for small noise—that is, ␦ << 1 in this setting. The goal is to obtain equations for the amplitude or envelope described by A(T) or B(T) on the
54
)(
+ f 22 ω 2 + 4κ 2 ψ A =
− f 2[( c1c ∆A + c12 A)(ω coss ωτ + 2κ sin ωτ )
x(t) ~ A(T)cos ␸t + B(T ) sin ␸t
y(t) ~ –␸A(T )sin ␸t + ␸B(T)cos ␸t ,
(11)
(f
2
1
)(
(12)
)
+ f 22 ω 2 + 4κ 2 ψ B =
f1[( c1c ∆A + c12 A)(ω cos ωτ + 2κ sin ωτ )
+ ( 2κω cos ωτ − ω sin ωτ )(c1c ∆B + c12B
− c12 (ω A + 2κ B ))]
+ f 2[( 2κω cos ωτ − ω sin ωτ )( c1c ∆A + c12 A)
− ( c1c ∆B + c12B )(ω cos(ωτ ) + 2κ sin(ωτ ))
− c12 ( −ω B + 2κ A)]
(13)
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APPENDIX
L
et’s combine two sets of equations for dx and dy, which
yields equations for the drift coefficients ␷A and ␷B and
diffusion coefficients ␴ij in the main text’s Equation 11. The
first uses Ito’s formula, which relates dx and dy to dA and dB,
dx =
∂x dx
dx
+ dA + dB
∂t dA
dB
= –␸Asin ␸t dt + cos ␸t[␷AdT + 11d␹1(T) + ␴12d␹2(T)]
+ sin ␸t[␷BdT + ␴21d␹1(T) + ␴22d␹2(T)],
(A)
and similarly for dy. Terms such as d2x/dA2 and d2x/dB2 don’t
appear here, due to Equation 10. Substitution of Equation
10 into Equation 9 gives a second expression involving dx
and dy:
dx = –␸(Asin␸t – Bcos␸t) dt
dy =
[2␬␸(Asin␸t – Bcos␸t) – (Acos␸t + Bsin␸t)
+ (A(T – ⑀2␶)cos␸(t – ⑀2␶) + B(T – 2)sin␸(t – ⑀2␶) – (Acos␸t +
Bsin␸t)]dt + ␦dw(s).
(B)
Only the additive noise is retained in these equations. As Figure 4 verifies, we have good agreement between the system-approximated density (with additive noise only) and
the density for the system with both additive and multiplicative noise. Thus we’ve confirmed that additive noise plays
the dominant role in the dynamics for the regime considered in the main text, so we can neglect the multiplicative
∆A(T ) =
∆B(T ) =
f1 = 1 +
A(T − ε 2τ ) − A(T )
ε2
B (T − ε τ ) − B (T )
ω 2 + 4κ 2
, f2 =
2κω
ω 2 + 4κ 2
(14)
(15)
The coefficients ␷A and ␷B give the decay of the
oscillations’ amplitude in the absence of noise for
c12 < 0. The noise terms are given by
f12 + f 22
2
ω + 4κ
σ
2 11
f12 + f 22
2
ω + 4κ
σ
2 12
MAY/JUNE 2006
=
p2 f1
δ
,
ε ω p2 + p2
1
2
=−
δ − p1 f1
ε ω p2 + p2
1
2
2π /ω
∫0
dx (Eq.A )u j + dy (Eq.A )v j dt =
dx (Eq.B )u j + dy (Eq.B )v j dt
j = 1, 2,
(C)
where subscripts refer to Equations A and B, and (uj, vj) for j
= 1, 2, are the two solutions to the adjoint linear problem for
Equation 9 with ␦ = 0,
continued on p. 57
(␸2 + 4␬2)p1 = ␸f1 + 2␬f2
2
ω2
2π /ω
∫0
␴21 = –␴12, ␴22 = ␴11
,
ε2
noise terms in the analysis.
To determine the equation for A and B, we equate dx and
dy from Equations A and B, write c1 = c1c + ⑀2c12 for c12 < 0 in
Equations A and B, and use a perturbation expansion for ⑀
<< 1 for the drift terms. The leading order (O(1)) terms cancel because we have the linear homogeneous equation with
c1 = c1c and Equation 10, meaning we treat A and B as constants to the leading order in the multiscale expansion. The
next order corrections are O(⑀2), leaving the drift terms with
coefficient ⑀2dt = dT. The leading order diffusion (noise)
terms have coefficient ␦ << 1.
The resulting system includes terms that vary on both the t
and T time scales. To obtain the coefficients in Equation 10,
which are functions of T only, we use a projection on the fast
oscillations with frequency ␸, defined as the inner product of
(dx, dy) with the solutions to the adjoint of the homogeneous
linear equation with c1 at the critical value of c1c for deterministic systems.1 As is typically used in multiple-scales analysis
for deterministic systems, the inner product has the form
(16)
(␸2 + 4␬2)p2 = ␸f2 + 2␬f1.
(17)
For the purposes of our discussion here, we show
only the additive noise terms. Note that these
terms have a coefficient of ␦/⑀, indicating the ⑀–1
amplification of the noise is due to parameters in
close proximity to the equilibrium position’s stability boundary. The drift coefficients ␷A and ␷B
include terms that depend on the delayed time T =
⑀2t, so, in general, we must determine the behavior
of A and B numerically from Equation 11. As described in the Appendix, the terms in Equation 11
with coefficient j are higher-order corrections, so
we don’t include them here.
Figure 4 compares the numerical calculation
for the stationary probability density p(x) obtained from Equation 9 (with both multiplicative
and additive noise) and the multiscale approxi-
55
1.5
Maximum amplitude
1.0
Large-amplitude
stable oscillation
x
0.5
0.0
–0.5
Small-amplitude
unstable oscillation
–1.0
Zero solution (stable)
(a)
Control parameter (c1)
–1.5
2,200
2,250
(b)
2,300
2,350
2,400
2,450
2,500
t
Figure 5. Implications of coherence resonance. (a) Branch for a subcritical bifurcation from the zero state. For subcritical
values of the control parameter, such as the material parameter c1, noise can cause a transition to stable, large-amplitude
oscillations through a resonance response that allows significant variation beyond unstable, small oscillations. (b)
Realization of the nonlinear system in Equation 5 with a small initial condition, ␶ = 65, and c1 =.104. Without noise (blue
line), the solution decays slowly to the state x = 0. For ␦ = .05 (pink line), the coherence resonance causes a transition
from small-amplitude oscillations to large-amplitude ones.
mation obtained from Equations 11 through 17
(that is, with only additive noise). The good
agreement shows that in this regime, the additive
noise plays the dominant role in amplifying the
oscillations of x, which are a percentage of the
equilibrium z0. The variance is an order of magnitude larger than the noise factor ␦ (in this case,
␦ = .02), reflecting amplification of the noise for
parameter values near the stability boundary in
Figure 3, in which the equilibrium steady state
loses stability to the chatter oscillation. Although
the parameter values are in the stability regime
for the deterministic equilibrium, the oscillations
are sustained and have an amplification factor related to ⑀–1, where ⑀ measures proximity to the
stability boundary.
We obtain the approximate stationary densities
by running simulations over a sufficiently long time
for 5,000 realizations. Note that the calculations
using the multiscale approximation are for A and B
on the T = ⑀–2t scale, so the computations for the
multiscale approximation are a factor of O(⑀–2)
faster than the full system simulation on the t scale.
The numerical simulations used both the EulerMaruyama equation and the multistep methods for
stochastic delay differential equations.28 By using
a higher-order method, we can verify that the oscillations aren’t a result of the numerical method.
56
Additional simulations appear elsewhere27 for
ranges of ␦ and c1; the behavior of the densities for
these other cases is similar.
Implications for Oscillations
in the Nonlinear System
Figure 5 illustrates the implications of coherence
resonance in the linear system for realizations of
the fully nonlinear system. A typical bifurcation
structure for the full system is that of a subcritical
bifurcation, which is found in several studies of deterministic models for machine tool vibrations.11,22,23 For this setting in the absence of noise,
small perturbations from the zero state are damped
because small oscillations are unstable. In the stochastic setting for parameter values near the bifurcation, analysis and computations of the linear
system show small oscillations that are amplified
through coherence resonance. The noise level ␦ is
amplified by a factor ⑀–1, related to the distance
from the critical bifurcation value. Once these oscillations are large enough, the nonlinear system
can make the transition to a branch of large-amplitude behavior, as shown in Figure 5a. The coherence resonance in the stochastic system thus
provides a mechanism by which the tool behavior
can shift to a nonlinear mode, and for large enough
x, the tool can even lose contact with the work
COMPUTING IN SCIENCE & ENGINEERING
continued from p. 55
For (u, v) = (u2, v2) in Equation C, we get
ω
⎛
⎞
cos ωt ⎟
⎜ sinωt , 2
2
ω
κ
+
4
⎟
(u1,v1) = ⎜
2κ
⎜
⎟
ω
+
sin
t
⎟⎠
⎝⎜ ω 2 + 4κ 2
[(␸2 + 4␬2)␷B + ␸2␷B + 2␸K␷A]dT ~
(c12[B(T)(2␬cos␸␶ – ␸sin␸␶) – A(T)(2␬sin␸␶ + ␸cos␸␶)
– 2␬B – ␸A]
+ c1c[B(2␬cos␸␶ – ␸sin␸␶) – A(T)(2␬sin␸␶ + ␸cos␸␶)])dT. (G)
2κ
⎛
⎞
cos ωt ⎟
⎜ cos ωt , 2
2
ω + 4κ
⎟
(u2 ,v 2 ) = ⎜
ω
⎜
⎟
+
sin
t
ω
⎟⎠ .
⎝⎜ ω 2 + 4κ 2
(D)
Under the assumption of multiple scales, we treat terms on
the slow scale T as independent of the fast time t in Equation
C, so the integration over t reduces to integrals of products
of cos␸t and sin␸t. We treat the drift and diffusion coefficients separately, starting first with the drift coefficients with
coefficient dT. To obtain the drift coefficients ␷A and ␷B for
(u, v) = (u1, v1) in Equation C, we get
[(␸ + 4␬ )␷A + ␸ ␷A + 2␸K␷B]dT ~
(c12[A(T)(2␬cos␸␶ – ␸sin␸␶) – B(T)(2␬sin␸␶ + ␸cos␸␶)
– 2␬A + ␸B]
+ c1c[A(2␬cos␸␶ – ␸sin␸␶) – B(T)(2␬sin␸␶ + ␸cos␸␶)])dT. (E)
2
2
2
2
dw =
δ
[cos ωtdW1(T ) + sinωtdW2 (T )] ,
ε
(H)
where W1 and W2 are independent Brownian motions. Then
the equations for the noise terms from Equation C are
2π /ω
∫0
[u j (σ 11d ξ1(T ) + σ 12d ξ2 (T ))
+v j (σ 21d ξ1(T ) + σ 22d ξ2 (T ))]dt =
⎡δ
v j ⎢ (cos ωtdW1(T )
⎣ε
+ sinωtdW2 (T ))]dt , j = 1, 2
2π /ω
∫0
.
(I)
Using Equation D in Equation I, we get equations from which
we can determine the diffusion coefficients in Equation 11.
Here, we’ve written ⑀2dt = dT and
A(T − ε 2τ ) = A(T ) + ε 2
Combining Equations E and G, we get the coefficients ␷A
and ␷B given in Equations D and E.
For the noise terms, we write dw in terms of T using properties of Brownian motion:
A(T − ε 2τ ) − A(T )
≡ A(T ) + ε ∆A(T )
Reference
ε2
,
(F)
1. J. Kevorkian and J.D. Cole, Perturbation Methods in Applied Mathematics,
Springer-Verlag, 1985.
treating A(T) = O(1) and using similar expressions for B(T –
⑀2␶).
piece intermittently. In this case, we must analyze
a more complex model that allows for periods during which the tool might skip.14
Figure 5b demonstrates the transition to a nonlinear mode with large oscillations for the full nonlinear system and compares the realization of the
system for parameters in the stability range for the
zero solution, with and without noise. In the absence of noise, the system’s oscillations damp out
to zero (red line), pushing the system to steady
state. However, even with small noise (in this case,
␦ = .05), we see that small oscillations are amplified
(green line), so that eventually the system makes
the transition to oscillations with O(1) amplitude.
Indeed, experiments show sustained oscillations for
parameters in the deterministic stable regime near
the stability boundary,14 suggesting that noise
could contribute to the dynamics.
Once the system makes a transition to large os-
MAY/JUNE 2006
cillations, the multiplicative noise contributes to
the irregularity; the additive noise doesn’t influence
the dynamics as much because it’s relatively small.
We leave a complete exploration of the fully nonlinear noisy system for future work; it depends on
the development of new approaches for analyzing
fully nonlinear systems with noise and memory.
W
e’ve seen that certain critical
combinations of noise levels and
parameter regimes near stability
boundaries can allow and even
promote transitions between different bistable
states. Even if the system is operating in a regime
in which we might expect the equilibrium state to
be linearly stable, small noisy perturbations can be
amplified through coherence resonance. This amplification is of central importance in practical set-
57
tings, where it can cause transitions to large-amplitude states or chatter.
A multiple-scales analysis yields a means for efficient computation of the probability density for a
machine tool’s position. This density is the basis for
quantities that can describe the system’s state: density, for example, can help us calculate the likelihood that the variations x exceed a threshold xc,
corresponding to the transition to chatter,25 or estimate the time in which such a transition might
occur. In this case, the amplification factor indicates
that this transition can occur on a O(1) time scale,
so we can use the analysis to predict parameter
regimes in which such a transition is likely to be
observed. Generalizations of the approach show
promise in biological modeling—in particular,
studies of periodic recurrence of epidemics due to
random variations in populations and in noise-induced neuronal activity and synchronization. It has
also been applied in nonlinear settings, such as logistic models and large-amplitude oscillations in
neuronal bursting dynamics.
Acknowledgments
I thank Gabor Stepan for many helpful discussions.
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Rachel Kuske is an associate professor in the Department
of Mathematics, University of British Columbia, Canada.
Her technical interests include stochastic models, applied
nonlinear dynamics, and mathematical modeling. Kuske
has a PhD in applied mathematics from Northwestern
University. She is a member of SIAM, the Canadian Applied and Industrial Mathematics Society, the Canadian
Mathematical Society, and the Association for Women in
Mathematics. Contact her at rachel@math.ubc.ca.
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