Abstract. It is well known that the Brusselator model reveals simple oscillations and damped oscillations by the suitable parameters. However there is no chemical reactor system that corresponds to the Brusselator model. We have a good real physical system that corresponds to the Brusselator model under some conditions. The system is a free damped oscillation of soft material and described by an equation of motion. In this paper, we show the relation between the Brusselator model and the physical system. We also estimate the experimental data of the physical system

Takashi Amemiya

Tomohiko Yamaguchi

Yoshitomi Morikawa

CiteSeerX

249
Forma, 15, 249–256, 2000Original  Paper
Analytical Approach for Oscillation Properties of Soft Materials
Yoshitomi MORIKAWA, Tomohiko YAMAGUCHI and Takashi AMEMIYA
National Institute of Materials and Chemical Research,
Higashi 1-1, Tsukuba, Ibaraki 305-8565, Japan
E-mail: morikawa@nimc.go.jp; tomo@nimc.go.jp; amemiya@nimc.go.jp
(Received April 19, 2000; Accepted May 19, 2000)
Keywords:  Modified Brusselator Model, The Equation of Motion, Soft Materials, Fabric
Pendulum, Free Damped Oscillations
Abstract.  It is well known that the Brusselator model reveals simple oscillations and
damped oscillations by the suitable parameters. However there is no chemical reactor
system that corresponds to the Brusselator model. We have a good real physical system
that corresponds to the Brusselator model under some conditions. The system is a free
damped oscillation of soft material and described by an equation of motion. In this paper,
we show the relation between the Brusselator model and the physical system. We also
estimate the experimental data of the physical system.
1.  Introduction
We consider a modified Brusselator model in an open system.
A → X,
B + X → Y + D,
2X + Y → (2 + ε)X, (1)
X → E.
The fluxes A and B are assumed to be constant. X and Y are the intermediates, and X
changes Y into X by the autocatalytic reaction. Free parameter ε reveals to a “strength “ of
the autocatalytic process and this modified Brusselator model is reduced to the original
Brusselator model (PRIGOGINE and LEFEVER, 1968; PRIGOGINE et al., 1969) when ε = 1. We
show briefly that this modified Brusselator model is related to a physical phenomenon
described by the equation of motion under some special conditions.
The rate equations concerning X and Y are
dX
dt
 = A – (B + 1)X + εX2Y,
250 Y. MORIKAWA et al.
dY
dt
 = BX – X2Y. (2)
Here, we take rate constants are equal to 1 for simplicity.
The stationary solution is given by the controllable positive parameters A and B, and
free parameter ε.
(X0, Y0) = 
A
B
B
B
A(1 ) +  1
 
 +  1
− 
−( )






ε
ε
, .
1
(3)
Time developments of the fluctuations (x, y) around this stationary solution are
approximated to the following linearized equations
dx
dt
 = L11x + L12y,
dy
dt
 = L21x + L22y, (4)
where Jacobian’s elements are
L11 = (2ε – 1)B – 1, L12 = εX02,
L21 = –B, L22 = –X0
2. (5)
We can choose L11 = 0 under the condition B = 1/(2ε – 1). We also get another
condition ε > 1/2 because B is positive. In this choice, trJ = –X02 < 0 and detJ = ε BX02
> 0. Therefore, if the condition (–trJ)2 – 4 detJ < 0 (or A < 2[ε /(2ε – 1)]3/2) is satisfied,
then a damped oscillation occurs.
After the normalization εX02 y ≡ y′ ,we can scale L12 = 1. Finally, by the replacements
εBX02 ≡ 
k
m
,  X0
2 ≡ 2γ, (6)
we get the equation of motion
dx
dt 
 = y′,
dy
dt
′
 = – 
k
m
x –2γy′ (7)
Analytical Approach for Oscillation Properties of Soft Materials 251
or
m
d x
dt
2
2 + 2mγ
dx
dt 
 + kx = 0. (8)
This is just the equation of motion at each position for the free damped oscillation of
soft material as shown in Sec. 2. The modified Brusselator model around the stationary
solution was related to the physical phenomenon described by the equation of motion under
some special conditions. The previous controllable parameters A and B correspond to the
damping coefficient γ and the elastic constant k. Here, m is the mass of soft material.
Therefore, we can also estimate the controllable parameters in the modified Brusselator
model by means of analyzing the free damped oscillation of soft material.
2.  The Equation of Motion for Soft Materials
Soft materials can be changed in the form by the external forces contrary to rigid
bodies and we have extended the equation of motion for soft material phenomenologically
from that for rigid body by considering this changeability in the form of soft material
(MORIKAWA, 1995).
The equation of motion for the free damped oscillation of rigid body is written by the
moment of inertia I under the assumption that the resistance is proportional to its velocity
I 
d
dt
2θ
2 + cIh
2 d
dt
θ
+ mghθ = 0, (9)
where h = l/2 and l is the length of rigid body. If rigid body has uniform mass distribution,
the moment of inertia is proportional to its mass I = m (l2/12 + h2). The displacement at each
position u on rigid body is approximated Xu  uθ (0 ≤ u ≤ l) within the small amplitude
oscillation. Then the equation of motion at each position u is derived
m
d Xu
dt
2
2  + 2mγ
dX
dt
u + kXu = 0,  (10)
k = mω02, ω0 = 2πν0.
We have extended this equation of motion to that of soft material phenomenologically
considering its changeability in the form
m
d X
dt
u
2
2 + 2mγu
dX
dt
u + kuXu = 0. (11)
The elastic coefficient ku = mω0u2 (ω0u = 2πν0u) and the damping coefficient γu have the
suffix u which reflect the changeability in the form of soft material. The proper frequency
252 Y. MORIKAWA et al.
ν0u has the relation with the oscillation period Tu and the damping coefficient γu
ν γ
π0
 =  
1
u
u
u
T




+ 










2 2
1 2
2
. (12)
Thus, we can analyze the oscillation characteristics by estimating the oscillation curves.
In the ideal case without damping, the proper frequency is given by
ν
π0
 =  
1
2
 
g
l




1 2
(simple pendulum), (13a)
ν
π0
1 2
2
1
2
= 



+






g
L
L
l
h
h
,    =
12
2
(physical pendulum). (13b)
We can compare the experimental values with these theoretical ones.
3.  Experiments
3.1.  Oscillation materials
We picked up several cotton fabric elements as soft materials and an acrylic plate as
rigid body. All fabric elements were cut out to be rectangular with the width 5 cm from the
cloth materials.
We cut out 4 samples (sample A–D) with the same length 20 cm from the same cloth
material 1. These samples were expected to have the same characteristics. We cut out the
fabric strips with the 5 types of length (5, 10, 20, 40, 60 cm) from the cloth material 1. We
also prepared the fabric strips with the same length 20 cm from 4 kinds of cloth materials
(1–4). Each thickness is 5.65, 3.20, 3.33, 6.56 × 10–2 cm and each weight is 1.50, 0.81, 0.65,
2.27 g respectively. The acrylic plate with the length 20 cm and the width 5 cm was prepared
to compare with the fabric strips.
3.2.  Oscillation experiments
The upper end of each fabric strip was fixed to a holding rod in a horizontal direction
that could rotate freely on its axis. Each fabric strip was hung down and oscillated freely
in the air centering round the rod just like the pendulum. Therefore, we call this as “fabric
pendulum”. In some cases, we put on several weights at the lower free end of the fabric
strips.
The acrylic plate was also fixed to the holding rod. We hung down and oscillated the
acrylic plate freely in the air as the fabric strips.
The oscillation amplitude was restricted within the narrow range in order to realize the
small angle approximation.
The control of the temperature and the humidity is crucial because cloth materials
contain the air inside and also water except for special cases. The temperature 21 ± 2°C and
the humidity 65 ± 5% were kept during the whole experiments.
Analytical Approach for Oscillation Properties of Soft Materials 253
4.  Results and Discussions
The oscillation characteristics depend on the position of soft material. In this paper,
we focus on the oscillation properties at the lower free end of soft material.
The discrepancy of the oscillation characteristics between 4 samples from the same
cloth material 1 became smaller as mass ratio between the weight and the fabric strip
increased. We pick up the sample D and exhibit the damping coefficient and the proper
frequency against mass ratio (Fig.1). Fabric pendulum was sensitively affected by the
weights. The oscillation characteristics became rapidly smaller and closer to those of
simple pendulum as mass ratio increased.
Figure 2 reveals the damping coefficient against the length of the fabric strip without
weights from the cloth material 1. Typical statistical error is shown for the material 1 with
the length 20 cm. The damping coefficient showed a trend to be smaller and closer to the
theoretical value of simple pendulum beyond the statistical errors as the length of the fabric
strip increased.
It was found that the changeability in the form and the size effect of soft material on
the oscillation characteristics decreased as the length of the fabric strip increased. This is
the similar effect to the mass ratio’s increase as shown above.
Figure 3 presents the damping coefficient against mass ratio for the fabric strips with
the length 20 cm from 4 kinds of cloth materials. The damping coefficient became smaller
and closer to the theoretical value of simple pendulum as mass ratio increased as shown in
Fig.1. The oscillation properties were slightly different between the heavier and thicker
fabric strips from the material 1 and 4 and the lighter and thinner strips from the material
2 and 3.
These are interpreted from the transition of the changeability in the form of the fabric
strip. The fabric strip has suffered some resistances due to the air existence and the inner
frictions, and the oscillation has been damped. Within the small mass ratio region, the inner
frictions dominate its damping and the oscillations of the thinner fabric strips were damped
faster than the thicker ones. The damping coefficients of the fabric strips were larger than
that of the acrylic plate. As mass ratio increases, the changeability in the form of the fabric
strip decreases and the air resistance dominates its damping. Therefore, the oscillations of
the acrylic plate were strongly damped than those of the fabric strips.
We can estimate the controllable parameters in the modified Brusselator model
corresponding to the free damped oscillation of soft material by Eq. (6) from the experimental
data. In the fabric strip with the length 20 cm without weights, ε  0.50, A  78, and B 
220. The fluxes are larger than those of the typical chemical reactor system by the order of
2. This is a reflection of the fact that k/m >> γ in fabric pendulum.
5.  Conclusions
It is well known that there is no chemical reactor system that corresponds to the
Brusselator model. The modified Brusselator model linearized by fluctuations corresponds
to the equation of motion for the free damped oscillation of soft material under the
conditions B = 1/(2ε – 1), A < 2 [ε/(2ε – 1)]3/2, ε BX02 = k/m, and X02 = 2γ.
We also estimated the oscillation properties of soft materials. Fabric pendulum was
254 Y. MORIKAWA et al.
Fig. 1.  Damping coefficient (a) and proper frequency (b) against mass ratio for the sample D from the cloth
material 1.
(a)
(b)
Analytical Approach for Oscillation Properties of Soft Materials 255
Fig. 2.  Damping coefficient against the length for the fabric strips without weights from the cloth material 1.
Fig. 3. Damping coefficient against mass ratio for the fabric strips with the length 20 cm from 4 kinds of cloth
materials.
256 Y. MORIKAWA et al.
found to be closer to simple pendulum when the weight and/or the length increased. The
weights improved the oscillation behaviors and the damped oscillation curves became
smoothly instead of decreasing the features of soft materials. The extrapolation of mass
ratio to 0 may be effective for soft materials accompanied with the large statistical errors
experimentally.
REFERENCES
MORIKAWA, Y. (1995) Sen’I Gakkaishi, 51, 333.
PRIGOGINE, I. and LEFEVER, R. (1968) J. Chem. Phys., 48, 1695.
PRIGOGINE, I., LEFEVER, R., GOLDBETER, A. and HERSCHKOWITZ-KAUFMAN, M. (1969) Nature, 223, 913.


Analytical Approach for Oscillation Properties of Soft Materials

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=26E3D4B411549C8AA89B179FD705E0AD?doi=10.1.1.1068.3043&rep=rep1&type=pdf

Analytical Approach for Oscillation Properties of Soft Materials

Abstract

Similar works

Full text

Available Versions

CiteSeerX