Local Complexity and Global
Nonlinear Modes in Large
Arrays of Elastic Rods in an Air
Cross-Flow
Masaharu Kuroda
Applied Complexity Engineering Group,
National Institute of Advanced Industrial Science and Technology,
Japan
m-kuroda@aist.go.jp
Francis C. Moon
Sibley School of Mechanical and Aerospace Engineering,
Cornell University,
USA
fcm3@cornell.edu
1.
Introduction
Through the ages in engineering fields, many researchers have been interested in flow
of a fluid around an object and vibration of the object. This report experimentally
investigates a periodic structure in a cross-flow, especially, an elastic structure such as
a rod array in an air cross-flow. Complex dynamics caused by coupling between the
fluid and periodic structure is famous as an outstanding problem in the research field
of fluid-related vibration or fluid-induced vibration [1-6]. Not only as a scientifically
prolific research target, but also as an experimental model such as a pipe array of a
heat exchanger system, it is very useful in engineering to investigate its dynamic
characteristics. From the viewpoint of a pattern-formation problem in spatio-temporal
dynamics of a complex system, we attempt to newly consider this kind of well-known
and persistent fluid-engineering problem.
In this study, we discuss experimental nonlinear dynamics of a large array of up to
1000 elements in a cross flow. In this paper we extend the results reported in the letter
of Moon and Kuroda [7] for a 1000 rod array. The experimental data presented here is
related to recent theoretical work of Homer and Hogan [8] who have presented an
impact dynamics model for a large number of pipes vibrating in a heat exchanger.
2.
Experimental Set-Up and Conditions
Figure 1 shows the experimental set-up. A rectangular array structure of
cantilevered rods placed in a wind tunnel is shown schematically. As shown in this
figure, rod-like structures are cantilevered at the base and are free to vibrate at the top.
Coupling between rods consists of fluid forces and contact when vibration amplitude
becomes too large. Fluid forces are of two types: fluid-elastic nearest-neighbor forces
equivalent to springs and dampers; and non-nearest neighbor forces produced by
vortices leaving the forward rows of cylinders and affecting dynamics of rearward
cylinder rows [9].
Elastic Rods
Figure. 1.
Schematic diagram of the experimental set-up
The wind tunnel is a low turbulence system with a 25.6 cm x 25.6-cm cross
section. Wind speeds ranged from 0.0-12.0 m/s. The Reynolds number based on rod
diameter ranges from 200-900. Photos show the entire view of an array structure in
the wind-tunnel observatory section. Equipped with small accelerometer probes, a test
rig is arranged inside the observation part and flood-lights, which are necessary for
videotaping, are located under the observation part. Also, photographic images with
shutter speed on the order of the vibration period of 10 Hz were taken. They also
show various rod arrays for differing materials, gap ratios, and numbers of elements
used in experiments.
3.
Steel-Rod Array Experiments
Each array used here is composed of steel rods with 1.59 mm diameter and 17.1cm
length. The first eigen frequency of the steel rod is 26 Hz.
3.1.
Local Complexity in an Array of 90 Steel-Rods
Observations of tip vibration dynamics in the 90 rod case reveal complex rod-motion
patterns. Some rods appear nearly stationary, others vibrate in a straight-line motion
at an angle to upstream flow, and the rest vibrate in elliptical patterns sometimes
associated with rod to rod contact. Figure 2 shows the ratio of rods in straight-line
motion and elliptical orbits as a function of wind speed. Clearly, this illustrates onset
of motion at a critical wind speed and growth of ratio of rods in linear motions after a
full turbulence regime. This may be related to greater incidence of rod impact at
higher wind velocities.
Percentage of Rods [%]
100
Percentage of Rods in Elliptical Motion
80
Percentage of Rods in Linear Motion
60
40
20
0
0.0
2.0
4.0
6.0
8.0
10.0
Upstream Flow Velocity [m/s]
Figure. 2.
Percentage of rods in straight-line and those in elliptical motion
While rod frequencies lie close to their eigen frequencies, phases and types of
motion, i.e., stationary, straight-line, and elliptical orbit, show no regular modal
pattern and change over time. It is important to emphasize that linear theory would
predict 2 x N x M eigen modes for an N x M array of rods. However, no clear pattern
emerges as wind velocity in the wind-tunnel increases. Thus, we will apply an
entropy measure of complexity to describe the dynamics [10].
3.1.1. Entropy Measure for Complexity
Entropy measures employed here are based on the information entropy
S = Σ pn log (1/ pn),
(1)
where pn is a probability measure for occurrence of a certain spatial pattern [11]. We
explored the possibility of defining entropy using previously obtained data expressed
by discrete variables. Two photographs, Figs. 3(a) and 3(b), show examples of
calculated entropies. Figures 4(a) and 4(b), which are matched with Figs. 3(a) and
3(b), respectively, are used for entropy calculation.
(a) A Photo at 7.7m/s flow velocity
by 2.0 sec. shutter speed
Figure. 3.
Photo examples of the array of 90 vibrating rods
(a) The spatial Pattern for Fig. 3(a)
Figure. 4.
(b) A Photo at 5.6m/s flow velocity
by 2.0 sec. shutter speed
(b) The spatial Pattern for Fig. 3(b)
Spatial pattern examples of the array of 90 vibrating rods
Entropy is calculated here for two-dimensional spatial patterns. We call this cluster
pattern entropy (CPE). From Fig. 5, it is understood that cluster pattern entropy took
the maximum value, but not in the full turbulence case.
Cluster Pattern Entropy
CPE(1 Cell = 2 x 2 Boxes) [-]
]
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0.0
2.0
4.0
6.0
8.0
10.0
Wind Speed [m/s]
White Noise Pattern
Gaussian Noise Pattern
CPE
Figure. 5.
3.2.
Cluster Pattern Entropy (CPE) as a function of wind speed
Spatio-Temporal Patterns Appearing in an Array of 300 Steel-Rods
The array has 30 columns and 10 rows with the gap ratio 1.0. Observed from the top
in the low wind-speed regime, rod tops seem to oscillate individually and do not show
significant difference in density distribution over time. If rod alignment were more
precise, we could have observed some kind of organized movement even in this wind
velocity regime.
Figure 6 exhibits videotape frames of rod-array behavior from above and
corresponding contour maps of rod-density distribution in the middle wind-speed
regime. The white area on the contour maps is manifestly larger than before; the blue
area appears as though it has torn itself and dense and scattered areas alternate
remarkably over time. Finally, clusters, i.e., collective movements of some rods,
emerge in the rod-density distribution. We confirmed that these clusters do not move
along a specific pattern at this stage, but rather just move and freely collide with one
another.
Figure 7 displays videotape frames of rod-array behavior from above and
corresponding contour maps of rod-density distribution in the high wind-speed regime.
The white area on contour maps continues to widen; rifts and gaps appear on the blue
area and the number of clusters displays continued increase. Furthermore, it is most
characteristic and important that those clusters are linked together as a chain and that
the chain moves in a diagonal direction to the center part of the backmost row
repeatedly, as a wave does. From the videotape data, it is clear that wave-like motions
originating at both sides collide at the center part on the back row.
T = T0
T = T0
T = T0
T = T0
T = T0 + ∆T
T = T0 + ∆T
T = T0 + ∆T
T = T0 + ∆T
T = T0 + 2∆T
T = T0 + 2∆T
T = T0 + 2∆T
T = T0 + 2∆T
T = T0 + 3∆T
T = T0 + 3∆T
T = T0 + 3∆T
T = T0 + 3∆T
T = T0 + 4∆T
0.0 -1.0
3.0 -4.0
1.0 -2.0
4.0 -5.0
2.0 -3.0
Figure. 6. A sequence of VTR frames
and rod-density distribution at 7.0 m/s
wind velocity
T = T0 + 4∆T
0.0 -1.0
1.0 -2.0
3.0 -4.0
4.0 -5.0
2.0 -3.0
Figure. 7. A sequence of VTR frames
and rod-density distribution at 10.5 m/s
wind velocity
3.2.1.
Discussion
Behavior of this wave-like motion looks like a soliton. This fact is the reason why
formulation of a mathematical model like the Toda-lattice, which is well-known in
the field of nonlinear lattice dynamics, prospectively leads to clarification of these
complex phenomena appearing in an array of fluid-elastic oscillators. Unfortunately,
it is not certain whether waves collapse or pass through each other without
soliton-like effect, or if they reflect each other at that point; the number of rows is
insufficient to make that determination.
Furthermore, complex behavior patterns of oscillating rods at each wind-velocity
regime seem to present an analog of cellular automata. Especially in the high-wind
velocity regime, the pattern seems to resemble Wolfram’s Class 4 in cellular automata
dynamics [12].
3.3.
Time Series Data from Accelerometers
On the other hand, time-series data on one rod, shown in Figs. 8 and 9, suggests that
bursting phenomena are probably related to impacts between rods as wind speed is
increased. These time-series data were obtained from a small accelerometer attached
1 cm up from the foot of the rod in the middle of the front row. In this location the
accelerometer was not sensitive to low frequency first mode vibrations of the rod but
was very sensitive to the higher modes in the rods produced by impacts. This meant
that the sensor did not respond to pre-impact flow induced vibrations.
(Magnification)
Figure. 8.
Acceleration at the front-center rod at 7.0 m/s wind speed
(Magnification)
Figure. 9.
Acceleration at the front-center rod at 11.2 m/s wind speed
In addition, Fig. 10 shows a plot of the number of bursting waveforms for 24 sec.
as a function of wind speed. Bursting phenomena take place due to rod collisions.
Bursting peaks were counted by the program coded on MATLAB.
Number of Bursts
2400
2000
1600
1200
800
400
0
1.4
2.8
4.2
5.6
7
8.4
9.8
11.2
Upstream Flow Velocity [m/s]
Averaged Number
Figure. 10.
103 m/s2
Number of bursts among rods as a function of wind speed at a threshold of 5.0 x
3.3.1. Relationship between Rod-to-Rod Collisions and Self-Organization
In the low wind-speed regime around 3.5 m/s, fluid-elastic forces govern rod
movement. Videotape records show that rods are free to move individually in this
wind speed regime. In the middle wind-speed regime of about 7.0 m/s, not only
fluid-elastic forces, but also impacts among rods start affecting rod behavior. It is
especially notable that boundaries appear among oscillating rods and that rod groups
behave collectively. In other words, rod clustering occurs. It can not be said that rod
clusters move with a clear temporal pattern, but they appear to be just pushing each
other. In the high wind speed regime of approximately 10.5 m/s, dominant forces
determining rod behavior shift to impact from fluid-elastic forces. This is understood
from the fact that bursting phenomena frequency at 10.5 m/s is 10 times higher than at
7.0 m/s, as shown in Fig. 10. As a result, global wave-like motion takes place.
Furthermore, slow-motion replay with a 1/2000 sec. shutter-speed captures this
wave-like motion superbly.
3.3.2. Power Law
Finally, we confirm that a scaling law exists in cluster generation and that the scaling
law is predicted to be described by a fractal dimension. The basic relationship
between the number of bursts and wind speed is
V2 = c N α,
(2)
where V represents wind velocity, N shows the number of bursts and c is a constant.
2
The term V is proportional to input energy into the rod array. By taking the
logarithm of both sides of Eq. (2), we obtain a log N-log V plot,
Y = const. + (α/2) X,
(3)
where X = Log N, Y = Log V and α is the fractal dimension. We obtain the
graph slope; Figure 11 indicates that α is almost 0.25. Therefore, energy input into
the rod array by wind and the number of bursts follow the power law with a fractal
dimension of 0.25.
L o g V [ V : W in d v e lo c it y ]
100
10
1
100
1000
Log N [N:Number ob bursts]
b
a
2
c
d
e
10000
Average
α
V =cN , α = 0.242
Figure. 11. Power law between the number of bursting signals and wind velocity at an
acceleration threshold of 5.0 x 103 m/s2
4.
Polycarbonate-Rod Array Experiments
Each array used here is composed of polycarbonate rods with 3.18 mm diameter and
20.0 cm length. The first eigen frequency of the polycarbonate rod is 18 Hz.
In these experiments, specially manufactured polycarbonate rods were made to
insure straight cylinders. These rods allowed greater frontal cross section without a
large increase in stiffness. With the transparent rods, we could shine light from
underneath the rod array so that the camera could see points of light moving as the
array vibrated, instead of using reflected light from the tops of the steel rods.
4.1.
Global Nonlinear
Polycarbonate-Rods
Modes
observed
in
an
Array
of
300
Global nonlinear modes in large arrays of elastic rod oscillators were discovered
while local complex behavior was revealed through videotaped experimental results.
Figures 12 and 13 show videotape frames capturing the behavior of a 300
polycarbonate rod array from the above at 11.20 m/s wind speed. The array has 25
columns and 12 rows with the gap ratio 1.5. Each two-picture set shows successive
images; the rod-array spatial pattern oscillates from that in the upper picture to that of
the lower picture.
Wind
Wind
Wind
Wind
Figure. 12.
Symmetrical mode of 25
column x 12 row array of 300 rods
Figure. 13. Asymmetrical mode of 25
column x 12 row array of 300 rods
Furthermore, it was clarified that observed nonlinear global modes are limited to
two types: a symmetrical mode and asymmetrical mode. Also, the overall vibrating
rod-array shape switches from one mode to another repeatedly over time [13].
4.2.
Global Nonlinear
Polycarbonate-Rods
Modes
observed
in
an
Array
of
1000
Figures 14 and 15 show videotape frames capturing behavior of a
1000-polycarbonate-rod array from above at 8.40 m/s wind speed. The array has 25
columns and 40 rows with the gap ratio 1.5. Each two-picture set shows successive
images, and the rod-array spatial pattern oscillates from that in the upper picture to
that of the lower picture. By examining videotaped rod motions, we found that several
characteristic frequencies of motion exist: the frequency of the switch between
symmetrical mode and asymmetrical mode, those of waves proceeding on boundary
edges of the array, and those of waves passing through the inner area of the array.
Wind
Wind
Wind
Wind
Figure. 14. Symmetrical mode of 25
column x 40 row array of 1000 rods
5.
Figure. 15. Asymmetrical mode of 25
column x 40 row array of 1000 rods
Conclusions
Non-stationary complex phenomena occurring in large arrays of up to 1000
vibrating-rods in the wind tunnel were investigated by experiment.
As the intensity of interaction between neighboring elements (frequency of
collisions among rods in this case) increases, a set of the elements (a rod-array in this
case) achieves globally better-organized behavior. Also, the organized behavior
produces a transfiguration in quality in a staircase pattern when it crosses over a
threshold as the phase transition of matter does. In this manner, specifically,
individual rods, clusters of rods, and a wave of a chain of clusters comprise the
central players of dynamic order-formation shifts at each stage. Finally, conformation
of the rod-array collective behaviors leads to two types of global nonlinear modes:
symmetrical mode and asymmetrical mode.
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