10
20
30
t [s] 40
50
60
70
0.07
0.02
-0.35
-0.25
-0.15
-0.05
0.05
0.15
0.25
0.35
-0.03
-0.08
Fig. 12. Deformation, orientation and trajectory of the centroid of the drop, using the
parameter lin = 50º . The mean deformation is DT = 0.1049, STD=0.0027, the mean orientation
angle is 41.8º.
Deformation and Orientation of the drop
0.12
100 le
0.1
n
80
gn
io
0.08
at
eter
60
An ]
0.06
io
eg
rm
am
40
ar 0.04
tat
[D
efo
P
D
20
0.02
rien
0
0
O
0
10
20
30 t [s]
40
50
60
0.12
0.08
0.04
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.04
-0.08
-0.12
Fig. 13. Deformation, orientation and trajectory of the centroid of the drop, using vertical
limits. The mean deformation is DT = 0.1039, STD=0.0020, the mean orientation angle is 41.8º.
Nonlinear Control Applied to the Rheology of Drops in Elongational Flows with Vorticity
51
6. Numerical results and comparisons with the experimental results
In order to study numerically the evolution of the drop on the proposed control strategy, we
use the results of Reyes et al. (2011). We just outline the main steps to introduce the
extensions used in this Chapter to study the effect of noise and the imperfections present in
the experimental situation.
We thus proceed as in Reyes et al. (2011) advancing the drop boundary using the equation
߲ܠ
߲ݐ = (ܞ௦ ∙ ܖ)ܖ
(6)
where the velocity ܞ௦ in Eq. 6 is determined solving a boundary integral equation on the
drop surface (See Pozrikidis, 1992).
The control is implemented as follows: From the new drop surface obtained from Eq. 6, we
determined the center of mass of the drop using the same integral of the experiments. Once
this is determined, we verified if it falls inside the tolerance region. When it is out of the
selected boundary, we applied the correcting flow obtained adjusting the angular velocities
using Eq. 1.
We considered the effect of noise and imperfections as follows. In the first place, we noted
from the experiments a systematic variation of the angular velocity due to the geometrical
imperfections of the cylinder. This was fitted with a single harmonic function with
frequency and amplitude determined from the experimental values for the observed flow
without drop. The random noise was taken to be white noise. With these new elements, we
used the same code described in Reyes et al. (2011) to calculate the trajectories of the drop’s
center of mass, the deformation parameter and the angle of alignment in order to compare
with the experimental results of the previous Section.
It is to be noted that the solution of the system are very sensitive in detail to the initial
conditions. However, the broad features which depend on the limit cycle nature of the
motion for the center of mass are very robust.
Because of this reason, we start the numerical solution with initial conditions for the drop’s
center of mass which are taken from the experimental data when the initial rapid transients
have subsided. Moreover, the numerical flow is started at nominal values since the
inhomogeneities presented prevent the analytical construction of the initial flow.
With this, we expect a very good agreement between the numerical and experimental values
of the deformation and orientation. This is shown in Figures 14-18. Although those figures
were generated with different initial conditions, the broad behavior is similar to the
experimental data. The trajectories are also compared. We see good agreement in the broad
features, in particular, when the control is operational.
The experimental data shows larger excursions from the nominal stagnation point. These
are due to the mismatch between the commercial worm gear and worm mechanism used to
reduce the angular velocity of the motors and transmit the motion to the rollers.
In Fig. 19 we display the X and Y component of the motion for the center of mass, the blue
lines shows the experimental values and the red lines the numerical solution. The
comparison is good considering the mismatch between the initial flows up to t = 10 s.
52
Applications of Nonlinear Control
0.2
0.1
0.0
− 0.1
DT
θ
− 0.2
0.108
42.4
0.106
0.104
42.2
0.102
42.0
− 0.3
0.100
41.8
0.098
.
.
t γ
t γ
5 10 15 20 25 30
5
10 15 20 25 30
− 0.4
− 0.4
− 0.2
0.0
0.2
0.4
Fig. 14. Trajectory of the centroid of a drop, using vertical limits. The insert graph shows the
deformation and the orientation angle. The mean deformation is D T = 0.1039,
STD = 0.000972 with a mean orientation angle of 41.8º, STD = 0.07626 .
0.2
0.1
0.0
− 0.1
DT
θ
− 0.2
0.108
42.4
0.106
0.104
42.2
0.102
42.0
− 0.3
0.100
41.8
0.098
.
.
t γ
t γ
5 10 15 20 25 30
5
10 15 20 25 30
− 0.4
− 0.4
− 0.2
0.0
0.2
0.4
Fig. 15. Trajectory of the centroid of a drop, using the parameter lin same as the incoming
axis (19.435 º). The insert graph shows the deformation and the orientation angle. The mean
deformation is D T = 0.10388, STD=0.00104017 with a mean orientation angle of 41.801º,
STD = 0.0764 .
Nonlinear Control Applied to the Rheology of Drops in Elongational Flows with Vorticity
53
0.2
0.1
0.0
− 0.1
DT
θ
− 0.2
0.108
42.4
0.106
42.2
0.104
0.102
42.0
− 0.3
0.100
41.8
0.098
.
.
t γ
t γ
0
5 10 15 20 25 30
5
10 15 20 25 30
− 0.4
− 0.4
− 0.2
0.0
0.2
0.4
Fig. 16. Trajectory of the centroid of a drop, using the parameter lin = 30 º. The insert graph
shows the deformation and the orientation angle. The mean deformation is D T = 0.103915,
STD=0.0009935 with a mean orientation angle of 41.7993º, STD = 0.0784447 .
0.2
0.1
0.0
− 0.1
DT
θ
− 0.2
0.108
42.4
0.106
0.104
42.2
0.102
42.0
− 0.3
0.100
41.8
0.098
.
.
t γ
t γ
5 10 15 20 25 30
5
10 15 20 25 30
− 0.4
− 0.4
− 0.2
0.0
0.2
0.4
Fig. 17. Trajectory of the centroid of a drop, using the parameter lin = 40 º. The insert graph
shows the deformation and the orientation angle. The mean deformation is D T = 0.103883,
STD=0.00092441 with a mean orientation angle of 41.7992º, STD = 0.0735669 .
54
Applications of Nonlinear Control
0.2
0.1
0.0
− 0.1
DT
θ
− 0.2
0.108
42.4
0.106
0.104
42.2
0.102
42.0
− 0.3
0.100
0.098
41.8
.
.
t γ
t γ
0
5 10 15 20 25 30
5
10 15 20 25 30
− 0.4
− 0.4
− 0.2
0.0
0.2
0.4
Fig. 18. Trajectory of the centroid of a drop, using the parameter lin = 50º. The insert graph
shows the deformation and the orientation angle. The mean deformation is D T = 0.103934,
STD=0.00096193 with a mean orientation angle of 41.7972º, STD = 0.0733545 .
X coordinate
0.3
0.2
0.1
.
0.0
γ t
5
10
15
20
25
30
− 0.1
− 0.2
− 0.3
Y coordinate
0.06
0.04
0.02
.
0.00
γ t
5
10
15
20
25
30
− 0.02
− 0.04
Fig. 19. Experimental and numerical comparisons of the X and Y coordinates of the center of
mass of a drop subjected to the control. The blue lines are for the experimental data whereas
the red lines are for the numerical data.
Nonlinear Control Applied to the Rheology of Drops in Elongational Flows with Vorticity
55
Beyond this time, the experimental data shows larger excursions. These are due to the
mismatch between the commercial worm gear and worm mechanism used to reduce the
angular velocity of the motors and transmit the motion to the rollers.
It is to be noted the remarkable agreement in the actual drop parameters which are the ones
of interest.
7. Conclusions
We have shown that it is possible to maintain the position of a drop about the unstable
stagnation point of the flow field generated by a TRM setup. This scheme is of the upmost
importance for studies of drops in elongational flow with significant vorticity, and of
relevance because its space of parameters is not accessible to FRM flows previously studied
since Bentley (1986a). Indeed, TRMs expand the family of 2D elongational flows amenable
with a four roll mill, albeit the former flows carry amounts of vorticity similar to that of
simple shear flows.
But because the TRMs configuration can only displace the stagnation point along the line
between the cylinder axes, the control scheme developed for FRMs or PBAs cannot be used
for studies of drop's dynamics in elongational flows of the TRM type. For TRMs the control
scheme is based upon features of Poincaré-Bendixson limiting cycles. However, these limit
cycles do not imply that the wandering trajectory of the drop has to be contained in a very
tight region about the nominal stagnation point. The control scheme appears capable of
working for a tolerance region with the cases tested experimentally.
The control implemented in the experimental has shown to be successful. At this point,
several complications have been seen in the implementation: nevertheless the control
scheme is robust enough to keep the drop inside a region where the parameters of interest
have a low variation, for a long times, enough to have reliable measures of the relevant
parameters. The figures 9-13 shown that despite the trajectory of the drop, the parameters of
interest in drop dynamics (DT and orientation angle) are the same.
It is important to mention that even when the comparisons are made just for only one flow-
type parameter, the results shown that it is reasonable to expect the same when we will use a
different flow-type parameter, i.e. a different geometry.
Important differences have been observed between the experimental and numerical
trajectories, this disagreement is due to mechanical imperfections, and could be fixed in the
future by using an system motion transmission with better precision.
8. Acknowledgment
IYR thanks to CEP-UNAM for funding his graduate fellowship, MAHR thanks DGAPA-
UNAM, AAM thanks the FENOMEC program. EG thanks CONACyT research grants.
9. References
Acrivos, A. (1983). The breakup of small drops and bubbles in shear flows. Ann. NY Acad.
Sci. , 404:1-11
56
Applications of Nonlinear Control
Arfken, G. (1971). Mathematical Methods for Physicists. 2nd Ed. Academic Press
Astarita, G. (1979). Objective and generally applicable criteria for flow classification. Journal
of Non-Newtonian Fluids Mechanics, 6:69-76
Bentley, B. J. and L. G. Leal. (1986a). A computer-controlled four-roll mill for investigations
of particle and drop dynamics in two-dimensional linear shear flows. J. Fluid Mech.
167:219-240
Bentley, B. J. and L. G. Leal. (1986b). An experimental investigation of drop deformation and
breakup in steady two-dimensional linear flows. J. Fluid Mech. 167:241-283
Birkhoffer, Beat H. et al. (2005). Computer-controlled flow cell for the study of particle and
drop dynamics in shear flow fields. Ind. Eng. Chem. Res. 44:6999-7009
Geffroy E. and L. G. Leal. (1992). Flow Birefringence of a Concentrated Polystyrene Solution
in a Two Roll-Mill 1. Steady Flow and Start-Up of Steady Flow. J. Polym. Sci. B:
Polym. Phys. 30(12):1329-1349
Gradshteyn, I. S. and I. M. Ryzhik. (1981). Tables of integrals, series and products. Academic
Press
Jeffery, G. B. (1922). The rotation of two circular cylinders in a viscous fluid. Proc Royal
Society A 101:169-174
Pozrikidis, C. (1992). Boundary Integral and singularity methods for linearized viscous flow.
Cambridge University Press
Pozrikidis, C. (2003). Modeling and simulation of capsules and biological cells. CRC Press
Rallison J. M. (1980). A note of the time-dependent deformation of a viscous drop which is
almost spherical. J. Fluid Mechanics 98(3):625-633
Reyes, M. A. H. and E. Geffroy. (2000). Study of low Reynolds number hydrodynamics
generated by symmetric co-rotating two-roll mills. Revista Mexicana de Física
46(2):135-147
Reyes, M. A. H. and E. Geffroy. (2000). A co-rotating two-roll mill for studies of two-
dimensional, elongational flows with vorticity . Phys. Fluids 12(10):2372-2376.
Reyes, M. A. H., A. A. Minzoni & E. Geffroy (2011), Numerical study of the effect of
nonlinear control on the behavior of a liquid drop in elongational flow with
vorticity”, Journal of Engineering Mathematics, 71(2):185-203.
Ross, S. (1984). Differential Equations. Wiley
Singh, P. and L. G. Leal. (1994). Computational studies of the FENE dumbbell model in a co-
rotating two-roll mill . J. Rheol. 38:485-517
Stone, H. A. (1994). Dynamics of drop deformation and breakup in viscous fluids. Ann. Rev.
Fluid Mech. 26:65-102
Taylor, G. I. (1932). The Viscosity of a Fluid Containing Small Drops of Another Fluid. Proc.
R. Soc. London, Ser. A, 138:41-48
Taylor, G. I. (1934). The Formation of Emulsions in Definable Fields of Flow. Proc. R. Soc.
London, Ser. A 146:501-523
Torza S., R. G. Cox and S. G. Mason. (1972). Particle motions in shear suspensions. J. Colloid
and Interface Sci. 38(2):395-411
Wang, J. J., D. Yavich, and L. G. Leal. (1994). Time resolved velocity gradient and optical
anisotropy in lineal flow by photon correlation spectroscopy. Phys. Fluids
6(11):3519-3534
Yang, H., C. C. Park, Y. T. Hu, and L. G. Leal. (2001). The coalescence of two equal--sized
drops in a two dimensional linear flow. Phys. Fluids 13(5):1087-1106
4
1Robust Control Research
of Chaos Phenomenon for
Diesel-Generator Set
on Parallel Connection
Man-lei Huang
School of Automation, Harbin Engineering University, Harbin,
China
1. Introduction
Several diesel-generator sets usually operate on parallel connection in ship power system,
which has altitudinal nonlinearity. When operating point of system changes, its dynamic
property will change markedly. The oscillation phenomenon of ship power system which is
acyclic, random and gusty or paroxysmal will occur on light load working condition, it can
result in system sectionalizing when it is serious, this phenomenon is called chaos. Chaos is
a very complicated phenomenon which is generated by the interaction of each parameter in
the nonlinear system. When it appears in ship power system, following the continuous and
random oscillation of system operating parameter, which endangers operation security of
system seriously, it must be prevented and eliminated effectively in the system. In order to
analyze the chaos phenomenon of ship power system, the nonlinear mathematical model of
two diesel-generator sets on parallel connection is built in this paper, which reflects the
variation law of ship power system. Then the light load working condition of two diesel-
generator sets on parallel connection in ship power station is analyzed by using Lyapunov
index method on the base of this, seeking the generating mechanism of chaos. A nonlinear
robust synthetic controller is designed which is based on the nonlinear mathematical model
of diesel-generator set, then a nonlinear robust synthetic control law is developed for the
diesel-generator set, it will be applied to control the chaos phenomena, thus providing
desirable stability for ship power system.
2. Mathematic model of diesel-generator set on parallel connection
The mathematical model of diesel-generator set include the mathematical model of
electromechanical transient process and electromagnetism transient process, first building
the mathematical model of electromechanical transient process, then the mathematical
1Foundation item: Supported by the National Natural Science Foundation of China under Grant
No.60774072; Fundamental Research Funds for the Central Universities of China under Grant
No.HEUCFT1005.
58
Applications of Nonlinear Control
model of electromagnetism transient process, the mathematical model of one diesel-
generator set is get on the base of this.
The mathematical model of diesel-generator set electromechanical transient process
describes the motion law of diesel-generator set[1-3], reflecting the dynamic change process of
power angle and angular speed, its expression is
d 1,
d t
2
d
T
1
c
E U
U X X
(1)
2
1
q
1
d
q
b
1
c
L
sin
sin 2 .
d t
T
T
T
T X
T 2
a 0
a 0
a 0
a 0
d
a 0
XdXq
In the equation: is power angle of diesel-generator set, electric angular speed of diesel-
generator set, U terminal voltage of generator stator winding, q
E<