Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications
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138
A.
A. Kolesnikov and
A.
A.
Kuzmenko
significantly exceed existing typical controllers, based on ideology
of
linear
control theory, by its dynamical properties, because
of
linear control theory
does not reflect real physical processes
in
turbogenerators.
Implementing
of
modified synergetics controllers for turbo generators
control will provide principal improvement
of
power systems static and
dynamic properties
in
emergency and extreme modes
of
operation.
References
[1]
Kolesnikov A.A., Kuzmenko A.A., at all. Synergetics Methods
of
Complex
Systems Control:
Power Systems. Moscow: URSS, 2006.
[2]
Koalov V.N., Shashihin V.Y. Synthesis
of
coordinating robust control
of
interconnected syncronous generators II "Electrichestvo" magazine, 2000, #9.
-Po
20-
26.
[3]
Michnevitch G.V. Synthesis
of
control system structure for synchronous machines
exitation. Moscow: Nauka, 1964.
Dynamic Stability Loss
of
Closed Circled
Cylindrical Shells Estimation Using Wavelets
V.
A. Krysko
l
,
J.
Awrejcewicz
2
,
M. Zhigalov
l
,
V.
Soldatov
l
,
E.
S.
Kuznetsova
l
,
and
S.
Mitskevich
l
I Saratov State Technical University, Department
of
Mathematics and
Modeling,
410054, Poly technical 77, Saratov, Russia
(e-mail:
tak@san.ru)
139
2 Technical University
of
Lodz, Department
of
Automatics and Biomechanics,
1/15 Stefanowski St.,
90-924 Lodz, Poland
(e-mail: awrejcew@p.lodz.pl)
Abstract:
In
this work dynamic stability loss
of
closed cylindrical shells
is
studied. The hybrid type PDEs governing cylindrical shells dynamics and regarding
deflection (Airy's) function and stresses are first derived. Then they are reduced to
ODEs and algebraic equations
(A
E) applying the Bubnov-Galerkin high order
approximations method. Dynamic stability loss
of
the cylindrical shells subject to
sign-changeable loading using wavelets is investigated.
In
addition, the convergence
of
the Bubnov-Galerkin method and results validation are addressed.
Keywords: chaos, closed cylindrical shells, wavelets, nonlinear dynamics, dynamic
stability loss.
1.
Introduction
Closed cylindrical shells are members
of
various constructions and
machines including airplanes, ships, civil engineering constructions,
measurement devices, etc. For many years thousands
of
researchers attacked
a key problem regarding safe and durable properties
of
thin- and thick-walled
constructions addressing and formulating the so called stability loss criteria
(see
[1]-[4]).
Note that even a particular case
of
this topic, i.e. analysis
of
free and forced nonlinear vibrations
of
circled cylindrical shells attracted and
still attracts many engineering oriented researchers (see for instance [5-10]
and the cited references therein).
In this work, among others, we propose a novel stability loss criterion
with a help
of
the wavelet-based analysis.
2. Problem statement
We study the closed circled cylindrical shell
of
finite length with constant
stiffness and density subject to action
of
non-uniform sign-changeable load
and being embedded in a temperature field (Fig.
1).
The following co-
ordinates are introduced: axis y refers to circled co-ordinate, whereas axis z
140
V.
A. Krysko et al.
overlaps a normal one
of
the shell's middle surface. Therefore, our shell
is
defined
as
Q =
{x,
y,z
I
(x,
y)
E [O;L] x
[0;2;r],-h:S;
z:S;
h}.
Fig.
1.
Computational scheme
Its governing dynamic non-dimensional equations have the following
form [9,10]:
I
[-2
a
4
w a
4
w 2 a
4
w]
-------:2c-
A
--4
+ 2
-2--2
+ A
--4
-
L(
w,
F)
12(1-
V )
ax ax ay ay
2 2
aF
aw
aw
2
-k,
-2-
-
-2-
-
E-
+
kyq(x,
y,t)
=
0',
.
ax
at at
.
(1)
{
a4F
a
4
F a
4
F 1 i }
~
2 W
A
-4-
+
2-
2
--
2
+ A
-4-
+
-L(w,
w)
+
ky
--2
=
o.
ax
ax
ay ay
2
ax
Equations
(I)
have been transformed to the non-dimensional form using the
following relations:
-
3-
RL
w=2hw
F=E
O
(2h)
F,
t=
c;:-t,
A=LIR;
x=Lx,
y=Ry;
2hVgEo
4
_
2h
_
Eo(2h)
ky
=
ky
-2
' q = q . Here
Land
R = R are shell length and
R
L2
R2
y
radius, respectively; t - time, E - damping coefficient
of
a shell surrounding
medium,
F - stress (Airy's) function, w - shell's deflection, h - shell's
thickness,
v - Poisson's coefficient, g - gravity acceleration,
Eo
- shell
material Young modulus,
ky
- shell's curvature regarding axis y and finally
Dynamic Stability Loss
of
Closed Circled Cylindrical Shells Estimation
141
iwiF iwiF
iw
iF
L(w
F)=
--
+ - -
-2
---
-'
, 2 2 2 2 '
dX
dy
dy
dX
dxdy dxdy
[
(
J
2]
iwiw
iw
L(w,
w) = 2
~2- -
~
2
---
ax
ay
axay
are widely known nonlinear operators (note that bars over non-dimensional
quantities are omitted).
Let us focus on investigation
of
one
of
the boundary conditions, i.e.
hinged
shell's
support with flexible ribs
on
the
shell's
edges:
w = M = N = c = 0 for x = 0'1 y =
o·
27C
x x y , , "
(2)
and with the following initial conditions:
dWI
w(x,
y)
It=O=
0,
a;
t=O
= o.
(3)
The
boundary value problem formulated so far is solved via the
Bubnov-Galerkin method with high order approximations. In this work the
being sought functions
wand
F are approximated by analytical expressions
having a finite number
of
parameters arbitrarily taken through the following
functions depending
on
time and spatial co-ordinates:
N,
N2
F = L
LB
..
(t)If/
"
(x
,
y)
,
;
=0
j=O
IJ IJ
(4)
where
rjJij
(x,
y)
and If/ij
(x,
y)
are certain functions
of
x,
y.
After the Bubnov-Galerkin method application to system (1), a system
of
ODE
s is obtained regarding functions
~/t)
and Bij
(t)
, and its matrix
form is:
(5)
where G =
Ilcijklll,
s =
IISijr
skl
ll,
D, =
IIDUjr
Skl
l1
,D
2
=
IID
2ijrski
li
,P
=
II~j
kl
ll
- squared matrices
of
dimensions
142
V.
A. Krysko
et
al.
dimensions
2·
N]
.
N2
xl.
Second equation
of
(5) is solved regarding matrix B
matrix method on each time step:
B =
[-P-I
D2A
- P-
1
C
2
]A.
using an inversed
(6)
Multiplying first equation
of
(5) by G
-1
and denoting A =
R,
one
achieves the Cauchy's problem for the following first order
ODEs:
{
~
=
-eR
+ G-1D1AB -
G-]SA
+
q(;)G-1Q
(7)
A=R
These transformations are allowed since matrices
G-]
and
p-l
exist
assuming linear independence
of
the co-ordinate functions.
We add to equations
(7) boundary and initial conditions and the yielded
Cauchy problem is solved using the fourth order Runge-Kutta algorithm
(computational step is chosen via the Runge principle).
In order to satisfy boundary conditions
(2), functions
9ij'
lfIij
appeared in (4) are sought
in
the form
of
a product
of
two functions, where
each
of
them depends only on one argument:
NI
N2
W = I I
A.(t)sin(i.7rx)cos(jy),
i=l
j=O
lJ
NI
N2
F = I I B
..
(t)sin(i.7rx)
cos(jy)
i=l
j=O
lJ
3. Numerical results
(8)
We study our shells' vibrations character using the following fixed
parameters:
k =112.5 for A =
2,
c =9, and subject to action
of
transversal
)'
load
q(t)
=
qo
sin(OJ/) distributed on the whole shell's surface
(0
~
x
~
1,
OJ
P = 2.3), and we take numbers
of
functions
N]
= 1, N 2 = 9 (note that
increase
of
those numbers do not increase the results accuracy).
Dynamic Stability Loss
of
Closed Circled Cylindrical Shells Estimation
143
Qualitative analysis
is
carried out using the well-known
characteristics:
w(0.5;
0;
t)
, wavelets in
2D
and 3D, phase portraits, Poincare
maps,
FFT
(Fast Fourier Transform), Lyapunov exponents a
nd
the
autocorrelation functions. In particular,
we
focus on dynamic stability lo
ss
estimation
of
our investigated shell. Further, only the following
characteristics will be reported: time histories, phase portraits, frequency
power spectra, wavelet-spectra, and shell buckling forms for
o
~
x
~
1; 0
~
y
~
21r
and for x = 0
.5;
0
~
y
~
21r
.
In order
to
construct the dependence w
(q),
the computations
m
ax
0
have been carried out for
qo
E
[0
; 0.4975] . In addition, for each element
qo
a maximum shell deflection in point (0,5; 0) has been estimated. Results are
reported in Fig.
2,
where one may observe the so-called shell dynamic
stability loss (small change
of
qo
is associated with a rapid shell deflection
increase). Inflexion point
of
the curve given in Fig. 2 allows
to
define the
critical value
q~r.
We study the vibration character
in
the shell pre-critical
state
(qo
~
0.225 , see Figures 3-8). Analysis of shell vibrations (for instance,
in
point
A)
allows
to
conclude that they are harmonic.
12
Fig. 2. Relation w
(q)
max
0
'''a
.n,
0.17
w(ll&5
0.16
('-155
0.
15
.,~
100 105 110
liS
120
,
Fig. 3.
w(0.5;0;t)
Goo
..
,(10
.:10
I
....
.,
XIII
Z»
Fig. 4. 2D Morlet
wavelet
A:
qo
= 0.1 - pre-critical
shell state;
B:
qo
= 0.3225 - after
first shell buckling;
C:
qo
= 0.3975 - after
second shell buckling.
Fig. 5. 3D Morlet wavelet
144
V.
A.
Kr
ysko
et
al.
Fig. 6. Phase portrait
w(O.5
;O;t) ,
w(O.5;O;t)
Fig. 7. Shell
deflection form
o
Fig. 8. Shell transversal
cross-section
I-I
Observe that when our dynamical system passed through the critical point
q~r
(we take
qo
= 0.3225 ), chaotic dynamics appears (see Figures 9-14).
:
cw
.
~
,
Fig. 9. Time history
w(0.5;
0;t)
qo
= 0.3225
Fig.
12. Phase portrait
w(0.5; 0;
t)
,
w(0.5; O;
t)
Fig. 10. 2D Morlet
wavelet
Fig. 13. Shell
deflection form
Fig. 11. 3D Morlet
wavelet
o
Fig. 14. Shell transversal
cross-section
I-I
In point C associated with second shell buckling (qo = 0.3975 ) the
wavelet-spectrum allows to detect
an
interesting qualitative change
of
the
shell dynamics in a very short time interval
(t
~
200
).
Dynamic Stability Loss 0/ Closed Circled Cylindrical Shells Estimation
145
Figure
15
. Time
history
w(0.5;0;t)
qo
= 0.3975
~
.
~
:~
'
I
•.
'r
r J
='l
~
, , " "
Figure 18. Phase
portrait w(0.
5;
0;
t),
W(O.5;
0;
t)
4. Conclusions
Figure 16. 2D Morlet
wavelet
Figure 19. Shell
de
fl
ection form
Figure
17
. 3D Morlet
wavelet
o
Figure 20. Shell
transversal cross-section
I-I
Owing to our computational technique used to study non-linear vibrations
of
closed circled cylindrical shells subject to action of transversal sign-
changeable load the following conclusions are carried out:
(i)
our
investigated shell exhibits dynamics stability loss during a transition from
regular to chaotic vibrations; (ii) we have detected the second type shell
buckling associated with chaotic intermittency behavior.
References
1.
A.
C.
Shian, T.
T.
Soong and D.
S.
Roth. Dynamic buckling
of
conical
shells with Imperfection.
AlAA Journal, 12(6):24-30,1974.
2.
B. Budiansky and D.
S.
Roth. Axisymmetric Dynamic Buckling
of
Clamped Shallow Spherical Shells. NASA
TN
D-151O, 597-606,
1962.
3.
B.
Ya
. Kanto
r.
Nonlinear Problems
of
Theory
of
Non-Homogenous
Shall
ow
Shells. Naukowa Dumka, Kiev, 1971, in Russian.
4. V. A. Krysko and A. V. Krysko. Problems
of
bifurcations and stiff
stability loss in nonlinear theory
of
plates. Mechanics
of
Plates and
146
V.
A.
Krysko et
al.
Shells
in
XXI century. The Saratov Technical University Press, 50-
67, 1999,
in
Russian.
5.
H.
N.
Chu. Influence
of
large amplitude on flexural vibrations
of
thin
cylindrical shell.
1.
Aerospace Sci., 28(8):602-609, 1961.
6.
J.
L. Novinski. Nonlinear transverse vibrations
of
orthotropic cylindrical
shells.
A1AA
J.,
1(3):617-620, 1963.
7.
D. A. Evensen. Some observations on the nonlinear vibrations
of
thin
cylindrical shells.
A1AA
J,
. 1(12):
2857-
2858, 1963.
8. M. Amabili,
F.
Pellicano and M.P. Paidoussis. Non-linear vibration
of
simply supported circular cylindrical shells coupled to quiescent
fluid. Journal
of
Fluids and Structures,
12
:883-918, 1998.
9.
V.
A.
Krysko,
J.
Awrejcewicz,
E.
S.
Kusnetsova and
A.
V.
Krysko.
Chaotic vibrations
of
closed cylindrical shells
in
a temperature field.
International Journal
of
Bifurcation and Chaos, 18(5):1515-1529,
2008.
10.
V.
A.
Krysko, J. Awrejcewicz,
E.
S. Kusnetsova and
A.
V.
Krysko
Chaotic vibrations
of
closed cylindrical shells in a temperature field.
Shock
and
Vibration, 15:335-343, 2008.
147
The Application
of
Multivariate Analysis Tools for
Non-Invasive Performance Analysis
of
Atmospheric
Pressure Plasma
V.
J.
Law!,
J.
Tynan
2
,
G.
Byrne
2
, D. P. Dowling2, and
S.
Daniels!
! Dublin City University, National Center
of
Plasma Science and Technology
Collins Avenue, Glasnevin, Dublin 9, Dublin, Ireland
2 School
of
Electrical, Electronic and Mechanical Engineering, UCD
Belfield, Dublin 4, Ireland
(e-mail:
vic.law@dcu.ie)
Abstract:
This paper describes the development and use
of
real-time non-invasive Multivariate
Analysis tools for the performance monitoring
of
atmospheric pressure plasma. The MVA
tools (acoustic spectrogram analysis, principal component analysis (PCA) and non-parametric
cluster analysis (NPCA) are embedded within a LabVIEW software program. The software program
is used
to
probe a parallel-plate atmospheric pressure process system.
It
is found that the acoustic
frequency spectrum distribution provides a signature
of
the plasma mode of operation. The
signatures are modeled
as
overtones
of
the fundamental drive frequency and combination signals
(intermodulation distortion). Within these spectrums syncopated patterns are observed. The acoustic
signatures are correlated with changing electrical parameters.
Using appropriate scaling factors, PCA
of
the current and voltage waveform are used
to
generate data set clusters that are deterministic of
the acoustic signals. Non-parametric cluster analysis is used
to
identify and classify the modes.
Keywords: multivariate analysis, frequency analysis, principal component analysis, non-parametric
cluster analysis, atmospheric pressure plasma, acoustic emission and electrical waveform
identification.
I. Introduction
The monitoring and control
of
plasma processes is essential during manufacture
to ensure uniform and predictable treatments. Many tools are available for
monitoring vacuum-based plasma systems; however, the use
of
atmospheric
pressure plasma systems is increasing due to their ability to process large areas
in a reel-to-reel manner [1, 2], or, where mobile hand-held plasma devices treat
small area surfaces [3]. With this increase in atmospheric pressure plasma
processing a new set
of
metrology tools must be developed to analysis the
plasma processing conditions and enable real-time decision on input parameters
to be made such that factors like coating thickness, morphology and chemistry
can be adequately controlled.
The plasma equipment used in these series
of
experiments was the SE-J 100
atmospheric pressure plasma system developed by Dow Corning. The
SEll
00 is
a stand-alone processing tool, which is capable
of
plasma treating and coating
flexible polymer webs.
It
consists
of
two chambers, each measuring 320mm x
320 mm and associated web handling equipment through which the polymer
web is passed. Using the frequency matching power supply, powers
of
up to
2000 W, or 1000 W per chamber,
is
applied.