Triplewave ensembles in a thin cylindrical shell
Triplewave ensembles in a thin cylindrical shell
TRIPLEWAVE ENSEMBLES IN A
THIN CYLINDRICAL SHELL
Kovriguine
DA, Potapov AI
Introduction
Primitive nonlinear quasiharmonic triplewave patterns in a
thinwalled cylindrical shell are investigated. This task is focused on the
resonant properties of the system. The main idea is to trace the propagation of
a quasiharmonic signal  is the wave stable or not? The stability prediction
is based on the iterative mathematical procedures. First, the lowestorder
nonlinear approximation model is derived and tested. If the wave is unstable
against small perturbations within this approximation, then the corresponding
instability mechanism is fixed and classified. Otherwise, the higherorder
iterations are continued up to obtaining some definite result.
The theory of thinwalled shells based on the KirhhoffLove
hypotheses is used to obtain equations governing nonlinear oscillations in a
shell. Then these equations are reduced to simplified mathematical models in the
form of modulation equations describing nonlinear coupling between
quasiharmonic modes. Physically, the propagation velocity of any mechanical
signal should not exceed the characteristic wave velocity inherent in the
material of the plate. This restriction allows one to define three main types
of elemental resonant ensembles  the triads of quasiharmonic modes of the
following kinds:
highfrequency
longitudinal and two lowfrequency bending waves (type triads);
highfrequency
shear and two lowfrequency bending waves ();
highfrequency
bending, lowfrequency bending and shear waves ();
highfrequency
bending and two lowfrequency bending waves ().
Here subscripts
identify the type of modes, namely () 
longitudinal, ()  bending, and () 
shear mode. The first one stands for the primary unstable highfrequency mode,
the other two subscripts denote secondary lowfrequency modes.
Triads of the
first three kinds (i  iii) can be observed in a flat plate (as the curvature
of the shell goes to zero), while the type
triads are inherent in cylindrical shells only.
Notice that the
known Karmantype dynamical governing equations can describe the type triplewave coupling only. The other triplewave
resonant ensembles, , and
, which refer to the nonlinear coupling between
highfrequency shear (longitudinal) mode and lowfrequency bending modes,
cannot be described by this model.
Quasiharmonic
bending waves, whose group velocities do not exceed the typical propagation
velocity of shear waves, are stable against small perturbations within the
lowestorder nonlinear approximation analysis. However amplitude envelopes of
these waves can be unstable with respect to small longwave perturbations in
the next approximation. Generally, such instability is associated with the
degenerated fourwave resonant interactions. In the present paper the
secondorder approximation effects is reduced to consideration of the selfaction
phenomenon only. The corresponding mathematical model in the form of
Zakharovtype equations is proposed to describe such highorder nonlinear wave
patterns.
Governing
equations
We consider a
deformed state of a thinwalled cylindrical shell of the length , thickness ,
radius , in the frame of references . The coordinate
belongs to a line beginning at the center of curvature, and passing
perpendicularly to the median surface of the shell, while and are
inplane coordinates on this surface ().
Since the cylindrical shell is an axisymmetric elastic structure, it is
convenient to pass from the actual frame of references to the cylindrical
coordinates, i.e. , where and
. Let the vector of displacements of a material point
lying on the median surface be .
Here , and
stand for the longitudinal, circumferential and
transversal components of displacements along the coordinates and ,
respectively, at the time . Then the spatial distribution of displacements reads
accordingly to the
geometrical paradigm of the KirhhoffLove hypotheses. From the viewpoint of
further mathematical rearrangements it is convenient to pass from the physical
sought variables to the corresponding dimensionless displacements . Let the radius and the length of the shell be
comparable values, i.e. , while the displacements be small enough, i.e. . Then the components of the deformation tensor can be
written in the form
where is the small parameter; ; and
. The expression for the spatial density of the
potential energy of the shell can be obtained using standard stressstraight
relationships accordingly to the dynamical part of the KirhhoffLove
hypotheses:
where is the Young modulus; denotes
the Poisson ratio; (the primes indicating the dimensionless variables
have been omitted). Neglecting the crosssection inertia of the shell, the
density of kinetic energy reads
where is the dimensionless time; is typical propagation velocity.
Let the Lagrangian
of the system be .
By using the
variational procedures of mechanics, one can obtain the following equations
governing the nonlinear vibrations of the cylindrical shell (the Donnell
model):
(1)
(2)
Equations (1) and
(2) are supplemented by the periodicity conditions
Dispersion
of linear waves
At the linear subset of eqs.(1)(2) describes a
superposition of harmonic waves
(3)
Here is the vector of complexvalued wave amplitudes of
the longitudinal, circumferential and bending component, respectively; is the phase, where are
the natural frequencies depending upon two integer numbers, namely (number of halfwaves in the longitudinal direction)
and (number of waves in the circumferential direction).
The dispersion relation defining this dependence has
the form
(4)
where
In the general
case this equation possesses three different roots () at fixed values of and
. Graphically, these solutions are represented by a
set of points occupied the three surfaces .
Their intersections with a plane passing the axis of frequencies are given by
fig.(1). Any natural frequency corresponds to the threedimensional vector of amplitudes . The components of this vector should be proportional
values, e.g. , where the ratios
and
are obeyed to the
orthogonality conditions
as .
Assume that , then the linearized subset of eqs.(1)(2) describes
planar oscillations in a thin ring. The lowfrequency branch corresponding
generally to bending waves is approximated by and , while the highfrequency azimuthal branch  and .
The bending and azimuthal modes are uncoupled with the shear modes. The shear
modes are polarized in the longitudinal direction and characterized by the
exact dispersion relation .
Consider now axisymmetric
waves (as ). The axisymmetric shear waves are polarized by
azimuth: , while the other two modes are uncoupled with the
shear mode. These high and lowfrequency branches are defined by the following
biquadratic equation
.
At the vicinity of
the highfrequency branch is approximated by
,
while the
lowfrequency branch is given by
.
Let , then the highfrequency asymptotic be
,
while the
lowfrequency asymptotic:
.
When neglecting
the inplane inertia of elastic waves, the governing equations (1)(2) can be
reduced to the following set (the Karman model):
(5)
Here and are
the differential operators; denotes
the Airy stress function defined by the relations , and
, where ,
while , and
stand for the components of the stress tensor. The
linearized subset of eqs.(5), at ,
is represented by a single equation
defining a single
variable , whose solutions satisfy the following dispersion
relation
(6)
Notice that the
expression (6) is a good approximation of the lowfrequency branch defined by
(4).
Evolution
equations
If , then the ansatz (3) to the eqs.(1)(2) can lead at
large times and spatial distances, ,
to a lack of the same order that the linearized solutions are themselves. To
compensate this defect, let us suppose that the amplitudes be now the slowly varying functions of independent
coordinates , and
, although the ansatz to the nonlinear governing
equations conserves formally the same form (3):
Obviously, both
the slow and the fast spatiotemporal
scales appear in the problem. The structure of the fast scales is fixed by the
fast rotating phases (), while the dependence of amplitudes upon the slow variables is unknown.
This dependence is
defined by the evolution equations describing the slow spatiotemporal
modulation of complex amplitudes.
There are many
routs to obtain the evolution equations. Let us consider a technique based on
the Lagrangian variational procedure. We pass from the density of Lagrangian
function to its average value
(7),
An advantage of
the transform (7) is that the average Lagrangian depends only upon the slowly
varying complex amplitudes and their derivatives on the slow spatiotemporal
scales , and
. In turn, the average Lagrangian does not depend upon
the fast variables.
The average
Lagrangian can be formally represented as power series in :
(8)
At the average Lagrangian (8) reads
where the
coefficient coincides exactly with the dispersion relation (3).
This means that .
The firstorder
approximation average Lagrangian depends
upon the slowly varying complex amplitudes and their first derivatives on the
slow spatiotemporal scales , and .
The corresponding evolution equations have the following form
(9)
Notice that the
secondorder approximation evolution equations cannot be directly obtained
using the formal expansion of the average Lagrangian , since some corrections of the term are necessary. These corrections are resulted from
unknown additional terms of order ,
which should generalize the ansatz (3):
provided that the
secondorder approximation nonlinear effects are of interest.
Triplewave
resonant ensembles
The lowestorder nonlinear analysis predicts that eqs.(9)
should describe the evolution of resonant triads in the cylindrical shell,
provided the following phase matching conditions
(10),
hold true, plus
the nonlinearity in eqs.(1)(2) possesses some appropriate structure. Here is a small phase detuning of order , i.e. .
The phase matching conditions (10) can be rewritten in the alternative form
where is a small frequency detuning; and are
the wave numbers of three resonantly coupled quasiharmonic nonlinear waves in
the circumferential and longitudinal directions, respectively. Then the
evolution equations (9) can be reduced to the form analogous to the classical
Euler equations, describing the motion of a gyro:
(11).
Here is the potential of the triplewave coupling; are the slowly varying amplitudes of three waves at
the frequencies and the wave numbers and
; are
the group velocities; is the differential operator; stand for the lengths of the polarization vectors ( and ); is the nonlinearity coefficient:
where .
Solutions to
eqs.(11) describe four main types of resonant triads in the cylindrical shell,
namely , ,  and type
triads. Here subscripts identify the type of modes, namely ()  longitudinal, () 
bending, and ()  shear mode. The first subscript stands for the
primary unstable highfrequency mode, the other two subscripts denote the
secondary lowfrequency modes.
A new type of the
nonlinear resonant wave coupling appears in the cylindrical shell, namely type triads, unlike similar processes in bars, rings
and plates. From the viewpoint of mathematical modeling, it is obvious that the
Karmantype equations cannot describe the triplewave coupling of , 
and types, but the type
triplewave coupling only. Since type
triads are inherent in both the Karman and Donnell models, these are of
interest in the present study.
triads
Highfrequency
azimuthal waves in the shell can be unstable with respect to small
perturbations of lowfrequency bending waves. Figure (2) depicts a projection
of the corresponding resonant manifold of the shell possessing the spatial
dimensions: and . The
primary highfrequency azimuthal mode is characterized by the spectral
parameters and (the
numerical values of and are
given in the captions to the figures). In the example presented the phase
detuning does not exceed one percent. Notice that the phase
detuning almost always approaches zero at some specially chosen ratios between and ,
i.e. at some special values of the parameter.
Almost all the exceptions correspond, as a rule, to the longwave processes,
since in such cases the parameter cannot
be small, e.g. .
NB Notice that type triads can be observed in a thin rectilinear
bar, circular ring and in a flat plate.
NBThe wave modes
entering type triads can propagate in the same spatial
direction.
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