Дипломная работа: Triple-wave ensembles in a thin cylindrical shell
Kovriguine DA, Potapov AI
Introduction
Primitive nonlinear quasi-harmonic triple-wave patterns in a thin-walled 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 quasi-harmonic signal — is the wave stable or not? The stability prediction is based on the iterative mathematical procedures. First, the lowest-order 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 higher-order iterations are continued up to obtaining some definite result.
The theory of thin-walled shells based on the Kirhhoff-Love 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 quasi-harmonic 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 quasi-harmonic modes of the following kinds:
high-frequency longitudinal and two low-frequency bending waves (
-type triads);
high-frequency shear and two low-frequency bending waves (
);
high-frequency bending, low-frequency bending and shear waves (
);
high-frequency bending and two low-frequency bending waves (
).
Here subscripts
identify the type of modes, namely (
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
Notice that the
known Karman-type dynamical governing equations can describe the
Quasi-harmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowest-order nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small long-wave perturbations in the next approximation. Generally, such instability is associated with the degenerated four-wave resonant interactions. In the present paper the second-order approximation effects is reduced to consideration of the self-action phenomenon only. The corresponding mathematical model in the form of Zakharov-type equations is proposed to describe such high-order nonlinear wave patterns.
Governing equations
We consider a
deformed state of a thin-walled cylindrical shell of the length
accordingly to the
geometrical paradigm of the Kirhhoff-Love hypotheses. From the
viewpoint of further mathematical rearrangements it is convenient to
pass from the physical sought variables
where
where
where
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
(3)
Here
is the vector of complex-valued 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 half-waves 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 three-dimensional 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 low-frequency branch corresponding
generally to bending waves is approximated by
and
,
while the high-frequency 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 low-frequency branches are defined by the following
biquadratic equation
.
At the vicinity of
the high-frequency branch is approximated by
,
while the low-frequency branch is given by
.
Let
,
then the high-frequency asymptotic be
,
while the low-frequency asymptotic:
.
When neglecting the in-plane 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 low-frequency 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
spatio-temporal 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 spatio-temporal 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
spatio-temporal 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 first-order
approximation average Lagrangian
depends upon the slowly varying complex amplitudes and their first
derivatives on the slow spatio-temporal scales
,
and
.
The corresponding evolution equations have the following form
(9)
Notice that the
second-order 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 second-order approximation nonlinear effects are of interest.
Triple-wave resonant ensembles
The lowest-order 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 quasi-harmonic
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 triple-wave 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 (
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
Karman-type equations cannot describe the triple-wave coupling of
-,
-
and
-types,
but the
-type
triple-wave coupling only. Since
-type
triads are inherent in both the Karman and Donnell models, these are
of interest in the present study.
-triads
High-frequency
azimuthal waves in the shell can be unstable with respect to small
perturbations of low-frequency bending waves. Figure (2) depicts a
projection of the corresponding resonant manifold of the shell
possessing the spatial dimensions:
and
.
The primary high-frequency 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 long-wave
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.
-triads
Analogously,
high-frequency shear waves in the shell can be unstable with respect
to small perturbations of low-frequency bending waves. Figure (3)
displays the projection of the
-type
resonant manifold of the shell with the same spatial sizes as in the
previous subsection. The wave parameters of primary high-frequency
shear mode are
and
.
The phase detuning does not exceed one percent. The triple-wave
resonant coupling cannot be observed in the case of long-wave
processes only, since in such cases the parameter
cannot be small.
NBThe wave modes
entering
-type
triads cannot propagate in the same spatial direction. Otherwise, the
nonlinearity parameter
in eqs.(11) goes to zero, as all the waves propagate in the same
direction. This means that such triads are essentially
two-dimensional dynamical objects.
-triads
High-frequency
bending waves in the shell can be unstable with respect to small
perturbations of low-frequency bending and shear waves. Figure (4)
displays an example of projection of the
-type
resonant manifold of the shell with the same sizes as in the previous
sections. The spectral parameters of the primary high-frequency
bending mode are
and
.
The phase detuning also does not exceed one percent. The triple-wave
resonant coupling can be observed only in the case when the group
velocity of the primary high-frequency bending mode exceeds the
typical velocity of shear waves.
NBEssentially, the
spectral parameters of
-type
triads fall near the boundary of the validity domain predicted by the
Kirhhoff-Love theory. This means that the real physical properties of
-type
triads can be different than theoretical ones.
NB-type
triads are essentially two-dimensional dynamical objects, since the
nonlinearity parameter goes to zero, as all the waves propagate in
the same direction.
-triads
High-frequency
bending waves in the shell can be unstable with respect to small
perturbations of low-frequency bending waves. Figure (5) displays an
example of the projection of the
-type
resonant manifold of the shell with the same sizes as in the previous
sections. The wave parameters of the primary high-frequency bending
mode are
and
.
The phase detuning does not exceed one percent. The triple-wave
resonant coupling cannot also be observed only in the case of
long-wave processes, since in such cases the parameter
cannot be small.
NBThe resonant
interactions of
-type
are inherent in cylindrical shells only.
Manly-Rawe relations
By multiplying each equation of the set (11) with the corresponding complex conjugate amplitude and then summing the result, one can reduce eqs.(11) to the following divergent laws
(12)
Notice that the
equations of the set (12) are always linearly dependent. Moreover,
these do not depend upon the nonlinearity potential
.
In the case of spatially uniform wave processes (
)
eqs.(12) are reduced to the well-known Manly-Rawe algebraic relations
(13)
where
are the portion of energy stored by the quasi-harmonic mode number
;
are the integration constants dependent only upon the initial
conditions. The Manly-Rawe relations (13) describe the laws of energy
partition between the modes of the triad. Equations (13), being
linearly dependent, can be always reduced to the law of energy
conservation
(14)
.
Equation (14)
predicts that the total energy of the resonant triad is always a
constant value
,
while the triad components can exchange by the portions of energy
,
accordingly to the laws (13). In turn, eqs.(13)-(14) represent the
two independent first integrals to the evolution equations (11) with
spatially uniform initial conditions. These first integrals describe
an unstable hyperbolic orbit behavior of triads near the stationary
point
,
or a stable motion near the two stationary elliptic points
,
and
.
In the case of
spatially uniform dynamical processes eqs.(11), with the help of the
first integrals, are integrated in terms of Jacobian elliptic
functions [1,2]. In the particular case, as
or
,
the general analytic solutions to eqs.(11), within an appropriate
Cauchy problem, can be obtained using a technique of the inverse
scattering transform [3]. In the general case eqs.(11) cannot be
integrated analytically.
Break-up instability of axisymmetric waves
Stability prediction of axisymmetric waves in cylindrical shells subject to small perturbations is of primary interest, since such waves are inherent in axisymmetric elastic structures. In the linear approximation the axisymmetric waves are of three types, namely bending, shear and longitudinal ones. These are the axisymmetric shear waves propagating without dispersion along the symmetry axis of the shell, i.e. modes polarized in the circumferential direction, and linearly coupled longitudinal and bending waves.
It was established experimentally and theoretically that axisymmetric waves lose the symmetry when propagating along the axis of the shell. From the theoretical viewpoint this phenomenon can be treated within several independent scenarios.
The simplest
scenario of the dynamical instability is associated with the
triple-wave resonant coupling, when the high-frequency mode breaks up
into some pairs of secondary waves. For instance, let us suppose that
an axisymmetric quasi-harmonic longitudinal wave (
and
)
travels along the shell. Figure (6) represents a projection of the
triple-wave resonant manifold of the shell, with the geometrical
sizes
m;
m;
m, on the plane of wave numbers. One can see the appearance of six
secondary wave pairs nonlinearly coupled with the primary wave.
Moreover, in the particular case the triple-wave phase matching is
reduced to the so-called resonance 2:1. This one can be proposed as
the main instability mechanism explaining some experimentally
observed patterns in shells subject to periodic cinematic excitations
[4].
It was pointed out
in the paper [5] that the resonance 2:1 is a rarely observed in
shells. The so-called resonance 1:1 was proposed instead as the
instability mechanism. This means that the primary axisymmetric mode
(with
)
can be unstable one with respect to small perturbations of the
asymmetric mode (with
)
possessing a natural frequency closed to that of the primary one.
From the viewpoint of theory of waves this situation is treated as
the degenerated four-wave resonant interaction.
In turn, one more mechanism explaining the loss of stability of axisymmetric waves in shells based on a paradigm of the so-called nonresonant interactions can be proposed [6,7,8]. By the way, it was underlined in the paper [6] that theoretical prognoses relevant to the modulation instability are extremely sensible upon the model explored. This means that the Karman-type equations and Donnell-type equations lead to different predictions related the stability properties of axisymmetric waves.
Self-action
The propagation of any intense bending waves in a long cylindrical shell is accompanied by the excitation of long-wave displacements related to the in-plane tensions and rotations. In turn, these long-wave fields can influence on the theoretically predicted dependence between the amplitude and frequency of the intense bending wave.
Moreover, quasi-harmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowest-order nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small long-wave perturbations in the next approximation.
Amplitude-frequency curve
Let us consider a stationary wave
traveling along the
single direction characterized by the ''companion'' coordinate
.
By substituting this expression into the first and second equations
of the set (1)-(2), one obtains the following differential relations
(15)
Here
while
where
and
.
Using (15) one can get the following nonlinear ordinary differential equation of the fourth order:
(16)
,
which describes simple stationary waves in the cylindrical shell (primes denote differentiation). Here
where
and
are the integration constants.
If the small
parameter
,
and
,
,
satisfies the dispersion relation (4), then a periodic solution to
the linearized equation (16) reads
where
are arbitrary constants, since
.
Let the parameter
be small enough, then a solution to eq.(16) can be represented in the
following form
(17)
where the amplitude
depends upon the slow variables
,
while
are small nonresonant corrections. After the substitution (17) into
eq.( 16) one obtains the expression of the first-order nonresonant
correction
and the following modulation equation
(18)
,
where the nonlinearity coefficient is given by
.
Suppose that the
wave vector
is conserved in the nonlinear solution. Taking into account that the
following relation
holds true for the stationary waves, one gets the following modulation equation instead of eq.(18):
or
,
where the point
denotes differentiation on the slow temporal scale
.
This equation has a simple solution for spatially uniform and
time-periodic waves of constant amplitude
:
,
which characterizes the amplitude-frequency response curve of the shell or the Stocks addition to the natural frequency of linear oscillations:
(19)
.
Spatio-temporal modulation of waves
Relation (19) cannot provide information related to the modulation instability of quasi-harmonic waves. To obtain this, one should slightly modify the ansatz (17):
(20)
where
and
denote the long-wave slowly varying fields being the functions of
arguments
and
(these turn in constants in the linear theory);
is the amplitude of the bending wave;
,
and
are small nonresonant corrections. By substituting the expression
(20) into the governing equations (1)-(2), one obtains, after some
rearranging, the following modulation equations
(21)
where
is the group velocity, and
.
Notice that eqs.(21) have a form of Zakharov-type equations.
Consider the
stationary quasi-harmonic bending wave packets. Let the propagation
velocity be
(22)
,
where the nonlinearity coefficient is equal to
,
while the non-oscillatory in-plane wave fields are defined by the following relations
and
.
The theory of modulated waves predicts that the amplitude envelope of a wavetrain governed by eq.(22) will be unstable one provided the following Lighthill criterion
(23)
is satisfied.
Envelope solitons
The experiments
described in the paper [7] arise from an effort to uncover wave
systems in solids which exhibit soliton behavior. The thin open-ended
nickel cylindrical shell, having the dimensions
cm,
cm and
cm, was made by an electroplating process. An acoustic beam generated
by a horn driver was aimed at the shell. The elastic waves generated
were flexural waves which propagated in the axial,
and
,
respectively, be the eigen numbers of the mode. The modes in which
is always one and
ranges from 6 to 32 were investigated. The only modes which we failed
to excite (for unknown reasons) were
=
9,10,19. A flexural wave pulse was generated by blasting the shell
with an acoustic wave train typically 15 waves long. At any given
frequency the displacement would be given by a standing wave
component and a traveling wave component. If the pickup transducer is
placed at a node in the standing wave its response will be limited to
the traveling wave whose amplitude is constant as it propagates.
The wave pulse at frequency of 1120 Hz was generated. The measured speed of the clockwise pulse was 23 m/s and that of the counter-clockwise pulse was 26 m/s, which are consistent with the value calculated from the dispersion curve (6) within ten percents. The experimentally observed bending wavetrains were best fitting plots of the theoretical hyperbolic functions, which characterizes the envelope solitons. The drop in amplitude, in 105/69 times, was believed due to attenuation of the wave. The shape was independent of the initial shape of the input pulse envelope.
The agreement
between the experimental data and the theoretical curve is excellent.
Figure 7 displays the dependence of the nonlinearity coefficient
and eigen frequencies
versus the wave number
of the cylindrical shell with the same geometrical dimensions as in
the work [7]. Evidently, the envelope solitons in the shell should
arise accordingly to the Lighthill criterion (23) in the range of
wave numbers
=6,7,..,32,
as
.
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