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Диплом Research of the primitive quasiharmonical triplewave patterns in thinwalled cylindrical shells Tracking of distribution of a stable wave. Reception of the equations of nonlinear fluctuations in an environment according to the hypothesis of Kirhoff. 
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Тип работы: Диплом.
Предмет: Математика.
Добавлен: 14.02.2010.
Сдан: 2010.
Уникальность по antiplagiat.ru: .



Описание (план): 


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.
triads
Analogously, highfrequency shear waves in the shell can be unstable with respect to small perturbations of lowfrequency 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 highfrequency shear mode are и т.д.................



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