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Дипломная работа: Triple-wave ensembles in a thin cylindrical shell

Название: Triple-wave ensembles in a thin cylindrical shell
Раздел: Рефераты по математике
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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:

(i)high-frequency longitudinal and two low-frequency bending waves (Triple-wave ensembles in a thin cylindrical shell-type triads);

(ii)high-frequency shear and two low-frequency bending waves (Triple-wave ensembles in a thin cylindrical shell);

(iii)high-frequency bending, low-frequency bending and shear waves (Triple-wave ensembles in a thin cylindrical shell);

(iv)high-frequency bending and two low-frequency bending waves (Triple-wave ensembles in a thin cylindrical shell).

Here subscripts identify the type of modes, namely (Triple-wave ensembles in a thin cylindrical shell) — longitudinal, (Triple-wave ensembles in a thin cylindrical shell) — bending, and (Triple-wave ensembles in a thin cylindrical shell) — shear mode. The first one stands for the primary unstable high-frequency mode, the other two subscripts denote secondary low-frequency 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 Triple-wave ensembles in a thin cylindrical shell-type triads are inherent in cylindrical shells only.

Notice that the known Karman-type dynamical governing equations can describe the Triple-wave ensembles in a thin cylindrical shell-type triple-wave coupling only. The other triple-wave resonant ensembles, Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, which refer to the nonlinear coupling between high-frequency shear (longitudinal) mode and low-frequency bending modes, cannot be described by this model.

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 Triple-wave ensembles in a thin cylindrical shell, thickness Triple-wave ensembles in a thin cylindrical shell, radius Triple-wave ensembles in a thin cylindrical shell, in the frame of references Triple-wave ensembles in a thin cylindrical shell. The Triple-wave ensembles in a thin cylindrical shell-coordinate belongs to a line beginning at the center of curvature, and passing perpendicularly to the median surface of the shell, while Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell are in-plane coordinates on this surface (Triple-wave ensembles in a thin cylindrical shell). 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. Triple-wave ensembles in a thin cylindrical shell, where Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. Let the vector of displacements of a material point lying on the median surface be Triple-wave ensembles in a thin cylindrical shell. Here Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell stand for the longitudinal, circumferential and transversal components of displacements along the coordinates Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, respectively, at the time Triple-wave ensembles in a thin cylindrical shell. Then the spatial distribution of displacements reads

Triple-wave ensembles in a thin cylindrical shell

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 Triple-wave ensembles in a thin cylindrical shell to the corresponding dimensionless displacements Triple-wave ensembles in a thin cylindrical shell. Let the radius and the length of the shell be comparable values, i.e. Triple-wave ensembles in a thin cylindrical shell, while the displacements be small enough, i.e. Triple-wave ensembles in a thin cylindrical shell. Then the components of the deformation tensor can be written in the form


Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell is the small parameter; Triple-wave ensembles in a thin cylindrical shell; Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. The expression for the spatial density of the potential energy of the shell can be obtained using standard stress-straight relationships accordingly to the dynamical part of the Kirhhoff-Love hypotheses:

Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell is the Young modulus; Triple-wave ensembles in a thin cylindrical shell denotes the Poisson ratio; Triple-wave ensembles in a thin cylindrical shell (the primes indicating the dimensionless variables have been omitted). Neglecting the cross-section inertia of the shell, the density of kinetic energy reads

Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell is the dimensionless time; Triple-wave ensembles in a thin cylindrical shell is typical propagation velocity.

Let the Lagrangian of the system be Triple-wave ensembles in a thin cylindrical shell.

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)Triple-wave ensembles in a thin cylindrical shell

(2)Triple-wave ensembles in a thin cylindrical shell

Equations (1) and (2) are supplemented by the periodicity conditions

Triple-wave ensembles in a thin cylindrical shell

Dispersion of linear waves

At Triple-wave ensembles in a thin cylindrical shell the linear subset of eqs.(1)-(2) describes a superposition of harmonic waves

(3)Triple-wave ensembles in a thin cylindrical shell


Here Triple-wave ensembles in a thin cylindrical shell is the vector of complex-valued wave amplitudes of the longitudinal, circumferential and bending component, respectively; Triple-wave ensembles in a thin cylindrical shell is the phase, where Triple-wave ensembles in a thin cylindrical shell are the natural frequencies depending upon two integer numbers, namely Triple-wave ensembles in a thin cylindrical shell (number of half-waves in the longitudinal direction) and Triple-wave ensembles in a thin cylindrical shell (number of waves in the circumferential direction). The dispersion relation defining this dependence Triple-wave ensembles in a thin cylindrical shell has the form

(4)Triple-wave ensembles in a thin cylindrical shell

where

Triple-wave ensembles in a thin cylindrical shell

In the general case this equation possesses three different roots (Triple-wave ensembles in a thin cylindrical shell) at fixed values of Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. Graphically, these solutions are represented by a set of points occupied the three surfaces Triple-wave ensembles in a thin cylindrical shell. Their intersections with a plane passing the axis of frequencies are given by fig.(1). Any natural frequency Triple-wave ensembles in a thin cylindrical shell corresponds to the three-dimensional vector of amplitudes Triple-wave ensembles in a thin cylindrical shell. The components of this vector should be proportional values, e.g. Triple-wave ensembles in a thin cylindrical shell, where the ratios


Triple-wave ensembles in a thin cylindrical shell

and

Triple-wave ensembles in a thin cylindrical shell

are obeyed to the orthogonality conditions

Triple-wave ensembles in a thin cylindrical shell

as Triple-wave ensembles in a thin cylindrical shellTriple-wave ensembles in a thin cylindrical shell.

Assume that Triple-wave ensembles in a thin cylindrical shell, 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 Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, while the high-frequency azimuthal branch — Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. 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 Triple-wave ensembles in a thin cylindrical shell.

Consider now axisymmetric waves (as Triple-wave ensembles in a thin cylindrical shell). The axisymmetric shear waves are polarized by azimuth: Triple-wave ensembles in a thin cylindrical shell, while the other two modes are uncoupled with the shear mode. These high- and low-frequency branches are defined by the following biquadratic equation

Triple-wave ensembles in a thin cylindrical shell.


At the vicinity of Triple-wave ensembles in a thin cylindrical shell the high-frequency branch is approximated by

Triple-wave ensembles in a thin cylindrical shell,

while the low-frequency branch is given by

Triple-wave ensembles in a thin cylindrical shell.

Let Triple-wave ensembles in a thin cylindrical shell, then the high-frequency asymptotic be

Triple-wave ensembles in a thin cylindrical shell,

while the low-frequency asymptotic:

Triple-wave ensembles in a thin cylindrical shell.

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)Triple-wave ensembles in a thin cylindrical shell

Here Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell are the differential operators; Triple-wave ensembles in a thin cylindrical shell denotes the Airy stress function defined by the relations Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, where Triple-wave ensembles in a thin cylindrical shell, while Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell stand for the components of the stress tensor. The linearized subset of eqs.(5), at Triple-wave ensembles in a thin cylindrical shell, is represented by a single equation

Triple-wave ensembles in a thin cylindrical shell

defining a single variable Triple-wave ensembles in a thin cylindrical shell, whose solutions satisfy the following dispersion relation

(6)Triple-wave ensembles in a thin cylindrical shell

Notice that the expression (6) is a good approximation of the low-frequency branch defined by (4).

Evolution equations

If Triple-wave ensembles in a thin cylindrical shell, then the ansatz (3) to the eqs.(1)-(2) can lead at large times and spatial distances, Triple-wave ensembles in a thin cylindrical shell, to a lack of the same order that the linearized solutions are themselves. To compensate this defect, let us suppose that the amplitudes Triple-wave ensembles in a thin cylindrical shell be now the slowly varying functions of independent coordinates Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, although the ansatz to the nonlinear governing equations conserves formally the same form (3):

Triple-wave ensembles in a thin cylindrical shell

Obviously, both the slow Triple-wave ensembles in a thin cylindrical shell and the fast Triple-wave ensembles in a thin cylindrical shell spatio-temporal scales appear in the problem. The structure of the fast scales is fixed by the fast rotating phases (Triple-wave ensembles in a thin cylindrical shell), while the dependence of amplitudes Triple-wave ensembles in a thin cylindrical shell 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 Triple-wave ensembles in a thin cylindrical shell to its average value

(7)Triple-wave ensembles in a thin cylindrical shell,

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 Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. In turn, the average Lagrangian does not depend upon the fast variables.

The average Lagrangian Triple-wave ensembles in a thin cylindrical shell can be formally represented as power series in Triple-wave ensembles in a thin cylindrical shell:

(8)Triple-wave ensembles in a thin cylindrical shell

At Triple-wave ensembles in a thin cylindrical shell the average Lagrangian (8) reads

Triple-wave ensembles in a thin cylindrical shell

where the coefficient Triple-wave ensembles in a thin cylindrical shell coincides exactly with the dispersion relation (3). This means that Triple-wave ensembles in a thin cylindrical shell.

The first-order approximation average Lagrangian Triple-wave ensembles in a thin cylindrical shell depends upon the slowly varying complex amplitudes and their first derivatives on the slow spatio-temporal scales Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. The corresponding evolution equations have the following form

(9)Triple-wave ensembles in a thin cylindrical shell

Notice that the second-order approximation evolution equations cannot be directly obtained using the formal expansion of the average Lagrangian Triple-wave ensembles in a thin cylindrical shell, since some corrections of the term Triple-wave ensembles in a thin cylindrical shell are necessary. These corrections are resulted from unknown additional terms Triple-wave ensembles in a thin cylindrical shell of order Triple-wave ensembles in a thin cylindrical shell, which should generalize the ansatz (3):

Triple-wave ensembles in a thin cylindrical shell

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)Triple-wave ensembles in a thin cylindrical shell,

hold true, plus the nonlinearity in eqs.(1)-(2) possesses some appropriate structure. Here Triple-wave ensembles in a thin cylindrical shell is a small phase detuning of order Triple-wave ensembles in a thin cylindrical shell, i.e. Triple-wave ensembles in a thin cylindrical shell. The phase matching conditions (10) can be rewritten in the alternative form

Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell is a small frequency detuning; Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell 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)Triple-wave ensembles in a thin cylindrical shell.

Here Triple-wave ensembles in a thin cylindrical shell is the potential of the triple-wave coupling; Triple-wave ensembles in a thin cylindrical shell are the slowly varying amplitudes of three waves at the frequencies Triple-wave ensembles in a thin cylindrical shell and the wave numbers Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell; Triple-wave ensembles in a thin cylindrical shellare the group velocities; Triple-wave ensembles in a thin cylindrical shell is the differential operator; Triple-wave ensembles in a thin cylindrical shell stand for the lengths of the polarization vectors (Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell); Triple-wave ensembles in a thin cylindrical shell is the nonlinearity coefficient:

Triple-wave ensembles in a thin cylindrical shell


where Triple-wave ensembles in a thin cylindrical shell.

Solutions to eqs.(11) describe four main types of resonant triads in the cylindrical shell, namely Triple-wave ensembles in a thin cylindrical shell-, Triple-wave ensembles in a thin cylindrical shell-, Triple-wave ensembles in a thin cylindrical shell- and Triple-wave ensembles in a thin cylindrical shell-type triads. Here subscripts identify the type of modes, namely (Triple-wave ensembles in a thin cylindrical shell) — longitudinal, (Triple-wave ensembles in a thin cylindrical shell) — bending, and (Triple-wave ensembles in a thin cylindrical shell) — shear mode. The first subscript stands for the primary unstable high-frequency mode, the other two subscripts denote the secondary low-frequency modes.

A new type of the nonlinear resonant wave coupling appears in the cylindrical shell, namely Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell-, Triple-wave ensembles in a thin cylindrical shell- and Triple-wave ensembles in a thin cylindrical shell-types, but the Triple-wave ensembles in a thin cylindrical shell-type triple-wave coupling only. Since Triple-wave ensembles in a thin cylindrical shell-type triads are inherent in both the Karman and Donnell models, these are of interest in the present study.

Triple-wave ensembles in a thin cylindrical shell-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: Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. The primary high-frequency azimuthal mode is characterized by the spectral parameters Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell (the numerical values of Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell are given in the captions to the figures). In the example presented the phase detuning Triple-wave ensembles in a thin cylindrical shelldoes not exceed one percent. Notice that the phase detuning almost always approaches zero at some specially chosen ratios between Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, i.e. at some special values of the parameterTriple-wave ensembles in a thin cylindrical shell. Almost all the exceptions correspond, as a rule, to the long-wave processes, since in such cases the parameter Triple-wave ensembles in a thin cylindrical shell cannot be small, e.g. Triple-wave ensembles in a thin cylindrical shell.

NB Notice that Triple-wave ensembles in a thin cylindrical shell-type triads can be observed in a thin rectilinear bar, circular ring and in a flat plate.

NBThe wave modes entering Triple-wave ensembles in a thin cylindrical shell-type triads can propagate in the same spatial direction.

Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. 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 Triple-wave ensembles in a thin cylindrical shell cannot be small.

NBThe wave modes entering Triple-wave ensembles in a thin cylindrical shell-type triads cannot propagate in the same spatial direction. Otherwise, the nonlinearity parameter Triple-wave ensembles in a thin cylindrical shell 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.

Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. 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 Triple-wave ensembles in a thin cylindrical shell-type triads fall near the boundary of the validity domain predicted by the Kirhhoff-Love theory. This means that the real physical properties of Triple-wave ensembles in a thin cylindrical shell-type triads can be different than theoretical ones.

NBTriple-wave ensembles in a thin cylindrical shell-type triads are essentially two-dimensional dynamical objects, since the nonlinearity parameter goes to zero, as all the waves propagate in the same direction.

Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell-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 Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell. 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 Triple-wave ensembles in a thin cylindrical shell cannot be small.

NBThe resonant interactions of Triple-wave ensembles in a thin cylindrical shell-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)Triple-wave ensembles in a thin cylindrical shell

Notice that the equations of the set (12) are always linearly dependent. Moreover, these do not depend upon the nonlinearity potential Triple-wave ensembles in a thin cylindrical shell. In the case of spatially uniform wave processes (Triple-wave ensembles in a thin cylindrical shell) eqs.(12) are reduced to the well-known Manly-Rawe algebraic relations

(13)Triple-wave ensembles in a thin cylindrical shell


where Triple-wave ensembles in a thin cylindrical shell are the portion of energy stored by the quasi-harmonic mode number Triple-wave ensembles in a thin cylindrical shell; Triple-wave ensembles in a thin cylindrical shell 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)Triple-wave ensembles in a thin cylindrical shell.

Equation (14) predicts that the total energy of the resonant triad is always a constant value Triple-wave ensembles in a thin cylindrical shell, while the triad components can exchange by the portions of energy Triple-wave ensembles in a thin cylindrical shell, 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 Triple-wave ensembles in a thin cylindrical shell, or a stable motion near the two stationary elliptic points Triple-wave ensembles in a thin cylindrical shell, and Triple-wave ensembles in a thin cylindrical shell.

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 Triple-wave ensembles in a thin cylindrical shell or Triple-wave ensembles in a thin cylindrical shell, 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 (Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell) travels along the shell. Figure (6) represents a projection of the triple-wave resonant manifold of the shell, with the geometrical sizes Triple-wave ensembles in a thin cylindrical shell m; Triple-wave ensembles in a thin cylindrical shell m; Triple-wave ensembles in a thin cylindrical shell 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 Triple-wave ensembles in a thin cylindrical shell) can be unstable one with respect to small perturbations of the asymmetric mode (with Triple-wave ensembles in a thin cylindrical shell) 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

Triple-wave ensembles in a thin cylindrical shell

traveling along the single direction characterized by the ''companion'' coordinate Triple-wave ensembles in a thin cylindrical shell. By substituting this expression into the first and second equations of the set (1)-(2), one obtains the following differential relations

(15)Triple-wave ensembles in a thin cylindrical shell


Here

Triple-wave ensembles in a thin cylindrical shell

while

Triple-wave ensembles in a thin cylindrical shell

whereTriple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell.

Using (15) one can get the following nonlinear ordinary differential equation of the fourth order:

(16)Triple-wave ensembles in a thin cylindrical shell,

which describes simple stationary waves in the cylindrical shell (primes denote differentiation). Here

Triple-wave ensembles in a thin cylindrical shell


whereTriple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell are the integration constants.

If the small parameter Triple-wave ensembles in a thin cylindrical shell, and Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell satisfies the dispersion relation (4), then a periodic solution to the linearized equation (16) reads

Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell are arbitrary constants, since Triple-wave ensembles in a thin cylindrical shell.

Let the parameter Triple-wave ensembles in a thin cylindrical shell be small enough, then a solution to eq.(16) can be represented in the following form

(17)Triple-wave ensembles in a thin cylindrical shell

where the amplitude Triple-wave ensembles in a thin cylindrical shell depends upon the slow variables Triple-wave ensembles in a thin cylindrical shell, while Triple-wave ensembles in a thin cylindrical shell are small nonresonant corrections. After the substitution (17) into eq.( 16) one obtains the expression of the first-order nonresonant correction

Triple-wave ensembles in a thin cylindrical shell

and the following modulation equation

(18)Triple-wave ensembles in a thin cylindrical shell,

where the nonlinearity coefficient is given by

Triple-wave ensembles in a thin cylindrical shell.


Suppose that the wave vector Triple-wave ensembles in a thin cylindrical shell is conserved in the nonlinear solution. Taking into account that the following relation

Triple-wave ensembles in a thin cylindrical shell

holds true for the stationary waves, one gets the following modulation equation instead of eq.(18):

Triple-wave ensembles in a thin cylindrical shell

or

Triple-wave ensembles in a thin cylindrical shell,

where the point denotes differentiation on the slow temporal scale Triple-wave ensembles in a thin cylindrical shell. This equation has a simple solution for spatially uniform and time-periodic waves of constant amplitude Triple-wave ensembles in a thin cylindrical shell:

Triple-wave ensembles in a thin cylindrical shell,

which characterizes the amplitude-frequency response curve of the shell or the Stocks addition to the natural frequency of linear oscillations:

(19)Triple-wave ensembles in a thin cylindrical shell.


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)Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell denote the long-wave slowly varying fields being the functions of arguments Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell (these turn in constants in the linear theory); Triple-wave ensembles in a thin cylindrical shell is the amplitude of the bending wave; Triple-wave ensembles in a thin cylindrical shell, Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell 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)Triple-wave ensembles in a thin cylindrical shell

where Triple-wave ensembles in a thin cylindrical shell is the group velocity, and Triple-wave ensembles in a thin cylindrical shell. Notice that eqs.(21) have a form of Zakharov-type equations.

Consider the stationary quasi-harmonic bending wave packets. Let the propagation velocity be Triple-wave ensembles in a thin cylindrical shell, then eqs.(21) can be reduced to the nonlinear Schrцdinger equation

(22)Triple-wave ensembles in a thin cylindrical shell,


where the nonlinearity coefficient is equal to

Triple-wave ensembles in a thin cylindrical shell,

while the non-oscillatory in-plane wave fields are defined by the following relations

Triple-wave ensembles in a thin cylindrical shell

and

Triple-wave ensembles in a thin cylindrical shell.

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)Triple-wave ensembles in a thin cylindrical shell

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 Triple-wave ensembles in a thin cylindrical shellcm, Triple-wave ensembles in a thin cylindrical shell cm and Triple-wave ensembles in a thin cylindrical shell 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, Triple-wave ensembles in a thin cylindrical shell, and circumferential, Triple-wave ensembles in a thin cylindrical shell, direction. Let Triple-wave ensembles in a thin cylindrical shell and Triple-wave ensembles in a thin cylindrical shell, respectively, be the eigen numbers of the mode. The modes in which Triple-wave ensembles in a thin cylindrical shell is always one and Triple-wave ensembles in a thin cylindrical shell ranges from 6 to 32 were investigated. The only modes which we failed to excite (for unknown reasons) were Triple-wave ensembles in a thin cylindrical shell= 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 Triple-wave ensembles in a thin cylindrical shell and eigen frequencies Triple-wave ensembles in a thin cylindrical shell versus the wave number Triple-wave ensembles in a thin cylindrical shell 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 Triple-wave ensembles in a thin cylindrical shell=6,7,..,32, as Triple-wave ensembles in a thin cylindrical shell.


REFERENCES

[1]Bretherton FP (1964), Resonant interactions between waves, J. Fluid Mech., 20, 457-472.

[2]Bloembergen K. (1965), Nonlinear optics, New York-Amsterdam.

[3]Ablowitz MJ, H Segur (1981), Solitons and the Inverse Scattering Transform, SIAM, Philadelphia.

[4]Kubenko VD, Kovalchuk PS, Krasnopolskaya TS (1984), Nonlinear interaction of flexible modes of oscillation in cylindrical shells, Kiev: Naukova dumka publisher (in Russian).

[5]Ginsberg JM (1974), Dynamic stability of transverse waves in circular cylindrical shell, Trans. ASME J. Appl. Mech., 41(1), 77-82.

[6]Bagdoev AG, Movsisyan LA (1980), Equations of modulation in nonlinear dispersive media and their application to waves in thin bodies, .Izv. AN Arm.SSR, 3, 29-40 (in Russian).

[7]Kovriguine DA, Potapov AI (1998), Nonlinear oscillations in a thin ring - I(II), Acta Mechanica, 126, 189-212.

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