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Triple-wave ensembles in a thin cylindrical shell  

Triple-wave ensembles in a thin cylindrical shell















Kovriguine DA, Potapov AI


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 () - longitudinal, () - bending, and () - 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 -type triads are inherent in cylindrical shells only.

Notice that the known Karman-type dynamical governing equations can describe the -type triple-wave coupling only. The other triple-wave resonant ensembles, ,  and , 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 , 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 in-plane 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 Kirhhoff-Love 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 stress-straight relationships accordingly to the dynamical part of the Kirhhoff-Love hypotheses:

where  is the Young modulus;  denotes the Poisson ratio;  (the primes indicating the dimensionless variables have been omitted). Neglecting the cross-section 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):



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


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



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


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):


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


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


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 :


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


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


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:


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 () - longitudinal, () - bending, and () - 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 -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.


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.

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