Triple-wave ensembles in a thin cylindrical shell (стр. 3 из 5)

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 (

) — 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.

-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.