Triple-wave ensembles in a thin cylindrical shell (стр. 2 из 5)

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

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

, 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)

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):