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

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

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):

(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

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.