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Bend effect on vibration pattern

5. 1 D homogeneous elastic line having inequal longitudinal and transverse stiffness coefficients

As we said in the introduction, the rocks with which we deal in geological and engineering problems are usually differently deformable in the longitudinal and transverse directions of a rock. So as the third example we will consider the vibration pattern variation in case of inequal longitudinal and transverse stiffness coefficients of an elastic line. In the item 2 we proved that in this case we cannot ignore the bend. Two parameters characterise now the constraints stiffness – the longitudinal s_lt and transverse s_tr stiffness coefficients. The modelling systems of equations for the semi-finite homogeneous elastic line takes the following form:

for the longitudinal component

The systems (28)-(29) differ from (1)-(4) by the kth and (k + 1)th equations of the modelling systems. The displacements of transient line elements x_k, y_k, _k+1, _k+1 are included in (28)-(29) with the coefficient

which disables us transforming the coordinates (, ) into (x, y) , as in case of equal stiffness coefficients. With it the solution considerably complicates. First of all, there arise two parameters

characterising the delay of along-the-line propagation of longitudinal and transverse components of the wave, and in general case these parameters differ. In this connection, the critical frequencies for the longitudinal and transverse components are also different and equal to

relatively. It causes the division of the frequency range not into two but into tree sections. In that first the vibrations of both longitudinal and transverse components relate to the periodical vibration regime. With the typical relationship of stiffnesses

this band has the bounds  0 < < _lt . In the second band  _lt < < _tr , when (33) was true, the longitudinal component vibrations relate to the critical regime of antiphase along-the-line damping vibrations, and for the transverse component the periodical regime is typical. It causes the fact that out of the region of external force action and of the bend, mainly transverse vibrations take place. Finally, at  > _tr  the vibrations are seen only in the region of an external force action, since in this band the aperiodical vibration regime relates to the transverse component.