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Bend effect on vibration pattern

With s_tr s_lt the solutions (34)-(35) transform to those described in previous item, as we see in Figures 12 and 13 and can easy show mathematically. Actually, with equal stiffness coefficients (37) transform as follows:

where _lt = _tr = . Substituting (38) sequentially to (36) and then to (34)-(35), we obtain (10), (11), (17) and (18). For example, for x_i:

For the rest solutions of (34)-(35) the reduction is similar.

Thus we see that in case of inequal longitudinal and transversal stiffness coefficients, the vibration pattern essentially changes. In the bend region the amplitude increases, and the external force to which the table-land part of a landscape responded elastically is able to cause the folding and even destruction on the mountain slope. The same, the mountain range sections before and after its bending can respond to the seismic wave very different.

The presented diagrams show that with the further development of this subject we can gain the possibility to judge by the rocks deformation of the phenomenon which caused it, not only qualitatively but to approach to the quantitative estimations. Besides, the rock behaviour under high-frequency excitation is also interesting, when the longitudinal component vibration relates to the aperiodical regime. As we said before, in this case the wave process longitudinal component reveals only in the regions of an external force action and of the bend. In the second case it will cause the rocks destruction only in the bend region, since, as is known, the most destruction takes place namely under joint action of longitudinal and transversal components of a wave.

6. Conclusions

We have proved that the bend angle influence on the vibration pattern in an elastic line reveals only if the longitudinal and transversal stiffnesses of an elastic line are inequal. Basing on the proved theorem, we have presented the solutions for a semi-finite elastic line having a bend at the kth element in cases of equal and inequal longitudinal and transversal stiffness coefficients and for a closed-loop elastic line. We have shown that for a semi-finite line the wave front inclination changes when transiting through the bend, and this change is proportional to the bend angle. This dependency is conditioned by the transfer regularities between the reference systems.

In a closed-loop elastic line under the concentrated force action, the standing waves arises. The line elements vibration direction coincides with the external force direction. When the distributed travelling-wave-like load acted on a closed-loop line, the vibration phase delays from the external force phase. With the growing frequency, the line elements vibration pattern acquires a complicated pattern. At all frequencies the pattern differs from the case of a concentrated force action.

With inequal stiffness coefficients, in the before-bend region the superposition of standing and progressive waves takes place and the finite resonance peaks arise; their amplitude depends on the relationship between the transversal and longitudinal stiffness coefficients, on the bend angle and on the external force inclination. The resonance frequencies for the longitudinal and transversal components do not coincide. In this connection, the progressive wave inclination does not correspond to the external force inclination, as in case of equal stiffness coefficients. With the retaining given parameters and external force inclination, as longitudinal as transversal as inclined waves can propagate in the line, dependently only on the external excitation frequency. When the bend angle vanishes, the resonance peaks disappear, and two waves propagate in the line, each having the velocity proportional to the related stiffness coefficient.

The revealed features cause the bend region in a line with inequal stiffness coefficients to be a stress concentrator. In this region the vibration amplitude abruptly increases, in the vibration pattern there arise the dynamical breakdowns able to cause the additional destruction. Due to different wave propagation velocity of longitudinal and transversal components, the wave inclination is inconstant along the line. In one and the same line at one and the same moment we can observe the waves with both positive and negative inclination angles.