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Bend effect on vibration pattern

First of all note that according to the conventional rule of coordinate transformation, we can substitute in (1) and (3) the parameters describing the shift of kth mass located at the bend

The similar relation will be true for the rest elements of a line located after the bend. Taking this into account, multiply (2) into cos , (4) into sin and subtract (4) from (2) term by term. After the manipulation we yield

Multiplying (2) into sin and (4) into cos and summing them term by term, we yield

Integrating (1), (3), (6) and (7) and noting (4), we see that the transformed system of differential equations is already independent of the angle . Consequently, the solutions in coordinates (x, y) will be invariable in the view of a bend.

As opposite to this, if the longitudinal and transverse stiffnesses of an elastic line were different, then in the initial modelling systems of differential equations we have to introduce the stiffness s_lt in (1) and (2), and s_tr,  s_tr s_lt   in (3) and (4). It is easy to check that after it, every attempt to transform the system  ( , )  into   (x, y) for the equations of the line elements located after the bend will not lead to such simplification. The angle   will remain in the modelling system of equations, therefore it will be present in the solutions describing the vibration process. In this case we have to consider the elastic system as that having an additional distinction at the bend point. The difference between the longitudinal and transverse stiffness values will lead to the different along-the-line wave propagation velocities, and the complicated vibration patterns will arise and require the additional study.

The proven theorem can be easily extended to the generalised coordinates of an elastic system. At the same time, the definition of generalised coordinates per se cannot substitute the essence of the proven assertion, since, according to the theorem, the modelling system much simplifies with the equal longitudinal and transverse stiffness, and instead two systems  (, )  and  (x, y) it can be reduced to the general system  (x, y) excluding the influence of the bend angle .

In this paper we will determine the solution for some models of elastic lines in the course of whose investigation this theorem is valid and which are widely applicable to the specific problems.