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Bend effect on vibration pattern

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

The similar relation will be true for the rest line elements 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 co-ordinates (x, y) will be invariable in the view of a bend.

As opposite to this, if the longitudinal and transversal stiffnesses of an elastic line are 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 transversal stiffness values will lead to the different along-the-line wave propagation velocities, and the complicated vibration patterns will arise needing the additional studying.

The proven theorem can be easy extended to the generalised co-ordinates of an elastic system. At the same time, the definition of generalised co-ordinates per se cannot substitute the essence of the proved assertion, since, according to the theorem, the modelling system much simplifies with the equal longitudinal and transversal 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 elastic line models in whose investigation this theorem is valid and which are widely applicable to the specific problems.