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S.B. Karavashkin, O.N. Karavashkina

Having now exact analytical solutions for elastic lumped and distributed systems [20] – [24] obtained without any matrix methods and fully determined in reference to the studied elastic systems parameters, we gain the possibility to find exact analytical solutions also for the lines having one or more bends. The theorem proof much simplifies the finding of solutions and the analysis of vibration pattern for a wide class of elastic systems. In this paper we will consider some of the most typical examples that can be investigated in frames of the theorem validity.

2. Theorem of bend influence on the pattern of processes

To investigate elastic bended lines, let us first prove the general theorem determining the degree of bend effect on the solution pattern.

THEOREM 1. A bend in an elastic line does not effect on the vibration pattern only in case, when the transversal and longitudinal stiffness coefficients of its constraints are equal.

To prove this theorem, consider some elastic line (see Fig. 1) having a bend at its kth element, and let in general case the masses of all mass elements be inequal. Suppose also that in the considered elastic line some wave process described by x and y components takes place. Conveniently introduce two reference systems (x,y)  and (, ) describing the vibration processes before and after the bend relatively. Their direction is shown in Fig. 1.

Then the modelling system of differential equations will have the following form:

for the x-component

where m_k is the mass of an elastic line related element; s is the stiffness coefficient, and _xk is the kth element displacement along the axis x;

for the -component

for the y-component

for the -component