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On solution for an infinite heteroheneous line

2. General solution for a heterogeneous ideal elastic line having one heterogeneity transition

As the base model, consider an infinite lumped line having one transition of mass heterogeneity under longitudinal external harmonic force action. Suppose that the external force acts on the kth line element, and k n , where n is the number of the boundary element of the heterogeneous section. The modelling system of differential equations for this line is

where _i is the longitudinal displacement of the ith mass of a line (- i ); s is the stiffness coefficient of a line; m₁ and m₂ are the element masses of related sections.

We can see from (1) that the external force application point and heterogeneity transition divide the line into three sections. According to it, the solution also divides into three intervals, and each has its distinctions. This solution has the following form:

for the first section i k

for the second section k i n

and for the third one i n + 1

where