V.2 No 1

61

On solution for an infinite heteroheneous line

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

As the basic model, consider an infinite lumped line with one transition of mass heterogeneity affected by the longitudinal external harmonic force. Suppose that the external force affects the k-th line element, and k n , where n   is the number of boundary element of the heterogeneous section. The modelling system of differential equations for this line is

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

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

for the first section i k

for the second section k i n

and for the third section i n + 1

where