The motion of the thread with a variable length

•Exact analytical solution of the problem of inertial tether deployment is developed.•The solution significantly depends on the analog of the Mach number.•In the limiting case the solution is received using two different methods.•The stability analysis of the received solution has been made.•Almost always the solution is unstable.

This paper is devoted to the peculiarities of a process of inertial tether system deployment of the great length (tens of kilometers). And one of the main practical issues of this article is the behavior of maximum oscillation amplitude.First of all the exact analytical solution of the original problem is received. Furthermore it is shown that ratio of transverse wave velocity of the tether to the velocity of deployment is one of the main parameters of the problem. In limiting case, when initial tether length equals zero, the solution is received using another method – the method of progressing waves. Also it is demonstrated that the limiting case agrees with the previously received results.Analysis of the solution has revealed that the maximum oscillation amplitude of the tether almost always rises with the time. This effect corresponds to asymptotic instability of the solution (in this article instability is regarded as rising of oscillation amplitude with the time). Only in specific degenerate conditions the solution is stable. The received results of amplitude increase have to be considered in simulation of the tether system behavior as far as small perturbative vibrations at the reel system always take place.