Exact solution of the problem for dynamic field of displacements in rods of variable length

Initial boundary value problems are considered for rods that change length over time. These problems are transformed to problems about rods with mobile ends. A special method which allows one to obtain exact solutions of such problems is developed. This method is a generalization of a method of reflections for rods and strings of constant length. As an application of this class of problems, the hoisting of ropes in mining lifts, including their problems, are considered. Exact expressions for displacements in rods of variable length are obtained. The same results can be applied to cross oscillations of strings of variable length.