Introduction. Design of bridges' constructional structures relates to the mathematical modeling of oscillatory processes that arise in such structures. Such mathematical models are described by differential equations of hyperbolic type. Methods of solving such problems are divided into direct and approximate. The basis of direct methods is the method of separating variables, the Green's method of function, the method of integral transformations.
Problem statement. An issue in these tasks is the problem of multiplying the generalized functions. In the proposed scheme, this problem with generalized functions is eliminated by reducing the differential equation to the system of differential equations and using the matrix calculation.
Purpose. The purpose of work is to obtain analytical form of solution of the problem of longitudinal vibrations of a rod consisting of four fragments of piecewise-constant section.
Methods. The proposed method of problem solving belongs to direct methods, which allow getting an analytical form of solution. The basis for solving the task includes the concept of quasi-derivatives, a modern theory of systems of linear differential equations, the classical Fourier method and a modified method of autofunctions. The advantage of this method is a possibility to examine a problem on each breakdown segment and then to combine the obtained solutions on the basis of matrix calculation. Such an approach allows the use of software tools for solving the problem.
Results. The main result of the work is analytical presentation the problem of longitudinal oscillations of the rod consisting of four parts of piecewise-stable section of cylindrical shape and obtaining the required number of its own values and its functions with the help of Software mathematical Package Maple.
Conclusion. The obtained results can be used in designing of building structures of bridges and supports.
