For the first time a new formal solving scheme of the general first boundary value problem for a hyperbolic type equation with piecewise constant coefficients and-singularities was proposed and justified. In the basis of the solving scheme is a concept of quasi-derivatives, a modern theory of systems of linear differential equations, the classical Fourier method and a reduction method. The advantage of this method is a possibility to examine a problem on each breakdown segment and then to combine obtained solutions on the basis of matrix calculation. Such an approach allows the use of software tools for solving the problem. MSC: 34B05.

In this paper, we present the results of modeling nonstationary oscillatory processes in rods consisting of an arbitrary number of pieces. When modeling oscillatory processes that occur in many technical objects (automotive shafts, rods) an important role is played by finding the amplitude and frequency of oscillations. Solving oscillatory problems is associated with various difficulties. Such difficulties are a consequence of the application of methods of operation calculus and methods of approximate calculations. The method of modeling of oscillatory processes offered in work is executed without application of operational methods and methods of approximate calculations. The method of oscillation process modeling proposed in this paper is a universal method. The work is based on the concept of quasi-derivatives. Applying the concept of quasi-derivatives helps to avoid the problem of multiplication of generalized functions. Analytical formulas for describing oscillatory processes in rods consisting of an arbitrary number of pieces are obtained. It can be applied in cases where pieces of rods consist of different materials, and also when in places of joints the masses are concentrated. The proposed method allows the use of computational software. An example of constructing eigenvalues and eigenfunctions for a rod consisting of two pieces is given.

The advantage of this method is a possibility to examine a problem on each breakdown segment and then to combine obtained solutions on the basis of matrix calculation. Such an approach allows the use of software tools for solving the problem and the graphic illustration of the solution. The received results have a direct application to applied problems in the theory of oscillation of the rods with piecewise variables by the distribution of parameters.