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S.B. Karavashkin, O.N. Karavashkina

2. Comparative analysis of conventional basic methods

With all diversity of the used methods, we can select a few being basic for them all.

To build the mathematical model per se, mainly two basic methods are used: the Lagrange method based on the energy balance and the method of force balance at the elastic system elements. The Lagrange method has the advantage that we can state the problem immediately in the generalised coordinates for the systems having many degrees of freedom; this is convenient in the basic research of complex vibration processes. The second method is more visual and simple in building the particular models of elastic systems, so it is used more often in the applied calculation. Both methods give identical results, so at this stage of mathematical modelling the difficulties are rare, they arise when seeking the solutions. Analysing them, we will confine ourselves to ideal 1D elastic lines related to the subject of this paper.

The conventional methods to find analytical solutions are usually based on one of three approaches to solving the problem of vibrant elastic systems:

- using directly the matrix methods (see e.g. [4]- [10]);

- using the integral equations method (see e.g. [11]);

- using the modified methods based on the matrix properties; the oscillatory, Voronoy, Toeplitz matrixes and others related to them (see e.g. [12], [13]);

- using the indirect methods based on the revealed regularities in the particular modelling systems of differential equations (see e.g. [4], [14], [15]).

In their turn, the problems of forced vibrations are usually based on the study of a homogeneous system of differential equations that relates to free vibrations in an elastic model, because the resonance frequencies coincide for the forced and free vibrations. So it will be efficient to begin our analysis with this type of methods.

For matrix methods the following structure of solutions is typical. Using the second-kind Lagrange equation

(where is the kinetic energy of the system; is the  potential energy of the system;  q_j and q_k are the generalised coordinates of system), we can yield the system of ordinary linear second-order differential equations

Its solution is sought in the form

where A_k, and are the constants that are to be determined.

Substituting (3) into (2) after an obvious reduction, there is yielded an algebraic system of a following type:

“This system to have a non-trivial solution, it is necessary and sufficient, its determinant to be zero:

[8, p. 529]. “The characteristic equation (5) is an algebraic equation of the 2s-order with respect to , hence, it has 2s solutions, i.e. 2s eigenvalues ( = 1, 2, ..., 2s). The system (4), after we substitute to it the solution , will determine the relationship between the ‘amplitudes’ C_k:

We can write the general solution of the system of differential equations (2) as a real (or imaginary) part of the sum of partial solutions, i.e., as

where the amplitudes (or rather their relationship) are determined by (6), and solutions - by (5)” [7, p. 263].