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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 emphasise a few being basic for them all.

To construct the mathematical model per se, one uses mainly two basic methods: the Lagrange method based on the energy balance, and the method of force balance on the elastic system elements. The first of them gives the scope to state the problem directly in the generalised co-ordinates for the systems having many degrees of freedom; this is advantageous in the fundamental research of complex vibration processes. The second method is more visual and simple, when constructing 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:

• the direct use of the matrix methods (see, e.g., [4]-[10]);

• the integral equations method (see, e.g., [11]);

• using the modified methods basing on the matrix properties; the oscillatory, Voronoy, Toeplitz matrixes etc. are related to them (see, e.g., [12], [13]);

• 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 investigation of a homogeneous system of differential equations corresponding to free vibrations in an elastic model, due to the fact that the resonance frequencies coincide for the forced and free vibrations. So it will be efficient to begin our analysis with this type of methods.

In the matrix methods, the following structure of solutions is inherent. Using the Lagrange equation of the second kind

(where is the system kinetic energy; is the system potential energy; q_j and q_k are the generalised co-ordinates of system) one yields the system of ordinary linear second-order differential equations

One seeks its solution in the form

where A_k, and are the constants that are to be determined. Substituting (3) into (2), after an obvious reduction, one obtains 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 . After one substitutes the solution to the system (4), this system will determine the relationship between the 'amplitudes' C_k:

One can write the general solution of (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].