SELF

76

S.B. Karavashkin, O.N. Karavashkina

“The problem is solvable, if all forces and moments are given. But for a rotating shaft neither forces nor moments can be given, there are known only their expressions through the masses and inertia moments of the pulleys and sags and inclinations at the places of their fixation” [6, p. 167]. In this case, considering the system of equations composed for the boundaries of all homogeneous sections of a shaft, Krylov comes to a conclusion: “We can see from the formulas that, beginning with y₀ , we can compose sequentially all other functions that always will have the form

where x₁ and x₂ are the functions composed by a definite way for the known functions of argument kx. Therefore it is obvious that the boundary conditions will have the form

and the equation that determines the critical frequencies will be

This expression will be so complicated that it will be possible to solve it only by sequential approximations for which we will need only a possibility to calculate the particular values (k) specified by the particular value k” [6, p. 173].

The similar results are yielded when using the method of integral equations. “If an integral equation of a vibrant system with continuously and discretely distributed masses has been composed, then, using the Fredholm method, we can find the eigenvalues. The essence of this method is that the eigenvalues are found as the roots of Fredholm series

where the coefficients of this series having (n - 1) discrete masses are calculated as

Actually, the eigenvalues calculation needs to construct preliminary the Green function and to find the complicated coefficients C_n , especially when calculating a high-order eigenvalue. Notice also that there exists a whole class of the boundary problems for which it is impossible to construct the so-named generalised Green function” [11, p. 38]. Using this method, Kuhta and Kravtchenko reduced a particular problem of forced vibrations to the numerical modelling. “To calculate the roots () = 0  (the frequencies of the considered problem), we estimate their lower boundary. Then, combining the chord technique with the selection technique which is suitably realised by computing in cases of both simple multiple roots, we determine the necessary number of frequencies in the increasing order” [11, p. 47].