6

S.B. Karavashkin

where

Then one assumes that  A_r can be presented as a product of some constant C into the sine of some angle  _p constant for the given value _p , i.e., for some allowed circular frequency of possible vibrations

where

In its turn, one determines the parameter _p from the boundary conditions (3), whence

Finally, proceeding from the obtained value _p, one finds the resolved circular frequencies _p, substituting (6) and (7) into (5):

One can immediately see from this brief analysis that the method of resolved modes can give the exact solutions only in case of free vibrations and only for a fixed-ends-line. If the forced vibrations were present in the line, the modelling system of differential equations cannot be reduced to the system of algebraic equations (4), because at least in one equation of this system there will be present the external force parameters breaking the uniformity of (4). And if at least one of the line ends is unfixed, then, as we will show below, we cannot indicate a priori the vibration amplitude value at the line ends, because in lumped lines, on distinct from the known solutions for those distributed, the vibration maximum will not take place at the free ends. Thus, as we see, the limitations inherent in the allowed modes method are quite essential, so we cannot think the problem of elastic lumped lines to be completely solved by this approach.

The second approach to this problem solution is Krylov method grounded on the matrix theory tool which bases on the energy conservation for the connected-bodies-system free vibrations that leads one to the system of differential equations similar to (1):

where a_jk , c_jk are the constant parameters characterising the investigated system of bodies; q_k   is the kth body location in the generalised co-ordinates; N is the bodies quantity in the investigated system.

Further, assuming that

where A_k ,   are some invariable values characterising the vibrations of a bodies structure, the system of equations (9) transforms to the determinant of a following type:

In the view of seeking the exact solutions, this method has the following complication. The characteristic equation (11) is an algebraic equation of 2N power with respect to ; hence, it has 2N roots, i.e. N eigenvalues _k(k = 1, 2, ..., N). With it, as is known, one can obtain the exact solution of an algebraic equation only up to N = 4 (in our case, the bi-N algebraic equation). Thus, the matrix approach to solve this problem for free vibrations is much limited by a small number of connected bodies. In case of forced vibrations, Krylov method is based on solving the problem for free vibrations, so the solution is sought by variation of the constant. With it one naturally obtains the line spectrum of forced vibrations. At the same time, as we will show below, in finite lines the forced vibrations spectrum has a continued pattern with the infinite resonances at the frequencies corresponding to the bodies structure natural vibrations. Furthermore, in a finite structure of connected bodies the forced vibrations have three regimes that cannot be determined on the basis of natural frequencies.

3. Analysis of exact complete solutions and checking the results obtained for finite lines

Above we indicated the substantial limitations inherent in existing approaches. We can easy overcome them in frames of method presented as the basic in [1], when obtaining the complete exact solutions for infinite elastic lumped lines. To show it, consider first the solutions obtained with this method for a finite unfixed-ends line, i.e., namely for the line whose exact solution one cannot obtain in general case, when the number of elements is large, neither by the resolved modes method nor by Krylov method.