SELF

74

S.B. Karavashkin, O.N. Karavashkina

With it the first condition of (11) is satisfied automatically. To satisfy the second one, the equality

or

has to be true. On the other hand, when substituting (13) into (9), we yield

As the values C = 0 and sin p = 0  are not of our interest, this condition can be true only when

or

[4, p.280].

Despite the effective transformation, this method does not resolve exhaustively the problems of seeking an analytical solution. Even if  _q  is determined exactly, the vibration amplitude has to be calculated on the basis of polynomials (12), with the calculation complication growing as n grows. And will be numerical all the same, so this method cannot be thought as mere analytical. Furthermore, the technique to find _q analytically is limited not only by homogeneous finite lines but also by the boundary conditions. As it was shown above, for the fixed ends the method gives a solution, but if at least one end is loaded or free, the whole structure of this technique will be destroyed. Actually, as we showed in [1], for a line with unfixed ends, “according to the allowed frequencies, the solution of this system has the form

where _i is the shift of i-th element, A is the first element vibration amplitude, _p =   is the parameter corresponding to the allowed frequencies _q  which are calculated analytically in the course of resolving, and s is the line stiffness coefficient. We can see from (19) that at  i = n, the vibration amplitude at the line end will not be maximal, as we used to think for a distributed line, but will be shifted by some phase _p depending on the allowed mode number. This affection will be especially perceptible at high frequencies, when _p is comparable with 1. And the denominator of (19) will also effect on the amplitude. It means, doing not knowing the exact analytical solutions presented in [1], we cannot state a priori the boundary conditions for a free-end line, while without it the whole Magnus method does not work.

True, for today it is practicable to extend the boundary conditions that are true for distributed lines to those lumped. In particular, this procedure is used in [6, pp. 149- 150] and [11, pp. 48- 49] when studying a shaft with n fitted disks and a beam having n lumped masses relatively. None the less, as is shown in [1], (19) fully satisfies the conventional modelling system of differential equations. Therefore, when calculating the exact analytical solutions for a lumped line, we have to consider this difference.

As regards to the known methods to describe the infinite models, this direction is even less clear. All solutions are mainly limited by finding the phase delay of a process along an infinite elastic line and by the dispersion characteristics (see, e.g., [14, p. 169] and [15, pp. 106- 107]). At the same time Pain writes: “The utmost ultrasonic frequency achieved for today is approximately 10 times less than = c₀/2 (the critical frequency). At the range from 5*10¹² to 1*10¹³   Hz we should expect many interesting experimental results” [14, p. 169]. Actually, the results presented in [2] show that in the infinite elastic systems both forced and free vibrations can exist. They can take place not only in the subcritical regime (less than critical frequency), as in finite lines, but also at the overcritical band, when the entire vibration energy localises at a small region but is not distributed along the whole infinite elastic line. In this case the energy accumulated in an infinite line can be finite. This conclusion is especially important since many real processes, e.g. in crystals, can be modelled with help of an ideal infinite model [15, p. 105]. Moreover, such modelling is valid also for finite elastic lines loaded by wave impedance. So this is a great disadvantage that for today in this area it is impossible to take into account the line features, the external force application point, the description of the pattern of processes etc. To a definite extent this gap was filled in [2]. In the present paper we will study one more important factor - affection of the external force application point. But the material interesting and important for the researchers and engineers is well more voluminous and requires a special attention to this research area.