SELF

50

S.B. Karavashkin, O.N. Karavashkina

2. The method to find the exact analytical solution

In Fig. 1 there is presented the studied semi-finite 1D elastic system with lumped masses M and elastic constraints s_g on whose start the harmonic force F(t) acts. As it was said in the introduction, the first and the last elements of its subsystems are connected rigidly.

Before we begin studying the system as a whole, consider a separate subsystem under some harmonic force F_i(t) acting on its start. This line consisting of n masses connected by constraints having stiffness s_s is shown in Fig. 2a. It is easy to see that this system can be presented by an equivalent diagram shown in Fig. 2b, where the rigid constraint is substituted by the second force acting on the end of elastic line. Note that such substitution is possible because the subsystem is perfectly symmetrical and the end elements are rigidly connected. In case of a heterogeneous subsystem or imperfectly rigid constraints of the end elements, the subsystem with its main system can be conveniently considered as a heterogeneous line. It naturally requires another technique being surplus for the present problem, so we will use the equivalent diagram, Fig. 2b.

To find the exact analytical solution for the diagram shown in Fig. 2b, conveniently use the results obtained in [22] for a homogeneous finite elastic line on whose interior element an harmonic force F(t) acts. In [22] three solutions were presented for forced vibrations  according  to  the   relationship  between  the  parameter  _s = (²m/4s_s)^1/2 and the unity. This regularity has the following form:

for the periodical regime, _s < 1

(1)

(2)

(3)

where _s   = arcsin _s  ; _s = (²m/4s_s)^1/2  ;  _s+ = _s + (_s² - 1)^1/2   ;  _{s -} = _s - (_s² - 1)^1/2 ; k is the number of element to which the external force is applied; p is the studied line element number and _p is the momentary displacement of the pth element of subsystem.