14

S.B. Karavashkin

Each of these systems is similar to those considered in [1]. Hence, we can write at once the exact analytical solutions for each of them, as follows:

for the x-component of vibration:

in the periodical regime ( < 1)

For the y-component of vibrations we yield correspondingly:

in the periodical regime ( < 1)

As a result of the superposition, there forms an inclined wave propagating in the positive direction of the axis x; this is corroborated by the vibration diagram shown in Fig. 2.

Typically, the inclined pattern of vibrations remains both with free vibrations in a lumped line and with the limiting process to a distributed line.

Basing on results presented in [1], the solution, e.g., for free vibrations has the following form:

for the x- component of vibrations

where X_k and Y_k are the x- and y- components of vibration amplitude of the kth element whose parameters are specified, and k is the element number whose vibration is specified.

In case of limiting process to a distributed line we can present

where   is the density; T is the line stiffness;   x₀  is the distance from the line start to the point of rest of the considered line element; m is the line elements mass. With it the solutions (3)-(8) transform into the following system:

The obtained system describes parametrically an inclined wave propagation along the axis x, as is shown in Fig. 3.