90

S.B. Karavashkin, O.N. Karavashkina

3. 1D homogeneous elastic lumped system with equal longitudinal and transverse stiffness coefficients

This model can be helpful in studying the problems reduced to the infinitesimally thin rods, homogeneous, isotropic materials and so on. The typical form of the model is shown in Fig. 2. As we proved in the theorem, in this case we can disregard the angle   in the modelling system of differential equations, so the system will take the following form:

Fig. 2. The calculation scheme of 1D homogeneous elastic line having a bend at the kth element with the bend angle and equal coefficients of longitudinal and transverse stiffnesses

and for the y-component

where is the angle of an external force inclination to the axis x, and F(t) = F₀ is the external force acting on the start of line.

Using the results presented in [20], we can write the solutions for (8) and (9).

At the subcritical band (periodical vibration regime) at < ₀

for the x-component

for the y-component

where , , i = 1, 2, 3, ....

At the overcritical band (aperiodical vibration regime) at  > ₀

for the x-component

for the y-component

where .

At the critical frequency (critical vibration regime) at  = ₀  

for the x-component

for the y-component