Bend effect on vibration pattern093

V.2 No 1	93
Bend effect on vibration pattern

In the periodical vibration regime ( < ₀ ), within the elastic line, the standing wave will form, since the solutions have the following form: for the x-component
	(21)
and for the y-component
	(22)
In the aperiodical regime ( > ₀ ), we see the antiphase vibrations damping along the line in the region of external force action, because the solutions have the following form: for the x-component
	(23)
and for the y-component
	(24)
where , . The solutions for critical regime ( = ₀ ) depend on the evenness of number n. With the even n the values _xi and _yi are infinite, and with the odd n the solutions take the following form: for the x-component
	(25)
and for the y-component
	(26)
It means, they practically coincide with the solutions (14)-(15).
Fig. 7. The diagrams of typical vibrations in a lumped homogeneous closed-loop elastic line under the harmonic external force inclined to the axis x at the angle = 60^{^o} , with respect to the circular frequency and external force momentary phase = t + ₀ . The elastic line parameters: n = 12; a = 0,02 m; s = 100 N/m; m = 0,01 kg; F₀ = 1 N; ₀ = 200 sec^{^-1}
In Figures 7 and 8, the typical diagrams are presented dependently on the external force frequency and its inclination angle to the axis x relatively. They were constructed on the basis of above solutions for the closed-loop elastic line.
Fig. 8. The typical vibrations in a lumped homogeneous closed-loop elastic line under the harmonic external force acting with the circular frequency = 20 sec^{^-1}, with respect to the angle of the external force inclination to the axis x. The line parameters: n = 12; a = 0,02 m; s = 100 N/m; m = 0,01 kg; F₀ = 1 N; ₀ = 200 sec^{^-1}