The instable parametric oscillation during the rope track transport

Abstract

The parametric oscillation appears if some parameter of the oscillating system, e.g. stiffness, changes its value in the process of oscillation. The two different frequencies have influence on the resulting motion. The natural frequency of the system and frequency of the stiffness change. The analytical solution for the harmonic change was performed by Mathieu. The resulting motion can have stable or instable form, depending on the value of two dimensionless constants. First represents the ratio between natural frequency a frequency of the stiffness change, second represents ratio between the variation and average value of stiffness. During the transport on the rope track in the horizontal mine opening two forms of the parametric oscillations can appear. The diagonal swinging in the horizontal direction (mathematical pendulum) and vertical oscillations due to rope stiffness are analyzed and the conditions of the parametric oscillations appearance are determined.