The vibrational energy and control of pendulum systems

The motion of a pendulum system with a finite number of degrees of freedom under the action of vibration, the frequency of which considerably exceeds the frequency of natural oscillations of the system, is considered. For the averaged motion, the effect of vibration leads to the occurrence of an additional term of the potential energy (the vibrational energy). A general expression for the vibrational energy is derived using canonical averaging of Hamilton's equations. Stable steady motion corresponds to a minimum of the effective potential energy. The general problem of determining the vibration parameters of the suspension point, for which periodic motions exist in the neighbourhood of a specified position of the pendulum system, is formulated. Its analytical solution is constructed from the condition for a minimum of the effective potential energy. It is shown that, for high-frequency vibrations of the suspension point, steady motion of a point mass of the pendulum is established, in which its velocity in the principal approximation is directed along the pendulum rod. The direction and amplitude of the vibrations of the suspension point, corresponding to equilibrium and stability, is determined for any specified position of the pendulum. (C) 2012 Elsevier Ltd. All rights reserved.