Suppression of Oscillation of a Certain Two-Mass System with the Help of the Generalized Gauss Principle

Abstract

The paper studies the suppression of oscillation of a certain two-mass system when it is transferred from the initial state of rest to the given state of rest during a time interval prescribed. The problem is solved by the two methods: the Pontryagin maximum principle (first method) and the generalized Gauss principle (second method). Computational results are presented and the solutions are compared to each other. When the time of motion is short the both methods give practically the same results, but when the time of motion is long the results differ widely. If the time of motion is long then the second method is more preferable than the first one, since the control obtained by the second method sways the mechanical system less than the control obtained by the classical approach. This can be explained by the fact that the first method contains the control including harmonics with the natural frequency of the system, and this seeks to put the system into resonance. In contrast to this, in the second method the control is sought in the form of time polynomial that provides relatively smooth motion of the system. It is noted that the first method always finds the control with jumps at the beginning and at the end of motion. The second method also gives the same jumps when the time of motion is short, but when the time of motion is long the similar jumps vanish when one uses the generalized Gauss principle.