Vibrational control of Mathieu´s equation

The vertically driven inverted pendulum-sometimes called the “Kapitza pendulum”-is a well-known example of an unstable system that can be stabilized by oscillatory forcing. Averaging methods and asymptotic stability results can be applied to develop a general framework for designing suitable inputs. Linearizing the Kapitza pendulum yields Mathieu´s equation, which is also extensively studied for its stability characteristics.In this paper, results from these two bodies of work are compared, from the point of view of open-loop control of mechanical systems. The averaging approaches applied to Mathieu´s equation are seen to access only a very limited portion of the stability map. Furthermore, stabilizing input signals exist that may be found from the linear stability map, but are not found by averaging methods. The results suggest that the linear stability map is a powerful and underutilized tool for design and analysis of open-loop oscillatory control.