Stability of the Vibrations of a Parameter-Excited System

The subject of this Brief Note is to introduce a special parameter-excited system with two degrees of freedom, which has its principal instability region at ω ≈ ω₀ instead at ω ≈ 2ω₀. It consists of a horizontal plate with a fixed pin and a slotted rigid bar on it, in which the bar can rotate and translate with respect to the plate. The stability of its vibrations is investigated for the case in which the plate is harmonicaly excited in its own plane. Starting with the Mathieu differential equation, which governs the rotational vibrations, it is possible to predict the excitation frequencies, which must be avoided, to ensure the stability.