Method of Stabilization of Resonant Parametric Vibrations

Abstract

From a practical point of view, the most important feature of parametric vibration systems is the phenomenon of so-called parametric resonance. Parametric resonance occurs when a point specified in a parameter space of a differential equation system describing parametric vibrations is inside one of the instability areas of that equation. There are vibrations with exponentially increasing amplitude. According to linear vibration theory, this increase is unlimited, and therefore the only possible variant of eliminating this, generally negative, phenomenon is to change the parameters of the system to be outside the area of instability. According to Floquet's theory, the stability or instability of the parametric system is determined by the absolute values of the system multipliers, which are complex eigenvalues of monodromy matrix (Floquet transient matrix). If the module of the largest multiplier is less than unity – the system is stable, if it is greater than unity – the system is unstable. The work presented a method of automatic stabilization of resonant parametric vibrations. A procedure based on the concept of directional derivative was used. The idea is that, of all possible options for changing the value of the design parameters of the system, the system should be automatically selected as it will ensure that the modular value of the largest multiplier decreases as soon as possible. The method also uses algorithms of first and second order sensitivity analysis of continuous and discontinuous parametric periodic systems presented in the previous two years at the RSP Seminar in two works by Wójcicki [1, 2].KeywordsStabilizationParametricResonanceVibrationsMultiplier