Fuzzy Phase Trajectories in Hemispherical Resonator Gyroscopes

Abstract

The paper considers elementary fuzzy oscillator models represented by hard and fuzzy second-order differential equations with hard and fuzzy initial conditions. Linear models describe wave processes in ring resonators of hemispherical resonator gyroscopes.We show that in the case 1 (a hard model with fuzzy initial conditions), when there is no internal friction (model 1), phase trajectories appear as a fuzzy centre shaped as an elliptical ring. When internal friction is present (model 2), phase trajectories appear as a fuzzy focus shaped as a circular logarithmic spiral. In the case 2, for a fuzzy hemispherical resonator gyroscope model with hard initial conditions, when there is no internal friction (model 1), a representative point of a fuzzy phase trajectory does not stop or increase its oscillations with time, meaning that the system is asymptotically unstable, while for the model 2 the origin singularity is a fuzzy stable focus. In the case 3, for a fuzzy hemispherical resonator gyroscope model with fuzzy initial conditions, when there is no internal friction (model 1), there is a fuzzy asymptotic instability in the model 1 of a hemispherical resonator gyroscope, while in the presence of internal friction (model 2), the phase trajectory is also a function of time and controls the asymptotic stability of the fuzzy model 2 of a hemispherical resonator gyroscope. Asymptotic stability is determined for all cases and models