Gimbal Suspension Gyro: Stability, Bifurcation, and Chaos

Abstract

In this paper the solutions of three problems, devoted to the investigation of the dynamics of balanced gimbal suspension gyro (GSG) which is installed at the immovable base are presented. In the first problem the stability of stationary motion (SM) of the GSG is analyzed under which the planes of the internal and external rings of GSG are orthogonal. The moment of viscous friction and the moment which is the function of the deviation angle of the internal ring act on the external axis. The analysis of stability of SM is carried out by means of construction of Lyapunov's function with the estimation of the attraction domain. In the second problem the mechanism of the stability loss of the SM under the transfer of the bifurcational parameter through the critical value is presented. In this case the periodic motion originates (Andronov-Hopf bifurcation). The orbital stability condition of the periodic motion is found The third problem investigates the forced vibration of the GSG under the action on the internal ring of the perturbed moment which is the sum of the small moment of viscous friction and moment with small amplitude and fixed frequency. Here we consider the case where the projection of the angular moment on the axis of the external ring is equal to zero. In case of absence of the perturbations the SMs under which the external and internal rings are orthogonal or lie in the common plane are stable and unstable, respectively. For the unperturbed system the equation of the separatrix which passes through the hyperbolic points is found. For the determination of the condition of intersection of the separatrices in the perturbed system, the distance between them is calculated (Melnikov's distance). In the space of parameters the domain in which this distance can change the sign is distinguished and it is the feature of the chaotic motion arising.