On stability of motion of unbalanced gyroscope

Abstract

In this paper we consider the stability of motion of an unbalanced gyroscope with a flexible shaft in gimbal suspension.Gyroscopes are used to control the stability of boom tower cranes. First, we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems arise in transportation problems. We prove several properties of the Hamiltonian matrix. Next, we consider the normalization of Hamiltonian matrix. We solve the system of matrix equations to find the generating function of the canonical transformation. Further, we find the symmetric solution of nonlinear algebraic matrix Riccati equation in one case. We propose the analytical method for solving this equation. Further, we investigate the motion of gyroscope. We consider the system of equations of motion of the gyroscope rotor and frames in the first approximation. We consider eigenvalues of characteristic equation corresponding to the motion equations. We obtain stability criterion in several cases.