On stability of motion of unbalanced gyroscope

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