A passive stabilization of permanent rotations of the rigid body around the principal axis

In the paper a stability problem of the Lagrange's gyroscope rotation, with attached passive-type damper, around the vertical axis is investigated. The question of optimum parameters choice of stabilizing device, which guarantee the maximal damping degree of perturbed motions, is studied. The novel approach to estimate the Lyapunov's exponents for characteristic equation of six's order is suggested. Some asymtotical formulas are given.