Indefinite damping in mechanical systems and gyroscopic stabilization

This paper deals with gyroscopic stabilization of the unstable system Mẍ + Dẋ + Kx = 0, with positive definite mass and stiffness matrices M and K, respectively, and an indefinite damping matrix D. The main question is for which skew-symmetric matrices G the system Mẍ + (D + G)ẋ + Kx = 0 can become stable? After investigating special cases we find an appropriate solution of the Lyapunov matrix equation for the general case. Examples show the deviation of the stability limit found by the Lyapunov method from the exact value.