Abstract The dynamics and stability of a freely spinning body carrying a viscous ring nutation damper is analyzed. A local nonlinear analysis around the equilibrium point is used here to determine the stability criterion, since a linear stability analysis fails due to the appearance of a zero eigenvalue. The center manifold theorem is employed to reduce the dimension of the system. The averaging method is used to obtain equations expanded up to third order in a small parameter. A stability criterion and an explicit equation for the decay time constant of the wobble motion of the spinning body are found. The analytical solution is verified by comparing analytical results with numerical solutions of the original nonlinear equations of motion. The stability criterion shows that the motion is stable if the spin axis is the principal axis of the maximum moment of inertia. The effect of the damper parameters on the stability of the motion is secondary.