In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.