The results of investigation of dynamic behaviour of mechanical systems which oscillations are described by a non-linear Duffing-type equation are presented in the paper. The outer periodic biharmonic excitation is applied to the mechanical system. The system state at each moment of time is defined by values of three phase co-ordinates. The sequence of points conforming to different system states forms phase trajectories in the extended phase space ?displacement - velocity ? acceleration?. Fundamentals of qualitative methods of dynamic systems investigation were developed by Poincar?. These methods have shown their efficiency in studies of autonomous oscillations of systems having one degree of freedom. The authors of this paper suggested application of projections of phase trajectories on a plane ?displacement ? acceleration? and ?velocity ? acceleration?. The application of these projections of phase trajectories allows to expand essentially the possibilities of qualitative methods of dynamic systems investigation. In particular, they can be used in procedures of structural identification.