In the paper, a pure nonlinear and damped two-mass oscillator excited with a periodical force is considered. The oscillator is modelled with a system of two coupled second order nonlinear and non-homogenous equations. Using the model, two problems are investigated: one, identification of the excitation force for the known vibrating response of the system, and the second, determination of vibrations of the system excited with the known periodical force. Using the steady-state motion of the nonlinear oscillator, a method for identification of the excitation force is developed. For the pure nonlinear oscillator, it is obtained that the forcing function has the form of the Ateb function. However, if the excitation force is known, the procedure for computing the steady-state vibration of the system is introduced. The solution corresponds to steady-state vibrations of the free oscillator, but the amplitude and phase are assumed to be time variable. The averaged solutions are obtained for the pure nonlinear oscillator with an additional linear elastic force and for the van der Pol oscillator. Analytically obtained solutions are compared with numerical ones. They are in good agreement.