Coupled shafts are considered as the main source of vibration in rotating systems. In this work, a mechanical system composed of an electric motor, an elastic coupling and one stage spur gear is investigated to analyze dynamic behavior of elastic coupling, its influence on the natural frequencies, the vibrations, and the response of the system. A lumped parameter model of the corresponding system is developed to identify the natural frequencies and vibration modes. In addition, distributions of modal kinetic and strain energies are studied.