"There are two equations of motion for a two-degree-of-freedom system, one for each mass (more precisely, for each degree of freedom). They are generally in the form of coupled differential equations that is, each equation involves all the coordinates. If a harmonic solution is assumed for each coordinate, the equations of motion lead to a frequency equation that gives two natural frequencies for the system. If we give suitable initial excitation, the system vibrates at one of these natural frequencies. During free vibration at one of the natural frequencies, the amplitudes of the two degrees of freedom (coordinates) are related in a specific manner and the configuration is called a normal mode, principal mode, or natural mode of vibration. Thus a two-degree-of-freedom system has two normal modes of vibration corresponding to the two natural frequencies. If we give an arbitrary initial excitation to the system, the resulting free vibration will be a superposition of the two normal modes of vibration. However, if the system vibrates under the action of an external harmonic force, the resulting forced harmonic vibration takes place at the frequency of the applied force. Under harmonic excitation, resonance occurs (i.e., the amplitudes of the two coordinates will be maximum) when the forcing frequency is equal to one of the natural frequencies of the system."
Mechanical Vibrations; Fifth Edition; pag 470, 471 , 472.
Rao, Singiresu S.