The free-oscillation method with decrement as a feature for vibroacoustical diagnostics of cracks is considered. For the first time theoretical bases of decrement usage for free-oscillation diagnosis method of cracks are developed. Three new generalizations of decrement of oscillations for non-linear systems are proposed. New expression of decrement of considered non-linear system is received. The above-mentioned system describes in detail the fatigue crack. Systematised investigations of decrement dependencies from the relative crack size and model natural frequency are carried out for cases of frequency-dependent and frequency-independent external and internal friction forces. For the first time the possibility of decrement increasing and decreasing with increment of the relative crack size is shown theoretically. The method adaptation is proposed. Estimate of testing detail natural frequency is proposed to be used as a measured interfering parameter for method adaptation. Experimental results with titanium compressor blade of aircraft gas turbine engine match the theoretical prognosis. The received results are new, general and can be used for diagnostics of objects described by non-linear one-mass model.