The concept of vibrational mechanics was pioneered in the works by Professor I.I. Blekhman and developed by his numerous disciples and coleagues. It is a powerful tool for the study of such systems with fast excitations, in which slow motion is of primary interest. One important application of this approach is the stochastic resonance, the phenomenon of resonance-like response of slow variables to intensity of stochastic excitation. This phenomenon is considered within the framework of vibrational mechanics as forced lowfrequency oscillations near the natural frequency, which evolves under the influence of changing high-frequency stochastic excitation. We propose a generalization of this approach to the case when the evolution of low-frequency properties of the system leads not to the equality of the natural frequency and the frequency of the external slow force, but to the loss of stability in a certain interval of the stochastic excitation intensity. Since in this case, as for stochastic resonance, the external manifestation of the process is the resonance-like response of the system, the considered effect can be called stochastic quasi-resonance, As an example, we consider a rotor with anisotropy of bending stiffness under the action of stochastic angular velocity oscillations.