Purpose. The aim of the paper is to study dynamic processes in nonlinear oscillatory systems with quasi-zero stiffness and with one or many degrees of freedom, which are widely used in industry for cargo and personnel vibration isolation during transportation. The previous studies of such systems were based only on the numerical approaches. In this paper, we propose to investigate thoroughly the dynamics of the above mentioned systems and the conditions for the occurrence of resonance phenomena in them using the asymptotic methods of nonlinear mechanics and applying the apparatus of special periodic functions. Methodology. The methods of studying resonance oscillations of vibration isolation equipment are based on the asymptotic methods of nonlinear mechanics, wave theory of motion and the use of special Ateb-functions. Findings. In this work, for the nonlinear quasi-zero stiffness vibration isolation systems with one and two degrees of freedom, we analytically obtained conditions of resonance oscillations, threshold values of resonance amplitudes depending on the system parameters. Originality. Scientific novelty lies in the fact that the dynamic processes in systems with concentrated masses and quasi-zero stiffness are analyzed on the basis of analytical approaches for the first time. In contrast to numerical approaches, analytical approaches allow investigating the features of the dynamics of such systems more precisely. Practical value. The proposed method can not only solve the problems of analysis, but also the equally important problem of oscillatory systems synthesis at the design stage, to choose such elastic properties of dynamical systems that prevent resonance phenomena. These modes of equipment operation will allow efficient and safe transportation.

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