A method of analytical study of the nonlinear transverse-angular oscillations influence of the sprung-unsprung part system on the traffic stability of a wheeled vehicle along curved road sections has been developed. A system of nonlinear differential equations describing the dynamics of the relative motion of this system has been obtained and a first approximation of its analytical solution has been constructed. The peculiarity of the oscillations of the specified parts is that their natural frequencies depend on the amplitudes. Therefore, the dynamic interaction between the sprung and unsprung parts, as well as the road surface and elastic tires are determined by the vibrations amplitude-frequency characteristics of the specified parts, which are caused by road irregularities. Using the kinetostatics equation of the unsprung-sprung part system, the forces of dynamic interaction of elastic wheels with the road surface were determined – the basic ratios for evaluating traffic stability. It has been established: the relative oscillations of the sprung and unsprung parts of a vehicle significantly reduce the critical speed values of stable motion for overturning; the critical speed value of stable motion for overturning is greater for larger static deformation values of elastic shock absorbers and tires, as well as the progressive characteristic of elastic shock absorbers. The obtained main results can serve as a basis for the creation of a software product of adaptive suspensions, and their reliability is confirmed by obtaining results in the extreme case, which apply to simpler - linear calculation models of vehicle dynamics.