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In the context of the objectives of and compliance with transportation of corridors for rolling stock and improving the technology of transportation of different modes of transport considered minimization of the geometric factor in positioning navigation, peculiarities of determination of errors of coordinate of the moving objects along physically fixed trajectories on the plane, it is proved the statement about the possibility of recovery of the trajectory of the mobile point on the plane with continuous or discrete measurements of one coordinate.

The hypothesis of formation of vibration conditions of a working environment based on representation that process of formation of interaction of elements in systems with not holding connections is considered as the process consisting of several phases is offered. Those are: contact phases at which reaction of connection is positive; a phase of a transition state when reaction is equal to zero; a gap phase in which the contacting surfaces can make the independent movements. Research problems are focused on specification of theoretical ideas of dynamic interactions of a solid body with the vibrating surface on the basis of the developed mathematical models received at introduction of additional external forces and elastic connections. Dynamic interactions of elements are considered; analytical approaches in an assessment of opportunities of vibration technological processes with continuous tossing of a working environment are defined and developed. The generalized approach for problems of dynamic synthesis which basis is introduction of concept about the function of a gap allowing to prove creation of trajectories of movements of material particles according to the certain criteria reflecting properties of trajectories of the movement of material particles in interaction with the vibrating surface is offered. The method of creation of mathematical models of formation of vibration interactions taking into account the factors having technological value or criteria to which conditions of realization of the multiple modes, influence of additional forces and elastic connections belong is developed. Within the generalized approach to the solution of problems of dynamic interaction of the loose environment with the vibrating surface possibilities of use of function of a gap for an assessment and research of forms of the movement of material particles according to influence of a number of the factors reflecting parameters of a vibration field, condition of interactions with environment and features of formation of the periodic modes of tossing are considered. The theorem of conditions and opportunities of realization of the steady modes of interaction with the vibrating surface is proved. The analytical ratios defining conditions of realization of forms of processes of interaction of a loose working environment with details in technologies of vibration hardening are received. Results of approbation of recommendations about creation of technological processes in the annex to an industrial technological complex are proposed.