~~~Railway rolling stock~~~
Subject of researches is determination and the justification of mathematical characteristics of motion of a railway wheel adequately reflecting the gear of its interaction with a rail. The activity purpose is adaptation of fundamental positions of the cycloidal curves reflecting motion of a circle to motion of conditional points of topological circles of section of a shroud of a railway wheel, having a difficult structure. The activity methodology is constructed on study and systematisation of fundamental investigations in the field of geometry, mathematics and mechanics of idealised motion of a circle in which basis mathematical properties of cycloidal curves are put. The model of motion of a wheel of a railway mounted axle (with immovable wheels on an axis) in which basis are put kinematic and mechanical properties golonomnykh systems is fetched. Following general golonomnykh, fundamental mathematical properties of system, analytical relations and ratio of kinematic parametres of a forward speed of railway vehicles and speed of interaction of wheels with rails which allow to supplement methodology of thrust calculations at us and in other countries are installed. By the current moment time in domestic practice use tens various formulas which actuate a component of linear speed of motion, instead of speed of interaction of wheels with rails, taking into account their physical and geometric parametres. Linear (translational) speed of motion of a wheel not absolutely adequately reflects the gear of interaction of wheels of railway vehicles with rails. The fundamental theory of cycloidal curves gives the chance is proved to normalise analytically speed motion of railway vehicles of various types, to designers and founders it a priori to form constructional and technology - economic characteristics warranting reliability, life and overall performance of the equipment dolly parts, reducing risk of a traffic security of trains.