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New interaction theory of a flexible cleaning vane with the head surface of root crop in the case when the vane installed on the driving shaft makes a plane motion is proposed. Based on the obtained differential equations of the vane motion some new mathematical dependencies, describing the basic parameters of this interaction are given.

The longitudinal vibrations theory of a continuous elastic body with one fixed extremity is designed. The Ostrogradskii-Hamilton principle of a stationary operation is applied. By Ritz's method the Ritz's equation of frequencies for viewed oscillatory process is obtained. In particular analytical expressions for definition of the first and second fundamental frequencies of body oscillations and forced oscillations amplitude of its any cross-section are obtained.