Purpose: Increasing the durability of the cutting tool when reprofiling the tread of the railway wheels by controlling the vibrations of the “machine-tool-workpiece” system. Methods: Vibrations arising in the process of turning the tread of car wheels under various cutting modes have been investigated. To do this, a matrix of experiment planning has been built using the Box-Wilson method, according to which all factors change in turn (cutting depth, cutting speed, feed). Based on the results of each series of the experiment, a mathematical model is selected and the numerical values of the coefficients of this equation are estimated. In accordance with the power dependences accepted in the theory of cutting, a mathematical model in the form of a logarithmic polynomial of a series of the first degree has been postulated for the experiments carried out. Mathematical processing of measurement results has been carried out using fractal dimension. The fractal dimension has been determined by the standardized range method. Results: A correspondence has been established between the durability of the cutting tool and the intensity of its vibrations when turning railway wheels. The mathematical dependence of the fractal dimension on the cutting modes is obtained, which allows to adjust the cutting speed or feed during the reprofiling the tread of the wheel when there are defects. Optimal cutting modes, which allow to achieve a given quality of the rolling surface of the wheel in the shortest time, give the highest value of the fractal dimension. The presence of such defects of the wheel tread, such as slid flat, buildups, shellings, leads to a decrease in the fractal dimension. Practical significance: The determination of the fractal dimension in the process of turning wheel pairs makes it possible to automatically adjust the cutting modes to ensure a given quality of the wheel tread and the required durability of the cutting tool.