Purpose. Finding the cause and effect relationships between the parameters of nonstationary longitudinal and torsional vibra tions of a drilling tool and developing a mathematical model to study their properties.Methodology. The solution to this problem is based on one of the basic principles of analytical mechanics -the Lagrange principle. The equation of motion of the studied rod mechanical system with four degrees of freedom is made using the Lagrange equation of the second kind, which is one of the next stages of its dynamic analysis.findings. In the process of solving this problem on the basis of industrial data, the analysis of axial load functions, angular rota tion speed and torque of the drilling tool for their nonstationarity was carried out. The frequency of change in these nonstation ary drilling mode functions is set by rapid Fourier transform. originality. For the first time, analytical dependences were obtained to determine the forces of cutting and friction on the bit from the second derivatives of the independent generalized coordinates of the mechanical system over time, as well as the relation ship between the generalized accelerations of the translational and rotational motions of the bodies of the mechanical system.Practical value. The practical value of the work is the obtained equations of motion of the system, taking into account the re ceived frequency compositions of the mode parameters of drilling until their subsequent numerical solution. The results of this task will further facilitate the choice of optimal modes of dynamic loading of the drilling tool in order to improve its energy efficiency and reliability. Analysis of the frequency dependences of the change in nonstationary functions of the drilling mode parameters will allow evaluating the level of wear of the rock destruction tool, as well as to predict the durability of the drill string elements.