Mechanisms consisting of two links, a leading (engine) and a slave (executive body), which are connected by long lines of force, are considered. With the use of the new method, general equations are derived in Laplace images describing the oscillations of the velocities of motion and stresses in mechanical and hydraulic systems. Obtained transfer functions. The functional coefficients in these equations take into account the properties of the mechanisms and the distribution of the parameters of the lines of force. As a result of the transition from equations in images to equations in the originals, expressions are obtained that describe in real-time oscillations of the velocities of motion. The criterion for the stability of the work is derived on the basis of Lyapunov's first method. As an example, a volumetric hydraulic drive with long hydraulic lines is considered. With the loss of stability in this mechanism, self-oscillations appeared. Areas of stable and unstable work are defined.

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The process of motion transmission in mechanisms consisting of two links connected by straight stems is considered. The Zener phenomelogic model was used, where the phenomena of relaxation and retardation are taken into account. It is shown that the elastic properties of a substance should be considered as a complex function. A new system of differential equations is presented, which explicitly describes the oscillations of the speeds of movement and stresses in the body. It is shown that the new method allows to take into account the degree of distribution of the parameters of force lines, and for short lines it makes it possible to describe the dynamics with simple expressions.

Abstract. The article describes the mathematical model and the method and results of simulation modeling of the drive of lifting of the downhole motor from the well. The system of differential equations takes into account the elastic properties of a long mechanical line (tubing string), as well as non-linearity: friction, variable elasticity, pressure limitation. Transient and frequency characteristics are obtained.

The problems of the loss of stable operation of two-link mechanisms consisting of the driving and driven members connected by long force lines are considered. The problem was solved on the basis of a new approach to the study of the dynamics of these systems with straight rods, based on the use of the equations of momentum (for longitudinal oscillations) and angular momentum (for torsional oscillations) in differential form. Stability was estimated by the first Lyapunov method, which consists in solving the resulting differential equations. As a result of modeling the work of a volumetric hydraulic drive, modes of self-oscillations are revealed. The areas of stability and instability are determined.