Construction of the discrete-continuous mathematical model of a hysteresis damper impact device

This study investigates the process of interaction between the impact device tool and its body elements during an impulse response from the processing medium in the presence of a hysteretic damper of mechanical vibrations. The task addressed is to build a mathematical model with hysteresis damping of oscillations of the impact device elements. In the mathematical model, the tool is represented by a rod of variable cross-section, and the body parts of the hydraulic hammer are represented by a discrete element with a reduced mass. To damp mechanical oscillations, a rheological model of the hysteresis type is used. The impact interaction of the device elements is modeled by the presence of rigid and dissipative connections. The motion of the impact device elements is described by a system of nonlinear differential equations. The combination of discrete and continuous types of models has made it possible to solve the task of synthesizing a mathematical model. A comparison for the discrete-continuous model and the discrete model of hysteresis curves justifies their correctness. The proposed model makes it possible to estimate the energy consumption for damping and the distribution of stresses along the length of the tool. When the recoil force changes in the range of 50–500 kN for 1 ms, the energy losses were up to 500 J, and the stress in the conical part of the tool was up to 560 MPa. To solve the initial-boundary problem, a numerical method is used, which includes the finite difference method and the Euler scheme with linearization. The parameters of the numerical method were determined using a discrete two-mass model. The length step is 0.005–0.01 of the tool length, the time step is 0.001–0.05 ms. The model could be used in the design of rock development devices and impact systems to increase hydrocarbon production in the oil industry