Mathematical model of piston–tool interaction in rock fracture by impact

The authors justify a computation model of a machine percussion system simulated by elastic cylindrical rods subjected to maximal axial load at minimal strain. Rock mass is assumed as a perfectly solid block. The piston and tool are described by the values of length, cross-section area, density and Young’s modulus. The model determines the force applied by the tool on the rock as function of time. It is assumed that transverse displacements and velocities of the rods are negligeable as compared with the axial displacements and velocities, while the rods are free from the action of the external forces different from the restraining forces. The variational equation expresses the principle of possible displacements. The variations are independent of time. The initial and boundary conditions are considered. The variational equation is solved using the method of straight lines, with replacement of a time differentiation operator by the finite difference operator. The problem reduces to the successive solving of boundary value problems with variable right-hand sides. The finite difference scheme is the approved implicit scheme of Crank–Nicolson. The boundary value problems are solved using the finite element method at each step of integrating. As a result, the variational equation transforms into a system of linear algebraic equations, and the reduced solution of this system yields the wanted force. The calculations are illustrated by the tool press force–time curve plotted with a step of 0.1 µs for hydropercussion machine G100 by Rammer, Finland. The relative calculation error of the impact duration and maximal force (in absolute magnitude) is not higher than 0.1%.

Keywords: rock, piston, percussive tool, mathematical model, force, velocity, displacements, initial and boundary conditions, variational equation, operator, matrix, time.
For citation:

Zhabin A. B., Lavit I. M., Polyakov A. V., Kerimov Z. E. Mathematical model of piston–tool interaction in rock fracture by impact. MIAB. Mining Inf. Anal. Bull. 2020;(7):94103. [In Russ]. DOI: 10.25018/0236-1493-2020-7-0-94-103.


The studies were supported by the Tula State University, R&D Project No. 2019-18.

Issue number: 7
Year: 2020
Page number: 94-103
ISBN: 0236-1493
UDK: 622.23.05:622.235
DOI: 10.25018/0236-1493-2020-7-0-94-103
Article receipt date: 26.03.2020
Date of review receipt: 21.04.2020
Date of the editorial board′s decision on the article′s publishing: 20.06.2020
About authors:

A.B. Zhabin1, Dr. Sci. (Eng.), Professor, e-mail:,
I.M. Lavit 1, Dr. Sci. (Phys. Mathem.), Professor, e-mail:,
A.V. Polyakov1, Dr. Sci. (Eng.), Professor, e-mail:,
Z.E. Kerimov1, Graduate Student, e-mail:,
1 Tula State University, 300012, Tula, Russia.


For contacts:

A.B. Zhabin, e-mail:


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