Учет трения в математических моделях диссипативных систем

Abstract

Obtaining models of mechanical processes with dissipation based on the Euler-Lagrange theory has undoubted advantages over Newton's theory due to the smaller size of the considered vector of variables included in the equations. However, the Euler-Lagrange variation theory is not applicable to the description of the motion of systems with dissipation. The aim of the work is to demonstrate the possibility of using the Euler-Lagrange theory in relation to dissipative systems with different types of friction. Mathematical models of systems with dissipation are based on the superposition of mechanical and thermodynamic Lagrangians. To obtain a mathematical description of dissipative systems it is proposed to use the field theory as applied to the thermodynamics of dissipative processes within the framework of the Lagrange formalism. The Euler-Lagrange equations are obtained for the Stokes and Coulomb friction models. As it was referred to the research results obtained there is possibility of accounting the energy dissipation in the Lagrange formalism. The mathematical models proposed describe dynamic processes in heterogeneous structures with friction based on the Euler-Lagrange theory. There are presented mathematical transformations that allow transition from models based on the Lagrange formalism to models based on Newtonian mechanics