Аналітико-числове обчислення ітераційними методами областей кон-такту для задачі про зношування пружної півплощини канонічними штампами

Abstract

The contact problems of contact interaction of canons of canonical forms (cylindrical, elliptical, hyperbolic, parabolic) with an elastic half-plane are considered in the work taking into account wear of the material. On the basis of the model of abrasive wear, a singular integro-differential equation with variable integration limits is obtained to determine the contact pressure, which depends on two parameters and is variable over time. Based on the method of splitting variables, the task was solved to solve a generalized problem on eigenvalues, including finding eigenvalues and eigenvectors. Also, when calculating the eigenvectors, the form of the stamp, the solution of an infinite system of linear algebraic equations, was taken into account.Based on the eigenvalues and vectors obtained, a functional series for contact pressure was constructed. An iterative analytic-numerical algorithm for finding contact areas is proposed for the problem of half-plane wear by canonical-shaped stamps, based on the use of extrapolation polynomials. The proposed iterative method is to modify the developed method of semi-direct search for wear regions, which are variable integration boundaries for the general integro-differential equation. Software modules have been developed to implement the proposed approach and an analysis of the effectiveness of using different methods of extrapolation in different software tools.For fixed moments, initial approximations of the wear region were found at a given constant force. The optimal number of initial wear areas is numerically established to extrapolate the following values at constant time intervals, so that the error can be neglected. The wear area (contact) of the described method for other time intervals in dimensionless quantities is numerically calculated. The graphs of contact pressures were constructed and the peculiarities of their distributions for different forms of dies were analyzed.