Wear cumulating process for friction pair implemented by first order recurrences

Wear determination engages a major part of our efforts in considered numerous tribology problems including animal teeth and joints, human joints, cams, piston, rings, tires, brakes, dirt seals, belts, shoes, fabrics, electrical contacts, tape and CD reader heads and others. At times it seems after numerous observation, that the rate of progress in the knowledge of wear is very slow. Modern design activities are mostly concern in the wear cumulative values determination rather than the wear values determination in successive time units of friction nods during the operation time. Therefore, presented paper determines in analytical way the cumulative wear values of friction nods after specified finite and infinite units of operation time. Therefore are considered the convergences and divergences of wear value cumulative processes for various friction nods. The kind of friction nods depends on type of wear process propagation. Such propagation is described in this paper by the recurrence first order equation. Mentioned equation is formulated with constant or variable parameters. This parameters are determined by virtue of experimental measurements and depend on: bearing materials properties, environmental conditions, roughness of cooperating surfaces, vibration effects, electrical conductivity of materials, hardness of materials, deformations mode of cooperating surfaces, temperature, air humidity, kind of the friction, sliding velocity, loading of the friction nod and others. The range of wearing influence components and devices is endless. Therefore, this paper presents the methods of solutions of numerous specific class of ordinary non-homogeneous recurrence equations of first order with variable coefficients occurring in hydrodynamic theory of bearing especially micro-bearing wear problems. The obtained solutions are examined by means of the UOS operator. The examples presented in this paper determine the abrasive wear cumulative values of micro-bearing system after the indicated time units of operating time. Moreover, after infinite time units of micro-bearing operation are considered the convergences and divergences of the obtained sums of cumulative wear process