We construct a mathematical model for studying the elastic deformations in a thermoelastic inhomogeneous solid of revolution applicable to sliding bearings. The method of numerical solution is based on the grid method and the relazation method.The development of efficient mathematical methods of studying the elastic deformations in thermoelastic inhomogeneous solids of revolution has great value for the reliable operation of rotating machine parts that function subject to temperature fields and force loads.Ya. S. Podstrigach et al. [5] gave the fundamental equations of thermoelasticity for solids of revolution. An efficient numerical method of solving the heat equation taking account of the dependence of the coefficient of thermal conductiv.ity on temperature was described in [1]. A method of numerical solution of the stationary equations of thermoelasticity for homogeneous solids was given in [3,4].In the present article, on the basis of [1,[3][4], we propose a method of solving a system of differential equations of nonstationary thermoelasticity for an inhomogeneous solid of revolution. We exhibit a connection with the study of deformations and thermal stresses in sliding bearings [2] in operation.A mathematical model of the thermoelastic problem for an inhomogeneous solid of revolution includes: the heat equation, three equations in the displacement, coupling conditions, boundary and initial conditions. The key (unknown) functions are taken to be the temperature and the components of the displacements.In a cylindrical coordinate system (r, qa, z) the equation for the temperature t(r, qo, z) for an inhomogeneous solid of revolution has the form [5] where At = At(r, qo, z) is the coefficient of thermal conductivity, co = co(r, ~, z) is the three-dimensional thermal capacity, and ~-is the time parameter.Suppose a boundary condition for the temperature is prescribed on the surface of the solid: t = e(r, ~, z).The initial condition when T = 0 has the form tl~.=o = to (r, ~, z).We write the equations in the displacements u, v, w in a solid of revolution in which the physicomechanical characteristics are functions of the cylindrical coordinates in the form