This article deals with the model of normal oscillations of the roller moving along the surface with a constant velocity in a presence of a liquid lubrication layer. Pressure distribution along the lubrication layer is obtained as a result of integration of the Reynolds equation taking into account both tangential and normal velocities of the roller with respect to the surface. A damping coefficient is determined as that of proportionality between the normal velocity and corresponding variation of the carrying capacity. After special normalizations, the problem is reduced to the stiff ordinary differential equation with small parameter multiplied on the highest order derivative term. For this equation, analytical solution is derived by method of asymptotic expansion on a singular small parameter. This solution contains regular terms of series expansion, as well as boundary layer functions decreasing rapidly with time. Characteristic decreasing time for these functions is proportional to the small parameter. The obtained analytical solutions is applied for the problem of roller relaxation to the new equilibrium state after sharp increase of the external loading. A peculiarity of this process is a rapid increase of the pressure peak just after the loading jump, which afterwards is gradually relaxing to the new stationary value corresponding to the increase external loading.
Иванов В.А., Еркаев Н.В. Аналитическая модель колебаний ролика, движущегося вдоль твердой поверхности в режиме гидродинамической. Математическое моделирование и численные методы, 2018, № 3, с. 49–66.