Application of the Spectral Technique to the Analysis of Dynamically Loaded Journal Bearings

Abstract

Successful use of a realistic bearing model in engineering practice depends heavily on the efficiency of the tool used. The analysis of dynamically loaded bearings requires repeated solutions of the Reynolds equation at each point of the journal orbit, therefore, the method used for this problem should be very fast. A new (spectral) method is presented, which allows the Reynolds equation to be solved sufficiently quickly and accurately. This method can be defined as a modification of the Galerkin method for partial differential equations as suggested by Kantorovich. The solution of the Reynolds equation is sought as a Fourier series with respect to the circumferential coordinate. This allows the original partial differential equation to be reduced to a system of ordinary differential equations which, in turn, is further approximated with a linear algebraic system. Since the matrix of this system is banded, the system can be solved efficiently with simple recursive relations. A novel method is introduced to determine the loaded zone corresponding to the Reynolds boundary conditions. The method uses ideas of functional analysis. Consideration of the loaded zone as a function, along with a condition for the pressure to be non-negative, allows a functional equation to be written and then solved with the Euler or Runge-Kutta methods. A comparison with published FE results for a steady-state case as well as for the dynamic load is shown.