A numerical scheme for solving a periodically forced Reynolds equation

A modified Reynolds equation is used to model the air‐film in a high‐speed squeeze‐film bearing. The axial position of the bearing stator is prescribed as a finite amplitude periodic oscillation. A numerical approach is considered for solving the uncoupled and coupled periodic problems associated with this model. The uncoupled problem requires the computation of the squeeze‐film dynamics when the rotor is held at a fixed axial position and the coupled problem incorporates the additional air–rotor interaction since the rotor position is unknown and modelled as a spring‐mass‐damper system. The details of a Fourier spectral collocation scheme are provided for the reduction of the modified Reynolds equation to a system of non‐linear, first‐order ordinary differential equations in space. Using the Matlab boundary value problem solver bvp4c this system of equations is solved to give the periodic pressure distributions and rotor heights. The high degree of accuracy in the spectral collocation scheme is demonstrated through comparison with an appropriate analytical solution. Further analysis indicates that the direct periodic solver is at least 10 times faster than the equivalent Crank–Nicholson finite‐difference scheme. For changing values of a selected physical parameter the method of arc‐length continuation is employed to track branches of solutions computed using the spectral collocation scheme. A selection of results is presented to demonstrate the range of accessible solutions and the robust nature of the numerical scheme. Copyright © 2010 John Wiley & Sons, Ltd.