The variational inequality principle originated from a class of elliptic obstacle problem applied to the prediction of cavitation in the fluid film bearing and squeeze film damper, and the finite element method has been used to discretize the resulting elliptic variational inequality. Based on the Taylor series, the Jacobian matrix of film force with respect to location and velocity of journal and bearing house has been constructed, and it can be obtained from resolving a set of partial differential equations. An iteration algorithm based on the complementary property of the elliptic variational inequality has been also constructed to determine the rupture boundary, namely, free boundary. On the basis of the fixed-interface eigen-mode and quasi-static modes, a method is constructed to condense the system. Furthermore, the Poincaré map is introduced to discretize the continuous flow, and the fixed point in the Poincaré section is the periodic solution and a method for calculating the Floquet multipliers is also constructed by resolving two sets of secondary ordinary differential equations in the form of matrix. The long-term dynamic behaviors and bifurcation of imbalance response of a rotor dynamic system, amounted in finite length fluid film bearings with squeeze film dampers in series, has been investigated based on the computation method mentioned above. In addition to the above, the influence of the truncation of mode series on the accuracy of solution in nonlinear dynamic system has been investigated tentatively by the comparison between mode series truncations, and the result obtained from the condensation method has been compared with the result obtained from the direct integration. All the results show that the method presented in this paper is effective in the prediction of cavitation in the fluid film and the bifurcation analysis of the system with fluid film.
H-solvability of the optimal control problem for a degenerate elliptic variational inequality
