A general fundamental mathematical form of unsteady nonlinear oil-film force acting on a journal bearing with unsteady motion is presented. It is pointed out the important role of the unsteady property of the free boundary position of the oil-film in the modeling of oil-film force. For short length bearing, it is called as "instantaneous π oil-film model". For finite width bearing, the free boundary position is determined by a variational equation. It is proved that three functions are needed to construct both the "instantaneous stiffness matrix" and "instantaneous damping matrix". The positive symmetric property of the "instantaneous damping matrix" is proved. Two types of practical oil-film journal bearing are discussed to demonstrate the theory. One is the exact solution of short width circular journal bearing, and the other is the approximate solution for finite width elliptical journal bearing.
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