Initially, both structure and bearing are concentric to each other with the common center located at Xc. During operation they move relative to each other. Let the centers of the structure and bearing now be located at xcs and xcb, respectively. The film thickness is obtained by calculating the height of the bearing surfaces from the structure surface. To evaluate the film thickness, consider a radial line from the current center of the structure xcs at an angle Θ from the local y-axis. This line intersects the bearing and the structural surfaces at points B and S, respectively. The radial line from the new center of the bearing xcb to B makes an angle Θb from the local y-axis. Let the normalized radial vectors from structure and bearing surfaces be er(Θ) and er(Θb), respectively. The position vector of the point B is given by

Here xb and xs are the position vectors of the points B and S, respectively. Rb(Θb) and Rs(Θ) are the radial positions of the bearing and structure surfaces from their respective centers. Since the points B and S are located along the radial vector er(Θ):

Here, e1 is the bearing axis direction, and e2 and e3 are the two transverse directions.

Substituting the expressions for xb, xs, er(Θ), er(Θb), and eθ(Θ) in Equation 8-1 and simplifying gives

Since the centers of both the bearing and the structural surfaces were initially at the reference surface center, we have xcb = Xc + ucb and xcs = Xc + ucs. Therefore,

This requires an evaluation of the radial position of the bearing with angular coordinate Θb. If the difference between Θb and Θ is assumed to be small, Equation 8-2 gives:

This can be generalized to include the change in Rb in the axial direction too in the following way:

If the surface of the structure is flat or cylindrical, the tangential gradient of the radial position on the structure surface will be zero. Using C = Rb − Rs the film thickness is