Perturbation Equations
Liquid Lubricated Bearing
The perturbation pressure pu and pv can be obtained by differentiating the governing equation for the pressure with respect to the journal displacement and velocity. For liquid lubricated bearings, Reynolds equation (see Equation 7-3) constitutes the governing equation. Differentiate the Reynolds equation with respect to the journal displacement:
This can be rewritten as follows
(7-11)
where Au and bu are defined as
(7-12)
and
with a being the mass flow field a = ρ h vav.
Similarly, the governing equations for the velocity perturbation pressure pv can be obtained by differentiating the Reynolds equation with respect to the journal velocity:
which can be rewritten as
(7-13)
with
(7-14)
and
Equation 7-11 and Equation 7-13 constitute the governing equations for the displacement and velocity perturbation pressure, respectively.
Gas Lubricated Bearing
In gas lubricated bearings, a modified Reynolds equation applies (see Equation 7-10). Differentiating the modified Reynolds equation yields a set of equations identical to Equation 7-11 and Equation 7-13 with a = pA h vav replacing the mass flow field in Equation 7-12 and Equation 7-14, and bu and bv given by
and