Long Hydrodynamic Journal Bearings
6.5
125
PRESSURE WAVE IN A LONG JOURNAL BEARING
For a common long journal bearing with a stationary sleeve, the pressure wave is derived by a double integration of Eq. (6.12). After the first integration, the following explicit expression for the pressure gradient is obtained: dp h þ C1 ¼ 6U m h3 dx
ð6-22Þ
Here, C1 is a constant of integration. In this equation, a regular derivative replaces the partial one, because in a long bearing, the pressure is a function of one variable, x, only. The constant, C1 , can be replaced by h0 , which is the film thickness at the point of peak pressure. At the point of peak pressure, dp ¼0 dx
at
h ¼ h0
ð6-23Þ
Substituting condition (6-23), in Eq. (6-22) results in C1 ¼ h0 , and Eq. (6-22) becomes dp h h0 ¼ 6U m h3 dx
ð6-24Þ
Equation (6-24) has one unknown, h0 , which is determined later from additional information about the pressure wave. The expression for the pressure distribution (pressure wave) around a journal bearing, along the x direction, is derived by integration of Eq. (6-24), and there will be an additional unknown—the constant of integration. By using the two boundary conditions of the pressure wave, we solve the two unknowns, h0 , and the second integration constant. The pressures at the start and at the end of the pressure wave are usually used as boundary conditions. However, in certain cases the locations of the start and the end of the pressure wave are not obvious. For example, the fluid film of a practical journal bearing involves a fluid cavitation, and other boundary conditions of the pressure wave are used for solving the two unknowns. These boundary conditions are discussed in this chapter. The solution method of two unknowns for these boundary conditions is more complex and requires computer iterations. Replacing x by an angular coordinate y, we get x ¼ Ry
ð6-25Þ