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Fluid-flow: Short BearingΒΆ

In the literature on bearings, several studies use the Reynolds equation,

\[ \dfrac{\partial}{\partial{x}}\left(h^3\dfrac{\partial{p}}{\partial{x}}\right)+\dfrac{\partial}{\partial{z}}\left(h^3\dfrac{\partial{p}}{\partial{z}}\right) = 6 \mu \left\{ \left(U_o + U_1\right) \dfrac{\partial{h}}{\partial{x}} + 2 V \right\}\]

after a series of simplifications, to find the pressure behavior in bearings. However, as it is an equation that has no analytical solution, they use the artifice of approximating the equation for cases of short bearings \(\left(L/D \rightarrow 0 \right)\) and infinitely long \(\left(L/D \rightarrow \infty \right)\) (L length, D diameter). Thus, one of the parts of the equation is neglected, and it is possible to find reduced models that can be solved analytically.

Most modern bearings in high performance turbomachinery applications have a small \(L/D\) ratio, rarely exceeding the unit. The author indicates that the short model provides accurate results for cylindrical bearings with the ratio \(L/D \leq 0.5\), being widely used for quick estimates of the performance characteristics of the static and dynamic forces of the bearing.

In this context, the bearing length is considered to be very small and, according to [Ishida and Yamamoto, 2013], the pressure variation in the \(z\) direction can be considered much greater than in the \( x \) direction, that is, \(\partial p/\partial x \ll \partial p/\partial z\). Thus, the first term of the Reynolds equation is neglected. Making the appropriate adjustments to the coordinate system adopted in this work, a formula is then obtained that describes the pressure behavior in the short bearing:

\[ p_{curto} = \dfrac{-3\mu \epsilon \omega \sin{\theta}}{\left(R_\theta - R_i\right)^2\left(1 + \epsilon \cos{\theta}\right)^3}\left[\left(z-\dfrac{L}{2}\right)^2 - \dfrac{L^2}{4}\right] \]

where \(\epsilon = \dfrac{e}{R_{o} - R_{i}}\) is the reason for eccentricity.

The numerical solution presented is verified with this approximation, which is used by the Fluid-Flow code if the bearing is classified as short (\(L/D \leq 1/4\))

import ross
from ross.fluid_flow.fluid_flow_graphics import (
    plot_pressure_theta_cylindrical,
    plot_pressure_z,
    plot_pressure_theta,
    plot_pressure_surface,
)
from ross.fluid_flow.fluid_flow import fluid_flow_example

my_fluid_flow_short = fluid_flow_example()
my_fluid_flow_short.calculate_pressure_matrix_analytical()

fig1 = plot_pressure_z(my_fluid_flow_short, theta=int(my_fluid_flow_short.ntheta / 2))
fig1.show()
fig2 = plot_pressure_theta(my_fluid_flow_short, z=int(my_fluid_flow_short.nz / 2))
fig2.show()
fig3 = plot_pressure_theta_cylindrical(
    my_fluid_flow_short, z=int(my_fluid_flow_short.nz / 2)
)
fig3.show()