Fluid-flow: Elliptical Bearing

Study below is based on [Mota, 2020], where is the complete theory used by FluidFlow:

The elliptical bearing or “lemon bearing”, as it is also known, is a variation of the cylindrical bearing with axial groove and reduced clearance in one direction.

For the inclusion of this new geometry, adaptations to the stator radius are necessary, as it will no longer be constant in \(\theta\). As seen in the figure above, the new stator is composed of the arc \(C_{1}\), with center in \(O_{1}\), joined to the arc \(C_{2}\), centered in \(O_{2}\), both with radius \(R_{o}\). In this new configuration, the centers are at a distance \(\epsilon\) from the origin, called ellipticity.

It is necessary to describe the stator from the origin. This new distance will be called \(R_{o}^{*} \) and it varies along the angular position:

\[R_o^* = \sqrt{R_o ^2 - \epsilon^2 \sin^2{\alpha}} + \epsilon \cos{\alpha}\]

where \(\alpha =\begin{cases} \pi/2 + \theta \text{,} &\mbox{if} \quad \theta \in 1^{\circ} \text{quadrant} \\ 3\pi/2 + \theta \text{,} &\mbox{if} \quad \theta \in 2^{\circ} \text{quadrant} \\ \theta - \pi/2 \text{,} &\mbox{if} \quad \theta \in 3^{\circ} \text{quadrant} \\ 5\pi/2 -\theta \text{,} &\mbox{if} \quad \theta \in 4^{\circ} \text{quadrant} \end{cases}\).

Another important parameter to be defined is the \( m \) preload which, in this text, will be established as:

\[m = \dfrac{\epsilon}{F}\]

where \(\epsilon\) is the ellipticity and \(F=R_{o}-R_{i}\) is the radial clearance.

For \(m=0\), the bearing becomes cylindrical, while for \(m \rightarrow 1\) the stator arcs tend to touch the axis.

import ross
from ross.fluid_flow.fluid_flow_graphics import (
    plot_pressure_theta,
    plot_pressure_surface,
)
from ross.fluid_flow.fluid_flow_coefficients import calculate_oil_film_force
from ross.fluid_flow.fluid_flow_coefficients import find_equilibrium_position
from ross.fluid_flow.fluid_flow_coefficients import (
    calculate_stiffness_and_damping_coefficients,
)

from ross.fluid_flow.fluid_flow import fluid_flow_example3
import plotly.graph_objects as go

# uncomment the lines below if you are having problems with plots not showing
# import plotly.io as pio
# pio.renderers.default = "notebook"

my_fluid_flow_eliptical = fluid_flow_example3()

fig1 = plot_pressure_theta(
    my_fluid_flow_eliptical, z=int(my_fluid_flow_eliptical.nz / 2)
)
fig1.show()
fig2 = plot_pressure_surface(my_fluid_flow_eliptical)
fig2.show()

radial_force, tangential_force, force_x, force_y = calculate_oil_film_force(
    my_fluid_flow_eliptical
)
print("N=", radial_force)
print("T=", tangential_force)
print("fx=", force_x)
print("fy=", force_y)

find_equilibrium_position(my_fluid_flow_eliptical)
print("(xi,yi)=", "(", my_fluid_flow_eliptical.xi, ",", my_fluid_flow_eliptical.yi, ")")
radial_force, tangential_force, force_x, force_y = calculate_oil_film_force(
    my_fluid_flow_eliptical
)
print("fx, fy=", force_x, ",", force_y)

K, C = calculate_stiffness_and_damping_coefficients(my_fluid_flow_eliptical)
kxx, kxy, kyx, kyy = K[0], K[1], K[2], K[3]
cxx, cxy, cyx, cyy = C[0], C[1], C[2], C[3]
print("Stiffness coefficients:")
print("kxx, kxy, kyx, kyy = ", kxx, kxy, kyx, kyy)
print("Damping coefficients:")
print("cxx, cxy, cyx, cyy", cxx, cxy, cyx, cyy)

N= 8.513259789914004
T= 99.63696300459412
fx= -2.1580153308775607e-07
fy= 99.99999994514712
(xi,yi)= ( 2.4436012185989533e-05 , -2.0878809269477477e-06 )
fx, fy= -0.0030146291641468537 , 99.99633144233852
Stiffness coefficients:
kxx, kxy, kyx, kyy =  1368871.3618627968 5531841.837720094 -4634423.575643063 6727562.385725975
Damping coefficients:
cxx, cxy, cyx, cyy 33119.81286832363 5289.458978360837 11573.160994142294 49189.00786101259

References

[Mot20]

Júlia de Araújo Mota. Estudo da teoria de lubrificação com parametrização diferenciada da geometria e aplicações em mancais hidrodinâmicos. Master's thesis, PPGI - Universidade Federal do Rio de Janeiro, 2020.