Fluid-flow: Wear Bearing

WEAR BEARINGS

Although lubrication reduces the friction between the metal surfaces of the bearing, these structures usually suffer wear after a long operating period or else due to a certain number of repetitions of the starting cycles.

The wear geometry that will be used in the FluidFlow has been adapted from the version presented by MACHADO; CAVALCA (2015) [1]. To include wear in the geometry, it is necessary to make some adaptations to the stator radius. Considering that the fault starts at the angular position \(\theta = \theta_{s}\) and ends at \(\theta = \theta_{f}\), the stator description from the origin is defined as:

\[R_o^* = R_o + d_{\theta}\]

where $\(d_{\theta} =\begin{cases} 0 \text{,} &\text{if} \quad 0 \leq \theta \leq \theta_s\text{,} \quad \theta_f \leq \theta \leq 2\pi \\ d_0 - F \left(1 + \cos{\left(\theta - \pi/2\right)} \right) \text{,} &\text{if} \quad \theta_s < \theta < \theta_f \end{cases}\)$ .

In \(\theta_{s}\) and \(\theta_{f}\), the wear depth is zero, so the location of the edges can be defined as follows:

\[\begin{split}\theta_s = \pi/2 + \cos^{-1}{\left(d_0/F -1\right)} + \gamma \nonumber\\ \theta_f = \pi/2 - \cos^{-1}{\left(d_0/F -1\right)} + \gamma \end{split}\]

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

from ross.bearings.fluid_flow import fluid_flow_example4
import plotly.graph_objects as go

# Make sure the default renderer is set to 'notebook' for inline plots in Jupyter
import plotly.io as pio

pio.renderers.default = "notebook"

my_fluid_flow_wear = fluid_flow_example4()

fig8 = plot_pressure_theta(my_fluid_flow_wear, z=int(my_fluid_flow_wear.nz / 2))
fig8.show()
fig9 = plot_pressure_surface(my_fluid_flow_wear)
fig9.show()

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

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

K, C = calculate_stiffness_and_damping_coefficients(my_fluid_flow_wear)
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= 17.525604312559775
T= 7.075534985965795
fx= -1.2555426920357604e-08
fy= 18.900000049154492
(xi,yi)= ( 1.7561016928289367e-05 , -4.349740818649866e-05 )
fx, fy= 1.1473699323971687e-07 , 18.899999954461517
Stiffness coefficients:
kxx, kxy, kyx, kyy =  1699540.7520544422 894454.7935271892 -1481116.9714871645 178901.44587230682
Damping coefficients:
cxx, cxy, cyx, cyy 26227.156604725995 -10888.76436102275 -5868.295288658701 21323.06468435377

Based on MOTA (2020), where is the complete theory used by FluidFlow:

Mota, J. A.; Estudo da teoria de lubrificação com parametrização diferenciada da geometria e aplicações em mancais hidrodinâmicos. Dissertação de Mestrado - Programa de Pós-Graduação em Informática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2020

REFERENCES CITED IN THE TEXT

[1] MACHADO, T. H.; CAVALCA, K. L. Modeling of hydrodynamic bearing wear in rotor-bearing systems. Mechanics Research Communications - Elsevier, [S.l.],v. 69, p. 15–23, 2015.