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Research Papers: Hydrodynamic Lubrication

# Modified Reynolds Equation for Aerostatic Porous Radial Bearings With Quadratic Forchheimer Pressure-Flow Assumption

[+] Author and Article Information
Rodrigo Nicoletti, Zilda C. Silveira, Benedito M. Purquerio

Department of Mechanical Engineering, Engineering School of São Carlos-EESC University of São Paulo-USP, Avenida Trabalhador São-Carlense, 400, São Carlos, SP, Brazil 13566-590

J. Tribol 130(3), 031701 (Jun 23, 2008) (12 pages) doi:10.1115/1.2919776 History: Received April 24, 2007; Revised October 30, 2007; Published June 23, 2008

## Abstract

Aerostatic porous bearings are becoming important elements in precision machines due to their inherent characteristics. The mathematical modeling of such bearings depends on the pressure-flow assumptions that are adopted for the flow in the porous medium. In this work, one proposes a nondimensional modified Reynolds equations based on the quadratic Forchheimer assumption. In this quadratic approach, the nondimensional parameter $Φ$ strongly affects the bearing load capacity, by defining the nonlinearity level of the system. For values of $Φ>10$, the results obtained with the modified Reynolds equation with quadratic Forchheimer assumption tend to those obtained with the linear Darcy model, thus showing that this is a more robust and global approach of the problem, and can be used for both pressure-flow assumptions (linear and quadratic). The threshold between linear and quadratic assumptions is numerically investigated for a bronze sintered porous bearing, and the effects of bearing geometry are discussed. Numerical results show that $Φ$ strongly affects the bearing loading capacity and stiffness coefficients.

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## Figures

Figure 1

Schematic of the porous medium with constant cross section area and thickness

Figure 2

Mass flow rate of air crossing the sintered bronze porous medium as a function of the pressure drop; (a) 52μm grain diameter; (b) 66μm grain diameter; (c) 114μm grain diameter

Figure 3

Schematic of the aerostatic porous bearing

Figure 4

Fluid flow kinematics, velocity profiles, and nonslip boundary conditions in the bearing gap

Figure 5

Schematic of a section of the porous medium (radial bearing)

Figure 6

Flow chart of the algorithm for solving the modified Reynolds equation

Figure 7

Aerostatic/aerodynamic forces acting on the shaft

Figure 8

Bearing load capacity as a function of the nondimensional parameters Γ and Φ (ε¯=0.5, L∕D=1.0); (a) surface; (b) contour map

Figure 9

Bearing load capacity as a function of the nondimensional parameter Γ and some values of Φ (ε¯=0.5, L∕D=1.0)

Figure 10

Bearing load capacity as a function of the nondimensional parameters Γ and Φ(L∕D=1.0); (a) ε¯=0.2; (b) ε¯=0.8

Figure 11

Maximum bearing load capacity as a function of the nondimensional parameters Γ, Φ, and rotor eccentricity ε¯(L∕D=1.0)

Figure 12

Bearing load capacity as a function of the nondimensional parameters Γ and Φ(ε¯=0.5); (a) L∕D=0.25; b) L∕D=0.5

Figure 13

Maximum bearing load capacity as a function of the nondimensional parameters Γ, Φ, and L∕D ratio (ε¯=0.5)

Figure 16

Bearing equivalent stiffness coefficients as a function of the nondimensional parameters Γ, Φ, and L∕D(ε¯=0.5); (a) K¯ξξ(L∕D=0.25); (b) K¯ηη(L∕D=0.25); (c) K¯ξξ(L∕D=0.5); (d) K¯ηη(L∕D=0.5)

Figure 17

Maximum bearing equivalent stiffness as a function of the nondimensional parameters Γ, Φ, and L∕D ratio (ε¯=0.5); (a) eccentricity direction; (b) normal to eccentricity direction

Figure 18

Rotor attitude angle as a function of the nondimensional parameters Γ, Φ, ε¯, and L∕D ratio; (a) variation of eccentricity; (b) variation of bearing ratio

Figure 19

Threshold between linear and nonlinear assumptions for different bearing geometries and supply pressures (Φ=10 isolines)—sintered bronze porous radial bearing; a) 52μm grain diameter; (b) 66μm grain diameter; (c) 114μm grain diameter

Figure 14

Bearing equivalent stiffness coefficients as a function of the nondimensional parameters Γ, Φ, and ε¯(L∕D=1.0); (a) K¯ξξ(ε¯=0.2); (b) K¯ηη(ε¯=0.2); (c) K¯ξξ(ε¯=0.5); (d) K¯ηη(ε¯=0.5); (e) K¯ξξ(ε¯=0.8); (f) K¯ηη(ε¯=0.8)

Figure 15

Maximum bearing equivalent stiffness as a function of the nondimensional parameters Γ, Φ, and rotor eccentricity ε¯(L∕D=1.0); (a) eccentricity direction; (b) normal to eccentricity direction

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