# Pressure Distributions Generated along a Self-Acting

Authors: Jun Liu; Yoshihiro Mochimaru
Journal: ISRN Tribology
 (2013)

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## Abstract

Numerical studies are carried out to investigate pressure distributions of a fluid-lubricated herringbone-grooved journal bearing with trapezoidal grooves of various angles. Additionally, the optimal trapezoidal groove geometry is discussed in terms of the radial load capacity and friction torque.


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## Body

## 1. Introduction

Recently, herringbone-grooved journal bearings have a lot of applications on small rotating mechanisms such as hard disk and turbo machinery. As for a herringbone-grooved gas journal bearing under a narrow groove theory, Vohr and Pan [1] obtained numerical solutions for a special case of small eccentricity. Cheng and Pan [2] gave time-dependent solution of a nonlinear Reynolds equation under stable operation parameters for gas-lubricated bearings. Hamrock and Fleming [3] investigated optimal conditions of radial load capacity for self-acting herringbone-grooved journal bearings. The film in an incompressible fluid was analyzed numerically by Murata et al. [4] based on the potential flow theory. Bonneau and Absi [5] applied a finite element method (FEM) to a compressible Reynolds equation to get aerodynamic characteristics at 4 through 16 grooves with moderate eccentricity. Rondonuwu and Winoto [6] measured pressure distributions along hydrodynamic herringbone-grooved journal bearing for several groove patterns.A fluid-lubricated herringbone-grooved journal bearing with trapezoidal grooves was introduced by Liu and Mochimaru [7], and the influence of viscous trapezoidal grooves on the bearing was evaluated later for various trapezoidal angles [8].In this paper, the pressure distributions generated along a self-acting fluid-lubricated herringbone-grooved journal bearing with trapezoidal grooves were numerically investigated, using a spectral finite difference scheme. In addition, the optimal trapezoidal groove geometry is discussed in terms of the radial load capacity and friction torque.


## 2. Analytical Model

Consider a fluid-lubricated journal bearing equipped with herringbone groove as shown in Figure1. The bearing length is 2l and the groove is symmetric with respect to its center of bearing. The shaft itself rotates around its center O* with an angular velocity ω in the counterclockwise direction and revolves around the center O of the bearing with an angular velocity Ω in the counterclockwise direction. The eccentricity of the shaft is given by OO*¯=e, and the outer bearing is fixed.Figure 1
Herringbone-grooved journal bearing.The radius of bearing isRb, the radius of shaft ignoring grooves is Rs0, the bearing clearance Cr is defined as Cr=Rb-Rs0, and the groove depth, the groove width, and ridge width, and the groove angle are denoted by δ, ag, ar, and β, respectively. For a trapezoidal groove, the groove geometry is characterized by the angles ψl and ψr as shown in Figure 2, where ψ~l and ψ~r are the apparent trapezoidal angles of groove in rθ cross-section.Figure 2
The trapezoidal groove.Two coordinate systems,S and S*, are introduced as follows.(1)
An inertial coordinate systemS(r,θ,z) is fixed in the center O of the outer bearing.(2)
A noninertial coordinate systemS*(r*,θ*,z*) is fixed at the center O* of the rotation shaft.Hereafter the superscript* is meant for the non-inertial coordinate system.


## 3. Pressure Distribution Equation and Spectral Finite Difference Method

The pressure distribution in fluid film,P, can be obtained through a modified Reynolds equation with curvature effect derived by Liu and Mochimaru [7] as
(1)∂∂Θ[(1+H)H3∂P∂Θ]+∂∂Z[H3∂P∂Z]=-6σUs(1-H2)+3σVs(1+H+54H2)∂H∂Θ+3σ(H+H22)∂Vs∂Θ,
where the dimensionless lubricant film thickness, H, is defined as H≡1.0-Rs/b. The dimensionless velocities, Us and Vs, at the surface of the rotating shaft are given by(2a)Us=dEdτcos(Θ-Φ)+EdΦdτsin(Θ-Φ)+2Rs/b*sin(Θ-Θ*-2τ),(2b)Vs=-dEdτsin(Θ-Φ)+EdΦdτcos(Θ-Φ)+2Rs/b*cos(Θ-Θ*-2τ),where(3)Θ=θ,Z=zRb,L=lRb,P=pPa,Rs/b=RsRb,σ=ωηpa,Us=2usRbω,Vs=2vsRbω,Rs/b*=Rs*Rb,τ=ωt2,E=eRb,Φ=ϕ.In the study, a spectral finite difference method is applied for solving the pressure distribution equation, the lubricated film thickness,H, is expressed in a Fourier series to the circumferential Θ-direction, and (1) is decomposed into each component of the Fourier series as shown in (4a)–(4d), so that (1) can be integrated with respect to time independently of each component to get a steady-state solution. This process results in high resolution in space and high speed in computation as follows:(4a)H(Z,Θ,τ)=∑n=0∞Hcn(Z,τ)cosnΘ+∑n=1∞Hsn(Z,τ)sinnΘ,(4b)Us(Z,Θ,τ)=∑n=0∞Ucn(Z,τ)cosnΘ+∑n=1∞Usn(Z,τ)sinnΘ,(4c)Vs(Z,Θ,τ)=∑n=0∞Vcn(Z,τ)cosnΘ+∑n=1∞Vsn(Z,τ)sinnΘ,(4d)P(Z,Θ,τ)=∑n=0∞Pcn(Z,τ)cosnΘ+∑n=1∞Psn(Z,τ)sinnΘ.AtZ=0, the groove shape is symmetric, the boundary conditions of pressure are(5a)∂Pcn∂Z=0(n≥0),(5b)∂Psn∂Z=0(n≥1),
and it is assumed that the fluid is open to the atmosphere at Z=L, so that boundary conditions of pressure are
(5c)Pcn=1.0(n=0),(5d)Pcn=0,Psn=0(n≥1).And in the numerical procedure, in order to accurately represent the geometry of lubricant film in the Fourier series, according to the sampling theorem [8], the Fourier series terms should take a half of the total number of geometry data in the circumferential direction as shown in Figure 3.Figure 3
Lubricant film geometry in the Fourier series.


## 4. Load Capacity, Attitude Angle, and Friction Torque

The dimensionless radial component of load force,Fr, tangential component of load Ft, and friction torque of shaft, Tf, are(6a)Fr=2∫0L∫02π[fR*dscos(Θ*+Φ)cosψRs/b*]dΘ*dZ*-2∫0L∫02π[fΘ*dssin(Θ*+Φ)cosψRs/b*]dΘ*dZ*,(6b)Ft=2∫0L∫02π[fR*dssin(Θ*+Φ)cosψRs/b*]dΘ*dZ*+2∫0L∫02π[fΘ*dscos(Θ*+Φ)cosψRs/b*]dΘ*dZ*,(7)Tf=-2∫0L∫02π[fΘ*dscosψRs/b*2]dΘ*dZ*,
where fR*ds and fΘ*ds are the R*- and Θ*-components of fluid force per unit area on the surface of shaft, which can be obtained as referenced in Liu and Mochimaru [9].Then the dimensionless load capacity,W, is given by
(8)W=Fr2+Ft2,
and the attitude angle, φ, is
(9)φ=tan-1(FtFr).


## 5. Results and Discussion

In case ofng=5, β=30 deg, ar/ag=1.0, δ=Cr, Cr/Rb=0.005, Φ=0, L=1.0, Ω=0, and Λ≡σ×(Rs/Cr)2=1.0, numerical results are shown in Figures 4–8.Figure 4
Isobar diagram with eccentricity.Figure 5
Isobar diagram with trapezoidal angle.Figure 6
Asymmetric trapezoidal groove.(a) Isobar diagram of pressure distribution. (b) Circumferential pressures distribution atZ*=0.
(a)
(b)(a) Load capacities with trapezoidal angle. (b) Attitude angles with trapezoidal angle. (c) Friction torque with trapezoidal angle.
(a)
(b)
(c)Figure4 displays an isobar diagram in the Θ*Z*-plane with eccentricity changes, it shows that the pressure gradient in Θ*- and Z*-direction is not zero even in the case of eccentricity equal to zero, which confirm the fact that the herringbone grooves have the pumping effect. And Figure 5 gives the pressure distribution with trapezoidal angle increases, as long as the trapezoidal angle is smaller than 30 deg, the high pressure zone is slightly increased with trapezoidal angle increase, but in cases that the groove angle is larger than 30 deg, the high pressure zone decreases and the lower pressure zone increases with an increase of trapezoidal angle.There are three types of asymmetric trapezoidal grooves as shown in Figure6. Figure 7(a) presents an isobar diagram in the Θ*Z*-plane, and the circumferential pressure distribution at the center of the shaft (at Z*=0) is shown in Figure 7(b), where the dashed line is for the pressure distribution of the plane journal bearing case. In addition, the load capacity, attitude angle, and friction torque with trapezoidal angle changes are evaluated as shown in Figure 8, it shows that the lower pressure zone is the least area in the case of type 3, which is a favorable status to prevent cavitation, but in the case of type 1, the radial load capacity is maximum and the friction torque is minimum; it is useful for application on the miniaturization electronic device.


## 6. The Optimal Parameters of Trapezoidal Groove Geometry

Comparing with plane journal bearing, groove journal bearing has some distinguished advantages such as lower noise, higher stability but also has disadvantages such as small radial load capacity and larger friction torque. The optimal trapezoidal groove geometry can be estimated by the value,Sr, which is given by the ratio of radial load capacity to friction torque as
(10)Sr≡Fr/ΛTf×(Cr/Rb).The common parameters of groove geometry are given asng=5, ar/ag=1.0, e=0.1Cr, Cr/Rb=0.005, Φ=0, L=1.0, Ω=0, and Λ=1.0. Figure 9 presents Sr with respect to trapezoidal angle; in case of β=30 deg, δ=Cr, and ψl=ψr, it shows the optimal trapezoidal angle is 20 deg ~50 deg.Figure 9
EstimationSr with trapezoidal angle.In case ofδ=Cr, and ψl=ψr=30 deg, Sr with the groove angle of values from 10 to 80 degrees are evaluated as a function of β in Figure 10, which shows that when grooves angle β is approximately 30 degree; Sr has a maximum value.Figure 10
EstimationSr with groove angle.Moreover, in case ofβ=30 deg, ψl=ψr=30 deg, the most suitable groove depth, δ, is 1.4 times of the bearing clearance Cr for the maximum value Sr as shown in Figure 11.Figure 11
EstimationSr with groove depth.


## 7. Conclusions

The pressure distributions generated along a fluid-lubricated herringbone-grooved journal bearing with trapezoidal groove are investigated at several cases of trapezoidal angles of groove. RegardingSr, which is given by the ratio of radial load capacity to friction torque, in the case of ng=5, ar/ag=1.0, e=0.1Cr, Cr/Rb=0.005, Φ=0, L=1.0, Ω=0, and Λ=1.0, the optimal parameters of trapezoidal grooves geometry are found as follows.(1)
The trapezoidal angle is 20 deg~50 deg.(2)
The groove depth is 1.4 times of the bearing clearance.(3)
The grooves angle is approximately 30 degree.



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*Source: 560917-2012-12-11.xml*