# The Effects of Trapezoidal Groove on a Self-Acting Fluid-Lubricated Herringbone Grooves Journal Bearing

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

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## Abstract

As a self-acting fluid-lubricated herringbone grooves journal bearing, a trapezoidal cross-sectional shape of grooves is considered. Trapezoidal groove shape effects on its bearing characteristics such as variations of load capacity, attitude, and friction torque for various trapezoidal angle of groove are determined.


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## Body

## 1. Introduction

Since a herringbone grooves journal bearing has high stability at half whirl speed, it is equipped in miniature rotating machines. Performances of herringbone-grooved journal bearing are investigated by many researchers, for example, Vohr and Pan [1], Hamrock and Fleming [2], Murata et al. [3], Bonneau and Absi [4], Jang and Chang [5], as well as Liu and Yoshihiro [6].In many cases numerical methods are applied to solve a pressure distribution equation (viz., a Reynolds equation of fluid thin film), evaluating fluid-lubricated herringbone-grooved journal bearing performance for load capacity and attitude angle.In this paper, attention is focused on trapezoidal groove shape of a self-acting fluid-lubricated herringbone grooves journal-bearing to investigate influences of a variation of trapezoidal angle of groove change on characteristics.


## 2. Analytical Model

Consider a fluid-lubricated journal bearing equipped with herringbone grooves as shown in Figure1. Let bearing length be 2l and groove be symmetric with respect to its center of bearing. The shaft itself rotates around its center O* with an angular velocity ω in the counter-clockwise direction and revolves around the center O of the bearing with an angular velocity Ω in the counter-clockwise direction. The eccentricity of the shaft is given by OO*¯=e, and the outer bearing is fixed.Figure 1
Herringbone-grooved journal bearing.The inner radius of the bearing isRb, the radius of the shaft corresponding to the plane without grooves is Rs0, the bearing clearance Cr is defined as  Cr=Rb-Rs0, and the groove depth, the groove width, ridge width, and grooves angle are denoted by   δ,ag,ar, and β,  respectively. For a trapezoidal type of grooves, the trapezoidal angle ψ is defined as shown in Figure 2.Figure 2
Cross-section perpendicular to the trapezoidal surface of groove.Here, two coordinate systems are used, that is,S(r,θ, z), an inertial cylinder coordinate system, is fixed at the center O of the outer bearing, and S*(r*,θ*,z*), noninertial cylinder coordinate system is fixed at the rotation shaft Relationship between the coordinates is given by
(1)rcosθ=r*cos(θ*+ωt)+ecosϕ,rsinθ=r*sin(θ*+ωt)+esinϕ,z=z*.Hereafter, the superscript * is meant for the noninertial coordinate system. The radial componentr or r* at the surface of the shaft is denoted by Rs in S or Rs* in S*.


## 3. Local Velocity and Viscous Stresses

The equation of motion of fluid in the bearing clearance is given under a lubrication approximation by(2)∂2vr∂r2+1r∂vr∂r-vrr2=0,-1ηr∂p∂θ+∂2vθ∂r2+1r∂vθ∂r-vθr2=0,-1η∂p∂z+∂2vz∂r2+1r∂vz∂r=0.Boundary conditions for velocity are(3)vr=us,vθ=vs,vz=0atr=Rs,vr=0,vθ=0,vz=0atr=Rb.Integrating (2) under the boundary conditions (3) gives the local velocity distribution as
(4)vr=-usRsRb2-Rs2(r-Rb2r),vθ=1η∂p∂θ[12rln(rRb)-12(r-Rb2r)×Rs2(Rb2-Rs2)ln(RsRb)]vθ=-vsRs(Rb2-Rs2)(r-Rb2r),vz=[r2-Rs24-Rb2-Rs24ln(Rs/r)ln(Rs/Rb)]1η∂p∂z.The physical components of viscous stress in the lubrication fluid are given by(5)τrr≡2η∂vr∂r,(6)τθθ≡2η(1r∂vθ∂θ+vrr),(7)τrθ≡η[r∂∂r(vθr)+1r∂vr∂θ],(8)τθz≡η(1r∂vz∂θ+∂vθ∂z),(9)τrz≡η(∂vr∂z+∂vz∂r),(10)τzz≡2η∂vz∂z.Substituting (4) into (5)–(7) at r=Rs gives
(11)TR=Rs/bRR=-UsRs/b(1+Rs/b21-Rs/b2),Tr=RsΘΘ=-[12Rs/b+lnRs/bRs/b(1-Rs/b2)]2σ∂P∂Θ∂Rs/b∂Θ+UsRs/b+1Rs/b∂Vs∂Θ+VsRs/b2(1+Rs/b21-Rs/b2)∂Rs/b∂Θ,TR=RsbRΘ=(12+lnRs/b1-Rs/b2)1σ∂P∂Θ+Us2Rs/b2(1+Rs/b21-Rs/b2)∂Rs/b∂Θ+12Rs/b∂Us∂Θ-VsRs/b(11-Rs/b2),TR=Rs/bZΘ=-(12+lnRs/b1-Rs/b2)1σ∂P∂Θ∂Rs/b∂Z+12∂Vs∂Z+Vs(1+Rs/b2)2Rs/b(1-Rs/b2)∂Rs/b∂Z-12σ∂P∂Z∂Rs/b∂Θ×(1+1-Rs/b22Rs/b2lnRs/b),TR=Rs/bRZ=12∂Us∂Z+Us2Rs/b(1+Rs/b21-Rs/b2)∂Rs/b∂Z+(Rs/b+1-Rs/b22Rs/blnRs/b)12σ∂P∂Z,TR=Rs/bzz=-(Rs/b+1-Rs/b22Rs/blnRs/b)1σ∂Rs/b∂Z∂P∂Z.The dimensionless velocities,Us and Vs, at the surface of the rotating shaft, are given by
(12)Us=dEdτcos(Θ-Φ)+EdΦdτsin(Θ-Φ)+2Rs/b*sin(Θ-Θ*-2τ),Vs=-dEdτsin(Θ-Φ)+EdΦdτcos(Θ-Φ)+2Rs/b*cos(Θ-Θ*-2τ),
where
(13)R=rRb,Θ=θ,Θ*=θ*,Z=zRb,P=pPa,Rs/b=RsRb,Rs/b*=Rs*Rb,Us=2usRbω,Vs=2vsRbω,E=eRb,σ=ηωPa,Φ=ϕ,τ=ωt2,TRR=τrrωη,TΘΘ=τθθωη,TRΘ=τrθωη.The dimensionless lubricant film thickness,H, is defined as H≡1-Rs/b. Since H is sufficiently small compared with unity, (11) reduce to
(14)TR=Rs/bRR≈-UsH,Tr=RsΘΘ≈-Hσ∂P∂Θ∂H∂Θ-Vs2H∂H∂Θ+∂Vs∂Θ+Us,TR=Rs/bRΘ≈-H2σ∂P∂Θ-Vs2H+12∂Us∂Θ-Us2H∂H∂Θ,TR=Rs/bRZ≈12∂Us∂Z-H2σ∂P∂Z-Us2H∂H∂Z+H2σ∂P∂Z,TR=Rs/bZΘ≈-1σ∂P∂Θ∂H∂Z-Vs2H∂H∂Z+12∂Vs∂Z-H2σ∂H∂Θ∂P∂Z,TR=Rs/bZZ≈-Hσ∂P∂Z∂H∂Z.The dimensionless pressure,P, can be obtained through a modified Reynolds equation with curvature effect derived by Liu and Yoshihiro [6] as
(15)∂∂Θ[(1+H)H3∂P∂Θ]+∂∂Z[H3∂P∂Z]=-6σUs(1-H2)+3σVs(1+H+54H2)∂H∂Θ+3σ(H+H22)∂Vs∂Θ.AtZ=0, the groove shape is symmetric, and at Z=L, it is assumed that the fluid is open to the atmosphere, so that boundary conditions of pressure are
(16)∂P∂Z=0atZ=0,P=1.0atZ=L.In the numerical analysis, a spectral finite-difference scheme is used, and (15) is decomposed into each component of the Fourier series to the circumferential Θ-direction.


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

In a non inertial cylindrical coordinate systemS*, physical components of total stresses at R*=Rs/b* are given by
(17)TR*=Rs/b*R*R*=(-Pσ+TR=Rs/bRR)cos2α-2TR=Rs/bRΘsinαcosα+(-Pσ+TR=Rs/bΘΘ)sin2α,TR*=Rs/b*R*Θ*=(TR=Rs/bRR-TR=Rs/bΘΘ)cosαsinα+TR=Rs/bRΘcos2α,TR*=Rs/b*Θ*Θ*=(-Pσ+TR=Rs/bRR)sin2α+2TR=Rs/bRΘsinαcosα+(-Pσ+TR=Rs/bΘΘ)cos2α,TR*=Rs/b*R*Z*=TR=Rs/bRZcosα-TR=Rs/bΘZsinα,TR*=Rs/b*Z*Θ*=TR=Rs/bRZsinα+TR=Rs/bΘZcosα,TR*=Rs/b*Z*Z*=-Pσ+TR=Rs/bZZ,
where α≈Θ-Θ*-2τ, since H≪1.For a trapezoidal surface of shaft in the coordinate systemS*(r*,θ*,z*) as shown in Figure 3, a unit vector, m, perpendicular to the trapezoidal surface is obtained as
(18)m=(cosψ,sinψ,-sinψcosβ),
where the apparent trapezoidal angle of groove in R*Θ* cross-section, ψ~, is given as
(19)cosψ~=cosψsin2β+cos2ψcos2β.
Then, through some mathematic calculations, the R*- and Θ*-components of fluid force per unit area on the side of the trapezoidal surface are given
(20)fR*ds=TR*=Rs/b*R*R*cosψ+TR*=Rs/b*R*Θ*sinψsinβ-TR*=Rs/b*R*Z*sinψcosβ,fΘ*ds=TR*=Rs/b*Θ*R*cosψ+TR*=Rs/b*Θ*Θ*sinψsinβ-TR*=Rs/b*Θ*Z*sinψcosβ,
respectively, on the left surface in Figure 3. In addition, on the right surface, in (18)–(20), ψ  is replaced by -ψ. And the dimensionless horizontal component of load force, Fr, vertical component of load Ft, and friction torque of shaft, Tf, are
(21)Fr=frPaR22=2σ∫0L∫02π[fR*dscos(Θ*+Φ)cosψRs/b*dΘ*]dZ*-2σ∫0L∫02π[fΘ*dssin(Θ*+Φ)cosψRs/b*dΘ*]dZ*,Ft=ftPaR22=2σ∫0L∫02π[fR*dssin(Θ*+Φ)cosψRs/b*dΘ*]dZ*+2σ∫0L∫02π[fΘ*dscos(Θ*+Φ)cosψRs/b*dΘ*]dZ*,Tf=tfPaRb3=2σ∫0L∫02π[fΘ*dscosψRs/b*dΘ*]×Rs/b*dZ*.Figure 3
Trapezoidal surface of shaft.Throughout (21), ψ  should be replaced as 0 if the location corresponds to groove or ridge.The dimensionless load capacity,W, is given by
(22)W=Fr2+Ft2,and the attitude angle,φ, is
(23)φ=tan-1(FtFr).


## 5. Results and Discussion

Figure4 shows the relation between load capacity and eccentricity in case of  ng=20,β=21.8degrees,ar/ag=1.0,L=Rb,Ω=0,Λ≡6σ×(Rb/Cr)2=0.029. The numerical results are in good agreement with the experimental data by Hirs [7]. The trapezoidal groove shape was not in the focus of Hirs. In the trapezoidal angle is fixed at 45 degrees, because the load capacity of Hirs model changes slight with trapezoidal angle from 10 to 80 degrees as shown in Figure 5.Figure 4
Comparison of load capacity with experimental data.Figure 5
Variations of load capacity with trapezoidal angle in case ofe/Cr=0.1.The effects of trapezoidal groove are estimated as(24)εw=W-WgWg,εφ=φ-φgφg,εT=Tf-TfgTfg,
where quantities with a superscript “g” correspond to those without trapezoidal surface, only pressure being taken into account, that is,
(25)Frg=2σ∫0L∫02πPcos(Θ*-Φ)dΘ*dZ*,Ftg=2σ∫0L∫02πPsin(Θ*-Φ)dΘ*dZ*,(26)Wg=(Frg)2+(Ftg)2,(27)φg=tan-1(FtgFrg),(28)Tfg=2σ∫0L∫02πTR*=Rs/b*R*Θ*×Rs/b*2dΘ*dZ*.In case ofδ=Cr,Cr/Rb=0.005,β=30degrees,ar/ag=1.0,Λ=1.0,L=Rb,Ω=0,e/Cr=0.3, the load capacity, attitude angle, and friction torque against 2 cases of groove number are evaluated as a function of trapezoidal angles as shown in Figures 6, 7, and 8  (ng=5andng=16). The characteristics of bearing for ng=16 are more sensitive than those for ng=5.  As long as trapezoidal angle is smaller than 60 degrees, the load capacity becomes large with trapezoidal angle, but in the case that trapezoidal angle is larger than 60 degrees, the load capacity decreases with the trapezoidal angle. The attitude angle is slightly decreased with the trapezoidal angle if the trapezoidal angle is smaller than 20 degrees, and if the trapezoidal angle is larger than 20 degrees, the attitude angle increases with the trapezoidal angle. The friction torque increases with the trapezoidal angle.Figure 6
Variations of load capacity with trapezoidal angle.Figure 7
Variations of attitude angle with trapezoidal angle.Figure 8
Variations of friction torque with trapezoidal angle.Figures9, 10, and 11 show effects of the trapezoidal groove shape on the journal bearing characteristics. The effects of trapezoidal shape on the load capacity and attitude angles are vanishingly small (less than 1%), where the trapezoidal shape has large effect on the friction torque.Figure 9
Effects of trapezoidal shape on the load capacity with trapezoidal angle.Figure 10
Effects of trapezoidal shape on the attitude angle with trapezoidal angle.Figure 11
Effects of trapezoidal shape on the friction torque with trapezoidal angle.


## 6. Conclusions

(1)
The characteristics of a self-acting fluid-lubricated herringbone grooves journal bearing with trapezoidal groove are calculated.
(2)
The influences of trapezoidal groove shape on the journal-bearing characteristics such as the variation of load capacity, attitude angle, and friction torque for various grooves geometry are discussed. It shows that the effects on the load capacity and attitude angles are vanishingly small, but the effects have unignorable function on the friction torque, and the effects of trapezoidal groove become larger with trapezoidal angle increase.



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