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

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.

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 Figure 1. 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 is 𝑅_𝑏, the radius of the shaft corresponding to the plane without grooves is 𝑅_𝑠0, the bearing clearance 𝐶_𝑟 is defined as  𝐶_𝑟=𝑅_𝑏−𝑅_𝑠0, and the groove depth, the groove width, ridge width, and grooves angle are denoted by   𝛿,𝑎_𝑔,𝑎_𝑟, 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 𝑟cos𝜃=𝑟^∗cos𝜃^∗+𝜔𝑡+𝑒cos𝜙,𝑟sin𝜃=𝑟^∗sin𝜃^∗+𝜔𝑡+𝑒sin𝜙,𝑧=𝑧^∗. (1)

Hereafter, the superscript * is meant for the noninertial coordinate system. The radial component 𝑟 or 𝑟^∗ at the surface of the shaft is denoted by 𝑅_𝑠 in 𝑆 or 𝑅^∗_𝑠 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 𝜕²𝑣_𝑟𝜕𝑟²+1𝑟𝜕𝑣_𝑟−𝑣𝜕𝑟𝑟^𝑟2−1=0,𝜂𝑟𝜕𝑝+𝜕𝜕𝜃²𝑣_𝜃𝜕𝑟²+1𝑟𝜕𝑣_𝜃−𝑣𝜕𝑟𝑟^𝜃2−1=0,𝜂𝜕𝑝+𝜕𝜕𝑧²𝑣_𝑧𝜕𝑟²+1𝑟𝜕𝑣_𝑧𝜕𝑟=0. (2)

Boundary conditions for velocity are 𝑣_𝑟=𝑢_𝑠,𝑣_𝜃=𝑣_𝑠,𝑣_𝑧=0at𝑟=𝑅_𝑠,𝑣_𝑟=0,𝑣_𝜃=0,𝑣_𝑧=0at𝑟=𝑅_𝑏. (3)

Integrating (2) under the boundary conditions (3) gives the local velocity distribution as 𝑣_𝑟𝑢=−^𝑠𝑅_𝑠𝑅²_𝑏−𝑅²_𝑠𝑅𝑟−^2𝑏𝑟,𝑣_𝜃=1𝜂𝜕𝑝1𝜕𝜃2𝑟𝑟ln𝑅_𝑏−12𝑅𝑟−^2𝑏𝑟×𝑅²_𝑠𝑅²_𝑏−𝑅²_𝑠𝑅ln^𝑠𝑅_𝑏−𝑣_𝑠𝑅_𝑠𝑅²_𝑏−𝑅²_𝑠𝑅𝑟−^2𝑏𝑟,𝑣_𝑧=𝑟²−𝑅²_𝑠4−𝑅²_𝑏−𝑅²_𝑠4ln𝑅_𝑠/𝑟ln𝑅_𝑠/𝑅_𝑏1𝜂𝜕𝑝.𝜕𝑧 (4)

The physical components of viscous stress in the lubrication fluid are given by 𝜏^𝑟𝑟≡2𝜂𝜕𝑣_𝑟,𝜏𝜕𝑟_𝜃𝜃1≡2𝜂𝑟𝜕𝑣_𝜃+𝑣𝜕𝜃𝑟^𝑟,𝜏^𝑟𝜃𝑟𝜕≡𝜂𝑣𝜕𝑟𝑟^𝜃+1𝑟𝜕𝑣_𝑟,𝜏𝜕𝜃_𝜃𝑧1≡𝜂𝑟𝜕𝑣_𝑧+𝜕𝜃𝜕𝑣_𝜃,𝜏𝜕𝑧_𝑟𝑧≡𝜂𝜕𝑣_𝑟+𝜕𝑧𝜕𝑣_𝑧,𝜏𝜕𝑟_zz≡2𝜂𝜕𝑣_𝑧.𝜕𝑧 (5)(6)(7)(8)(9)(10)

Substituting (4) into (5)–(7) at 𝑟=𝑅_𝑠 gives 𝑇^𝑅𝑅_{𝑅=𝑅𝑠/𝑏}𝑈=−^𝑠𝑅_𝑠/𝑏1+𝑅²_𝑠/𝑏1−𝑅²_𝑠/𝑏,𝑇^ΘΘ_{𝑟=𝑅𝑠}1=−2𝑅_𝑠/𝑏+ln𝑅_𝑠/𝑏𝑅_𝑠/𝑏1−𝑅²_𝑠/𝑏2𝜎𝜕𝑃𝜕Θ𝜕𝑅_𝑠/𝑏+𝑈𝜕Θ𝑅^𝑠_𝑠/𝑏+1𝑅_𝑠/𝑏𝜕𝑉_𝑠+𝑉𝜕Θ𝑅^𝑠2_𝑠/𝑏1+𝑅²_𝑠/𝑏1−𝑅²_𝑠/𝑏𝜕𝑅_𝑠/𝑏,𝑇𝜕Θ_{𝑅Θ𝑅=𝑅}𝑠𝑏=12+ln𝑅_𝑠/𝑏1−𝑅²_𝑠/𝑏1𝜎𝜕𝑃+𝑈𝜕Θ2𝑅^𝑠2_𝑠/𝑏1+𝑅²_𝑠/𝑏1−𝑅²_𝑠/𝑏𝜕𝑅_𝑠/𝑏+1𝜕Θ2𝑅_𝑠/𝑏𝜕𝑈_𝑠−𝑉𝜕Θ𝑅^𝑠_𝑠/𝑏11−𝑅²_𝑠/𝑏,𝑇^𝑍Θ_{𝑅=𝑅𝑠/𝑏}1=−2+ln𝑅_𝑠/𝑏1−𝑅²_𝑠/𝑏1𝜎𝜕𝑃𝜕Θ𝜕𝑅_𝑠/𝑏+1𝜕𝑍2𝜕𝑉_𝑠+𝑉𝜕𝑍_𝑠1+𝑅²_𝑠/𝑏2𝑅_𝑠/𝑏1−𝑅²_𝑠/𝑏𝜕𝑅_𝑠/𝑏−1𝜕𝑍2𝜎𝜕𝑃𝜕𝑍𝜕𝑅_𝑠/𝑏×𝜕Θ1+1−𝑅²_𝑠/𝑏2𝑅²_𝑠/𝑏ln𝑅_𝑠/𝑏,𝑇^𝑅𝑍_{𝑅=𝑅𝑠/𝑏}=12𝜕𝑈_𝑠+𝑈𝜕𝑍2𝑅^𝑠_𝑠/𝑏1+𝑅²_𝑠/𝑏1−𝑅²_𝑠/𝑏𝜕𝑅_𝑠/𝑏+𝑅𝜕𝑍_𝑠/𝑏+1−𝑅²_𝑠/𝑏2𝑅_𝑠/𝑏ln𝑅_𝑠/𝑏12𝜎𝜕𝑃,𝑇𝜕𝑍_{𝑧𝑧𝑅=𝑅𝑠/𝑏}=−𝑅_𝑠/𝑏+1−𝑅²_𝑠/𝑏2𝑅_𝑠/𝑏ln𝑅_𝑠/𝑏1𝜎𝜕𝑅_𝑠/𝑏𝜕𝑍𝜕𝑃.𝜕𝑍 (11)

The dimensionless velocities, 𝑈_𝑠 and 𝑉_𝑠, at the surface of the rotating shaft, are given by 𝑈_𝑠=𝑑𝐸()𝑑𝜏cosΘ−Φ+𝐸𝑑Φ()𝑑𝜏sinΘ−Φ+2𝑅^∗_𝑠/𝑏sinΘ−Θ^∗,𝑉−2𝜏_𝑠=−𝑑𝐸()𝑑𝜏sinΘ−Φ+𝐸𝑑Φ()𝑑𝜏cosΘ−Φ+2𝑅^∗_𝑠/𝑏cosΘ−Θ^∗−2𝜏, (12) where 𝑟𝑅=𝑅_𝑏,Θ=𝜃,Θ^∗=𝜃^∗𝑧,𝑍=𝑅_𝑏𝑝,𝑃=𝑃_𝑎,𝑅_𝑠/𝑏=𝑅_𝑠𝑅_𝑏,𝑅^∗_𝑠/𝑏=𝑅^∗_𝑠𝑅_𝑏,𝑈_𝑠=2𝑢_𝑠𝑅_𝑏𝜔,𝑉_𝑠=2𝑣_𝑠𝑅_𝑏𝜔,𝑒𝐸=𝑅_𝑏,𝜎=𝜂𝜔𝑃_𝑎,Φ=𝜙,𝜏=𝜔𝑡2,𝑇^𝑅𝑅=𝜏^𝑟𝑟,𝑇𝜔𝜂_ΘΘ=𝜏^𝜃𝜃𝜔𝜂,𝑇^𝑅Θ=𝜏^𝑟𝜃.𝜔𝜂 (13)

The dimensionless lubricant film thickness, 𝐻, is defined as 𝐻≡1−𝑅_𝑠/𝑏. Since 𝐻 is sufficiently small compared with unity, (11) reduce to 𝑇^𝑅𝑅_{𝑅=𝑅𝑠/𝑏}𝑈≈−^𝑠𝐻,𝑇^ΘΘ_{𝑟=𝑅𝑠}𝐻≈−𝜎𝜕𝑃𝜕Θ𝜕𝐻−𝑉𝜕Θ2𝐻^𝑠𝜕𝐻+𝜕Θ𝜕𝑉_𝑠𝜕Θ+𝑈_𝑠,𝑇^𝑅Θ_{𝑅=𝑅𝑠/𝑏}𝐻≈−2𝜎𝜕𝑃−𝑉𝜕Θ^𝑠+12𝐻2𝜕𝑈_𝑠−𝑈𝜕Θ2𝐻^𝑠𝜕𝐻,𝑇𝜕Θ_{𝑅𝑍𝑅=𝑅𝑠/𝑏}≈12𝜕𝑈_𝑠−𝐻𝜕𝑍2𝜎𝜕𝑃−𝑈𝜕𝑍2𝐻^𝑠𝜕𝐻+𝐻𝜕𝑍2𝜎𝜕𝑃,𝑇𝜕𝑍_{𝑍Θ𝑅=𝑅𝑠/𝑏}1≈−𝜎𝜕𝑃𝜕Θ𝜕𝐻−𝑉𝜕𝑍2𝐻^𝑠𝜕𝐻+1𝜕𝑍2𝜕𝑉_𝑠−𝐻𝜕𝑍2𝜎𝜕𝐻𝜕Θ𝜕𝑃,𝑇𝜕𝑍_{𝑍𝑍𝑅=𝑅𝑠/𝑏}𝐻≈−𝜎𝜕𝑃𝜕𝑍𝜕𝐻.𝜕𝑍 (14)

The dimensionless pressure, 𝑃, can be obtained through a modified Reynolds equation with curvature effect derived by Liu and Yoshihiro [6] as 𝜕)𝐻𝜕Θ(1+𝐻³𝜕𝑃+𝜕𝜕Θ𝐻𝜕𝑍³𝜕𝑃𝜕𝑍=−6𝜎𝑈_𝑠𝐻1−2+3𝜎𝑉_𝑠51+𝐻+4𝐻²𝜕𝐻𝐻𝜕Θ+3𝜎𝐻+²2𝜕𝑉_𝑠.𝜕Θ (15)

At 𝑍=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 𝜕𝑃𝜕𝑍=0at𝑍=0,𝑃=1.0at𝑍=𝐿. (16)

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 system 𝑆^∗, physical components of total stresses at 𝑅^∗=𝑅^∗_𝑠/𝑏 are given by 𝑇^{𝑅∗𝑅∗}_𝑅^∗_=𝑅^∗_𝑠/𝑏=−𝑃𝜎+𝑇^𝑅𝑅_{𝑅=𝑅𝑠/𝑏}cos²𝛼−2𝑇^𝑅Θ_{𝑅=𝑅𝑠/𝑏}+−𝑃sin𝛼cos𝛼𝜎+𝑇^ΘΘ_{𝑅=𝑅𝑠/𝑏}sin²𝑇𝛼,_{𝑅∗Θ∗𝑅∗=𝑅∗𝑠/𝑏}=𝑇^𝑅𝑅_{𝑅=𝑅𝑠/𝑏}−𝑇^ΘΘ_{𝑅=𝑅𝑠/𝑏}cos𝛼sin𝛼+𝑇^𝑅Θ_{𝑅=𝑅𝑠/𝑏}𝑇cos2𝛼,_{Θ∗Θ∗𝑅∗=𝑅∗𝑠/𝑏}=−𝑃𝜎+𝑇^𝑅𝑅_{𝑅=𝑅𝑠/𝑏}sin²𝛼+2𝑇^𝑅Θ_{𝑅=𝑅𝑠/𝑏}+−𝑃sin𝛼cos𝛼𝜎+𝑇^ΘΘ_{𝑅=𝑅𝑠/𝑏}cos²𝑇𝛼,_{𝑅∗𝑍∗𝑅∗=𝑅∗𝑠/𝑏}=𝑇^𝑅𝑍_{𝑅=𝑅𝑠/𝑏}cos𝛼−𝑇^Θ𝑍_{𝑅=𝑅𝑠/𝑏}𝑇sin𝛼,_{𝑍∗Θ∗𝑅∗=𝑅∗𝑠/𝑏}=𝑇^𝑅𝑍_{𝑅=𝑅𝑠/𝑏}sin𝛼+𝑇^Θ𝑍_{𝑅=𝑅𝑠/𝑏}𝑇cos𝛼,_{𝑍∗𝑍∗𝑅∗=𝑅∗𝑠/𝑏}=−𝑃𝜎+𝑇^𝑍𝑍_{𝑅=𝑅𝑠/𝑏}, (17) where 𝛼≈Θ−Θ^∗−2𝜏, since 𝐻≪1.

For a trapezoidal surface of shaft in the coordinate system 𝑆^∗(𝑟^∗,𝜃^∗,𝑧^∗) as shown in Figure 3, a unit vector, m, perpendicular to the trapezoidal surface is obtained as 𝐦=cos𝜓,sin𝜓,−sin𝜓cos𝛽, (18) where the apparent trapezoidal angle of groove in 𝑅^∗Θ^∗ cross-section, 𝜓, is given as cos𝜓=cos𝜓sin²𝛽+cos²𝜓cos²𝛽. (19) 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 𝑓_𝑅^𝑑𝑠_∗=𝑇^{𝑅∗𝑅∗}_𝑅^∗_=𝑅^∗_𝑠/𝑏cos𝜓+𝑇^𝑅∗Θ∗_𝑅^∗_=𝑅^∗_𝑠/𝑏sin𝜓sin𝛽−𝑇^{𝑅∗𝑍∗}_𝑅^∗_=𝑅^∗_𝑠/𝑏𝑓sin𝜓cos𝛽,_{Θ𝑑𝑠∗}=𝑇^Θ∗𝑅∗_𝑅^∗_=𝑅^∗_𝑠/𝑏cos𝜓+𝑇^Θ∗Θ∗_𝑅^∗_=𝑅^∗_𝑠/𝑏sin𝜓sin𝛽−𝑇^Θ∗𝑍∗_𝑅^∗_=𝑅^∗_𝑠/𝑏sin𝜓cos𝛽, (20) 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, 𝐹_𝑟, vertical component of load 𝐹_𝑡, and friction torque of shaft, 𝑇_𝑓, are 𝐹_𝑟=𝑓_𝑟𝑃_𝑎𝑅²₂=2𝜎^𝐿₀₀^2𝜋𝑓_𝑅^𝑑𝑠_∗cosΘ^∗+Φ𝑅cos𝜓^∗𝑠/𝑏𝑑Θ^∗𝑑𝑍^∗−2𝜎^𝐿₀₀^2𝜋𝑓_Θ^𝑑𝑠_∗sinΘ^∗+Φ𝑅cos𝜓^∗𝑠/𝑏𝑑Θ^∗𝑑𝑍^∗,𝐹_𝑡=𝑓_𝑡𝑃_𝑎𝑅²₂=2𝜎^𝐿₀₀^2𝜋𝑓_𝑅^𝑑𝑠_∗sinΘ^∗+Φ𝑅cos𝜓^∗𝑠/𝑏𝑑Θ^∗𝑑𝑍^∗+2𝜎^𝐿₀₀^2𝜋𝑓_Θ^𝑑𝑠_∗cosΘ^∗+Φ𝑅cos𝜓^∗𝑠/𝑏𝑑Θ^∗𝑑𝑍^∗,𝑇_𝑓=𝑡_𝑓𝑃_𝑎𝑅³_𝑏=2𝜎^𝐿₀₀^2𝜋𝑓_Θ^𝑑𝑠_∗𝑅cos𝜓^∗𝑠/𝑏𝑑Θ^∗×𝑅^∗_𝑠/𝑏𝑑𝑍^∗. (21)

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 𝑊=𝐹²_𝑟+𝐹²_𝑡, (22)

and the attitude angle,  𝜑, is 𝜑=tan⁻¹𝐹_𝑡𝐹_𝑟. (23)

5. Results and Discussion

Figure 4 shows the relation between load capacity and eccentricity in case of  𝑛_𝑔=20,𝛽=21.8degrees,𝑎_𝑟/𝑎_𝑔=1.0,𝐿=𝑅_𝑏,Ω=0,Λ≡6𝜎×(𝑅_𝑏/𝐶_𝑟)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 of 𝑒/𝐶𝑟=0.1.

The effects of trapezoidal groove are estimated as 𝜀_𝑤=𝑊−𝑊^𝑔𝑊^𝑔,𝜀_𝜑=𝜑−𝜑^𝑔𝜑^𝑔,𝜀_𝑇=𝑇_𝑓−𝑇^𝑔_𝑓𝑇^𝑔_𝑓, (24) where quantities with a superscript “𝑔” correspond to those without trapezoidal surface, only pressure being taken into account, that is, 𝐹^𝑔_𝑟=2𝜎^𝐿₀₀^2𝜋𝑃cosΘ^∗−Φ𝑑Θ^∗𝑑𝑍^∗,𝐹^𝑔_𝑡=2𝜎^𝐿₀₀^2𝜋𝑃sinΘ^∗−Φ𝑑Θ^∗𝑑𝑍^∗,𝑊^𝑔=𝐹^𝑔_𝑟²+𝐹^𝑔_𝑡²,𝜑^𝑔=tan⁻¹𝐹^𝑔_𝑡𝐹^𝑔_𝑟,𝑇^𝑔_𝑓=2𝜎^𝐿₀₀^2𝜋𝑇^𝑅∗Θ∗_𝑅^∗_=𝑅^∗_𝑠/𝑏×𝑅^∗2_𝑠/𝑏𝑑Θ^∗𝑑𝑍^∗. (25)(26)(27)(28)

In case of   𝛿=𝐶_𝑟,𝐶_𝑟/𝑅_𝑏=0.005,𝛽=30degrees,𝑎_𝑟/𝑎_𝑔=1.0,Λ=1.0,𝐿=𝑅_𝑏,Ω=0,𝑒/𝐶_𝑟=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  (𝑛_𝑔=5and𝑛_𝑔=16). The characteristics of bearing for 𝑛_𝑔=16 are more sensitive than those for 𝑛_𝑔=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.

Figures 9, 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.

Nomenclature

𝑎𝑔,𝑎𝑟:	Grooves width, ridge width
𝐶𝑟:	Bearing clearance
𝑒,𝐸:	Eccentricity, dimensionless eccentricity
𝑓𝑅𝑑𝑠∗,𝑓Θ𝑑𝑠∗:	Dimensionless the 𝑅∗-, and Θ∗-components of fluid force per unit area which acts on surface of shaft
𝐹𝑟,𝐹𝑡:	Dimensionless horizontal component, vertical component of load
𝐻:	Dimensionless fluid film thickness
𝑙,𝐿∶	Bearing length, dimensionless bearing length
𝑛𝑔:	Number of grooves
𝐦:	A unit vector which is perpendicular to the trapezoidal surface
𝑝,𝑃:	Pressure, dimensionless pressure
𝑃𝑎:	Atmospheric pressure
𝑟,𝜃,𝑧:	Inertial coordinates
𝑅𝑏∶	Radius of bearing
𝑅𝑠0∶	Radius of shaft without grooves
𝑅𝑠:	Radial component of coordinate at surface of shaft
𝑅𝑠/𝑏:	Dimensionless radial component of coordinate at surface of shaft
𝑡,𝜏:	Time, dimensionless time
𝑇𝑅𝑅𝑅=𝑅𝑠/𝑏:	Dimensionless 𝑅𝑅-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇𝑅Θ𝑅=𝑅𝑠/𝑏:	Dimensionless 𝑅Θ-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇ΘΘ𝑅=𝑅𝑠/𝑏:	Dimensionless ΘΘ-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇𝑍Θ𝑅=𝑅𝑠/𝑏:	Dimensionless Θ𝑅-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇𝑅𝑍𝑅=𝑅𝑠/𝑏:	Dimensionless 𝑅𝑍-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇𝑍𝑍𝑅=𝑅𝑠/𝑏:	Dimensionless 𝑍𝑍-component of viscous stress on the surface of shaft (at𝑅=𝑅𝑠/𝑏)
𝑇𝑓:	Dimensionless friction torque of rotating shaft
𝑢𝑠:	Circumferential velocity at surface of rotating shaft
𝑈𝑠:	Dimensionless circumferential velocity at surface of rotating shaft
𝑣𝑠∶	Radial velocity at surface of rotating shaft
𝑉𝑠∶	Dimensionless radial velocity at surface of rotating shaft
𝑣𝑟,𝑣𝜃,𝑣𝑧:	Velocity components of lubricant fluid
𝑊:	Dimensionless load capacity of bearing
𝛽:	Groove angle
𝛿,Δ:	Groove depth, dimensionless groove depth
𝜂:	Viscosity of fluid
Λ:	Bearing number
𝜎:	Dimensionless number
𝜏𝑖𝑗, 𝑇𝑖𝑗:	Viscous stress, dimensionless viscous stress
𝜑:	Attitude angle of shaft
𝜓:	Trapezoidal angle of groove
𝜓:	Apparent trapezoidal angle of groove in 𝑅∗Θ∗  cross-section
𝜙,Φ:	Angle between the fixed axis of abscissa (𝜃=0) and the axis of eccentricity, dimensionless angle
𝜔:	Rotation velocity of shaft
Ω:	Rwirl velocity of shaft
superscript∗:	Noninertial coordinate.