Analysis of oil-lubricated herringbone grooved journal bearing with trapezoidal cross-section, using a spectral finite difference method

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

397

2010, 22(5), supplement :408-412 DOI: 10.1016/S1001-6058(09)60228-6

Analysis of oil-lubricated herringbone grooved journal bearing with trapezoidal cross-section, using a spectral finite difference method Jun Liu *, Mochimaru Yoshihiro** *Tochigi R&D Center, Keihin-Corp, 2021-8, Hoshakuji Takanezawa-machi,Shioya-gun, Tochigi, Japan **Department of International Development Engineering, Graduate School of Science and Engineering, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan E-mail: [email protected] ; [email protected] ABSTRACT: A modified Reynolds equation for an oillubricated journal bearing is derived, including curvature effect under a lubrication theory, and the curvature effect on characteristics of the journal bearing is discussed. In addition, in case of a herringbone-grooved journal bearing with trapezoidal cross-sections, the modified Reynolds equation is solved, using a spectral finite difference method to get characteristics such as variation of load capacity and attitude angle for trapezoidal angles of from 1.8 through 7.2 degrees. KEY WORDS: fluid lubrication, modified Reynolds equation, curvature effect, herringbone grooves, trapezoid groove, journal bearing

1 INTRODUCTION Recently, fluid dynamic grooved bearings are used extensively in audio visual or office-automation instruments. As for a herringbone grooved gas journal bearing under a narrow groove theory, Vohr and Pan [1] got numerical solutions, and Vohr and Chow [2] treated a special case at small eccentricity. Hamrock and Fleming [3] investigated optimal parameters for self-acting herringbone journal bearings at the maximum radial load capacity. Cheng and Pan [4] gave a time dependent solution of a nonlinear Reynolds equation, with the assessment of stable operation parameters for gas lubricated bearings. The film in an incompressible fluid was analyzed numerically by Murata et al [5] based on a potential flow theory. Kawabata, Ashino, and Tachibana[6] treated a case of large eccentricity, using a narrow grooved theory. Bonneau & Absi [7] applied a finite element method (FEM) to a compressible Reynolds equation to get aerodynamic characteristics for 4 through 16 grooves with moderate eccentricity. In this paper, a herringbone grooved oil-lubricated

journal bearing with trapezoid cross-sections is analyzed to get static characteristics such as variation of its load capacity and attitude angle for various trapezoidal angles including curvature effects. 2 ANALYSIS 2.1 Modified reynolds equation including a curvature effect Consider an oil-lubricated journal bearing equipped with herringbone grooves as shown in Fig.1. Let bearing length be 2L and grooves be symmetric with respect to the center of bearing. The shaft itself rotates around the center O* with 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.

Fig.1 Herringbone-grooved journal bearing

The radius of bearing is Rb; 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, ridge width, and grooves angle are denoted by δ, ag, ar, β respectively.

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

398

Two coordinate systems, S and S*, are introduced as follows: (1) An inertial coordinate system S (r,θ,z) is fixed in the center O of the outer bearing. (2) A non-inertial coordinate system S*(r*,θ*,z*) is fixed at the center O* of the rotation shaft. Relationship between coordinates is given by r cos θ = r * cos(θ * + ωt ) + e cos φ , (1a) * * (1b) r sin θ = r sin(θ + ωt ) + e sin φ , (1c) z = z* . Hereafter the superscript * is meant for the noninertial coordinate system. The radial component of coordinate at the surface of the shaft is denoted by Rs in S or R*s in S*. In a Cartesian coordinate system S, the equation of motion of lubricant oil in the bearing clearance is given under a lubrication approximation by ∂ 2 v r 1 ∂v r v r 0= − , + (2a) ∂r 2 r ∂r r 2 2 1 ∂p ∂ vθ 1 ∂vθ vθ + 0=− + − 2 , (2b) r ∂r η r∂θ ∂r 2 r 2 1 ∂p ∂ v z 1 ∂v z (2c) + + 0=− . η ∂z ∂r 2 r ∂r The equation of continuity is 1 ∂ (rv r ) ∂vθ ∂v z + + =0 r ∂r r∂θ ∂z

.

(3)

Boundary conditions of velocity are v r = u s , vθ = v s , v z = 0 at r = R s ,

(4a) (4b)

v r = 0, vθ = 0, v z = 0 at r = Rb .

Integrating Eqs.2a-2c under a lubrication approximation gives the velocity components as u R ⎛ R2 ⎞ , v r = − 2 s s 2 ⎜⎜ r − b ⎟⎟ (5a) r ⎠ Rb − R s ⎝

vθ =

1 ∂p η ∂θ

⎡⎛ 1 ⎞ R ln Rb − R ln Rs 1R ln Rb ⎟⎟ − ⎢⎜⎜ r ln r − 2 r 2(Rb2 − R ) ⎠ ⎣⎢⎝ 2 2 b

⎛ R2 ⎞⎤ v R ⎛ R2 ⎞ × ⎜⎜ r − b ⎟⎟⎥ − 2 s s 2 ⎜⎜ r − b ⎟⎟ r ⎠⎥⎦ (Rb − Rs ) ⎝ r ⎠ ⎝

2 b

2 s 2 s

,

(5b)

⎛ r 2 − R s2 Rb2 − R s2 ln R s − ln r ⎞ 1 ∂p ⎟ . (5c) v z = ⎜⎜ + 4 4 ln Rb − ln R s ⎟⎠ η ∂z ⎝ Substituting Eqs.5a~5c into Eq.3 and integrating it over the region R s ≤ r ≤ Rb leads to Rs / b ∂ ⎛ ∂P ⎞ 1 ∂ ⎛ ∂P ⎞ ⎜ F1 ⎟ = 2σU s ⎜ F2 ⎟+ 1 + Rs / b ∂Θ ⎝ ∂Θ ⎠ 2 ∂Z ⎝ ∂Z ⎠

+ σV s

(1 + R ) 2 s/b

R s / b (1 + R s / b )

2

(1 − R s / b ) ∂V s ∂R s / b −σ (1 + R s / b ) ∂Θ ∂Θ

F2 =

−2 R s / b ln R s / b −1 + Rs / b 1 + Rs / b

,

where

Θ =θ , Z =

ωη

R p z , P= , Rs / b = s , Rb Pa Rb

2u s 2v s , Vs = . pa Rb ω Rb ω The dimensionless velocities, Us and Vs, at the surface of the rotating shaft are given by

σ=

, Us =

dE dΦ cos( Θ − Φ ) + E sin( Θ − Φ ) dτ dτ (7a) + 2 Rs*/ b sin( Θ − Θ* − 2τ ) , dE dΦ Vs = − sin( Θ − Φ ) + E cos( Θ − Φ ) dτ dτ (7b) + 2 Rs*/ b cos(*Θ − Θ* − 2τ ) , R e ωt where R s* / b = s , τ = ,E= , Φ = φ. 2 Rb Rb The dimensionless lubricant film thickness, H, is defined as H≡1-Rs/b. Since H<<1, neglecting higher order small quantities gives Rs / b 1 − Rs / b H H 2 1 H ≈ − , ≈ + , 1 + Rs / b 2 4 1 + Rs / b 2 4 Us =

1 + R s2/ b 1 H 5H 2 ≈ + + , 8 R s / b (1 + R s / b ) 2 2 2 1 1 1 F1 ≈ − H 3 , F2 ≈ − H 3 − H 4 . 3 6 6 Substituting these approximations into Eq.6 leads to a modified Reynolds equation including curvature effect ∂ ⎡ (1 + H )H 3 ∂P ⎤⎥ + ∂ ⎡⎢ H 3 ∂P ⎤⎥ = −6σU s ⎛⎜1 − H ⎞⎟ ⎢ 2⎠ ∂Θ ⎣ ∂Θ ⎦ ∂Z ⎣ ∂Z ⎦ ⎝

⎛ 5 H 2 ⎞ ∂V s ⎛ ⎞ ∂H ⎟ + 3σV s ⎜1 + H + H 2 ⎟ + 3σ ⎜⎜ H + . (8) 4 2 ⎟⎠ ∂Θ ⎝ ⎠ ∂Θ ⎝ In case of a herringbone-grooved journal bearing with trapezoidal cross-sections, 3 types of trapezoidal grooves are considered, and the trapezoidal angle α is defined through circumferential components θ* of the trapezoid in the non-inertial coordinate S* as shown in Fig.2.

, (6)

2

F1 =

2 3 2 1 − Rs / b (1 − R s / b ) Rs / b − − 3 3 ln R s / b

,

Fig.2 Cross-sectional geometry of groove

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China The trapezoidal grooves surface profile of shaft is given by ( f > c1 : ridge) Rs0 ⎧ ⎪⎪ ⎛ ⎞ f c − 2 ⎟⎟ (c1 ≥ f ≥ c 2 : trapezoid ) , (9) R s* = ⎨ R s 0 − δ ⎜⎜ c c − 2 ⎠ ⎝ 1 ⎪ ⎪⎩ ( f < c 2 : groove) Rs0 − δ where

⎡ ng f = sin ⎢ ⎢⎣ 2

* ⎞⎤ ⎛ * ⎜ Θ − tan β Z ⎟⎥ ⎜ L ⎟⎠⎥ ⎝ ⎦

Table 1.The grooves shape parameters (ng=5, β=30deg) c1=0.747 c1=0.805 c1=0.719 c2=0.695 c2=0.665 c2=0.593 ar/ag 1.0 1.0 1.0 α 1.8 deg 3.6 deg 7.2 deg

2.2 Load capacity and attitude angle of journal bearing

In case grooved shaft rotates, the following two cases are possible: (1) If the load to the shaft is given, E and Φ will be evaluated at each position of shaft. (2) If E and Φ is provided, the load components are calculated at each position of shaft. Here, only the latter case is considered. The dimensionless radial component of load capacity, Fr, is L 2π (10a)

Fr = 2 ∫

0

∫

0

P cos( Θ − Φ )dΘdZ .

The dimensionless tangential component of load capacity, Ft , is L 2π (10b)

Ft = 2 ∫

0

∫

0

P sin(Θ − Φ )dΘdZ

.

The dimensionless bearing load capacity, F, is given by F = Fr2 + Ft 2 .

The attitude angle, ψ, is ⎛F ⎞ ψ = tan −1 ⎜⎜ t ⎟⎟ . ⎝ Fr ⎠

(10c)

(11)

2.3 A spectral finite difference scheme

In the analysis the variables in Eq.8 are expressed in terms of a Fourier series in the circumferential direction, and the modified Reynolds equation is decomposed into each component of Fourier series, so that Eq.8 can be integrated with respect to time independently of each component to get a steady-state solution.

∞

∞

n =0

n =1

H ( Z , Θ,τ ) = ∑ H cn (Z ,τ ) cos nΘ + ∑ H sn (Z ,τ )sin nΘ , (12a) ∞

∞

n =0

n =1

U s ( Z , Θ,τ ) = ∑ U cn (Z ,τ ) cos nΘ + ∑ U sn (Z ,τ )sin nΘ ,

(12b) ∞

∞

n =0

n =1

∞

∞

n =0

n =1

Vs ( Z , Θ,τ ) = ∑ Vcn (Z ,τ ) cos nΘ + ∑ Vsn (Z ,τ )sin nΘ ,

, c1 and c2 (0
<1.0) are parameters corresponding to grooves width ag, ridge width ar, and trapezoidal angle α.

399

(12c) P ( Z , Θ,τ ) = ∑ Pcn (Z ,τ ) cos nΘ + ∑ Psn (Z ,τ )sin nΘ ,

(12d) At Z=0, the groove shape is symmetric, and at Z=L, it estimated the z-direction velocity vz<< vΘ, so that boundary conditions of pressure are ∂Pcn ∂Psn = 0 (n ≥ 0) , = 0 (n ≥ 1) at Z = 0 , (13a) ∂Z ∂Z Pcn = 1.0 (n = 0) at Z = L ,

Pcn = 0 , Psn = 0 (n ≥ 1) at Z = L .

(13b)

3 RESULTS

Figure 3 and Fig.4 present curvature effects in the steady-state velocity components of lubricant oil in the clearance of the journal bearing. The curvature effects are defined as follows v − v0 v − v0 v − v0 ε r = r 0 r , εθ = θ 0 θ , ε z = z 0 z , vr vθ vz where the velocity components with superscript “0” are those of the velocity without curvature effect , ⎛ r − Rs ⎞ ⎟ , v r0 = u s ⎜⎜1 − (14a) Rb − R s ⎟⎠ ⎝ vθ0 =

⎛ r − Rs 1 ∂p (r − Rs )(r − Rb ) + v s ⎜⎜1 − 2ηRb ∂θ Rb − R s ⎝

v z0 =

1 ∂p (r − Rs )(r − Rb ) 2η ∂z

.

⎞ ⎟ ,(14b) ⎟ ⎠

(14c)

Equations.14a-14c can be derived from Eq.2 under neglecting the first derivative terms with respect to r. These figures show that the curvature effects in velocity components become large as curvature factor χ (i.e. inverse of Rb) increases. In particular, in case of thick lubricant oil films, increment of curvature effects increase rapidly with film thickness increases. So it can be predicted that the curvature effect in grooved journal bearing can not be ignored in cases of large eccentricity and large groove depth. In case of α=1.8 deg., ng=5, β=30 deg., ar/ag=1.0, L=Rb, Ω=0, two types of grooves depth (δ＝0.1Cr and δ＝Cr) are computed, and the radial load component, Fr, with including curvature and that without considering curvature is shown in Fig.5.

400

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

Fig.3 Curvature effect in radial and circumferential component of velocity of lubricant oil

Fig.6 Load capacity with respect to eccentricities

In case of ng=5, ar/ag=1.0, δ=0.5Cr, E=0.1Cr, L=Rb, Ω=0, the radial load capacity, Fr, and attitude angle, ψ, of groove angle of from 10 to 80 degrees are evaluated as a function of β in Fig.7 and Fig.8. They show that when grooves angle β is approximately 32.8 degree, the radial load component is largest. It is in good agreement with the most suitable groove angle which is obtained from narrow groove theory by Vohr and Chow (1965). Moreover, as for three types of trapezoidal angle of grooves, it shows the radial component of load capacity decreases with increasing trapezoidal angle, and the attitude angle tends to increase as trapezoidal angle increases. Fig.4 Curvature effect in z- direction component of velocity of lubricant oil

Fig.7 Radial load capacity versus grooves angle β Fig.5 Curvature effect in journal bearing

In case of α=1.8 deg., ng=5, β=30 deg., ar/ag=1.0, L=Rb, Ω=0, the load capacity of groove depth δ of from 0.1 to 1.0 times the bearing clearance is computed. Figure 6 shows the load capacity with respect to the eccentricities. The results are in good agreement with those by Murata (1980) and Kawabata (1983) in case of small eccentricities and small groove depth.

Fig.8 Attitude angle versus grooves angle β

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China 4 CONCLUSIONS

(1) A modified Reynolds equation is derived, including curvature effect under a lubrication approximation. (2) The modified Reynolds equation is solved to investigate characteristics of a herringbone grooved journal bearing with trapezoid crosssections. NOMENCALATURE

ag , ar, : grooves width, ridge width Cr : bearing clearance e , E : eccentricity, dimensionless eccentricity F, Fr, Ft : dimensionless load capacity, radial component, tangential component h , H : oil film thickness, dimensionless oil film thickness L : bearing length ng : number of grooves 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 us : circumferential velocity at surface of rotating shaft U : dimensionless circumferential velocity at surface of rotating shaft vs : radial velocity at surface of rotating shaft V : dimensionless radial velocity at surface of rotating shaft v r , vθ , v z : velocity components of lubricant oil with curvature v r0 , vθ0 , v z0 : velocity components of lubricant oil

401

without curvature α : angle of trapezoidal groove shape β : groove angle δ : groove depth η : viscosity of fluid σ : dimensionless number φ , Φ : angle between the fixed axis of abscissa (θ= 0) and the axis of eccentricity, dimensionless angle ψ : attitude angle between eccentricity direction and load capacity direction ω : rotation velocity of shaft Ω : swirl velocity of shaft superscript * : non-inertial coordinate subscript cn and sn : cosine and sine coefficients of Fourier series REFERENCES [1] Vohr J H, Pan C H T. On the spiral-grooved, self-acting gas bearings. MTI technical report, MTI63TR52, 1963. [2] Vohr J H, Chow C Y. Characteristics of herringbonegrooved, gas-lubricated journal bearings. ASME J. Basic Eng., 1965, 87: 568-578. [3] Hamrock B J, Fleming D P. Optimization of self-acting herringbone grooved journal bearings for maximum radial load capacity. 5th Gas Bearing. Symp.. 1971,1(13). [4] Cheng H S, Pan C H T. Stability analysis of gas-lubricated, self-acting, plain, cylindrical, journal bearings of finite length, using Galerkin’s method. J. Basic Engineering., 1965, 87(1): 185-192. [5] Murata S, Miyake Y, Kawabata N. Two-dimensional analysis of herringbone groove journal bearing. Bull. JSME, 1980, 23(181): 1980–1987 [6] Kawabata N, Ashino I, Tachibana M, et al. Analysis of the load carrying capacity of herringbone-grooved journal bearings. Trans. JSME. Ser.C. (in Japanese), 1983, 49(440): 648-656. [7] Bonneau D, Absi J. Analysis of aerodynamic journal bearing with small number of herringbone grooves by finite element method. ASME J. Tribol., 1994, 116: 698-704.