_{1}

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A spectral difference method is applied to get numerical solutions for a fluid-lubricated herringbone grooved journal bearing with trapezoidal grooves by previous work of the authors. However, an inexpedience in which Fourier series of the film profile does not converge at jump points of groove start or groove end in the case of rectangle groove was still remained. In the paper, an inexpedience of numerical analysis under a special case at rectangle groove is challenged to solve. As a result, for compensation of which Fourier series does not converge at jump points in a special case of rectangle groove, Fourier coefficient of fluid film thickness is proposed as taking the limit of which in a case trapezoidal groove at trapezoidal angle approaches 0.

Recently, herringbone grooves journal bearing is used extensively in small high-speed rotating mechanisms, due to their outstanding higher stability and lower leakage, in comparison with a plain journal bearing.

Until now, numerical studies are carried out to investigate characteristics of herringbone groove journal bearing by many researchers. Vohr and Pan [

In 1980, the film in an incompressible fluid was analyzed numerically by Murata et al. [

In 2010, the authors employed a spectral difference scheme to analyze an oil-lubricated herringbone grooved journal bearing with trapezoidal groove [

Consider a fluid-lubricated journal bearing equipped with herringbone grooves as shown in _{s} with an angular velocity ω in the counter-clockwise direction, and revolves around the center O_{b} of the fixed-bearing with an angular velocity Ω in the counter-clockwise direction, the inner radius of the bearing is R_{b}_{0}, the bearing clearance C_{r} is defined as C r = R b 0 − R s 0 , and the radius of the shaft corresponding to the plane without grooves is R_{s}_{0}, the groove depth, the groove width, ridge width, and grooves angle are denoted by C_{g}, a_{g}, a_{r}, and β respectively.

The eccentricity of the shaft is given by O b O s ¯ = e , and the fluid film thickness, h, is defined as h ≡ R b − R s .

Here two coordinate systems ( θ , z ) and ( ζ , φ ) which are fixed at the rotation shaft, are introduced as shown in

θ = ζ + φ cos ( β ) , (1a)

z = φ sin ( β ) . (1b)

In coordinate system ( ζ , φ ) , assuming a steady state, e = c o n s t and Ω = 0 , the dimensionless Reynolds equation can be written as

∇ Φ ( H 3 ∇ Φ P ) + ∇ Ψ ( H 3 ∇ Ψ P ) = − 6 σ V s ∇ Φ H − 6 σ H ∇ Φ V s , (2)

where

∇ Φ = ∂ ∂ Φ , ∇ Ψ = − 1 tan ( β ) ∂ ∂ Φ + 1 sin ( β ) ∂ ∂ ψ

Φ = ζ , Ψ = φ R b 0 , P = p P a , H = h R b 0 , σ = ω η p a ,

the dimensionless velocities, V_{s}, at the surface of the rotating shaft are given by

V s ≡ R s ω R b 0 ω = R s / b . (3)

In a spectral finite difference scheme, the Equation (2b) is decomposed into each component of the Fourier series to the circumferential Φ -direction.

H ( Ψ , Φ ) = ∑ n = 0 ∞ H c n ( Ψ ) cos ( n Φ ) + ∑ n = 1 ∞ H s n ( Ψ ) sin ( n Φ ) , (4a)

P ( Ψ , Φ ) = ∑ n = 0 ∞ P c n ( Ψ ) cos ( n Φ ) + ∑ n = 1 ∞ P s n ( Ψ ) sin ( n Φ ) , (4b)

at Ψ = 0 , the groove shape is symmetric, the boundary conditions of pressure are

∂ P c n ∂ Ψ = n P s n cos ( β ) ( n ≥ 0 ) , (5a)

∂ P s n ∂ Ψ = − n P c n cos ( β ) ( n ≥ 1 ) , (5b)

and it is assumed that the fluid is open to the atmosphere at Ψ = L / tan β , so that boundary conditions of pressure are

P c n = 1.0 ( n = 0 ) , (5c)

P c n = 0 , P s n = 0 ( n ≥ 1 ) . (5d)

The dimensionless fluid film thickness of herringbone grooves journal bearing can be rewritten as

H ( Ψ , Φ ) = H 0 ( Φ ) + E cos [ Φ + Ψ cos ( β ) ] , (6)

where H 0 , is fluid film thickness without eccentricity, E is dimensionless eccentricity as E = e / R b 0 .

1) In the Case of Trapezoidal Groove

As for a trapezoidal groove which was assumed as shown in

In case of a r / a g = 1.0 , the fluid film thickness with trapezoidal grooves, H 0 , is given by

H 0 ( Φ ) = { C r / b ridge C r / b + C g / b Δ Φ ( Φ − Φ i ) ridge → groove C r / b + C g / b − C g / b Δ Φ ( Φ − Φ i ) groove → ridge C r / b + C g / b groove , (7)

where, C r / b = C r R b 0 , C g / b = C g R b 0 .

The Fourier cosine and sine coefficient of fluid film thickness are obtained as

H 0 _ c 0 = C r / s + C g / b 2 , (8a)

H 0 _ c n = 2 π C g / b n 2 Δ Φ ∑ k = 1 2 N g [ ( − 1 ) k sin ( n k π N g − n Δ Φ 2 ) sin ( n Δ Φ 2 ) ] , (8b)

H 0 _ s n = − 2 π C g / b n 2 Δ Φ ∑ k = 1 2 N g [ ( − 1 ) k cos ( k n π N g − n Δ Φ 2 ) sin ( n Δ Φ 2 ) ] , (8c)

where N g is number of grooves.

The “error” in the partial Fourier series at n = N in case of trapezoidal grooves is given as follows,

error = ∑ n = N + 1 ∞ H c n ( Ψ ) cos ( n Φ ) + ∑ n = N + 1 ∞ H s n ( Ψ ) sin ( n Φ ) = 1 π C g / b Δ Φ ∑ k = 1 2 N g ( − 1 ) k { cos ( N k π N g ) N − cos ( N k π N g − N Δ Φ − N δ Φ ) N + Δ Φ [ S i ( N k π N g − N Δ δ Φ ) ] + ( k π N g − δ Φ ) [ S i ( N k π N g − N δ Φ ) − S i ( N k π N g − N Δ Φ − N δ Φ ) ] }

= 1 π C g / b Δ Φ ∑ k = 1 2 N g ( − 1 ) k { − 2 N sin ( N k π N g − N Δ Φ 2 ) sin N Δ Φ 2 + Δ Φ [ S i ( N k π N g ) ] + ( k π N g ) [ S i ( N k π N g ) − S i ( N k π N g − N Δ Φ ) ]

where S i ( x ) is sinc function. _{g} = 5 with the partial Fourier series term number increases, and obviously, the maximum error decays rapidly in case of trapezoidal grooves.

2) In a Special Case at Rectangle Groove

More generally, for the Fourier series expansion of fluid film thickness in case of rectangle groove, the nth partial Fourier series will overshoot this jump by

approximately at a point of groove start or groove end (Gibbs phenomenon), the “error” in the partial Fourier series will be about 8.95% and 14.11% of groove depth larger than the jump in the original fluid film thickness in two cases, in the limits of increasing many terms and of increasingly high node densities, respectively, see

How to reduce the Gibbs phenomenon, is an interesting and important topic in mathematics, several mathematician deal with this topic, and obtained some method for its field, e.g. filtering and spectral re-projection. Filtering is a classical tool for mitigating the Gibbs phenomenon in Fourier expansions, however filtering does not completely remove the Gibbs phenomenon. To completely remove the Gibbs phenomenon, one can re-expand the function in a carefully chosen different basis, it is spectral re-projection method, which is given by Gottlieb and Shu [_{N}(x) in a different basis (“Gibbs complementary”). But understanding that needs high mathematic knowledge, and it is not easy for industrial applications, an easy analysis method always requested from engineer, and so for compensation of its defect, the following method which is proposed.

Since Equation (8) are continuity and differentiable at a point Δ Φ = 0 , then Fourier coefficient of fluid film thickness in the case of rectangle groove can been replaced as taking the limit of Equation (8) at trapezoidal angle Δ Φ approaches 0, which are given as

lim Δ Φ → 0 H 0 _ c 0 = C r / b + C g / b 2 , (9a)

lim Δ Φ → 0 H 0 _ c n = C g / b π n ∑ k = 1 2 N g [ ( − 1 ) k sin ( n k π N g ) ] , (9b)

lim Δ Φ → 0 H 0 _ s n = − C g / b π n ∑ k = 1 2 N g [ ( − 1 ) k cos ( n k π N g ) ] , (9c)

and the maximum “error” at n = N of proposed method which is Equation (9) can be obtained

error = ∑ n = N + 1 ∞ H c n ( Ψ ) cos ( n Φ ) + ∑ n = N + 1 ∞ H s n ( Ψ ) sin ( n Φ ) ≈ C g / b π ∑ k = 1 2 N g ( − 1 ) k ∫ N + 1 ∞ 1 n { sin ( n k π N g − n δ Φ ) } d n = C g / b π { ∑ k = 1 2 N g ( − 1 ) k S i ( N ( k π N g − δ Φ ) ) } (10)

and since k π / N g ≫ δ Φ , thus

e r r o r = C g / b π ∑ k = 1 2 N g ( − 1 ) k S i ( N π k N g ) . (11)

_{g} = 5 with the partial Fourier series term number increases, and obviously, the maximum error decays rapidly in case of trapezoidal grooves.

Let shows an example of the film thickness at rectangle groove by Fourier’s series using the limit of trapezoidal angle Δ Φ approaches 0 in

To confirm the applicability of the above method, Hirs model [_{g} = 20, β = 21.8 deg., a_{r}/a_{g} = 1.0, L = R_{b}, Ω = 0, Λ ≡ 6 σ × ( R b / C r ) 2 = 0. 21 was picked, the relation between load capacity and eccentricity are numerical analyzed, and which compare to experiments of Hirs as shown in

As analysis of a fluid-lubricated herringbone grooved journal bearing under a spectral difference scheme, for compensation of which Fourier series does not converge at jump points in a special case of rectangle groove. Fourier coefficient of fluid film thickness is proposed as taking the limit of which in a case trapezoidal groove at trapezoidal angle approaches 0. In addition, the difference of the film thickness and the number of terms of its Fourier series are investigated with the ratio of the Fourier’s series terms number to grooves number.

A spectral difference method is applied to get numerical solutions for a fluid-lubricated herringbone grooved journal bearing with trapezoidal grooves, and then the numerical analysis scheme will be extended to be suitable for a special case of rectangle groove.

Fourier coefficients of fluid film can be replaced as taking the limit of the trapezoidal angle approaches 0.

The author gratefully acknowledges Emeritus Professor Y. Mochimaru of the Tokyo Institute of Technology for advice.

The author declares no conflicts of interest regarding the publication of this paper.

Liu, J. (2019) A Spectral Finite Difference Method for Analysis of a Fluid-Lubricated Herringbone Grooves Journal Bearing under a Special Case at Rectangle Groove. Applied Mathematics, 10, 1029-1038. https://doi.org/10.4236/am.2019.1012071

a_{g}, a_{r}_{,}= grooves width, ridge width

C_{g}, C_{g/b} = groove depth, dimensionless groove depth

C_{r} = bearing clearance

e, E = eccentricity, dimensionless eccentricity

H = dimensionless fluid film thickness

l, L = bearing length, dimensionless bearing length

N_{g} = number of grooves

p, P = pressure, dimensionless pressure

P_{a} = atmospheric pressure

r, θ, z = inertial coordinates

R_{b} = radius of bearing_{ }

R_{s} = radial component of coordinate at surface of shaft

R_{s}_{0} = radius of shaft without grooves

R_{s}_{/b} = dimensionless radial component of coordinate at surface of shaft

t, τ= time, dimensionless time

u_{s} = circumferential velocity at surface of rotating shaft

U_{s} = dimensionless circumferential velocity at surface of rotating shaft

v_{s} = radial velocity at surface of rotating shaft

V_{s} = dimensionless radial velocity at surface of rotating shaft

v r , v θ , v z = velocity components of lubricant fluid

W = dimensionless load capacity of bearing

β = groove angle

η = viscosity of fluid

Λ = bearing number

σ = dimensionless number

φ = attitude angle of shaft

Δ Φ = trapezoidal angle of groove

ϕ , Φ = angle between the fixed axis of abscissa (θ = 0) and the axis of eccentricity, dimensionless angle

ω = rotation velocity of shaft

Ω = swirl velocity of shaft

superscript*: non-inertial coordinate