Journal Bearing and Wear: Method

Journal Bearing and Wear

What the tool does

A plain journal bearing carries a radial load on a thin film of oil that the rotating journal drags into the converging gap between the two eccentric cylinders. The tool solves the Reynolds equation for that film numerically on the unwrapped bearing surface, for the entered load (the eccentricity is found) or for an entered eccentricity ratio (the load is found), and reports the position of the journal, the minimum film thickness, the pressure field and its peak, the friction force, torque and power, the oil flow in and out of the film, and the temperature rise of the oil. Beside the numerical result it prints the classical closed forms an analyst would otherwise use: Petroff's no-load friction, the short-bearing approximation of NACA Report 1157 (DuBois and Ocvirk 1953) and the infinitely long bearing of Sommerfeld (1904) in the half-Sommerfeld form, all at the same Sommerfeld number. The design charts at the end are the numerical solution over the whole eccentricity range for the entered length-diameter ratio, which is what the Raimondi–Boyd charts (1958) tabulate for a few ratios; the tool computes them instead of reading them. A separate section estimates sliding wear with Archard's law and the pressure-velocity product for the starts, stops and boundary-lubricated running that the full film does not cover.

Geometry and conventions

The journal of radius r turns at N′ revolutions per second inside a bearing of radius r + c_r and length l; c_r is the radial clearance and c_d = 2c_r the diametral clearance. Under load the journal centre moves off the bearing centre by the eccentricity e, and the film thickness around the circumference is h = c_r(1 + ε cos θ), with the eccentricity ratio ε = e/c_r and θ counted from the position of maximum film thickness in the direction of rotation (Report 1157, Figure 7). The film is thinnest at θ = 180°, where h₀ = c_r(1 − ε). The applied load does not act along the line of centres: the film resultant is skewed, and the line of centres lies at the attitude angle φ from the load line, the minimum film thickness being downstream of the load in the sense of rotation. The dimensionless group that fixes the whole solution for a given length-diameter ratio is the Sommerfeld number

S = (r/c_r)² μ N′/p′, with p′ = W/(l d) the load on the projected area;

Report 1157 uses its reciprocal times (l/d)², the load number 1/C_n with C_n = S (l/d)², because the short-bearing results then plot as single lines for every l/d.

Reynolds equation and the numerical solution

For a thin isoviscous film with no slip, the pressure p obeys the Reynolds equation on the unwrapped surface (x = rθ circumferential, z axial),

∂/∂x (h³ ∂p/∂x) + ∂/∂z (h³ ∂p/∂z) = 6 μ U ∂h/∂x, U = 2π N′ r.

With H = h/c_r, Z = z/(l/2) and p = 6 μ ω (r/c_r)² P (ω = 2πN′) it becomes ∂/∂θ(H³ ∂P/∂θ) + (d/l)² ∂/∂Z(H³ ∂P/∂Z) = −ε sin θ, which depends on ε and l/d only. The tool discretises it with second-order central differences on a grid of N_θ nodes around the circumference (periodic) and N_z nodes over the half length (symmetry at the midplane, p = 0 at the ends), 120 × 24 by default. The linear system of every pressurised node is solved directly, column by column, as a block-tridiagonal system whose periodic coupling is removed by expressing every column in terms of the first; the cost is a fraction of a second per solution.

The diverging half of the film cannot sustain the large negative pressures the equation would give there: the oil film ruptures into streamers of air and oil. Two treatments are offered. The half-Sommerfeld (Gümbel) boundary solves the full periodic problem and sets every negative pressure to zero; it is simple and it is the boundary the Report 1157 closed forms use, but it violates flow continuity at θ = 180°. The Reynolds (Swift–Stieber) boundary, the default, requires p ≥ 0 everywhere with both p and its normal gradient zero along the rupture line, which the solution locates itself. Christopherson (1941) showed that this is obtained by relaxing the difference equations with the pressures clamped at zero; the tool runs that projected over-relaxation from the clamped full-film solution until the field stops changing (to 10⁻¹⁰ of the peak) and then makes one direct solve on the set of pressurised nodes, checking that no pressure on the set is negative and that no clamped node would rise. The result is the exact solution of the discrete complementarity problem; the mesh-convergence table shows how far the discrete solution is from the continuous one (the Sommerfeld number converges at second order, the side-flow at first order because it is an end gradient).

For an entered load the tool first solves at twenty-two eccentricity ratios from 0.05 to 0.98 (the curve the charts need anyway), interpolates the eccentricity for the required Sommerfeld number on that curve and refines it with secant iterations on log S until S is matched to one part in 10⁷. With the two-point viscosity route the whole inversion sits inside a second iteration on the film temperature (below).

Load, attitude angle and peak pressure

Integrating the pressure over the pressurised film gives the two components of the film force along and across the line of centres (Report 1157 eqs 14 and 14a, Figure 7); their resultant balances the load and their ratio gives the attitude angle, tan φ = P_Y/P_X. In the tool's nondimensional variables the resultant is R = |(∫∫P cos θ, ∫∫P sin θ) dθ dZ| and S = 1/(3πR). The peak pressure is taken at the midplane with a parabolic refinement between nodes; its position θ_max from the maximum film thickness and the peak-pressure angle α = φ − (180° − θ_max) from the load line (eq. 28) are reported, together with the rupture angle of the Reynolds solution and the peak-pressure ratio k = p_max/p′ (eq. 27). Figure 2 shows the midplane pressure against the angle from the load line with the short-bearing curve for comparison, Figure 3 the axial profile through the peak.

Friction

The shear stress on the journal is τ = μU/h + (h/2) ∂p/∂x. The first (Couette) term is integrated over the full circumference, the cavitated region being treated as a full laminar film: Report 1157 found its 360° shear integral in agreement with its friction measurements, and Cameron and Wood had made the same observation. The pressure term is integrated where the film is complete. The friction force F gives the coefficient f = F/W, the torque F r and the power F U, which is the heat released in the film. Petroff's no-load value F_o = 2π μ U r l/c_r = 2π² μ d l N′ (d/c_d) is printed beside it, so that the friction ratio F/F_o of Report 1157 shows the increase of friction with load; in chart form the friction variable (r/c_r) f tends to Petroff's line 2π² S at light load.

Oil flow

The journal drags oil into the converging film at the maximum film thickness: Q = ∫(Uh/2 − h³/(12μ) ∂p/∂x) dz, the inlet or total flow. Part of it leaks out of the two ends of the pressurised film, Q_s = 2∫(−h³/(12μ) ∂p/∂z)_{z = l/2} r dθ, and the rest recirculates through the cavitated zone. Both are reported in volume units and as the flow variable Q/(r c_r N′ l) of the charts, with the side-flow ratio Q_s/Q. Report 1157's required oil flow Q_r = π d l c_d N′ ε/2 (eq. 23), the side leakage of the short-bearing film, is printed in the comparison table.

Temperature rise and viscosity

The friction heat is taken as carried away entirely by the oil (adiabatic film, no conduction to shaft or housing, which is conservative for the temperature rise). The side leakage leaves at the mean film temperature T_in + ΔT/2 and the recirculating flow at the outlet temperature T_in + ΔT, so F U = ρ c_p Q ΔT (1 − Q_s/(2Q)). Divided by p′ this is the dimensionless temperature-rise variable ρ c_p ΔT/p′ = 4π (r/c_r) f / [Q/(r c_r N′ l) (1 − Q_s/(2Q))], a function of S and l/d alone, which is Figure 7.

With the constant-viscosity route the entered viscosity is the film viscosity and the balance is evaluated once. With the two-point route the Walther form of ASTM D341, log₁₀ log₁₀(ν + 0.7) = A − B log₁₀ T (ν the kinematic viscosity in mm²/s, T in kelvin), is fitted through the two entered data-sheet points (converted from the entered absolute viscosities with the entered density, taken constant), and the film viscosity is evaluated at T_m = T_in + ΔT/2. Since ΔT depends on the viscosity, the tool iterates: it solves the bearing at μ(T_m), recomputes ΔT, and corrects T_m by the secant method until it repeats to 10⁻⁶ K; the passes are tabulated. An ISO VG grade is defined by its kinematic viscosity at 40 °C (the grade number, ±10 %); its data sheet gives the 40 °C and 100 °C values that make the two points. The default two-point oil is the SAE 10 oil of NACA TN 3491 (Figure 9 of that report, read from the figure).

Short-bearing approximation (NACA Report 1157)

Michell and Cardullo noticed, and Ocvirk and DuBois developed, that for short bearings the circumferential pressure flow h³/(12μ) ∂p/∂x is small beside the Couette flow Uh/2, so the first term of the Reynolds equation may be dropped and the equation integrates in closed form: p = (3μU/(r c_r²))(l²/4 − z²) ε sin θ/(1 + ε cos θ)³ (eq. 13), parabolic across the length and positive over the converging half only. From it follow the capacity number C_n = (1 − ε²)²/(πε) [π²(1 − ε²) + 16ε²]^−1/2 (eq. 17), tan φ = π(1 − ε²)^1/2/(4ε) (eq. 18), the friction variable f_v = f (d/c_d)(l/d)² = 2π² C_n/(1 − ε²)^1/2 (eq. 20), the friction ratio F/F_o = (1 − ε²)^−1/2 (eq. 21), the required flow number q_n = ε (eq. 24), the peak-pressure position cos θ_max = (1 − √(1 + 24ε²))/(4ε) (eq. 26), the peak-pressure ratio (eq. 27) and the peak-pressure angle (eq. 28). The report's tests on a 1⅜-inch journal at l/d = ¼, ½ and 1 agreed with these curves within the scatter of the data; the approximation overestimates the load capacity as l/d and ε grow (the neglected term scales like (l/d)² times the circumferential curvature of the pressure, which sharpens at high eccentricity), and the tool's comparison table shows the difference for the case at hand.

Infinitely long bearing (Sommerfeld)

Dropping the axial term instead gives the two-dimensional problem Sommerfeld solved in 1904 with his substitution 1 + ε cos θ = (1 − ε²)/(1 − ε cos γ). The full solution has a skew-symmetric pressure with equal negative pressures in the diverging film, a load at right angles to the line of centres (φ = 90°) and S = (2 + ε²)(1 − ε²)^1/2/(12π²ε) (Report 1157 eq. 32); discarding the negative half (half Sommerfeld) gives the load per unit length W(c/r)²/(μ ω r l) = 6ε[π² − ε²(π² − 4)]^1/2/((2 + ε²)(1 − ε²)) and tan φ = π(1 − ε²)^1/2/(2ε) (NASA RP-1255 eqs 10-40 and 10-41), with the maximum pressure at cos θ_m = −3ε/(2 + ε²). Because the long bearing has no side leakage it carries far more load at a given eccentricity than any finite bearing; the table prints its eccentricity and peak pressure at the entered Sommerfeld number as the other bound on the numerical result.

Design charts

Figures 4 to 7 plot, against the Sommerfeld number on a logarithmic axis, the eccentricity ratio and the film-thickness variable h₀/c_r = 1 − ε, the friction variable (r/c_r) f with Petroff's line, the flow variable and the side-flow ratio, and the temperature-rise variable, all from the numerical solution at the entered l/d and cavitation boundary, with the Report 1157 short-bearing curves dashed and the solved case marked. They answer the design questions directly: how the film thickness and the friction change with speed, viscosity, load or clearance (all of which enter through S), and where the chosen clearance sits between the thin-film and the high-friction ends of the curve.

Supply groove

When a supply pressure is entered, an axial groove of the entered length is placed opposite the load (at 180° + φ from the line of centres, upstream of the maximum film thickness) and held at that pressure in the Reynolds solution, which raises the inlet flow and slightly the load capacity. The eccentricity is set from the load without the groove and the groove solution is then an add-on; the page reports the resulting mismatch between film resultant and load when it exceeds two percent.

Sliding wear

The hydrodynamic film separates the surfaces at running speed; wear occurs when the film is too thin or absent, at starts and stops, under boundary lubrication or in a dry bushing. Archard's law (1953) gives the volume removed as V = K W s/H, with K a dimensionless wear coefficient from tests on the material pair, W the normal load, s the sliding distance and H the hardness of the softer surface expressed as a pressure (1 HB = 9.807 MPa). The tool reports the volume, the mean depth over the projected area 2 r l, the specific wear rate K/H in mm³ per N·m, and the pressure-velocity product PV = p_w U, which bushing materials are rated by; the limit is an entered material value because it belongs to the supplier's data sheet.

Limits

Steady load and speed, rigid aligned surfaces, isoviscous Newtonian oil, laminar film, full 360° bearing with plain bore (no lobes, grooves other than the optional supply groove, or misalignment), no thermal or elastic distortion of the clearance, no inlet-pressure load. The temperature rise is an adiabatic upper estimate; the two-point viscosity route takes the density constant. The length-diameter ratio is limited to 0.05 to 4 and the clearance to below a tenth of the radius. Dynamic loads, whirl and stability are not treated.

Checks

The library's closed forms reproduce every defining integral of Report 1157 and of the Sommerfeld solutions by numerical quadrature to machine precision, the report's printed cases (model 2 of its reference 13 and the four McKee and McKee bearings on page 20, the sample calculation sheet of Table IV), and the numerical solver agrees to nine digits with an independent numpy/scipy solution of the same discrete problem by two other methods, tends to the short-bearing forms as l/d → 0 and to the long-bearing forms as l/d → ∞, and recovers Petroff's friction at zero eccentricity; the default case is re-solved dimensionally in Python. The checks are in test-scripts/test_journal_bearing.php.