10.2 Rolling Element & Hydrodynamic Journal Bearings

10.2 Rolling Element & Hydrodynamic Journal Bearings

Quick Summary: Mechanical bearings support rotating shafts while minimizing friction losses and wear. Rolling element bearings rely on concentrated point or line contact governed by fatigue life ($L_{10}$ rating), whereas hydrodynamic journal bearings support loads on a self-pressurized fluid film governed by Petroff's equation and the Sommerfeld number.

Bearings Classification & Operating Principles

Bearings are broadly categorized into rolling contact (antifriction) bearings and sliding contact (journal/sleeve) bearings. Each type exhibits distinct load capabilities, friction characteristics, and design life methodologies.

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1. Rolling Element Bearing Types & Kinematics

1. Deep Groove Ball Bearings: Versatile, handles high radial loads and moderate double-direction axial loads at high rotational speeds.
2. Angular Contact Ball Bearings: High contact angle ($\alpha = 15^\circ - 40^\circ$); handles combined radial and heavy unidirectional thrust loads.
3. Cylindrical Roller Bearings: Line contact delivers extremely high radial load capacity but zero axial capacity.
4. Spherical Roller Bearings: Double-row barrel rollers with a spherical outer race; accommodates severe angular shaft misalignment ($2^\circ - 3^\circ$).
5. Tapered Roller Bearings: Conical rollers arranged along pitch cones; supports heavy combined radial and thrust loads in transmissions and vehicle hubs.

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2. Dynamic Load Rating & $L_{10}$ Life Equation

Basic Dynamic Load Rating ($C$)

The basic dynamic load rating $C$ is the constant radial load that $90\%$ of a group of identical bearings will endure for $1,000,000$ revolutions ($10^6$ revs) without structural fatigue failure (spalling).

Equivalent Dynamic Radial Load ($P_e$)

When a bearing experiences both radial load $F_r$ and thrust load $F_a$, the combined equivalent load $P_e$ is:

$$P_e = X V F_r + Y F_a$$

Where:

• $V$ = Rotation factor ($1.0$ for inner ring rotation, $1.2$ for outer ring rotation)
• $X$ = Radial load factor
• $Y$ = Axial (thrust) load factor
• Factors $X$ and $Y$ depend on the ratio $F_a / (V F_r)$ relative to threshold parameter $e$.

$L_{10}$ Rating Life Formula

Rating life in millions of revolutions:

$$L_{10} = \left(\frac{C}{P_e}\right)^p \times 10^6 \text{ revolutions}$$

Where life exponent $p$ is:

• $p = 3.0$ for ball bearings (point contact)
• $p = \frac{10}{3} \approx 3.333$ for roller bearings (line contact)

Rating life in operating hours $L_{10h}$ at rotational speed $N$ (rpm):

$$L_{10h} = \frac{10^6}{60 N} \left(\frac{C}{P_e}\right)^p = \frac{L_{10} \times 10^6}{60 N} \text{ hours}$$

Reliability Adjustment Factor ($a_1$)

For survival probabilities higher than $90\%$, the adjusted rating life is $L_{na} = a_1 a_2 a_3 L_{10}$. For $95\%$ reliability, $a_1 = 0.62$; for $99\%$ reliability, $a_1 = 0.31$.

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3. Hydrodynamic Journal Bearings & Petroff's Equation

In a hydrodynamic journal bearing, a rotating journal of radius $r$ creates a converging oil wedge within radial clearance $c = R - r$, generating fluid dynamic pressure that floats the shaft without metal contact.

Petroff's Concentric Friction Equation

Assuming a concentric shaft under light radial load, viscous shearing stress according to Newton's law of viscosity is $\tau = \mu \frac{d v}{d y} = \mu \frac{2 \pi r N'}{c}$, where $N'$ is rotational speed in rev/s.

Integrating shear stress over journal surface area $A = 2 \pi r L$ yields Petroff's friction torque formula:

$$T_f = \frac{2 \pi^2 \mu N' L r^3}{c}$$

Where:

• $\mu$ = Dynamic viscosity of lubricant ($\text{Pa}\cdot\text{s} = \text{N}\cdot\text{s/m}^2$)
• $N'$ = Journal rotational speed (rev/sec = $N/60$)
• $L$ = Bearing length (m)
• $r$ = Journal radius (m)
• $c$ = Radial clearance (m)

Friction power loss: $P_{loss} = 2 \pi N' T_f = \frac{4 \pi^3 \mu (N')^2 L r^3}{c}$.

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4. Sommerfeld Number & Stribeck Lubrication Regimes

Sommerfeld Number ($S$)

The Sommerfeld Number (Bearing Characteristic Number) is the master dimensionless group governing hydrodynamic bearing performance:

$$S = \left(\frac{r}{c}\right)^2 \frac{\mu N'}{P}$$

Where $P = \frac{W}{2 r L}$ is the projected bearing pressure (Pa), and $W$ is total radial load (N).

Minimum Oil Film Thickness ($h_0$)

Under load, the journal shifts eccentrically by distance $e$, defining eccentricity ratio $\epsilon = e/c$. Minimum film thickness occurs at the line of centers:

$$h_0 = c (1 - \epsilon)$$

Design guidelines require $h_0 \ge 0.005 - 0.010\text{ mm}$ ($5 - 10\ \mu\text{m}$) to prevent surface contact under thermal deflection.

Stribeck Curve Regimes

• Boundary Lubrication ($S < 0.001$): Asperity contact occurs; high friction ($f = 0.08 - 0.15$).
• Mixed Lubrication ($0.001 < S < 0.01$): Partial fluid film; friction drops to a minimum ($f_{min} \approx 0.001$).
• Hydrodynamic Lubrication ($S > 0.01$): Complete fluid film separation ($h_0 > 5\ \mu\text{m}$); coefficient of friction increases linearly with $S$ due to viscous shearing ($f = 0.002 - 0.01$).

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5. Step-by-Step Worked Bearing Calculation

Problem Statement

A 6308 deep groove ball bearing ($C = 40.5\text{ kN}$) operates at $N = 1800\text{ rpm}$ supporting radial load $F_r = 5.0\text{ kN}$ and axial thrust load $F_a = 2.5\text{ kN}$. Inner ring rotates ($V = 1.0$). Radial factor $X = 0.56$ and thrust factor $Y = 1.45$.

Calculate: (1) Equivalent dynamic load $P_e$, (2) $L_{10}$ life in millions of revolutions, (3) Operating life $L_{10h}$ in hours.

Solution Steps

1. Equivalent Dynamic Radial Load:

   $$P_e = X V F_r + Y F_a = (0.56 \times 1.0 \times 5.0\text{ kN}) + (1.45 \times 2.5\text{ kN}) = 2.80 + 3.625 = 6.425\text{ kN} = 6425\text{ N}$$

2. $L_{10}$ Rating Life (Ball Bearing $p=3$):

   $$L_{10} = \left(\frac{C}{P_e}\right)^3 = \left(\frac{40.5\text{ kN}}{6.425\text{ kN}}\right)^3 = (6.3035)^3 = 250.46\text{ million revolutions}$$

3. Operating Life in Hours:

   $$L_{10h} = \frac{L_{10} \times 10^6}{60 \times N} = \frac{250.46 \times 10^6}{60 \times 1800} = \frac{250,460,000}{108,000} = 2319\text{ hours}$$