PE Exam Exam > PE Exam Notes > Mechanical Engineering for PE > Cheatsheet: Bearings, Gears, Shafts
Cheatsheet: Bearings, Gears, Shafts
1. Bearings
1.1 Types and Selection
| Bearing Type | Characteristics and Applications |
|---|---|
| Rolling Contact (Ball) | Low friction, high speed, radial and thrust loads, limited life (L10), AFBMA ratings |
| Rolling Contact (Roller) | Higher load capacity than ball, cylindrical/tapered/spherical configurations, lower speed limit |
| Plain/Journal | Hydrodynamic lubrication, infinite life if properly maintained, quiet operation, high damping |
| Thrust Bearings | Axial loads only, ball or tapered roller types for combined loads |
| Needle Bearings | High load capacity in small radial space, length/diameter ratio ≥ 2.5 |
1.2 Rolling Element Bearing Life
| Parameter | Formula/Value |
|---|---|
| Basic Life Equation | L10 = (C/P)^a × 10^6 revolutions, where a = 3 for ball bearings, a = 10/3 for roller bearings |
| Life in Hours | LH = (10^6/60N) × (C/P)^a hours, where N = rpm |
| L10 Life | 90% of bearings survive, 10% fail |
| Dynamic Load Rating (C) | Load for 1 million revolutions of L10 life, provided by manufacturer |
| Equivalent Load (P) | P = XFr + YFa, where X = radial factor, Y = thrust factor, Fr = radial load, Fa = axial load |
| Reliability Factor (a1) | L10 = 1.0, L5 = 0.62, L1 = 0.21 |
| Application Factor (a2) | Smooth: 1.0-1.2, Normal: 1.2-1.5, Heavy shock: 1.5-3.0 |
1.3 Journal Bearing Analysis
| Parameter | Formula/Description |
|---|---|
| Sommerfeld Number | S = (μN/P)(r/c)^2, dimensionless performance parameter |
| Bearing Pressure | P = W/(LD), where W = load, L = length, D = diameter |
| Clearance Ratio | c/r, where c = radial clearance, r = shaft radius |
| Eccentricity Ratio | ε = e/c, where e = eccentricity, ranges 0 to 1 |
| Minimum Film Thickness | h0 = c(1 - ε) |
| Coefficient of Friction | f = function of S and L/D ratio, from charts |
| Heat Generated | Q = fWV, where V = surface velocity = πDN/60 |
| Viscosity-Temperature | μ decreases exponentially with temperature increase |
1.4 Lubrication Regimes
- Hydrodynamic: Full fluid film separation, no metal contact, requires sufficient speed and viscosity
- Boundary: Partial contact, lubrication by surface films, occurs at low speeds or high loads
- Mixed: Transition between hydrodynamic and boundary
- ZN/P Criterion: Z = viscosity (cP), N = speed (rpm), P = pressure (psi); ZN/P > 30 for hydrodynamic
2. Gears
2.1 Gear Types and Geometry
| Gear Type | Configuration and Application |
|---|---|
| Spur | Parallel shafts, teeth parallel to axis, high efficiency (98-99%), noisy at high speeds |
| Helical | Parallel shafts, teeth at helix angle, smoother/quieter, axial thrust present, efficiency 96-98% |
| Bevel | Intersecting shafts (90° common), straight or spiral teeth, efficiency 93-97% |
| Worm | Non-intersecting perpendicular shafts, high ratio (up to 100:1), self-locking possible, efficiency 40-90% |
| Planetary | Compact high-ratio systems, multiple load paths, coaxial input/output |
2.2 Fundamental Gear Parameters
| Parameter | Formula/Definition |
|---|---|
| Diametral Pitch (Pd) | Pd = N/d = π/p, teeth per inch of diameter, US standard |
| Module (m) | m = d/N = p/π, mm of diameter per tooth, metric standard |
| Circular Pitch (p) | p = πd/N = π/Pd, distance between adjacent teeth on pitch circle |
| Pitch Diameter (d) | d = N/Pd = mN, diameter of pitch circle |
| Center Distance (C) | C = (d1 + d2)/2 = (N1 + N2)/(2Pd) |
| Velocity Ratio (VR) | VR = ω1/ω2 = N2/N1 = d2/d1 |
| Addendum (a) | a = 1/Pd = m (standard full-depth) |
| Dedendum (b) | b = 1.25/Pd = 1.25m (standard full-depth) |
| Pressure Angle (φ) | Standard values: 20° (most common), 25°, 14.5° (obsolete) |
| Base Circle Diameter | db = d cos φ |
2.3 Contact Ratio and Interference
| Concept | Formula/Criteria |
|---|---|
| Contact Ratio (mc) | mc = length of action / base pitch, must be ≥ 1.2 for smooth operation |
| Minimum Teeth (No Undercut) | Nmin = 2k/sin²φ, where k = 1 for full-depth; Nmin = 18 for φ = 20°, k = 1 |
| Interference Condition | Occurs when addendum circle exceeds base circle tangent point |
2.4 AGMA Bending Stress (Lewis Equation)
| Parameter | Formula |
|---|---|
| Bending Stress | σ = Wt/(F × m × Y) or σ = Wt × Pd/(F × Y), where Wt = tangential load |
| Tangential Load | Wt = T/r = 2T/d = 33,000 × HP/(π × d × N/12) = 63,025 × HP/(N × d) |
| Lewis Form Factor (Y) | Function of tooth number and pressure angle, from AGMA tables |
| Face Width (F) | Recommended: 3p ≤ F ≤ 5p or 8/Pd ≤ F ≤ 16/Pd |
2.5 AGMA Surface Durability (Pitting)
| Parameter | Formula |
|---|---|
| Contact Stress | σc = Cp × √(Wt × Ka × Ks × Km × Cv)/(d × F × I) |
| Elastic Coefficient (Cp) | Cp = √(1/[π((1-ν1²)/E1 + (1-ν2²)/E2)]), steel: Cp = 2,300 √psi |
| Geometry Factor (I) | I = (cos φ sin φ)/2 × (mg/(mg + 1)), for external gears, mg = gear ratio |
| Application Factor (Ka) | Uniform: 1.0, Light shock: 1.25, Medium shock: 1.5, Heavy shock: 1.75 |
| Size Factor (Ks) | 1.0 for Pd ≥ 5, increases for larger teeth |
| Load Distribution (Km) | 1.0 to 2.0, accounts for misalignment and deflection |
| Dynamic Factor (Cv) | Accounts for inaccuracies and dynamic loads, 0.5 to 1.0, lower for higher speeds |
2.6 Helical Gear Factors
| Parameter | Formula |
|---|---|
| Normal Pitch | pn = pt cos ψ, where ψ = helix angle |
| Normal Pressure Angle | tan φn = tan φt cos ψ |
| Axial Pitch | px = p/tan ψ |
| Axial Thrust | Fa = Wt tan ψ |
| Center Distance | C = (N1 + N2)/(2Pd cos ψ) |
2.7 Worm Gear Relationships
| Parameter | Formula |
|---|---|
| Lead | L = Nw × px, where Nw = number of threads (starts) |
| Lead Angle | tan λ = L/(π dw), where dw = worm pitch diameter |
| Velocity Ratio | VR = Ng/Nw, where Ng = gear teeth, Nw = worm threads |
| Efficiency | η = (cos φn - f tan λ)/(cos φn + f cot λ), where f = coefficient of friction |
| Self-Locking | Occurs when f ≥ tan λ, worm can drive gear but not reverse |
2.8 Gear Materials and Allowable Stresses
| Material | Bending Strength (ksi) |
|---|---|
| Steel, Through-Hardened (180 BHN) | 25-30 |
| Steel, Through-Hardened (300 BHN) | 35-40 |
| Steel, Case-Hardened (50-60 HRC) | 55-65 |
| Cast Iron, Grade 40 | 8-10 |
| Bronze | 12-18 |
3. Shafts
3.1 Shaft Design Criteria
- Strength: Combined stress from bending, torsion, axial loads must not exceed material limits
- Rigidity: Deflection and slope must be within limits for bearing and gear alignment
- Critical Speed: Operating speed must be away from natural frequencies (typically N < 0.5ncr="" or="" n=""> 2Ncr)
- Corrosion and Wear: Material selection and surface treatment for environment
3.2 Combined Stress Theories
| Theory | Formula |
|---|---|
| Maximum Shear Stress (Tresca) | τmax = √(σ²/4 + τ²) ≤ Sy/(2 × N), where N = safety factor |
| Distortion Energy (von Mises) | σ' = √(σ² + 3τ²) ≤ Sy/N |
| ASME Code for Shafts | σ' = √((Km M)² + ¾(Kt T)²)/(π d³/32) ≤ Sy/N |
3.3 Shaft Stress Calculations
| Stress Type | Formula |
|---|---|
| Bending Stress | σ = Mc/I = 32M/(π d³) for solid circular shaft |
| Torsional Shear Stress | τ = Tc/J = 16T/(π d³) for solid circular shaft |
| Axial Stress | σa = F/A = 4F/(π d²) |
| Transverse Shear | τ = VQ/(It) = 4V/(3A) for circular section, usually negligible |
3.4 Stress Concentration Factors
| Feature | Typical Kt Range |
|---|---|
| Sharp Shoulder (r/d = 0.02) | 2.5-3.0 |
| Well-Rounded Fillet (r/d = 0.1) | 1.5-1.7 |
| Keyway (profile) | 2.0-3.0 |
| Keyway (end-milled) | 3.0-4.0 |
| Thread (cut) | 2.5-3.8 |
| Thread (rolled) | 2.2-3.0 |
| Press Fit | 2.0-3.0 |
| Retaining Ring Groove | 4.0-5.0 |
3.5 Fatigue Considerations
| Parameter | Formula/Value |
|---|---|
| Endurance Limit (Steel) | Se' = 0.5 Sut for Sut ≤ 200 ksi; Se' = 100 ksi for Sut > 200 ksi |
| Surface Factor (ka) | Ground: 1.0, Machined: 0.8-0.9, Hot-rolled: 0.5-0.7, As-forged: 0.3-0.5 |
| Size Factor (kb) | kb = (d/0.3)^(-0.107) for 0.3 ≤ d ≤ 10 inches; kb = 1 for d ≤ 0.3 inches |
| Reliability Factor (kc) | 50%: 1.0, 90%: 0.89, 95%: 0.87, 99%: 0.81, 99.9%: 0.75 |
| Modified Endurance | Se = ka × kb × kc × Se' |
| Fatigue Stress Concentration | Kf = 1 + q(Kt - 1), where q = notch sensitivity (0 to 1) |
| Alternating Stress | σa = Kf × (σmax - σmin)/2 |
| Mean Stress | σm = (σmax + σmin)/2 |
3.6 Fatigue Failure Criteria
| Criterion | Formula |
|---|---|
| Soderberg (Conservative) | σa/Se + σm/Sy = 1/N |
| Goodman (Common) | σa/Se + σm/Sut = 1/N |
| Gerber (Less Conservative) | σa/Se + (σm/Sut)² = 1/N |
| ASME Elliptic | (σa/Se)² + (σm/Sy)² = 1/N² |
3.7 Shaft Deflection
| Parameter | Formula |
|---|---|
| Bending Deflection | y = f(loading, supports, geometry), use beam tables or integration |
| Maximum Slope | θ = dy/dx, critical for bearing alignment (limit: 0.001 rad) |
| Torsional Deflection | θ = TL/(GJ), where G = shear modulus, J = polar moment |
| Polar Moment (Solid) | J = π d⁴/32 |
| Area Moment (Solid) | I = π d⁴/64 |
3.8 Critical Speed
| Parameter | Formula |
|---|---|
| First Critical Speed | Ncr = (60/2π) × √(g/δst), where δst = static deflection |
| Rayleigh's Method | ωn = √[g Σ(Wi yi)/Σ(Wi yi²)], where yi = deflection at location i |
| Safety Margin | Operating speed: N < 0.5ncr="" or="" n=""> 2Ncr |
3.9 Keys and Splines
| Type | Design Formula |
|---|---|
| Square/Rectangular Key | Shear: τ = 2T/(d × w × L), Bearing: σ = 4T/(d × h × L) |
| Key Dimensions | Width (w) and height (h) based on shaft diameter, ANSI B17.1 |
| Spline Torque Capacity | T = (N × D × L × h × P)/2, where N = teeth, P = allowable pressure |
| Woodruff Key | Semi-circular, self-aligning, higher stress concentration than rectangular |
3.10 Shaft-Hub Connections
| Method | Characteristics |
|---|---|
| Setscrew | Simple, low torque, point or cup end, stress concentration concern |
| Pin | Positive drive, shear and bending in pin, τ = F/(d × L) |
| Press/Shrink Fit | Interference fit, high torque capacity, difficult assembly/disassembly |
| Tapered Fit | Self-locking, axial force creates friction, common for pulleys |
3.11 Materials and Properties
| Material | Typical Sut (ksi) |
|---|---|
| AISI 1020 (Cold-Drawn) | 75 |
| AISI 1045 (Q&T) | 110-120 |
| AISI 4340 (Q&T) | 150-200 |
| Stainless 304 | 85 |
| Stainless 17-4PH | 180-200 |
3.12 Design Guidelines
- Minimum shaft diameter at stress concentration: d = ∛(32N/π) × √((Km M)² + ¾(Kt T)²)/Sy
- Provide generous fillets at shoulders: r/d ≥ 0.1 to minimize Kt
- Locate stress concentrations away from high moment regions when possible
- Standard diameters: Use preferred sizes for availability (0.5, 0.625, 0.75, 1.0, 1.25, 1.5, 2.0 inches)
- Shoulder heights: 1.5× key height or 0.1× shaft diameter for positive component location
- Thread relief and undercuts: Provide clearance for grinding and assembly
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