Formula Sheet: Bearings, Gears, Shafts

Bearings

Bearing Life and Reliability

Basic Dynamic Load Rating (L10 Life) \[L_{10} = \left(\frac{C}{P}\right)^k\]

• L₁₀ = rated bearing life in millions of revolutions (90% reliability)
• C = basic dynamic load rating (catalog rating), lbf or N
• P = equivalent radial load, lbf or N
• k = 3 for ball bearings, 10/3 for roller bearings

Bearing Life in Hours \[L_{10h} = \frac{L_{10} \times 10^6}{60n}\]

• L_10h = rated life in hours
• L₁₀ = rated life in millions of revolutions
• n = rotational speed, rpm

Adjusted Rating Life (with reliability factor) \[L_R = a_1 \times L_{10}\]

• L_R = life at specified reliability
• a₁ = reliability factor
• Common values: a₁ = 1 (90% reliability), a₁ = 0.62 (95%), a₁ = 0.21 (99%)

Relationship Between Lives at Different Reliabilities \[\frac{L_2}{L_1} = \left(\frac{R_1}{R_2}\right)^{1/b}\]

• L₁, L₂ = bearing lives at reliabilities R₁ and R₂
• b = Weibull shape parameter ≈ 1.5 for ball bearings, 1.11 for roller bearings

Bearing Load Analysis

Equivalent Radial Load (Radial and Thrust Combined) \[P = XVF_r + YF_a\]

• P = equivalent radial load, lbf or N
• F_r = radial load, lbf or N
• F_a = thrust (axial) load, lbf or N
• X = radial load factor (from bearing manufacturer)
• Y = thrust load factor (from bearing manufacturer)
• V = rotation factor (V = 1 if inner ring rotates, V = 1.2 if outer ring rotates)

Minimum Equivalent Load Condition \[P = F_r \text{ when } F_a/F_r \leq e\] \[P = XF_r + YF_a \text{ when } F_a/F_r > e\]

• e = thrust factor threshold (from bearing manufacturer tables)

Journal Bearings

Sommerfeld Number \[S = \left(\frac{r}{c}\right)^2 \frac{\mu N}{P}\]

• S = Sommerfeld number (dimensionless)
• r = journal radius, in or m
• c = radial clearance, in or m
• μ = absolute viscosity, lbf·s/in² or Pa·s
• N = rotational speed, rev/s
• P = bearing pressure = W/(LD), psi or Pa

Bearing Pressure \[P = \frac{W}{LD}\]

• W = radial load on bearing, lbf or N
• L = bearing length, in or m
• D = bearing diameter, in or m

Coefficient of Friction (Journal Bearing) \[f = \phi\left(\frac{r}{c}\right)\frac{\mu N}{P}\]

• f = coefficient of friction (dimensionless)
• φ = friction variable (function of L/D ratio and Sommerfeld number)

Frictional Power Loss \[H_f = \frac{2\pi fWrN}{J}\]

• H_f = heat generated, Btu/min or W
• f = coefficient of friction
• W = radial load, lbf or N
• r = journal radius, in or m
• N = rotational speed, rev/s
• J = mechanical equivalent of heat (778 ft·lbf/Btu in US units)

Simplified Frictional Power Loss \[P_{loss} = 2\pi \tau N W r\]

• τ = torque, ft·lbf or N·m
• Alternative: \(P_{loss} = f \times W \times V\) where V = surface velocity

Minimum Film Thickness \[h_0 = c(1 - \epsilon)\]

• h₀ = minimum film thickness, in or m
• c = radial clearance, in or m
• ε = eccentricity ratio (dimensionless, 0 to 1)

Petroff's Equation (Lightly Loaded Bearings) \[f = 2\pi^2\left(\frac{r}{c}\right)\frac{\mu N}{P}\]

• Valid for concentric journal operation (low loads)
• Provides approximate coefficient of friction

Hydrostatic Bearings

Load Capacity \[W = P_s A_{eff}\]

• W = load capacity, lbf or N
• P_s = supply pressure, psi or Pa
• A_eff = effective bearing area, in² or m²

Gears

Gear Geometry and Kinematics

Pitch Diameter \[d = \frac{N}{P_d}\]

• d = pitch diameter, in
• N = number of teeth
• P_d = diametral pitch, teeth/in

Module (SI System) \[m = \frac{d}{N}\]

• m = module, mm
• d = pitch diameter, mm
• N = number of teeth

Relationship Between Module and Diametral Pitch \[m = \frac{25.4}{P_d}\]

• m = module, mm
• P_d = diametral pitch, teeth/in

Circular Pitch \[p = \frac{\pi d}{N} = \frac{\pi}{P_d}\]

• p = circular pitch, in or mm

Addendum \[a = \frac{1}{P_d}\]

• a = addendum, in
• In SI: a = m

Dedendum \[b = \frac{1.25}{P_d} = 1.25a\]

• b = dedendum, in
• In SI: b = 1.25m

Whole Depth \[h_t = a + b = \frac{2.25}{P_d}\]

• h_t = whole depth, in

Outside Diameter \[d_o = d + 2a = \frac{N + 2}{P_d}\]

• d_o = outside diameter, in or mm

Root Diameter \[d_r = d - 2b\]

• d_r = root diameter, in or mm

Center Distance \[C = \frac{d_P + d_G}{2} = \frac{N_P + N_G}{2P_d}\]

• C = center distance, in or mm
• d_P = pitch diameter of pinion
• d_G = pitch diameter of gear
• N_P = number of teeth on pinion
• N_G = number of teeth on gear

Velocity Ratio (Gear Ratio) \[VR = \frac{n_P}{n_G} = \frac{N_G}{N_P} = \frac{d_G}{d_P}\]

• VR = velocity ratio (dimensionless)
• n_P = pinion speed, rpm
• n_G = gear speed, rpm

Pitch Line Velocity \[V = \frac{\pi d n}{12} \text{ (US units, ft/min)}\] \[V = \frac{\pi d n}{60,000} \text{ (SI units, m/s)}\]

• V = pitch line velocity, ft/min or m/s
• d = pitch diameter, in or mm
• n = rotational speed, rpm

Gear Force Analysis (Spur Gears)

Transmitted Load (Tangential Force) \[W^t = \frac{33,000 \times HP}{V}\]

• W^t = tangential force, lbf
• HP = transmitted power, horsepower
• V = pitch line velocity, ft/min

SI Version \[W^t = \frac{P}{V}\]

• W^t = tangential force, N
• P = transmitted power, W
• V = pitch line velocity, m/s

Radial Force (Spur Gears) \[W^r = W^t \tan \phi\]

• W^r = radial (separating) force, lbf or N
• φ = pressure angle (typically 20° or 25°)

Total Force on Gear Tooth \[W = \frac{W^t}{\cos \phi}\]

• W = total normal force, lbf or N

Gear Force Analysis (Helical Gears)

Tangential Force \[W^t = \frac{33,000 \times HP}{V}\]

• Same as spur gears

Radial Force (Helical Gears) \[W^r = W^t \frac{\tan \phi_n}{\cos \psi}\]

• φ_n = normal pressure angle, degrees
• ψ = helix angle, degrees

Axial (Thrust) Force (Helical Gears) \[W^a = W^t \tan \psi\]

• W^a = axial force, lbf or N

Normal Pressure Angle Relationship \[\tan \phi_t = \frac{\tan \phi_n}{\cos \psi}\]

• φ_t = transverse pressure angle
• φ_n = normal pressure angle

Gear Force Analysis (Bevel Gears)

Tangential Force \[W^t = \frac{33,000 \times HP}{V}\] Radial Force (Bevel Gears) \[W^r = W^t \tan \phi \cos \gamma\]

• γ = pitch angle, degrees

Axial Force (Bevel Gears) \[W^a = W^t \tan \phi \sin \gamma\] Pitch Angle (Straight Bevel Gears) \[\tan \gamma = \frac{N_P}{N_G}\]

• γ = pitch angle of pinion
• For gear: γ_G = 90° - γ_P

Worm Gears

Velocity Ratio \[VR = \frac{N_W}{N_G}\]

• N_W = number of threads on worm
• N_G = number of teeth on worm gear

Lead \[L = N_W \times p\]

• L = lead, in or mm
• p = axial pitch, in or mm

Lead Angle \[\tan \lambda = \frac{L}{\pi d_W}\]

• λ = lead angle, degrees
• d_W = pitch diameter of worm, in or mm

Center Distance \[C = \frac{d_W + d_G}{2}\] Efficiency \[\eta = \frac{\cos \phi - f \tan \lambda}{\cos \phi + f \cot \lambda}\]

• η = efficiency (dimensionless)
• f = coefficient of friction
• φ = normal pressure angle
• λ = lead angle

Self-Locking Condition \[\lambda < \arctan(f)\]="">

• Worm gear is self-locking if lead angle is less than friction angle

AGMA Bending Stress (Lewis Equation Modified)

Bending Stress Number \[s_t = \frac{W^t P_d}{FJ} K_o K_v K_s K_m K_B\]

• s_t = bending stress, psi
• W^t = tangential transmitted load, lbf
• P_d = diametral pitch, teeth/in
• F = face width, in
• J = geometry factor (from AGMA charts)
• K_o = overload factor
• K_v = dynamic factor
• K_s = size factor
• K_m = load distribution factor
• K_B = rim thickness factor

Allowable Bending Stress \[s_{at} = \frac{S_t Y_N}{K_T K_R}\]

• s_at = allowable bending stress, psi
• S_t = allowable bending stress number (from AGMA tables), psi
• Y_N = stress cycle factor for bending
• K_T = temperature factor
• K_R = reliability factor

Factor of Safety (Bending) \[SF = \frac{s_{at}}{s_t}\]

• SF = factor of safety (should be ≥ 1)

AGMA Contact Stress (Pitting Resistance)

Contact Stress Number (Spur Gears) \[s_c = C_p \sqrt{\frac{W^t K_o K_v K_s K_m}{F d_P I}}\]

• s_c = contact stress, psi
• C_p = elastic coefficient, √psi
• d_P = pitch diameter of pinion, in
• I = geometry factor for pitting (external gears)

Geometry Factor for Pitting \[I = \frac{\cos \phi \sin \phi}{2 m_N} \frac{m_G}{m_G + 1}\]

• m_N = load sharing ratio (typically 1)
• m_G = gear ratio = N_G/N_P

Elastic Coefficient \[C_p = \sqrt{\frac{1}{\pi\left[\frac{1-\nu_P^2}{E_P} + \frac{1-\nu_G^2}{E_G}\right]}}\]

• E_P, E_G = modulus of elasticity for pinion and gear, psi
• ν_P, ν_G = Poisson's ratio for pinion and gear
• For steel on steel: C_p ≈ 2300 √psi

Allowable Contact Stress \[s_{ac} = \frac{S_c Z_N C_H}{K_T K_R}\]

• s_ac = allowable contact stress, psi
• S_c = allowable contact stress number (from AGMA), psi
• Z_N = stress cycle factor for pitting
• C_H = hardness ratio factor

Factor of Safety (Contact) \[SF = \frac{s_{ac}}{s_c}\]

Dynamic Factor (K_v)

AGMA Dynamic Factor \[K_v = \left(\frac{A + \sqrt{V}}{A}\right)^B\]

• V = pitch line velocity, ft/min
• A, B = constants based on quality number Q_v
• A = 50 + 56(1 - B)
• B = 0.25(12 - Q_v)^2/3
• Q_v = quality number (3 to 11, higher = better quality)

Shafts

Shaft Design for Stress

Maximum Shear Stress Theory (Tresca) \[\tau_{max} = \frac{16}{\pi d^3}\sqrt{(K_f M)^2 + (K_{fs} T)^2}\]

• τ_max = maximum shear stress, psi or Pa
• d = shaft diameter, in or m
• M = bending moment, lbf·in or N·m
• T = torque, lbf·in or N·m
• K_f = fatigue stress concentration factor for bending
• K_fs = fatigue stress concentration factor for torsion

Distortion Energy Theory (von Mises) \[\sigma' = \sqrt{\sigma^2 + 3\tau^2}\]

• σ' = equivalent stress (von Mises stress), psi or Pa
• σ = normal stress, psi or Pa
• τ = shear stress, psi or Pa

ASME Code for Shaft Design (Steady Loading) \[d = \sqrt[3]{\frac{16n}{\pi S_y}\sqrt{(K_f M)^2 + (K_{fs} T)^2}}\]

• n = factor of safety
• S_y = yield strength, psi or Pa

ASME Code for Shaft Design (Fatigue Loading) \[d = \sqrt[3]{\frac{16n}{\pi}\sqrt{\left(\frac{K_f M_a}{S_e} + \frac{K_{fc} M_m}{S_y}\right)^2 + 3\left(\frac{K_{fs} T_a}{S_e} + \frac{K_{fsc} T_m}{S_y}\right)^2}}\]

• M_a = alternating bending moment, lbf·in or N·m
• M_m = mean (steady) bending moment, lbf·in or N·m
• T_a = alternating torque, lbf·in or N·m
• T_m = mean (steady) torque, lbf·in or N·m
• S_e = endurance limit, psi or Pa
• K_fc = stress concentration factor for mean bending
• K_fsc = stress concentration factor for mean torsion

Simplified ASME Code (Fluctuating Stress) \[d = \sqrt[3]{\frac{16n}{\pi S_e}\sqrt{4(K_f M_a)^2 + 3(K_{fs} T_m)^2}}\]

• Assumes fully reversed bending and steady torsion

Torque and Power Relationships

Torque-Power Relationship \[T = \frac{63,025 \times HP}{n}\]

• T = torque, lbf·in
• HP = power, horsepower
• n = rotational speed, rpm

SI Version \[T = \frac{P \times 60}{2\pi n} = \frac{9549 \times P}{n}\]

• T = torque, N·m
• P = power, kW
• n = rotational speed, rpm

Bending and Torsional Stress

Bending Stress (Solid Circular Shaft) \[\sigma = \frac{Mc}{I} = \frac{32M}{\pi d^3}\]

• σ = bending stress, psi or Pa
• M = bending moment, lbf·in or N·m
• c = distance from neutral axis to outer fiber = d/2, in or m
• I = second moment of area, in⁴ or m⁴
• d = diameter, in or m

Torsional Shear Stress (Solid Circular Shaft) \[\tau = \frac{Tc}{J} = \frac{16T}{\pi d^3}\]

• τ = torsional shear stress, psi or Pa
• T = torque, lbf·in or N·m
• J = polar second moment of area, in⁴ or m⁴

Second Moment of Area (Solid Circular) \[I = \frac{\pi d^4}{64}\] Polar Second Moment of Area (Solid Circular) \[J = \frac{\pi d^4}{32}\] Hollow Circular Shaft - Second Moment \[I = \frac{\pi (d_o^4 - d_i^4)}{64}\]

• d_o = outer diameter
• d_i = inner diameter

Hollow Circular Shaft - Polar Second Moment \[J = \frac{\pi (d_o^4 - d_i^4)}{32}\]

Shaft Deflection

Angle of Twist \[\theta = \frac{TL}{GJ}\]

• θ = angle of twist, radians
• T = torque, lbf·in or N·m
• L = length, in or m
• G = shear modulus, psi or Pa
• J = polar second moment of area, in⁴ or m⁴

Angular Deflection (Degrees per Unit Length) \[\theta_{deg} = \frac{TL \times 180}{\pi GJ} = \frac{584TL}{Gd^4}\]

• θ_deg = angle of twist, degrees
• Second form valid for solid circular shafts (US units)

Maximum Slope (Cantilever Beam with End Load) \[\theta_{max} = \frac{FL^3}{3EI}\]

• θ_max = slope at free end, radians
• F = load at end, lbf or N
• E = modulus of elasticity, psi or Pa

Maximum Deflection (Cantilever Beam with End Load) \[y_{max} = \frac{FL^3}{3EI}\]

• y_max = deflection at free end, in or m

Maximum Deflection (Simply Supported Beam, Center Load) \[y_{max} = \frac{FL^3}{48EI}\]

• y_max = deflection at center, in or m

Maximum Deflection (Simply Supported Beam, Uniform Load) \[y_{max} = \frac{5wL^4}{384EI}\]

• w = uniform load per unit length, lbf/in or N/m

Critical Speed

First Critical Speed (Rayleigh's Method) \[n_{cr} = \frac{60}{2\pi}\sqrt{\frac{g}{\delta_{st}}}\]

• n_cr = first critical speed, rpm
• g = gravitational acceleration (386 in/s² or 9.81 m/s²)
• δ_st = static deflection at location of rotating mass, in or m

Simplified Formula (US units) \[n_{cr} = \frac{187.7}{\sqrt{\delta_{st}}}\]

• δ_st in inches
• n_cr in rpm

Multiple Masses (Rayleigh's Method) \[n_{cr} = \frac{60}{2\pi}\sqrt{\frac{g\sum W_i y_i}{\sum W_i y_i^2}}\]

• W_i = weight of mass i, lbf or N
• y_i = static deflection at location of mass i, in or m

Fatigue Considerations

Endurance Limit (Rotating Beam) \[S_e = k_a k_b k_c k_d k_e k_f S'_e\]

• S_e = endurance limit, psi or Pa
• S'_e = endurance limit of test specimen (uncorrected), psi or Pa
• k_a = surface finish factor
• k_b = size factor
• k_c = load factor
• k_d = temperature factor
• k_e = reliability factor
• k_f = miscellaneous effects factor

Uncorrected Endurance Limit (Steel) \[S'_e = 0.5 S_{ut} \text{ (for } S_{ut} \leq 200 \text{ ksi)}\] \[S'_e = 100 \text{ ksi (for } S_{ut} > 200 \text{ ksi)}\]

• S_ut = ultimate tensile strength, psi

Surface Finish Factor \[k_a = a S_{ut}^b\]

• a, b = constants depending on surface finish
• Ground: a = 1.58, b = -0.085 (US units, S_ut in ksi)
• Machined: a = 2.70, b = -0.265
• Hot-rolled: a = 14.4, b = -0.718
• As-forged: a = 39.9, b = -0.995

Size Factor (Rotating Shaft) \[k_b = 0.879 d^{-0.107} \text{ for } 0.11 \leq d \leq 2 \text{ in}\] \[k_b = 0.91 d^{-0.157} \text{ for } 2 < d="" \leq="" 10="" \text{="" in}\]="">

• d = diameter, in

Load Factor

• k_c = 1.0 for bending
• k_c = 0.85 for axial loading
• k_c = 0.59 for torsion

Temperature Factor

• k_d = 1.0 for T ≤ 450°C
• k_d = 0.975 + 0.432 × 10^-3T - 0.115 × 10^-5T² + 0.104 × 10^-8T³ - 0.595 × 10^-12T⁴ for 450°C < t=""><>

Reliability Factor

• k_e = 1.000 for 50% reliability
• k_e = 0.897 for 90% reliability
• k_e = 0.868 for 95% reliability
• k_e = 0.814 for 99% reliability
• k_e = 0.753 for 99.9% reliability

Fatigue Stress Concentration Factor \[K_f = 1 + q(K_t - 1)\]

• K_f = fatigue stress concentration factor (dimensionless)
• K_t = theoretical stress concentration factor (from charts)
• q = notch sensitivity (0 to 1, from charts)

Goodman Criterion (Fatigue Analysis)

Modified Goodman Diagram \[\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n}\]

• σ_a = alternating stress amplitude, psi or Pa
• σ_m = mean stress, psi or Pa
• n = factor of safety

For Shafts with Bending and Torsion \[\frac{1}{n} = \frac{16}{\pi d^3}\left[\frac{1}{S_e}\sqrt{4(K_f M_a)^2 + 3(K_{fs} T_a)^2} + \frac{1}{S_{ut}}\sqrt{4(K_{fc} M_m)^2 + 3(K_{fsc} T_m)^2}\right]\] Soderberg Criterion (Conservative) \[\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n}\]

• Uses yield strength instead of ultimate strength

Gerber Criterion \[\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_{ut}}\right)^2 = \frac{1}{n}\]

• Less conservative than Goodman

Keys and Keyways

Shear Stress in Key \[\tau = \frac{2T}{dlw}\]

• τ = shear stress in key, psi or Pa
• T = torque, lbf·in or N·m
• d = shaft diameter, in or m
• l = key length, in or m
• w = key width, in or m

Bearing Stress in Key \[\sigma_b = \frac{4T}{dlh}\]

• σ_b = bearing (compressive) stress, psi or Pa
• h = key height, in or m

Key Proportions (Square Key)

• w = h = d/4 (approximate)
• Standard keys follow ANSI B17.1

Interference Fits

Pressure at Interface \[p = \frac{\delta E}{d}\left[\frac{(d_o^2 - d^2)(d^2 - d_i^2)}{(d_o^2 - d_i^2)d^2}\right]\]

• p = interface pressure, psi or Pa
• δ = total diametral interference, in or m
• E = modulus of elasticity, psi or Pa
• d = nominal diameter, in or m
• d_o = outer diameter of hub, in or m
• d_i = inner diameter of shaft (solid shaft: d_i = 0), in or m

Torque Capacity \[T = \frac{\pi f p d^2 l}{2}\]

• T = torque capacity, lbf·in or N·m
• f = coefficient of friction
• l = length of engagement, in or m

Shaft Materials

Common Shaft Material Properties

• AISI 1020 (cold-drawn): S_y = 57 ksi, S_ut = 68 ksi
• AISI 1040 (as-rolled): S_y = 42 ksi, S_ut = 76 ksi
• AISI 4140 (Q&T): S_y = 165 ksi, S_ut = 185 ksi
• Steel: E = 30 × 10⁶ psi, G = 11.5 × 10⁶ psi