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GuidePublished 14 Aug 202623 min readBy Kevin JoginMachine DesignMachine ElementsPulley Groove Geometry — Key DimensionsKey Terms Glossary

Engineering · Machine Design · Machine Elements

Mechanical Design Data and Machine-Element Reference: Pulley Groove Geometry

Engineering handbook for mechanical design data and machine-element reference, covering pulley groove geometry — key dimensions, key terms glossary, shafts,...

Executive summary

This handbook section converts the supplied engineering material into a practical, source-controlled reference. It concentrates on the following learning outcomes.

Pulley Groove Geometry — Key Dimensions
Key Terms Glossary
SHAFTS, KEYS, BEARINGS & SEALS
Shafts, Keys, Circlips, Seals & Rolled Steel Sections
Shaft Definition and Characteristics
Shaft Loads and Stresses

Pulley Groove Geometry — Key Dimensions

flowchart LR
    A[Belt Section Identified] --> B[Determine PCD Range]
    B --> C{PCD ≤ Threshold?}
    C -->|Yes| D["Single Groove Angle: 34°"]
    C -->|No| E["Dual Groove Angle: 38°"]
    D --> F[Apply Standard Groove Dimensions]
    E --> F
    F --> G["Key Dimensions:
    A' = Groove Angle
    D = Groove Depth
    e = Groove Pitch
    l = Belt Seat Width
    b = Tolerance Band
    lp = Datum Length
    W = Top Width
    R = Nominal Radius"]


Key Terms Glossary

  • Wedge Belt: A V-shaped belt with a trapezoidal cross-section that wedges into the pulley groove for friction-based power transmission
  • Narrow-Section Wedge Belt: An optimised V-belt profile (SPB, SPC) with a higher power-to-width ratio compared to classical sections
  • CRE Belt: A classical/conventional wedge belt cross-section (SPZ, SPA, SPB) — smaller and less power-dense than narrow-section equivalents
  • Pitch Diameter (PCD): The effective working diameter of the pulley at the belt's neutral axis — used for all speed and power calculations
  • Outside Diameter (O): The overall outer diameter of the pulley including the groove lands
  • Taper Lock Bush: A split, tapered sleeve that locks a pulley concentrically onto a shaft using clamping bolts — allows keyless or keyed mounting
  • Bush Number: A standardised code identifying the taper lock bush dimensions, bore range, and bolt pattern
  • Speed Ratio: The ratio of driven pulley PCD to driver pulley PCD — determines the torque multiplication and speed reduction
  • Belt Speed (m/s): The linear speed of the belt, calculated as π × PCD × RPM / 60,000 — critical for power rating selection and centrifugal load limits
  • Groove Angle (A'): The included angle of the V-groove — typically 34° for smaller pulleys and 38° for larger pulleys within the same belt section
  • Groove Pitch (e): The centre-to-centre distance between adjacent grooves on a multi-groove pulley
  • Face Width (F): The total width across the pulley face, encompassing all grooves and edge margins
  • Non-Preferred Pulley Size: A catalogue entry marked with an asterisk (*) indicating it is not the standard/recommended size — may have longer lead times or limited availability
  • Type 6NR: A pulley construction variant using a non-standard retaining method — typically found in older or transitional designs


Quick Revision

  • SPB wedge belts cover PCDs from 140–315 mm and can transmit up to approximately 31 kW per belt — used for medium-to-heavy industrial drives
  • SPC wedge belts cover PCDs from 224–560 mm and can transmit up to approximately 60 kW per belt — the largest standard narrow-section profile
  • CRE belts (SPZ, SPA, SPB) are smaller classical profiles suited for light to medium duty — max power per belt ranges from ~5 kW (SPZ) to ~13 kW (SPB CRE)
  • Belt speed must not exceed 40 m/s — for speeds between 30–40 m/s, confirm pulley suitability with the manufacturer
  • Additional power is added per belt when the speed ratio exceeds 1.0 — this accounts for improved belt wrap on the smaller pulley
  • Taper lock pulleys use a split taper bush for shaft mounting — bush number determines max bore and shaft compatibility
  • Groove angles change with pulley PCD: 34° for smaller pulleys, 38° for larger pulleys within each belt section
  • Minimum groove counts vary by belt section: SPZ/Z = 1, SPA/A = 1, SPB/B = 2, SPC/C = 3
  • Pulley types progress from solid/plate construction at small diameters to spoked construction at large diameters for weight reduction
  • Non-preferred sizes (marked with *) should be avoided in new designs — use standard catalogue entries for availability and cost efficiency
  • Always verify: belt section → power rating → pulley PCD → number of grooves → bush size → bore compatibility → groove dimensions



SHAFTS, KEYS, BEARINGS & SEALS



Shafts, Keys, Circlips, Seals & Rolled Steel Sections



Overview

  • This chapter covers the mechanical design of shafts — rotating members supported by bearings that transmit torque and power
  • Topics include shaft load classification, failure modes, design approaches, stress formulas, load estimation methods, and design procedures for determining minimum shaft diameter
  • Also covered: circlip and seal sizing (metric long-life series), and rolled steel section selection for beams and columns
  • Key design philosophy: calculate equivalent torque and moment from combined loading, apply shock/fatigue factors, then determine shaft diameter using strength-of-materials formulas
  • Two primary design approaches are presented — one based on endurance limit and another based on basic strength of materials with generous safety factors


Key Concepts

  • Shaft: A rotating member supported by bearings that transmits torque and power
  • Steady Loads: Torsional, bending, and axial loads that occur continuously during operation
  • Shock Loads: Intermittent, sudden increases in load (e.g., initial engagement in rolling or pressing operations)
  • Inertia Loads: Loads arising from acceleration or deceleration of the shaft and attached equipment
  • Equivalent Torque (T_E): A combined measure incorporating both torque and bending moment — used to find shaft diameter from shear stress
  • Equivalent Moment (M_E): A combined measure used to find shaft diameter from bending stress
  • Shock/Fatigue Factors (K_T, K_M): Multipliers applied to steady torque and moment to account for dynamic loading effects
  • Stress Concentration: Localised increase in stress at geometric discontinuities such as keyways, shoulders, grooves, or holes
  • Endurance Limit: The stress level below which a material can theoretically sustain an infinite number of load cycles without fatigue failure
  • Drive Application Factor (f): A multiplier used to account for the difference between actual transverse shaft load and the simplified calculated value


Shaft Definition and Characteristics

  • Definition: A rotating member supported by bearings that transmits torque and power
  • Rotation may be continuous, intermittent, uni-directional, or reversing
  • Shafts attached to wheels are often called axles
  • Usually circular in cross-section — solid or hollow; sometimes square for specific applications
  • Typically rigid; flexible shafts (cables) exist for specialised applications but are not covered here
  • Usually long relative to diameter
  • Commonly made from steel or other metals; non-metallic shafts are sometimes used
  • Torque and power are transmitted from an input location (e.g., a prime mover) to an output location (e.g., a driven load)
  • Input may come from a motor, engine, or intermediate shaft via gears, belts, or chains
  • Output may be transmitted directly to a load or via power transmission devices

Shaft Loads and Stresses

Steady Loads
  • Primary design loads that occur relatively continuously during operation
  • Three types act on shafts, often simultaneously:
Load Type Example Source Resultant Stress
Torsional Motor, gear, belt, or chain drive Torsional shear stress
Bending Transverse load from weight, gear forces, belt/chain tension Bending stress (axial tension and compression)
Axial Propeller or weight load on a vertical shaft Axial tension and compression
Shock Loads
  • Intermittent in nature, causing sudden increases in load
  • Common in rolling mills, punching/cropping presses, and similar applications
  • Shafts must be designed to withstand shock loads even if they occur only momentarily
  • Load-limiting devices (shear pins, overload protection) are often fitted to protect the shaft
Inertia Loads
  • Occur during acceleration or deceleration (speed changes)
  • Magnitude depends on:
    • Rate of acceleration/deceleration (magnitude)
    • Mass moment of inertia of the shaft and coupled equipment/transmission devices
  • Typically occur during start-up and shut-down phases
  • For electric motors:
    • Soft-start (current-limiting device fitted): starting torque ≤ 1.5–2× rated load torque
    • Hard-start (no current limiter): starting torque can be 3–5× rated load torque

Shaft Failure Modes

Failure Due to Excessive Load
  • Occurs when shaft stress exceeds the yield stress
  • Relatively rare due to load-limiting devices (shear pins, overload protection) in most systems
  • If overload causes yielding without fracture, the shaft may remain serviceable
  • Design goal: prevent stress from exceeding yield stress; permanent deformation = failure
Failure Due to Fatigue
  • Most common failure mode for shafts with high revolution counts
  • Can occur even when stresses are well below yield point
  • Most likely when loads continually fluctuate, especially with stress reversal
  • Two mechanisms of stress reversal:
    • Change in load direction: Most common for torsional stress reversal (e.g., vehicle transmission shafts reversing between drive and braking)
    • Rotation of the shaft: Most common for bending stress reversal (e.g., a horizontal shaft with a downward load — top and bottom alternate between tension and compression each half-revolution)

Shaft Design Approaches

Approach 1: Endurance-Based
  • Calculate peak loads (including inertia and shock) and stress concentrations as accurately as possible
  • Allowable shaft stresses include an allowance for shaft size — larger diameter = lower allowable stress
  • For shafts with many revolutions: design to prevent fatigue failure using the endurance limit
  • Endurance limit is determined from standardised fatigue tests on polished specimens (typically 8–10 mm diameter)
  • A small factor of safety (typically ~1.2) is applied, based on the endurance limit
  • Based on relevant national rotating shaft design standards
  • Calculate the maximum design load likely under operating conditions
  • Apply shock/fatigue factors and a relatively generous factor of safety
  • Account for inertia loads and non-uniformity of material properties
  • Use basic strength of materials formulas to calculate shaft stresses or diameter
  • More fundamental approach; provides better understanding of stresses involved
  • Avoids complex formulas with unstated assumptions
  • Based on recognised engineering code methodology

Core Stress Formulas

Combined Shear Stress (Formula 1)

fs=16TEπd3f_s = \frac{16 \, T_E}{\pi \, d^3}

  • Used to find shaft diameter from torsional shear stress
Combined Axial Stress (Formula 2)

fs=32TEπd3f_s = \frac{32 \, T_E}{\pi \, d^3}

  • Used to find shaft diameter from bending (axial) stress
Equivalent Torque (Formula 3)

TE=T2+M2T_E = \sqrt{T^2 + M^2}

  • Combines torque (T) and bending moment (M) into a single equivalent value
Equivalent Moment (Formula 4)

ME=0.5(TE+M)M_E = 0.5 \, (T_E + M)

  • Used for bending stress calculations
Design Torque and Moment (with shock/fatigue factors)

T=KTTSM=KMMST = K_T \cdot T_S \qquad M = K_M \cdot M_S

  • Where T_S and M_S are the steady torque and moment
  • K_T = shock/fatigue factor in torsion
  • K_M = shock/fatigue factor in bending

Shock/Fatigue Factor Values

Loading Condition K_T (Torsion) K_M (Bending)
Static or gradually applied load 1.0 1.5
Suddenly applied with minor shock 1.0–1.5 1.5–2.0
Suddenly applied with heavy shock 1.5–3.0 2.0–3.0
  • K_M = 1.5 minimum even for static loads — accounts for bending stress reversal due to shaft rotation (constant load direction and magnitude)
  • These factors apply to bending, torsion, or combined bending and torsion (the most common loading)
  • For significant axial loads, more complex formulas (e.g., from relevant engineering codes) should be used, including column effects for compression

Allowable Stresses and Factors of Safety

  • For steel shafts using the fundamental design approach:
    • Bending (tension or compression): the smaller of 40% f_y or 24% f_ult
    • Torsion (shear): the smaller of 30% f_y or 18% f_ult
  • Where:
    • f_y = yield strength
    • f_ult = ultimate tensile strength
  • The allowable shear stress is based on the assumption that shear strength ≈ 75% of tensile strength

Stress Concentration at Keyways

  • Keyways are one of the most important sources of stress concentration in shafts
  • Located where gears, sprockets, or pulleys are fitted — usually the most highly stressed locations
  • Design rule of thumb: allowable stresses with a keyway are 75% of allowable stresses without the keyway
  • For other stress concentrations (steps, holes), consult relevant engineering design standards and handbooks

Estimating Shaft Loads

1. Weight (Gravitational Load)
  • Applies when a heavy pulley, flywheel, or similar component is mounted on a non-vertical shaft
  • Causes a transverse bending force:

F=mgF = m \cdot g

  • Assumed: shaft supported by low-friction bearings (frictional torque negligible)
2. Chain Drive
  • Chain tension creates a transverse force on the shaft at the sprocket
  • Force at sprocket: F=T2+T1F = T_2 + T_1 — (Formula 5) — tight side + slack side tension
  • Torque at sprocket: T=(T2T1)d/2T = (T_2 - T_1) \cdot d/2 — (Formula 6)
  • Where d = pitch circle diameter (PCD) of the sprocket (in metres)
  • When transmitting power, slack side tension is usually negligible → T_1 ≈ 0
3. Belt Drive (Vee or Wedge)
  • Same approach as chain drive, but slack side tension is NOT zero (friction-dependent)
  • Formulas 5 and 6 apply for belt drives as well
  • For parallel belts: F = T_2 + T_1 (correct); for non-parallel belts: use vector sum (but scalar sum errs on the safe side)
  • When both T_2 and T_1 are unknown, use one of three methods:
Method (a): Assume Slack Side Tension
Belt Section Slack Side Tension (N)
SPZ 100
SPA 150
SPB 350
SPC 750
  • Based on mid-load power at 1000 rev/min, tension ratio 12:1, smallest recommended PCD for belt size
Method (b): Assume Belt Tension Ratio
Drive Ratio Belt Tension Ratio
1 16.3
2 12
3 10.4
4 9.5
5 9
6 8.6
  • Based on 90% of tension ratio at slip point on smaller pulley; wedge angle 38°, friction coefficient 0.3, centre distance = sum of pulley PCDs; centrifugal effects excluded
  • Linear interpolation can be used for intermediate values
Method (c): Assume a Drive Application Factor
  • If slack side tension were zero: F=2T/dF = 2T/d
  • Actual force is greater because T_1 ≠ 0 → apply factor f:

F=2fTdF = \frac{2 \, f \, T}{d}

  • For vee or wedge belt drives, f is typically taken as 1.5
  • For chain drives: if T_1 = 0, then f = 1
  • Note: f = 1.5 gives a result ~20% higher than other methods; f = 1.25 gives closer correlation
4. Gear Drive
  • Force on the shaft = resultant transverse force at the gear tooth contact point
  • Three force components:
    • F_t (tangential force): produces the torque; Ft=2T/dF_t = 2T/d — (Formula 8)
    • F_s (separating/radial force): keeps gears in mesh; acts through gear centrelines
    • F (resultant transverse force): vector sum of F_t and F_s
  • Pressure angle (θ): angle between F and F_t — typically 20° unless stated otherwise
Spur Gear Formulas

Fs=Fttanθ(Formula 9)F_s = F_t \cdot \tan\theta \qquad \text{(Formula 9)}

F=Ft2+Fs2(Formula 10)F = \sqrt{F_t^2 + F_s^2} \qquad \text{(Formula 10)}

Helical Gear Formulas
  • Helical gears have teeth cut at an angle (helix angle α) to the shaft axis
  • Stronger and quieter than spur gears, but produce an additional axial force

Fs=Fttanθcosα(Formula 11 — separating force)F_s = \frac{F_t \cdot \tan\theta}{\cos\alpha} \qquad \text{(Formula 11 — separating force)}

Fa=Fttanα(Formula 12 — axial force)F_a = F_t \cdot \tan\alpha \qquad \text{(Formula 12 — axial force)}

  • The resultant transverse force is still F=Ft2+Fs2F = \sqrt{F_t^2 + F_s^2}

Design Procedure

  1. Estimate all loads acting on the shaft (weight, drive forces, gear forces, etc.)
  2. Draw torque, shear force, and bending moment diagrams
    • Shear force diagram is optional but useful to draw before the bending moment diagram
  3. Identify the critical location — position of maximum combined stress (usually where torque and bending moment are both at maximum, typically at gear/sprocket/pulley locations)
  4. Determine the design torque and moment by applying shock/fatigue factors (K_T, K_M)
  5. Calculate equivalent torque (T_E) and equivalent moment (M_E)
  6. Calculate allowable stresses (with keyway reduction if applicable)
  7. Determine minimum shaft diameter using Formulas 1 and 2
  8. Select the closest standard shaft size (round up)
Single-Plane vs Multi-Plane Bending
  • Single-plane: all resultant transverse forces act in the same plane → one bending moment diagram needed
  • Multi-plane: transverse forces act in different planes (e.g., horizontal and vertical) → draw bending moment diagrams for each plane, then combine:

M=Mv2+Mh2(Formula 13 — resultant bending moment)M = \sqrt{M_v^2 + M_h^2} \qquad \text{(Formula 13 — resultant bending moment)}

  • Where M_v = vertical plane moment, M_h = horizontal plane moment
Design Notes
  • Treatment excludes significant axial loads — in most shafts, direct axial stress is small relative to bending and torsional stresses
  • Examples use single-diameter shafts; the same principles apply to stepped shafts — each step diameter is determined from the maximum stress at that section
  • For stepped shafts: apply a stress-concentration factor at each step (depends on ratio of diameters and internal radius)

Rolled Steel Sections

Overview
  • Hot rolled sections are available in standard profiles: universal beams, universal columns, parallel flange channels, equal/unequal angles, and merchant bar (rounds, squares, flats)
  • Hot rolled sections have a commercial finish — not suitable for rotating shafts (use bright steel for shafts)
  • Available in several grades:
Grade Minimum Yield (MPa) Minimum UTS (MPa)
250 250 410
300 plus 300 440
350 350 480
Beam Selection Procedure
  1. Determine the maximum bending moment (M) from loading and span
  2. Calculate the allowable bending stress using the design factor:
    • e.g., for a design factor of 2 on yield: fb=fy/2f_b = f_y / 2
  3. Calculate the required section modulus: Z=M/fbZ = M / f_b
  4. Select the smallest standard section with Z ≥ required Z from beam tables
  5. Check self-weight: recalculate reactions, moment, and Z including beam self-weight
  6. Verify the selected section is still adequate
Column Selection Procedure
  1. Determine the effective length (L_e) based on end conditions:
    • Both ends pinned: L_e = L
    • One fixed, one free (cantilever): L_e = 2L
    • One fixed, one pinned: L_e = 0.7L
    • Both ends fixed: L_e = 0.5L
  2. Calculate the design critical load = applied load × design factor
  3. Calculate the limiting slenderness ratio:

(Ler)lim=2π2Efy\left(\frac{L_e}{r}\right)_{lim} = \sqrt{\frac{2\pi^2 E}{f_y}}

  1. Select a trial section from column tables; use the smaller radius of gyration (r_y) for buckling analysis
  2. Calculate actual Le/rL_e/r and compare with limiting value
  3. If Le/rL_e/r > limiting value → slender column → use Euler formula:

Fcr=π2EA(Le/r)2F_{cr} = \frac{\pi^2 E A}{(L_e/r)^2}

  1. Check that FcrF_{cr} ≥ design critical load; iterate if necessary


Shaft Design Approaches Compared

Feature Approach 1 (Endurance-Based) Approach 2 (Strength-Based)
Basis Endurance limit from fatigue testing Basic strength of materials
Factor of Safety Small (~1.2) Relatively generous
Load Handling Peak loads calculated accurately Maximum likely operating loads + factors
Complexity Complex formulas Simpler, more transparent formulas
Understanding May use formulas with unstated assumptions Better understanding of stress state
Standards Based on rotating shaft design standards Based on engineering code methodology
Best For High-cycle fatigue-critical applications General shaft design

Shaft Failure Modes Compared

Failure Mode Cause Likelihood Prevention
Excessive Load Stress exceeds yield Rare (load limiters fitted) Shear pins, overload protection
Fatigue Cyclic stress reversal below yield Most common Design below endurance limit; minimise stress concentrations

Belt Load Estimation Methods Compared

Method Input Required Accuracy Notes
(a) Assume T_1 Belt section type Moderate Uses standard slack side tension values
(b) Assume tension ratio Drive ratio Moderate Based on near-slip conditions
(c) Application factor Factor f Approximate f = 1.5 typical; overstates by ~20% vs other methods

Spur vs Helical Gear Forces

Parameter Spur Gear Helical Gear
Tangential force (F_t) 2T/d 2T/d
Separating force (F_s) F_t · tan θ F_t · tan θ / cos α
Axial force (F_a) None F_t · tan α
Resultant transverse (F) √(F_t² + F_s²) √(F_t² + F_s²) — very similar to spur
Noise Higher Lower
Strength Lower Higher


Shaft Design Process

flowchart TD
    A[Identify All Shaft Loads] --> B[Estimate Load Magnitudes]
    B --> C{Load Type?}
    C -->|Weight| D[F = mg]
    C -->|Chain Drive| E["F = T₂ + T₁ <br/> T = (T₂ - T₁) · d/2"]
    C -->|Belt Drive| F[Use Method a, b, or c]
    C -->|Gear Drive| G["F_t = 2T/d <br/> F_s = F_t · tan θ <br/> F = √(F_t² + F_s²)"]
    D --> H[Draw Shear Force & Bending Moment Diagrams]
    E --> H
    F --> H
    G --> H
    H --> I[Identify Critical Location]
    I --> J["Apply Shock/Fatigue Factors <br/> T = K_T · T_S <br/> M = K_M · M_S"]
    J --> K["Calculate T_E = √(T² + M²) <br/> M_E = 0.5(T_E + M)"]
    K --> L["Calculate Allowable Stresses <br/> Bending: min(0.4·f_y, 0.24·f_ult) <br/> Shear: min(0.3·f_y, 0.18·f_ult)"]
    L --> M{Keyway Present?}
    M -->|Yes| N[Multiply Allowable Stresses × 0.75]
    M -->|No| O[Use Full Allowable Stresses]
    N --> P["Solve for d from: <br/> f_s = 16·T_E / (π·d³) <br/> f = 32·M_E / (π·d³)"]
    O --> P
    P --> Q[Select Larger Diameter <br/> Round Up to Standard Size]

Shaft Load Classification

flowchart LR
    A[Shaft Loads] --> B[Steady Loads]
    A --> C[Shock Loads]
    A --> D[Inertia Loads]
    B --> B1[Torsional]
    B --> B2[Bending]
    B --> B3[Axial]
    C --> C1[Intermittent <br/> Sudden Increase]
    D --> D1[Start-Up / Shut-Down]
    D1 --> D2["Soft-Start: 1.5–2× rated"]
    D1 --> D3["Hard-Start: 3–5× rated"]

Shaft Failure Decision Tree

flowchart TD
    A[Shaft Under Load] --> B{Stress > Yield?}
    B -->|Yes| C[Excessive Load Failure]
    C --> C1[Permanent Deformation]
    C1 --> C2{Shaft Broken?}
    C2 -->|No| C3[May Still Be Serviceable]
    C2 -->|Yes| C4[Replace Shaft]
    B -->|No| D{Cyclic Stress Reversal?}
    D -->|Yes| E{Stress > Endurance Limit?}
    E -->|Yes| F[Fatigue Failure Over Time]
    E -->|No| G[Infinite Life — No Failure]
    D -->|No| G

Multi-Plane Bending Resolution

flowchart TD
    A[Forces on Shaft in Multiple Planes] --> B[Resolve into Vertical & Horizontal Components]
    B --> C[Draw Vertical Plane BM Diagram → M_v]
    B --> D[Draw Horizontal Plane BM Diagram → M_h]
    C --> E["Resultant: M = √(M_v² + M_h²)"]
    D --> E
    E --> F[Proceed with Design Using Resultant M]

Beam Selection Flowchart

flowchart TD
    A[Given: Span, Loading, Grade, Design Factor] --> B[Calculate Max Bending Moment M]
    B --> C["Allowable Stress f_b = f_y / Design Factor"]
    C --> D["Required Z = M / f_b"]
    D --> E[Select Smallest Section with Z ≥ Required]
    E --> F[Check Self-Weight]
    F --> G{Z Still Adequate?}
    G -->|Yes| H[Section Confirmed]
    G -->|No| I[Select Next Larger Section]
    I --> F


Key Terms Glossary

Term Definition
Shaft A rotating member supported by bearings that transmits torque and power
Axle A shaft to which wheels are attached
Steady Load A primary design load occurring continuously during operation
Shock Load An intermittent, sudden increase in load
Inertia Load A load arising from acceleration or deceleration of the shaft
Equivalent Torque (T_E) √(T² + M²) — combines torque and bending moment for shear stress calculation
Equivalent Moment (M_E) 0.5(T_E + M) — combines torque and bending moment for bending stress calculation
K_T Shock/fatigue factor applied to torsion
K_M Shock/fatigue factor applied to bending
Endurance Limit Maximum stress for infinite fatigue life under cyclic loading
Stress Concentration Localised stress increase at geometric discontinuities
Keyway A groove cut in the shaft to accept a key for torque transmission; major source of stress concentration
PCD (Pitch Circle Diameter) The effective diameter of a gear, sprocket, or pulley used in force/torque calculations
Pressure Angle (θ) Angle between the tangential and resultant forces at a gear tooth; typically 20°
Helix Angle (α) Angle of tooth cut relative to the shaft axis in helical gears
Tangential Force (F_t) Force at the gear tooth that produces torque; F_t = 2T/d
Separating Force (F_s) Radial force keeping meshing gears engaged
Drive Application Factor (f) Multiplier accounting for actual vs simplified transverse shaft load; typically 1.5 for belt drives
Yield Strength (f_y) Stress at which permanent deformation begins
Ultimate Tensile Strength (f_ult) Maximum stress a material can sustain before fracture
Section Modulus (Z) A geometric property of a cross-section relating bending moment to bending stress; Z = M/f_b
Radius of Gyration (r) A geometric property used in column buckling analysis; relates moment of inertia to cross-sectional area
Slenderness Ratio (L_e/r) Ratio of effective column length to radius of gyration; determines buckling behaviour
Euler Formula Critical buckling load formula for slender columns: F_cr = π²EA/(L_e/r)²
Universal Beam (UB) An I-shaped hot rolled section optimised for bending (deep, narrow flanges)
Universal Column (UC) An I-shaped hot rolled section optimised for axial compression (square-ish profile, wide flanges)


Shaft Design — Must-Know Formulas

  • Combined shear stress: fs=16TE/(πd3)f_s = 16 T_E / (\pi d^3)
  • Combined bending stress: f=32ME/(πd3)f = 32 M_E / (\pi d^3)
  • Equivalent torque: TE=T2+M2T_E = \sqrt{T^2 + M^2}
  • Equivalent moment: ME=0.5(TE+M)M_E = 0.5(T_E + M)
  • Design torque: T=KTTST = K_T \cdot T_S
  • Design moment: M=KMMSM = K_M \cdot M_S

Allowable Stresses (Steel Shafts)

  • Bending: smaller of 40% f_y or 24% f_ult
  • Shear: smaller of 30% f_y or 18% f_ult
  • With keyway: multiply both by 0.75

Shock/Fatigue Factors — Quick Reference

  • Static/gradual: K_T = 1.0, K_M = 1.5
  • Sudden + minor shock: K_T = 1.0–1.5, K_M = 1.5–2.0
  • Sudden + heavy shock: K_T = 1.5–3.0, K_M = 2.0–3.0

Shaft Load Formulas

  • Weight: F = mg
  • Chain/belt drive force: F = T_2 + T_1
  • Chain/belt torque: T = (T_2 - T_1) · d/2
  • Belt drive with factor: F = 2fT/d (f ≈ 1.5 for belt drives)
  • Gear tangential force: F_t = 2T/d
  • Spur gear separating force: F_s = F_t · tan θ
  • Helical gear separating force: F_s = F_t · tan θ / cos α
  • Helical gear axial force: F_a = F_t · tan α
  • Resultant gear force: F = √(F_t² + F_s²)
  • Multi-plane resultant moment: M = √(M_v² + M_h²)

Column Design — Quick Reference

  • Limiting slenderness ratio: (Le/r)lim=2π2E/fy(L_e/r)_{lim} = \sqrt{2\pi^2 E / f_y}
  • Euler critical load: Fcr=π2EA/(Le/r)2F_{cr} = \pi^2 E A / (L_e/r)^2
  • Use the smaller radius of gyration for buckling checks
  • If Le/rL_e/r > limiting value → slender → Euler applies

Key Design Reminders

  • K_M is never less than 1.5 (even for static loads) due to bending stress reversal from rotation
  • Fatigue is the most common shaft failure mode — not overload
  • Soft-start motors: 1.5–2× rated torque; hard-start: 3–5× rated torque
  • Shaft diameter is determined by the more critical of shear stress and bending stress — check both
  • For stepped shafts: apply stress concentration factors at each step
  • Hot rolled sections → not for rotating shafts; use bright steel instead



Rolling Element Bearings



Overview

  • Topic: Rolling element bearing selection, life calculation, and design methodology
  • Scope: Covers bearing types, load rating systems, life prediction methods (basic, adjusted, and advanced), bearing selection procedures for multiple bearing categories, and supporting reference data
  • Core Principle: Bearing life is a statistical estimate influenced by load, lubrication, cleanliness, and material — the more factors accounted for, the more accurate the prediction
  • Key Takeaway: Three progressively refined methods exist for calculating bearing life, each incorporating additional real-world factors beyond basic load capacity


Dynamic Load Rating and Bearing Life

  • Basic Dynamic Load Rating (C): The constant radial load under which a group of identical bearings will achieve a basic rating life of 1 million revolutions
  • L₁₀ Life: The life that 90% of a sufficiently large group of identical bearings can be expected to attain or exceed under given operating loads
  • The average life of a bearing is approximately 5 times the L₁₀ life
  • L₁₀ life is expressed in millions of revolutions

Three Methods for Determining Bearing Life

  • Method 1 — Basic L₁₀ Life: Accounts only for the loads on the bearing; simplest and most conservative
  • Method 2 — Adjusted Rating Life (Lna): Extends Method 1 by incorporating reliability, material type, and lubricant viscosity
  • Method 3 — New Life Theory (Lnaa): Further extends Method 2 by adding the concept of a fatigue load limit and contamination factor, enabling infinite life prediction under ideal conditions

Factors Affecting Bearing Life

  • Steel type: Standard bearing steels meet international specifications; premium steels can exceed standard life properties
  • Lubrication: Encompasses lubricant type (oil/grease), additives, viscosity, temperature, circulation method, and change intervals
  • Cleanliness: Encompasses environmental contaminants, particulate ingress, water contamination, lubricant filtration, and sealing method


Bearing Types Overview

  • 37 common types of rolling element bearings exist, categorised into radial and thrust families
  • Selection depends on factors such as: load direction, shaft speed, accuracy, noise, friction, self-alignment capability
Radial Bearings
  • Deep groove ball bearings — Single row, double row, with shields or seals, with snap ring groove in outer ring
  • Self-aligning ball bearings — Cylindrical or tapered bore, with seals, with extended inner ring
  • Angular contact ball bearings — Single row, paired mounting, precision, double row, four-point contact
  • Cylindrical roller bearings — Multiple sub-types (NJ, NJP, NNU, NN), single/double/four row, full complement
  • Needle roller bearings — Drawn cup, open/closed ends, with flanges, with/without inner ring, with seals
  • Spherical roller bearings — Cylindrical or tapered bore
  • Taper roller bearings — Single row, four row, crossed
Thrust Bearings
  • Thrust ball bearings — Single/double direction, with flat/spherical housing washers, with sealing rings
  • Angular contact thrust ball bearings — Single/double direction
  • Cylindrical roller thrust bearings
  • Needle roller thrust bearings
  • Spherical roller thrust bearings
  • Taper roller thrust bearings — Single/double direction

Engineering use and verification

Begin with load paths, motion, interfaces and credible failure modes. Define duty cycle, environment, alignment, lubrication, manufacturing variation and maintenance access before choosing a component. Check static strength, fatigue, stiffness, heat, wear and fastening together because improving one constraint can worsen another. Record assumptions and verify the assembled system, not just catalogue ratings for isolated parts.

  • Confirm scope, assumptions, interfaces and required outcome.
  • Use one controlled unit system and show every conversion.
  • Identify current project, customer and regulatory requirements.
  • Separate source examples from mandatory acceptance criteria.
  • Check calculations, tables and selections by an independent method.
  • Verify safety, maintainability and credible failure modes.
  • Record evidence, revisions, approvals and unresolved limitations.
  • Validate the result under representative operating conditions.

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