Mechanical Design Data and Machine-Element Reference: . Overhung Load Calculation

. Overhung Load Calculation — Derivation

The overhung load formula is derived from belt/chain tension analysis:

• For a belt or chain drive on the gearbox output shaft with slack-side tension T₁ and tight-side tension T₂:
  • Overhung load: F = T₁ + T₂
  • Torque: T = (T₂ − T₁) × d/2
• If T₁ = 0 (zero slack-side tension): F = 2T/d
• If T₁ ≠ 0, introduce the drive application factor (f): F = 2fT/d
  • f accounts for the fact that the total belt/chain force exceeds the net tangential force
• Substituting power (P = Tω = T × πN/30):
  • F = (60 × f × P) / (π × d × N)

Drive application factors: | Drive Type | Factor (f) | |---|---| | Chain drive or toothed belt | 1.0 | | Gear drive | 1.25 | | Vee belt | 1.5 | | Flat friction belt | 2.0 |

For gear drives, the factor of 1.25 accounts for the pressure angle creating a separating force in addition to the tangential force, resulting in a resultant overhung load greater than the tangential component alone.

. Worked Example — Worm Gearbox Selection

Problem: An electric motor drives an overdriven worm gearbox. The output shaft has a chain pinion keyed to it, transmitting power via roller chain to a chain wheel on a non-uniformly loaded oven conveyor.

Given Data:

• Required power at oven conveyor chain wheel: 20 kW
• Chain wheel speed: 15 ± 0.5 rev/min
• Chain drive reduction ratio: 2:1
• Chain pinion PCD (keyed to gearbox output shaft): 270 mm
• Electric motor: 4-pole, full load speed 1460 rev/min
• Operating hours: 16 h/day average, continuous
• Maximum ambient temperature: 45°C
• Assumed chain drive efficiency: 96%
• Gearbox lubricant: Mineral oil

Solution (Step-by-Step):

1. Design output power = 20 / 0.96 = 20.83 kW; required gearbox output speed = 30 ± 1 rev/min (due to 2:1 chain reduction)
2. Reduction ratio = 1460 / 30 = 48.7:1 → single-reduction gearbox (ratio < 70)
3. Closest nominal ratio = 50
4. Nominal output speed = 1460 / 50 = 29.2 rev/min
5. Load classification = M (medium impulsive) — non-uniformly loaded oven conveyor
6. Service factor = 1.33 (interpolated for 16 h/day continuous operation with electric motor and medium impulsive load)
7. Output torque (mechanical) = P/ω = 20,830 / (π × 29.2/30) = 6813 Nm; Selection output torque = 1.33 × 6813 = 9084 Nm
8. Preliminary selection: From ratio 50/1 table at 1500 rev/min input → W12 (output torque mechanical = 9838 Nm > 9084 Nm)
9. Actual ratio = 50/1 (same as nominal for W12); output speed = 29.2 rev/min → within 30 ± 1 range ✓
10. Thermal check: Gearbox operates continuously at 45°C → thermal service factor = 1.485 (interpolated)
11. Selection output torque (thermal) = 6813 × 1.485 = 10,117 Nm; W12 thermal rating = 8156 Nm → W12 FAILS thermal check
    • Upgrade to W14 (thermal rating = 11,696 Nm > 10,117 Nm) ✓
    • Actual ratio for W14 at 50/1 is the same → output speed unchanged
12. Overhung load check (chain pinion on output shaft): f = 1 (chain drive), d = 0.270 m
    • OH load = (2 × 1 × 6813) / 0.270 = 50,470 N
    • Allowable OH load for W14 at ratio 50/1 and 1450 rev/min input = 99,100 N → OK ✓
13. Thrust load: No thrust load (chain drive, not helical gear) ✓
14. Efficiency (from W14 data table at 1500 rev/min, ratio 50): 84%
    • Design output torque = 6813 Nm; Design output power = 20.83 kW
    • Input power = 20.83 / 0.84 = 24.8 kW
    • Input torque = 6813 / (50 × 0.84) = 162 Nm (or equivalently: 24,800 / (π × 1460/30) = 162 Nm)

Summary Table:

15. Specification: Overdriven type, model designation TWO 14 (two-worm overdriven, size 14)
    • Nominal input shaft diameter: 75 mm
    • Nominal output shaft diameter: 120 mm
    • Side bolt hole centre distance: 597 mm (2 × dimension H)
    • End bolt hole centre distance: 431.8 mm (2 × dimension J)

Coupling Type Comparison

Single vs Double Reduction Gearboxes

Worm Gearbox Selection Flowchart

flowchart TD
    A[Step 1: Establish Mechanical Data<br/>Input/output torque, power, speed,<br/>duty cycle, ambient temp] --> B[Step 2: Calculate Reduction Ratio<br/>Ratio = Input Speed ÷ Output Speed]
    B --> C{Step 3: Ratio > 70?}
    C -- Yes --> D[Use Double-Reduction Gearbox]
    C -- No --> E[Use Single-Reduction Gearbox]
    D --> F[Step 3: Select Closest Nominal Ratio]
    E --> F
    F --> G[Step 4: Nominal Output Speed<br/>= Input Speed ÷ Nominal Ratio]
    G --> H[Step 5: Determine Load Classification<br/>S / M / H]
    H --> I[Step 6: Determine Service Factor<br/>from Table 2]
    I --> J[Step 7: Calculate Selection Capacity<br/>= Design Value × Service Factor]
    J --> K[Step 8: Preliminary Gearbox Selection<br/>Smallest gearbox exceeding selection capacity]
    K --> L[Step 9: Verify Actual Ratio & Output Speed<br/>Check tolerance range]
    L --> M{Step 10: Continuous<br/>operation?}
    M -- Yes --> N[Get Thermal Service Factor<br/>from Table 3]
    N --> O[Step 11: Check Thermal Rating<br/>Selection thermal ≤ Gearbox thermal?]
    O -- Fail --> P[Select Larger Gearbox<br/>or Add Auxiliary Cooling]
    P --> L
    O -- Pass --> Q{Step 12: External drive<br/>on output shaft?}
    M -- No --> Q
    Q -- Yes --> R[Calculate Overhung Load<br/>F = 2fT/d]
    R --> S{OH Load ≤<br/>Allowable?}
    S -- Fail --> T[Select Larger Gearbox<br/>or Use Layshaft]
    T --> L
    S -- Pass --> U{Step 13: Thrust<br/>load present?}
    Q -- No --> U
    U -- Yes --> V[Check Thrust Load<br/>≤ Allowable?]
    V -- Fail --> T
    V -- Pass --> W[Step 14: Determine Efficiency<br/>Calculate remaining unknowns]
    U -- No --> W
    W --> X[Step 15: Specify Gearbox<br/>Type, size, shaft diameters,<br/>mounting dimensions]

Coupling Selection Decision Tree

flowchart TD
    A[Start: Coupling Selection] --> B{Primary<br/>Requirement?}
    B -- High misalignment<br/>& vibration damping --> C[Tyre-Type Coupling]
    B -- Torsional rigidity<br/>& precision --> D[Disc-Type Coupling]
    B -- General purpose<br/>moderate loads --> E[Pin-Type Coupling]
    B -- Easy assembly<br/>& moderate loads --> F[Chain-Type Coupling]
    C --> G{Max Torque<br/>≤ 6270 Nm?}
    G -- Yes --> H[Select from Tyre<br/>Coupling Range]
    G -- No --> I[Consider Pin-Type<br/>or alternative]
    D --> J{Max Speed<br/>≤ 2900 RPM?}
    J -- Yes --> K[Select from Disc<br/>Coupling Range]
    J -- No --> L[Consider Pin-Type<br/>for higher speeds]
    E --> M[Select from Pin-Type<br/>Coupling Range]
    F --> N[Select from Chain-Type<br/>Coupling Range]
    H --> O[Select Taper Bush<br/>& Verify Bore Range]
    K --> O
    M --> O
    N --> O
    I --> E
    L --> E
    O --> P[Verify Key & Keyway<br/>Dimensions]

Power Transmission System Overview

flowchart LR
    A[Prime Mover<br/>Electric Motor /<br/>IC Engine] -->|Input Shaft| B[Worm Gearbox<br/>Speed Reduction<br/>Torque Multiplication]
    B -->|Output Shaft| C[Coupling<br/>Misalignment<br/>Compensation]
    C --> D[Driven Machine<br/>Conveyor / Pump /<br/>Mixer / etc.]

    B -.->|Overhung Load| E[Chain / Belt /<br/>Gear Drive]
    E --> D

    style A fill:#f0f0f0,stroke:#333
    style B fill:#f0f0f0,stroke:#333
    style C fill:#f0f0f0,stroke:#333
    style D fill:#f0f0f0,stroke:#333
    style E fill:#f0f0f0,stroke:#333

Key Terms Glossary

• Bore: The internal diameter of a coupling hub or bush that fits onto the shaft
• Centre Distance: The distance between the centreline of the worm shaft and the centreline of the wheel shaft in a gearbox; used as the gearbox size designation
• Double Reduction: A gearbox arrangement using two stages of worm/wheel reduction to achieve very high ratios (75:1 and above)
• Drive Application Factor (f): A multiplier applied in overhung load calculations to account for the type of drive mechanism (chain, gear, belt, etc.)
• End Float: The maximum permissible axial movement of a coupling hub relative to its mating half
• Force-Feed Lubrication: A pressurised oil supply system required when gearbox ratings exceed the sump-lubrication capacity (indicated by shaded areas in data tables)
• Keyway: A machined slot in a shaft and hub into which a key is fitted to transmit torque and prevent relative rotation
• Load Classification: Categorisation of the driven machine as Steady (S), Medium Impulsive (M), or Highly Impulsive (H), used to determine the service factor
• Nominal Ratio: The catalogue or labelled reduction ratio of a gearbox; may differ slightly from the actual (exact) ratio
• Overhung Load: The radial force acting on the gearbox output shaft due to belt/chain/gear tension from an externally mounted drive mechanism
• PCD (Pitch Circle Diameter): The effective diameter of a sprocket, pulley, or gear at which the driving force acts
• Service Factor: A multiplier applied to the design load to account for operating conditions (load type, duty cycle, prime mover characteristics)
• Setting Width: The axial distance between the two half-bodies of a coupling at the correct installed position
• Single Reduction: A gearbox arrangement using one worm/wheel stage, suitable for ratios up to 70:1
• Taper Bush: A standardised conical locking device used to secure hubs (couplings, sprockets, pulleys) onto shafts using a keyway and set screws
• Thermal Rating: The maximum continuous power or torque a gearbox can transmit without overheating, limited by heat dissipation capacity
• Thermal Service Factor: A multiplier that adjusts the thermal rating requirement based on the ambient temperature around the gearbox
• Thrust Load: An axial force acting along the output shaft, typically caused by helical gears or other mechanisms
• Torsional Stiffness: The resistance of a coupling to angular deflection under torque, measured in Nm/° — higher values indicate a more rigid coupling
• Wormshaft: The screw-like input element of a worm gearbox
• Wormwheel: The gear element of a worm gearbox that meshes with the worm; typically made of phosphor bronze

Quick Revision

• Four coupling types covered: Pin-type (highest torque/speed range), Tyre-type (best misalignment/damping), Disc-type (best torsional rigidity), Chain-type (easiest assembly)
• Taper bushes provide the standardised shaft-to-hub interface; always specify both bush number and bore size
• Key dimensions are determined by shaft diameter according to the standard table
• Worm gearboxes: Single reduction up to 70:1; double reduction from 75:1 to 4900:1
• Seven gear sizes available, designated by nominal centre distance
• Selection is a 15-step process: Data → Ratio → Nominal ratio → Output speed → Load class → Service factor → Selection capacity → Preliminary selection → Verify ratio/speed → Thermal check → OH load check → Thrust check → Efficiency → Calculate unknowns → Specify
• Service factor depends on three variables: prime mover type, load classification (S/M/H), and duty duration
• Thermal check is critical for continuously operating gearboxes — the thermal rating often governs selection over the mechanical rating
• Overhung load formula: F = 2fT/d = 60fP/(πdN) — always check against allowable values
• Drive application factors: Chain = 1.0, Gear = 1.25, Vee belt = 1.5, Flat belt = 2.0
• Efficiency ranges from ~84% (high ratio, small gearbox) to ~96% (low ratio, large gearbox)
• If OH load exceeds allowable: Use an intermediate layshaft with its own bearings and a flexible coupling to the gearbox to decouple the radial load
• Higher gearbox ratings achievable via synthetic oils, oil coolers, high-tensile shafts, and dual keys

POWER TRANSMISSION: BELTS & CHAINS

Chain Drives & Couplings

Overview

• This note covers the design, selection, and maintenance of two critical mechanical power transmission systems: chain drives and shaft couplings
• Chain drives transmit power between parallel shafts using a roller chain engaged with toothed sprockets
• Couplings connect two shafts end-to-end, transmitting torque while accommodating various degrees of misalignment
• Both systems are fundamental to industrial machinery, conveyor systems, and general mechanical power transmission
• The note also briefly covers taper lock bushes used to secure sprockets and couplings to shafts
• Content is drawn from a mechanical design data manual and includes selection procedures, rating charts, sprocket data, lubrication methods, and coupling comparison tables

Key Concepts

• Chain Pitch — the distance between adjacent chain link pins; the primary dimension used to classify chain size
• Drive Ratio — the ratio of driven sprocket teeth to driver sprocket teeth (i = Z₂ / Z₁)
• Selection Power — the adjusted power value used to select a chain from rating charts, calculated by applying service factors to the actual transmitted power
• Application Factor (f₁) — accounts for dynamic overloads based on driver and driven machine characteristics
• Tooth Factor (f₂) — modifies selection power based on the number of teeth on the driver sprocket (referenced to a 19-tooth baseline)
• Bearing Pressure — the contact pressure between pin and bush surfaces; a key indicator of chain wear performance
• Taper Lock Bush — a tapered sleeve that grips a shaft when tightened, providing a secure and re-usable mounting for sprockets and couplings
• Misalignment — the deviation from perfect shaft alignment; classified as angular, axial (parallel), end float, or torsional
• Equivalent Selection Power (Pₑ) — the adjusted power value used to select a coupling, normalised to a reference speed

Taper Lock Bushes

• Taper lock bushes are the quickest and simplest method of securing sprockets and couplings to shafts (both imperial and metric)
• The tapered surface of both bush and sprocket/coupling hub combine to create a load-bearing connection via the lock action of hardened high-tensile screws
• Taper bush ranges from different manufacturers are generally fully interchangeable
• When ordering, specify both the bush number and bore size required
• Available in metric bore sizes ranging from small (e.g., 9 mm) up to very large bores depending on bush series
• Standard bush series include: TB1008, TB1210, TB1215, TB1610, TB1615, TB2012, TB2017, TB2517, TB2525, TB3020, TB3030, TB3535, TB4040
• Each bush series accommodates a specific range of bore sizes and is matched to corresponding sprocket or coupling hub dimensions

Chain Drives — Introduction

• Two primary chain standards exist: national standard (metric) and imperial standard (inch-based)
• Chains manufactured to both standards are available; the metric standard is more commonly used in many regions
• There is no single unified national standard for roller chain in all countries — chains are made to comply with one or both international standards
• Chain size is specified by the chain pitch (distance between adjacent link pins)
• Both standards use imperial (inch) pitch sizes; metric pitch equivalents are listed in catalogues
• Chain may be specified by inch or mm pitch; all other catalogue dimensions are given in metric (mm) units

Types of Roller Chain

• Simple (Simplex) — single strand; most common for general power transmission
• Duplex — two parallel strands; higher load capacity
• Triplex — three parallel strands; highest standard load capacity
• Quadruplex — four parallel strands; for very heavy-duty applications
• Other varieties include: straight-sided chain, double pitch chain, cranked link chain, plastic bush chain, O-ring chain, hollow bearing pin chain, and sidebow chain
• The most common type for power transmission is the high-waisted precision steel roller chain
• Chains used for conveying rather than power transmission have different data and attachments

Roller Chain Construction

• A precision steel roller chain consists of a series of journal bearings held in precise relationship by constraining link plates
• Each bearing consists of a bearing pin and bush on which the chain roller revolves
• The bearing pin and bush are case-hardened to allow articulation under high pressures
• The design permits both load-carrying pressures and gearing action to be transmitted via the chain rollers
• Chains are classified according to pitch, roller diameter, and width between inner plates
• These dimensions collectively determine the form and width of the sprocket teeth

Chain Drive Expected Life

• When properly selected, installed, lubricated, and maintained with loads not exceeding design values, a quality roller chain is expected to have an operating life of 8 million cycles or 15,000 hours
• The rating charts in selection procedures assume a minimum life expectancy of 15,000 hours with proper installation and lubrication

Standards Reference Guide

Chain Drive Selection Method

Symbols, Terms, and Units

• Z₁ = Number of teeth on driver sprocket (pinion)
• Z₂ = Number of teeth on driven sprocket (wheel)
• C = Centre distance (mm)
• P = Chain pitch (mm)
• i = Drive ratio
• L = Chain length (pitches)

Selection Summary (Step-by-Step)

Step 1 — Select Drive Ratio and Sprockets

• Use standard sprocket sizes to choose a ratio based on available tooth counts
• Best practice: use an odd number of teeth combined with an even number of chain pitches
• Ideally, sprockets should have a minimum of 19 teeth
• If operating at high speed or subject to impulsive loads, the smaller sprocket should have at least 25 teeth and should be hardened
• Maximum recommended teeth on any sprocket: 114 teeth
• Drive ratio formula: i = Z₂ / Z₁
• For large ratio drives, check that the angle of lap on Z₁ is not less than 120°

Step 2 — Establish Application Factor (f₁)

• f₁ accounts for dynamic overloads depending on the characteristics of both the driver and driven machines
• Can be selected directly or by analogy using the application factor chart

Driver classifications:

• Smooth Running — electric motors, steam & gas turbines, internal combustion engines with hydraulic coupling
• Slight Shocks — internal combustion engines with 6+ cylinders, or engines with mechanical coupling, electric motors with frequent starts
• Moderate Shocks — internal combustion engines with fewer than 6 cylinders, with mechanical coupling

Step 3 — Tooth Factor (f₂)

• The tooth factor modifies the final power selection based on the size of the driver sprocket
• Formula: f₂ = 19 / Z₁
• The rating curves in standard charts are based on a 19-tooth sprocket; smaller sprockets increase the chain load per tooth

Step 4 — Calculate Selection Power

$$\text{Selection Power} = \text{Power (kW)} \times f_1 \times f_2$$

• This adjusted power value is then used with the rating charts

Step 5 — Select Chain Drive

• From the rating chart, select the smallest pitch of simple chain that can transmit the selection power at the speed of the driver sprocket Z₁
• This normally results in the most economical drive
• If the selection power exceeds the capacity of the simple chain, consider a multiplex chain (duplex or triplex) of the same pitch size

Step 6 — Calculate Chain Length

• Chain length in pitches (L) for a two-point drive at any centre distance:

$$L = \frac{Z_1 + Z_2}{2} + \frac{2C}{P} + \frac{\left(\frac{Z_2 - Z_1}{2\pi}\right)^2 \times P}{C}$$

• Round up the calculated number of pitches to a whole number of even pitches
• Odd numbers of pitches require a cranked link (offset link), which is not recommended
• If a jockey sprocket is used for adjustment, add two pitches to the chain length
• C is the contemplated centre distance in mm; should generally be between 30–50 pitches
• Example: For 1/2″ pitch chain, C = 1.5 × 25.4 × 40 = 1524 mm

Step 7 — Calculate Exact Centre Distance

• The actual centre distance for the chain length (L) calculated above will generally be greater than originally contemplated
• Revised centre distance formula:

$$C = \frac{P}{8} \left[ 2L - Z_1 - Z_2 + \sqrt{(2L - Z_2 - Z_1)^2 - \frac{\pi}{3.88}(Z_2 - Z_1)^2} \right]$$

Where:

• P = Chain pitch (mm)
• L = Chain length (pitches)
• Z₁ = Number of teeth in driver sprocket
• Z₂ = Number of teeth in driven sprocket

Step 8 — Choose Lubrication Method

• The recommended lubrication method is based on chain speed and power transmitted, found in the rating charts

Rating Chart Construction

• Rating charts appear complex but are constructed from three simple lines:
  • Link plate fatigue line — dominates at lower speeds (failure if maximum power recommendation is exceeded)
  • Pin galling line — occurs due to boundary lubrication breakdown at very high speeds
  • Bush and roller fatigue curve — at the intersection of the fatigue and galling lines, this curve dominates
• The rounded tops of each selection curve account for these intersections
• For driver sprocket speeds less than 10 rpm: multiply the transmitted power by 10/n and read from the 10 rpm column (where n = driver sprocket speed)
• 1 Kilowatt = 1.34 hp

Chain Suspension Force

• The force between one link and the next due to chain mass is small and is internally balanced within the chain
• This causes the chain to adopt a sagging catenary shape between the sprockets
• Allowance must be made in installation for slightly different postures adopted by the chain between zero and maximum load

Lubrication

• Chain drives must be protected against dirt and moisture
• Use good quality non-detergent mineral-based oil
• A periodic change of oil is desirable
• Heavy oils and greases are generally too stiff to enter chain working surfaces and should not be used
• The lubricant must reach the bearing areas of the chain by being directed between the inner and outer link plates, preferably at the point where the chain enters the sprocket on the bottom strand

Lubricant Viscosity by Temperature

• For the majority of applications, a multigrade oil (e.g., SAE 20/50) would be suitable

Use of Grease

• Grease lubrication is not recommended; if used, the following conditions apply:
  • Limit chain speed to 4 m/s
  • Normal greases applied to the outside surfaces only seal the bearing surfaces and will not work — causes premature failure
  • Grease must be heated until fluid and the chain immersed and allowed to soak until all air bubbles cease
  • Regular cleaning and regreasing at intervals is required depending on power and speed
  • Temperatures above 80°C will cause damage to many greases and reduce effectiveness

Abnormal Ambient Temperatures

• For elevated temperatures up to 250°C, use dry lubricants such as colloidal graphite, MoS₂ in white spirit, or poly-alkaline glycol carriers
• For low temperatures between −5°C and −40°C, special low-temperature initial greases and subsequent oil lubricants are necessary

Four Lubrication Methods

Effect of Temperature on Chain Drives

• Chain and chaincase temperatures during operation are an important control factor
• Depending on severity of use, special attention to lubrication method may be required
• Chain temperatures above 100°C should be avoided if possible
• Chain can generally give acceptable performance up to around 250°C in some circumstances
• Improving lubrication cooling effectiveness: increase oil volume to up to 4.5 litres per minute per chain strand and incorporate external cooling for the oil

Bearing Pressures

• When a chain has been correctly selected, the expected failure mode over a very long period is wear
• A key indicator of likely wear performance is bearing pressure — the magnitude of contact pressure between the key mating surfaces (pin and bush)
• Bearing pressure is calculated by dividing the working load by the bearing area
• Bearing areas for standard chains are quoted in manufacturer designer data

Bearing Pressure vs Chain Velocity (General Guidance)

• These values give an indication only and should not replace the standard chain selection methods

Transmission Sprockets — Data Summary

• Sprocket data is provided for various standard pitches
• Materials available: Steel (standard) and Heavy Duty Cast Iron
• Bore types: Plain bore and Taper bore (using taper lock bushes)
• Chain configurations: Simple, Duplex, and Triplex

Sprocket Data in the supplied reference

• Sprockets are available with welded hubs for certain larger sizes
• Rebore, keyway, and setscrew modification services are available from manufacturers