Machine Design — Bearings and Gears

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Bearings: Ball bearings (contact angle) and roller bearings (line contact). Bearing life L₁₀ = (C/P)^p where C = basic dynamic capacity, P = equivalent load, p = 3 for ball, 10/3 for roller. Higher C means longer life.

Gear Terminology: Module m = d/T = pitch diameter / teeth count. Always use module (not diametral pitch) in GATE. Pressure angle typically 20° (standard).

Gear Types: Spur (parallel axes, no thrust), Helical (thrust component), Bevel (intersecting axes), Worm (crossed axes, high reduction ratio).

Gear Trains: Train value = product of teeth ratios. For compound gear train, output direction is determined by number of intermediate (idler) gears. Train value > 1 means speed reduction.

Belts: Flat belt — V > V_flat due to friction (V-belt wraps tighter). V-belt service factor depends on load type.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Bearings

Types and Selection

Ball bearings carry combined radial and thrust loads. Key parameters:

• Contact angle (α): 0° = deep groove, 15–40° = angular contact
• Basic dynamic capacity (C): Load for 10⁶ revolutions to 10% failure (from manufacturer catalog)
• Equivalents load (P): X × F_r + Y × F_a (X = radial factor, Y = thrust factor)

Roller bearings carry higher loads at line contact:

• Cylindrical roller: Radial only, high capacity
• Tapered roller: Combined radial + thrust
• Needle roller: Maximum radial capacity per diameter

⚠️ GATE trap: Confusing static capacity (C₀) with dynamic capacity (C). C₀ is for non-rotating or slow-moving applications. C is for dynamically loaded rotating bearings.

Bearing Life Calculation

Basic rated life (L₁₀): 10⁶ revolutions at 90% survival rate.

L₁₀ = (C/P)^p

Bearing Type	Exponent p
Ball bearing	3
Roller bearing	10/3 ≈ 3.33

Equivalent load P: P = X × F_r + Y × F_a

For deep groove ball bearings:

• If F_a/F_r ≤ e (load ratio threshold): X=1, Y=0 → P = F_r
• If F_a/F_r > e: X=0.56, Y depends on F_a/C₀

Example: C = 22 kN, F_r = 5 kN, F_a = 2 kN Check ratio: 2/5 = 0.4. If e = 0.4 (typical), ratio equals e → use radial only. P = F_r = 5 kN. L₁₀ = (22/5)³ = (4.4)³ = 85.2 million revolutions

Bearing Mounting

Fixed-fixed: Maximum rigidity, both ends locked against axial movement. For long shafts. Fixed-free: One end fixed (against rotation and axial), other free to move axially. For short shafts. Floating-floating: Both ends allow axial float. Requires preloading.

Abutment dimensions: Shoulder diameter > inner ring bore, fillet radius must be accommodated.

Gear Geometry and Terminology

Standard Gear Terms

Term	Symbol	Formula	Description
Circular pitch	p	πm	Distance between adjacent teeth along pitch circle
Module	m	d/T	Size parameter (mm) — use this in GATE
Diametral pitch	P_d	T/d	Inverse of module — avoid in GATE
Pitch diameter	d	m × T	Reference circle diameter
Addendum	a	m	Tip of tooth above pitch circle
Dedendum	b	1.25m	Root below pitch circle
Whole depth	h	2.25m	a + b
Pressure angle	φ	20° (std)	Normal to tooth profile
Base circle	d_b	d cos φ	For involute generation

Gear ratio (i): i = T₂/T₁ = d₂/d₁ = N₁/N₂ (always > 1 for reduction)

Gear Tooth Geometry

For standard full-depth teeth:

• Addendum = m
• Dedendum = 1.25m (standard clearance = 0.25m)
• Working depth = 2m
• Whole depth = 2.25m

To avoid interference (in spur gears): Minimum teeth for rack with no interference: T_min = 2 × (1 + √(1+sin φ)/sin φ) × (for standard 20° gears, approximately 17 teeth at pinion)

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Gear Types

Spur Gears

• Teeth parallel to axis of rotation
• Line contact along full width (actually point contact due to elastic deflection)
• Only transmit motion between parallel shafts
• No axial thrust (no helical angle)
• Highest load capacity for given size (simple geometry)
• Noise and vibration at high speeds

Helical Gears

• Teeth cut at helix angle β to axis
• Gradual tooth contact — quieter, smoother
• Creates axial thrust force F_a = F_t × tan(β)
• Can transmit motion between parallel or crossed axes (spiral bevel)
• Higher load capacity due to larger contact ratio

Double helical (herringbone): Two helical gears mirrored to cancel axial thrust — but complex to manufacture.

Bevel Gears

• Teeth cut on a cone surface
• Transmit motion between intersecting shafts (typically 90°)
• Straight bevel: No overlap, simpler, for low-speed
• Spiral bevel: Curved teeth, smoother, for higher speeds
• Zerol bevel: Zero helix angle but curved — compromise

Force analysis for bevel gear (at pitch cone angle δ):

• Tangential: F_t = 2T/d_m (at mean diameter)
• Radial: F_r = F_t × tan(φ) × cos(δ)
• Thrust: F_a = F_t × tan(φ) × sin(δ)

Worm Gears

• Worm (screw) meshes with worm wheel
• High reduction ratio in single stage (up to 100:1)
• Sliding contact — low efficiency (typically 40–90%)
• Self-locking: Worm can drive wheel but not vice versa (if lead angle < friction angle)
• High heat generation due to sliding

Worm geometry: Lead = π × d_worm × tan(λ) where λ = lead angle Gear ratio = T_wheel / T_worm = 1 / tan(λ) = number of starts of worm

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Gear Trains

Simple Gear Train

All gears mounted on fixed shafts. Train value (velocity ratio): VR = N_input / N_output = T_driven / T_driving = product of driven/teeth of drivers

For gears 1-2-3-4 in sequence: VR = (T₂/T₁) × (T₃/T₂) × (T₄/T₃) = T₄/T₁ The intermediate gears cancel out (they rotate but don’t affect VR).

Compound Gear Train

For stepped reduction where shafts are fixed: VR = (T₂/T₁) × (T₄/T₃) for two stages.

Example: T₁=20, T₂=40, T₃=25, T₄=75 Stage 1: T₂/T₁ = 40/20 = 2 Stage 2: T₄/T₃ = 75/25 = 3 Total VR = 2 × 3 = 6 (speed reduced 6×)

Direction: Compound train with odd number of mesh points → output rotates opposite to input. Even number → same direction.

Epicyclic (Planetary) Gear Trains

This is the most challenging gear train type in GATE.

Key formula (Willis equation): (N_sun – N_arm)/(N_ring – N_arm) = (–T_ring/T_sun)

Where typically:

• N_sun = N₁ (sun gear)
• N_ring = N₃ (ring gear / internal gear)
• N_arm = N₄ (carrier arm)

When arm is fixed (holding): Use relative speeds. Let N_arm = 0. Then: (N_sun – 0)/(N_ring – 0) = –T_ring/T_sun

Step-by-step for epicyclic:

1. Identify which members are fixed, input, output
2. Apply Willis equation with arm as reference
3. Solve for unknown speed

⚠️ Common GATE mistake: Forgetting the negative sign in Willis equation. The ratio includes a minus because sun and ring rotate in opposite directions (internal mesh).

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Belt Drives

Flat Belts

Velocity ratio: V = π × d₁ × N₁ / 60 = π × d₂ × N₂ / 60 VR = N₁/N₂ = d₂/d₁ (approximately, for no slip)

Belt length (open belt): L = 2C + (π/2)(d₁+d₂) + (d₂–d₁)²/(4C) Where C = center distance

Tensions: Tight side T₁, Slack side T₂ Power P = (T₁ – T₂) × V where V is belt velocity (m/s) Maximum tension ratio: T₁/T₂ = e^(μθ) where θ = wrap angle in radians, μ = coefficient of friction

V-Belts

Advantage over flat belt: Higher power capacity, tighter wrap angle (up to 180° per sheave), self-centering.

V-shape increases normal force and friction: effective μ_eff = μ / sin(β) where β = V-groove angle (typically 20° for standard V-belts, so μ_eff ≈ 2.9 × μ)

Belt type selection from manufacturer catalogs based on:

• Design power = rated power × service factor
• Small sheave diameter (limits speed)
• Center distance and length

Service factors (typical):

Load Type	Service Factor
Uniform (fans, pumps)	1.2–1.3
Moderate shock (compressors)	1.4–1.6
Heavy shock (crushers)	1.7–2.0

💡 GATE Tip: Belt drive questions often ask about VR with slip, or about tension ratio. Remember that belt length formulas are rarely needed in GATE — focus on VR and tension calculations.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Bearing Steels and Heat Treatment

SAE 52100 (AISI 52100) — standard bearing steel:

• 1% Cr, 1% C
• Through-hardened to 60–65 HRC
• Used for balls, rollers, rings

Heat treatment stages:

1. Austenitizing: Heat to ~850°C
2. Quenching: Rapid cool to transform to martensite
3. Tempering: 150–200°C to relieve stresses and achieve target hardness

Failure modes:

• Spalling: Fatigue failure starting at subsurface — most common
• Brinelling: Static indentation from振动 or incorrect handling
• Abrasive wear: Contamination in lubricant
• Corrosion: Water or chemical contamination
• Electrical pitting: EDM damage from electric motors

Surface Fatigue in Gears

Gear tooth failure by fatigue (pitting) follows the Hertzian contact stress model:

σ_H = Z_E × √(F_t × K / (b × d × q)) [contact/pitting stress]

Where:

• Z_E = elastic coefficient (for steel–steel, Z_E ≈ 1898 MPa√mm)
• F_t = tangential load
• b = face width
• d = pinion pitch diameter
• q = contact ratio

Bending stress (Lewis equation): σ_b = F_t × K / (b × m × Y) where Y = Lewis form factor (depends on number of teeth and pressure angle)

Standard 20° full-depth teeth have Lewis form factor:

Teeth	Y (20° FD)
20	0.326
30	0.358
50	0.384

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Epicyclic Gear Train — Detailed Problem

Example: Sun gear (T=24) meshes with planet gears (T=12 each, 3 planets) which mesh with ring gear (T=48). Arm is fixed. Find output speed of ring gear when sun gear is input at 100 RPM.

Solution using Willis equation: Let arm fixed (N_arm = 0). Then Willis becomes: (N_sun – 0)/(N_ring – 0) = –T_ring/T_sun N_sun/N_ring = –48/24 = –2 N_ring = –N_sun/2 = –100/2 = –50 RPM

Negative sign means ring rotates opposite to sun. ✓

Alternative method — tabulation:

1. Column 1: Sun, Planet, Ring (fixed arm reference)
2. For each gear: teeth engagement = –(T_mesh/T_driver)
3. Solve systematically

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