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🔩 1. Helical Springs (Close-Coiled Compression Springs)

Purpose: Energy absorption, vibration damping, force maintenance.

Key Formulas:

1. Deflection (δ):

   $$\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}{\mathrm{<span\; style="background-color:\; white;">D</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">G</span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}$$

   • $F$ = Load, $D$ = Mean coil diameter, $d$ = Wire diameter.

   • $n$ = Active coils, $G$ = Modulus of rigidity.

2. Stiffness (k):

   $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">G</span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}{\mathrm{<span\; style="background-color:\; white;">8</span>}{\mathrm{<span\; style="background-color:\; white;">D</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\mathrm{<span\; style="background-color:\; white;">n</span>}}$$

3. Shear Stress (τ):

   $$\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}\mathrm{<span\; style="background-color:\; white;">D</span>}{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">w</span>}}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$

   • $K_w$ = Wahl’s factor:

     $${\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">w</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">C</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}\mathrm{<span\; style="background-color:\; white;">C</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">4</span>}}<span\; style="background-color:\; white;">+</span>\frac{\mathrm{<span\; style="background-color:\; white;">0.615</span>}}{\mathrm{<span\; style="background-color:\; white;">C</span>}}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">C</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">D</span>}}{\mathrm{<span\; style="background-color:\; white;">d</span>}}$$

Design Considerations:

• Spring Index (C): Optimal range = 5–12.

• End Types: Closed & ground (for stability), open ends (for minimal friction).

• Fatigue Life: Shot peening improves durability.

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🔧 2. Welded Joints

Types: Fillet, Butt, Lap, T-joint.

Key Concepts:

1. Fillet Weld Throat Thickness (t):

   $$\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">0.707</span>}<span\; style="background-color:\; white;">\times </span>\text{<span style="background-color: white;">leg size (a)</span>}$$

2. Weld Strength:

   $$\text{<span style="background-color: white;">Strength</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">t</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">L</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">\sigma </span>}}_{\text{<span style="background-color: white;">allowable</span>}}$$

   • $L$ = Weld length.

Design Tips:

• Static vs. Dynamic Loads: Use higher safety factors for dynamic loads.

• Avoid Eccentricity: Minimize bending stresses.

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🔩 3. Riveted Joints

Types: Lap Joint, Butt Joint (Single/Double strap).

Failure Modes & Formulas:

1. Shear Strength (per rivet):

   $$\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">n</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">A</span>}}_{\mathrm{<span\; style="background-color:\; white;">s</span>}}}$$

   • $A_s$ = Shear area of rivet.

2. Crushing Strength:

   $${\mathrm{<span\; style="background-color:\; white;">\sigma </span>}}_{\mathrm{<span\; style="background-color:\; white;">c</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">n</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">d</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">t</span>}}$$

   • $t$ = Plate thickness.

3. Efficiency (η):

   $$\mathrm{<span\; style="background-color:\; white;">\eta </span>}<span\; style="background-color:\; white;">=</span>\frac{\text{<span style="background-color: white;">Weakest of </span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">P</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">,</span>{\mathrm{<span\; style="background-color:\; white;">P</span>}}_{\mathrm{<span\; style="background-color:\; white;">s</span>}}<span\; style="background-color:\; white;">,</span>{\mathrm{<span\; style="background-color:\; white;">P</span>}}_{\mathrm{<span\; style="background-color:\; white;">c</span>}}<span\; style="background-color:\; white;">)</span>}{\text{<span style="background-color: white;">Strength of solid plate</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">100</span>}\mathrm{<span\; style="background-color:\; white;">\%</span>}$$

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⚙️ 4. Clutches (Friction Type)

Types: Single Plate, Multi-Plate, Cone, Centrifugal.

Key Formulas:

1. Torque Capacity (Uniform Pressure):

   $$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">W</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">R</span>}}_{\mathrm{<span\; style="background-color:\; white;">o</span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">R</span>}}_{\mathrm{<span\; style="background-color:\; white;">i</span>}}<span\; style="background-color:\; white;">)</span>$$

2. Torque Capacity (Uniform Wear):

   $$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">W</span>}}{\mathrm{<span\; style="background-color:\; white;">4</span>}}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">R</span>}}_{\mathrm{<span\; style="background-color:\; white;">o</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">R</span>}}_{\mathrm{<span\; style="background-color:\; white;">i</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span>$$

   • $R_o,R_i$ = Outer/Inner radii.

Design Focus:

• Heat Dissipation: Critical for high-speed applications.

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🛑 5. Brakes (Drum & Disc)

Types: Shoe, Band, Disc.

Key Formulas:

1. Braking Torque (Drum):

   $$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">W</span>}\mathrm{<span\; style="background-color:\; white;">R</span>}$$

2. Energy Dissipated:

   $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">I</span>}{\mathrm{<span\; style="background-color:\; white;">\omega </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

Design Tips:

• Self-Energizing: Leading shoe increases braking force.

• Material: Asbestos-free linings for heat resistance.

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🔗 6. Knuckle Joint

Purpose: Connect rods under tension with angular flexibility.

Failure Checks:

1. Pin Shear:

   $$\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}_{\mathrm{<span\; style="background-color:\; white;">p</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}$$

2. Crushing of Eye/Fork:

   $${\mathrm{<span\; style="background-color:\; white;">\sigma </span>}}_{\mathrm{<span\; style="background-color:\; white;">c</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{{\mathrm{<span\; style="background-color:\; white;">d</span>}}_{\mathrm{<span\; style="background-color:\; white;">p</span>}}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">t</span>}}$$

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🔩 7. Spigot & Socket Cotter Joint

Purpose: Axial force transmission (e.g., piston rods).

Failure Modes:

1. Tension in Rod:

   $${\mathrm{<span\; style="background-color:\; white;">\sigma </span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">4</span>}}$$

2. Cotter Shear:

   $$\mathrm{<span\; style="background-color:\; white;">\tau </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">F</span>}}{\mathrm{<span\; style="background-color:\; white;">b</span>}<span\; style="background-color:\; white;">\times </span>\mathrm{<span\; style="background-color:\; white;">t</span>}}$$

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⚙️ 8. Power Screw (Screw Jack)

Purpose: Convert rotary to linear motion.

Key Formulas:

1. Torque (Square Thread):

   $$\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">W</span>}{\mathrm{<span\; style="background-color:\; white;">d</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">tan</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">)</span>$$

2. Efficiency (η):

   $$\mathrm{<span\; style="background-color:\; white;">\eta </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">tan</span>}<span\; style="background-color:\; white;"></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}}{\mathrm{<span\; style="background-color:\; white;">tan</span>}<span\; style="background-color:\; white;"></span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}<span\; style="background-color:\; white;">+</span>\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">)</span>}$$

Self-Locking Condition:

$$\mathrm{<span\; style="background-color:\; white;">\varphi </span>}<span\; style="background-color:\; white;">></span>\mathrm{<span\; style="background-color:\; white;">\alpha </span>}$$

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🍃 9. Leaf Springs (Semi-Elliptical)

Purpose: Vehicle suspension.

Key Formulas:

1. Bending Stress:

   $${\mathrm{<span\; style="background-color:\; white;">\sigma </span>}}_{\mathrm{<span\; style="background-color:\; white;">b</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">6</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}\mathrm{<span\; style="background-color:\; white;">L</span>}}{\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">b</span>}{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

2. Deflection:

   $$\mathrm{<span\; style="background-color:\; white;">\delta </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">3</span>}\mathrm{<span\; style="background-color:\; white;">F</span>}{\mathrm{<span\; style="background-color:\; white;">L</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">E</span>}\mathrm{<span\; style="background-color:\; white;">b</span>}{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$

Design Tips:

• Pre-Stressing (Nip): Ensures even load distribution.

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📌 Summary Table: Critical Parameters

Component	Key Formula	Design Focus
Helical Spring	$\delta =\frac{8F{D}^{3}n}{G{d}^4}$	Fatigue life, spring index
Welded Joint	$\tau =\frac{F}{tL}$	Weld orientation, throat size
Clutch	$T=\mu W{R}_m$	Heat dissipation
Power Screw	$\eta =\frac{\tan \alpha }{\tan (\alpha +\varphi )}$	Self-locking condition

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🎯 Practical Tips:

• Springs: Use shot peening to enhance fatigue resistance.

• Joints: Avoid stress concentrations with smooth transitions.

• Brakes/Clutches: Prioritize materials with high $\mu$ and heat tolerance.