1. Introduction

In machine design, keys, cotters, and couplings are essential elements used for power transmission and joining machine parts.

• Keys → Connect shaft and hub (gear, pulley)
• Cotters → Connect co-axial rods under axial load
• Couplings → Connect two shafts for power transmission

👉 Key idea:
These elements ensure safe transmission of torque and force without slipping or failure.

---

2. Design of Keys

2.1 What is a Key?

A key is a small metal piece inserted between a shaft and hub to prevent relative rotation and transmit torque.

---

2.2 Types of Keys

4

• Sunk Key (most common)
• Saddle Key
• Woodruff Key
• Feather Key
• Gib-head Key

---

2.3 Forces Acting on Key

• Shear force due to torque
• Crushing (compressive) force between key and hub

---

2.4 Design of Key

(a) Shear Failure

$$\tau = \frac{2T}{d \cdot l \cdot b}$$τ=d⋅l⋅b2T​

(b) Crushing Failure

$$\sigma_c = \frac{4T}{d \cdot l \cdot t}$$σc​=d⋅l⋅t4T​

Where:

• $T$T = Torque
• $d$d = Shaft diameter
• $l$l = Length of key
• $b$b = Width of key
• $t$t = Thickness of key

👉 Design condition:

• Shear stress ≤ allowable shear
• Crushing stress ≤ allowable compressive

---

2.5 Standard Proportions of Key

• Width $b = \frac{d}{4}$b=4d​
• Thickness $t = \frac{d}{6}$t=6d​

---

2.6 Failure of Keys

• Shear failure
• Crushing failure

---

3. Design of Cotters

3.1 What is a Cotter Joint?

A cotter joint is used to connect two co-axial rods subjected to axial tensile or compressive forces.

---

3.2 Types of Cotter Joints

4

• Socket and Spigot Joint
• Sleeve and Cotter Joint

---

3.3 Forces Acting

• Axial tensile or compressive force

---

3.4 Design of Cotter

(a) Tensile Failure of Rod

$$\sigma_t = \frac{P}{A}$$σt​=AP​

---

(b) Shear Failure of Cotter

$$\tau = \frac{P}{2bt}$$τ=2btP​

---

(c) Crushing Failure

$$\sigma_c = \frac{P}{bt}$$σc​=btP​

Where:

• $P$P = Load
• $b$b = Width of cotter
• $t$t = Thickness

---

3.5 Proportions of Cotter

• Thickness $t \approx 0.25d$t≈0.25d
• Width $b \approx 1.25d$b≈1.25d

---

3.6 Failure of Cotter Joint

• Shear of cotter
• Crushing of cotter
• Tearing of rod
• Failure of spigot/socket

---

4. Design of Couplings

4.1 What is a Coupling?

A coupling is used to connect two shafts for transmitting power.

---

4.2 Types of Couplings

4

(a) Rigid Coupling

• Muff (sleeve) coupling
• Flange coupling

(b) Flexible Coupling

• Bush pin coupling
• Oldham coupling
• Universal coupling

---

4.3 Design of Muff Coupling

• Torque transmitted:

$$T = \frac{\pi}{16} \tau d^3$$T=16π​τd3

• Key design same as shaft key

---

4.4 Design of Flange Coupling

(a) Shaft Design

$$T = \frac{\pi}{16} \tau d^3$$T=16π​τd3

---

(b) Bolt Design

$$T = n \cdot F \cdot R$$T=n⋅F⋅R

Where:

• $n$n = Number of bolts
• $F$F = Force on each bolt
• $R$R = Radius of bolt circle

---

(c) Flange Design

• Based on shear and crushing

4.5 Failure of Couplings

• Shear failure of bolts
• Key failure
• Shaft failure
• Flange failure

5. Design Considerations

• Strength and rigidity
• Material selection
• Safety factor
• Alignment
• Ease of assembly and maintenance

6. Applications

• Power transmission systems
• Automobiles
• Machine tools
• Marine and industrial machinery

7. Advantages

• Efficient power transmission
• Easy assembly/disassembly
• Reliable performance

---