What is angle of twist?

Short Answer:

The angle of twist is the angular rotation of one end of a shaft relative to the other when it is subjected to a torque or twisting moment. It represents how much the shaft twists along its length due to applied torque.

In simple words, the angle of twist shows the degree of twisting or turning of a shaft when a torque acts on it. It depends on the applied torque, the shaft’s length and geometry, and the material’s modulus of rigidity. The higher the torque or the longer the shaft, the greater the angle of twist.

Angle of Twist

Detailed Explanation :

The angle of twist is an important concept in torsional analysis of circular shafts. It measures the rotational deformation that occurs when a torque acts on a shaft. Engineers use this concept to ensure that shafts, axles, and similar components do not twist excessively, which could cause misalignment, vibration, or failure in machines.

Definition of Angle of Twist

The angle of twist is defined as:

“The angle through which one end of a shaft rotates relative to the other end when it is subjected to a torque.”

It is denoted by the Greek letter θ (theta) and is measured in radians.

Mathematically, the angle of twist is expressed as:

where,

  •  = angle of twist (radians)
  •  = applied torque (N·m)
  •  = length of the shaft (m)
  •  = modulus of rigidity (N/m²)
  •  = polar moment of inertia of the cross-section (m⁴)

This formula is derived from the torsion equation and represents the elastic angular deformation of the shaft.

Physical Meaning

When a circular shaft is subjected to torque, every cross-section rotates about the axis of the shaft. However, the rotation is not the same along the length — one end may remain fixed while the other end rotates. The relative angular displacement between the two ends is called the angle of twist.

It is important to note that:

  • The angle of twist increases with applied torque (T) and length (L).
  • It decreases with higher rigidity (G) or larger polar moment of inertia (J).
    Thus, a stiffer or thicker shaft twists less for the same applied torque.

Derivation of Angle of Twist

From the torsion equation,

Rearranging for , we get:

This relation shows that the angle of twist is directly proportional to the applied torque  and shaft length , and inversely proportional to the material property  and cross-sectional property .

Hence,

  • Stronger materials with higher modulus of rigidity twist less.
  • Larger diameters (higher ) make the shaft more resistant to twisting.
  • Longer shafts experience greater angular deformation.

Units of Angle of Twist

The angle of twist is dimensionless because it is measured in radians, which is the ratio of arc length to radius. In engineering applications, it is often expressed in degrees or radians per meter for convenience.

Factors Affecting Angle of Twist

  1. Applied Torque (T):
    Greater torque produces greater twisting.
  2. Length of Shaft (L):
    The longer the shaft, the higher the total angular displacement.
  3. Material Property (G):
    Modulus of rigidity indicates material stiffness. Materials like steel (high G) twist less, while softer materials like aluminum (low G) twist more.
  4. Cross-sectional Shape and Size (J):
    Polar moment of inertia  depends on the geometry. A solid shaft has lower  than a hollow shaft of the same outer diameter, hence a greater twist for the same torque.

Angle of Twist for Different Sections

  1. Solid Circular Shaft:
    For a solid circular shaft of diameter :

Substituting in the main formula:

  1. Hollow Circular Shaft:
    For a hollow shaft with outer diameter  and inner diameter :

Hence,

This shows that for the same outer diameter, a hollow shaft can have almost the same strength with less material and less weight, making it more efficient.

Example (Conceptual)

Consider a steel shaft of 2 m length and 40 mm diameter transmitting a torque of 100 N·m.
Given ,

 

Thus, the shaft twists by a very small angle, showing how strong steel is in resisting torsion.

Importance of Angle of Twist

  1. Design Limitation:
    Every machine shaft has a maximum allowable angle of twist to prevent misalignment. For example, drive shafts and axles must not twist beyond permissible limits.
  2. Material Selection:
    Helps select materials with suitable rigidity (high ) for particular applications.
  3. Torsional Stiffness:
    The torsional stiffness of a shaft is given by .
    Larger  means higher resistance to twisting.
  4. Performance in Machinery:
    Reduces chances of vibration, noise, and fatigue in rotating components.
Conclusion

In conclusion, the angle of twist is the measure of rotational deformation in a shaft when torque is applied. It depends on torque, shaft length, cross-sectional geometry, and material stiffness. The equation  helps engineers design shafts that can safely resist twisting and maintain alignment in machinery. By controlling the angle of twist, the performance and durability of mechanical systems are significantly improved.