Define angle of twist.

Short Answer:

The angle of twist is the angle through which one end of a shaft rotates relative to the other end when the shaft is subjected to a torque or twisting moment. It represents the amount of rotational deformation produced due to torsion. The greater the torque or the longer the shaft, the larger the angle of twist, depending on the material and cross-section of the shaft.

The angle of twist is an important factor in torsion analysis because it helps in determining the shaft’s stiffness and strength. It is directly proportional to the applied torque and length of the shaft and inversely proportional to the shaft’s rigidity and polar moment of inertia.

Detailed Explanation:

Angle of Twist

The angle of twist is defined as the angular displacement between two cross-sections of a shaft due to the application of a torque. When a shaft is subjected to a twisting moment, each cross-section of the shaft tends to rotate relative to the adjacent section, producing a helical deformation along its length. The amount of this rotation, measured in radians or degrees, is known as the angle of twist.

It represents how much a shaft twists under a given load and helps to evaluate whether the shaft can safely transmit the required torque without excessive deformation. The angle of twist plays a crucial role in the design of mechanical shafts used in engines, machines, and power transmission systems.

Mathematical Expression

For a circular shaft subjected to torsion, the angle of twist (θ) is given by:

Where:

  • θ = Angle of twist (radians)
  • T = Applied torque (N·m)
  • L = Length of the shaft (m)
  • J = Polar moment of inertia of the shaft’s cross-section (m⁴)
  • G = Modulus of rigidity or shear modulus of the material (N/m²)

This formula shows that the angle of twist increases with the torque (T) and length (L) of the shaft and decreases with the material rigidity (G) and cross-sectional strength (J).

Derivation Concept

Consider a circular shaft of radius r and length L fixed at one end and subjected to a torque T at the other. Due to this torque, shear stress and strain are induced in the shaft material. The shear strain (γ) at a radius r is related to the angle of twist by:

From the theory of elasticity, shear stress (τ) is given by:

Since torsional shear stress is also related to torque as:

Combining the two equations gives:

Rearranging, we obtain the final expression for the angle of twist:

This relationship forms the basis of torsion theory in solid mechanics.

Factors Affecting Angle of Twist

  1. Torque (T):
    A higher applied torque produces a greater angle of twist for the same shaft.
  2. Length of Shaft (L):
    The angle of twist increases with the length of the shaft, as twisting accumulates along the shaft’s length.
  3. Material Property (G):
    The modulus of rigidity (G) represents the material’s resistance to shear deformation. Materials with higher G (like steel) exhibit smaller twists.
  4. Cross-Section (J):
    The polar moment of inertia (J) depends on the shaft’s geometry. A larger J value (thicker or hollow shaft) results in a smaller angle of twist.

Significance of Angle of Twist

  • It helps in checking whether a shaft will perform satisfactorily under given working conditions.
  • In design, a maximum permissible angle of twist is specified to prevent misalignment or vibration in rotating machines.
  • It also provides an understanding of torsional stiffness, which is a measure of a shaft’s ability to resist twisting.

Practical Example

Suppose a steel shaft of 1.2 m length and 40 mm diameter is subjected to a torque of 150 N·m. The modulus of rigidity (G) for steel is 80 GPa. Using the formula, engineers can calculate the angle of twist to ensure that it remains within safe design limits for efficient operation.

Conclusion

The angle of twist is a fundamental concept in torsion mechanics that indicates how much a shaft twists when subjected to torque. It depends on the shaft’s geometry, material properties, and the applied load. Controlling the angle of twist ensures proper functioning of mechanical systems, preventing excessive deformation, vibration, and failure in rotating components.