Short Answer:

Stiffness in torsion is the ability of a shaft or structural member to resist twisting when a torque is applied. It defines how much torque is needed to produce a certain angle of twist in the shaft. A shaft with high torsional stiffness twists less for the same applied torque, showing greater resistance to deformation.
In simple words, torsional stiffness indicates the rigidity of a shaft in twisting. It is an important property in mechanical design because it affects how efficiently torque and power are transmitted without excessive angular deflection.

Detailed Explanation :

Stiffness in Torsion

When a mechanical shaft or any circular component is subjected to torque, it tends to twist about its longitudinal axis. The resistance offered by the shaft to this twisting action is called its stiffness in torsion. It is a fundamental property in torsional mechanics that helps determine how much the shaft will deform when transmitting torque.

1. Meaning of Torsional Stiffness
   Torsional stiffness can be defined as the torque required to produce a unit angle of twist in a shaft of given length. It represents the ability of a material or structure to resist angular deformation when subjected to torsional load. Mathematically, it is expressed as:

where,
= Torque applied (in N·m),
= Angle of twist (in radians).

A shaft with greater torsional stiffness requires a higher torque to achieve the same amount of twist compared to one with lower stiffness. Hence, a stiffer shaft experiences less angular deformation and maintains better alignment in mechanical systems.

2. Relation Between Torque, Angle of Twist, and Shaft Properties
   From the torsion equation, we know:

Rearranging for torsional stiffness,

Here,
= Modulus of rigidity of the material,
= Polar moment of inertia of the shaft cross-section,
= Length of the shaft.

Thus, Torsional stiffness = .

This formula shows that torsional stiffness depends on three key factors:

• Material property (G): The higher the modulus of rigidity, the greater the stiffness.
• Geometrical property (J): The larger the polar moment of inertia, the greater the resistance to twist.
• Length (L): Longer shafts have lower stiffness because the same torque causes more twist over a greater length.

3. Factors Affecting Torsional Stiffness

• Material Used:
  Materials with high modulus of rigidity such as steel and titanium provide higher torsional stiffness compared to materials like aluminum or plastic.
• Shaft Diameter:
  Since the polar moment of inertia  for a circular shaft depends on the fourth power of the diameter, even a small increase in shaft diameter greatly improves torsional stiffness. For a solid shaft, .
• Length of Shaft:
  Longer shafts are more flexible and show less stiffness because twist increases proportionally with length.
• Cross-sectional Shape:
  A hollow circular shaft has a higher stiffness-to-weight ratio compared to a solid shaft because it retains the outer, highly stressed region while removing the less stressed core.

4. Physical Significance of Torsional Stiffness
   Torsional stiffness plays a crucial role in mechanical systems such as automobile drive shafts, turbine rotors, machine spindles, and power transmission components. High torsional stiffness ensures that:

• The shaft does not twist excessively during operation.
• The angular deflection between connected components remains minimal.
• Power is transmitted efficiently without delay or vibration.
• The mechanical system maintains precision and stability.

In contrast, low torsional stiffness can cause problems like misalignment, vibrations, fatigue, and mechanical failure, especially in high-speed rotating machines.

5. Practical Examples

• Automobile Driveshafts: Shafts must have high torsional stiffness to avoid delay in power delivery from the engine to the wheels.
• Machine Tool Spindles: These require high stiffness to maintain precision during machining.
• Propeller Shafts in Ships or Aircraft: They are designed for an optimal balance between strength, stiffness, and weight.

For example, if a driveshaft twists too much, the engine power may not be effectively transmitted to the wheels, leading to inefficient operation and wear.

6. Design Considerations
   When designing a shaft, both strengthand stiffnessare considered.

• Strength ensures the shaft does not fail under maximum torque.
• Stiffness ensures the shaft does not deform excessively, maintaining proper functionality.

Engineers often design shafts with a safety margin that ensures both adequate strength and sufficient torsional rigidity. Sometimes hollow shafts are used to increase stiffness-to-weight ratio, especially in applications where weight reduction is important.

7. Unit of Torsional Stiffness
   The unit of torsional stiffness is N·m per radian(Newton-meter per rad). It indicates the torque needed to cause a twist of one radian in the shaft.

For example, if the torsional stiffness of a shaft is , it means that a torque of 800 N·m is required to twist the shaft by 1 radian.

Conclusion:

Stiffness in torsion defines how strongly a shaft resists twisting when torque is applied. It depends on the material’s modulus of rigidity, shaft geometry, and length. Higher torsional stiffness means the shaft can transmit torque with minimal angular deflection, ensuring better performance and durability. It is a key parameter in designing mechanical components like shafts, axles, and couplings, where rigidity and accuracy are essential for efficient and safe operation.