4. TORSION IN SHAFTS AND SPRINGS

For 4th Semester Polytechnic ME Students
Written by Garima Kanwar | Blog: Rajasthan Polytechnic

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Course Code ME 4002 (Same in MA/MP/MT 4002)
Course Title STRENGTH OF MATERIALS

4. Torsion in Shafts and Springs

This chapter covers the concepts related to torsion in shafts and the design and analysis of springs. The main focus is on the behavior of shafts under torsional load, and the design of springs under various loading conditions.

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4.1 Definition and Function of Shaft

• Shaft Definition: A shaft is a mechanical component that transmits power, torque, or rotational motion between machine elements such as gears, pulleys, and flywheels. It is typically a long, cylindrical member subjected to torsion (twisting), bending, or axial loads.

• Function of Shaft:

  • To transmit torque from one part of the machine to another.
  • To support and rotate machine elements (e.g., gears, pulleys).
  • To provide rotational motion between two parts, such as in a gearbox or motor.

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4.2 Calculation of Polar Moment of Inertia (M.I.) for Solid and Hollow Shafts

• The polar moment of inertia $J$ is a geometric property that measures an object's ability to resist torsional deformation.

1. For a Solid Shaft: The polar moment of inertia for a solid circular shaft of radius $r$ is given by:

   $$J = \frac{\pi r^4}{2}$$

   where $r$ is the radius of the shaft.

2. For a Hollow Shaft: For a hollow shaft with an outer radius $r_o$ and an inner radius $r_i$, the polar moment of inertia is:

   $$J = \frac{\pi (r_o^4 - r_i^4)}{2}$$

   where $r_o$ is the outer radius and $r_i$ is the inner radius.

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4.3 Assumptions in Simple Torsion

• Simple Torsion Assumptions:
  1. The shaft material is homogeneous and isotropic (having uniform properties in all directions).
  2. The cross-section of the shaft remains plane and undistorted (the shaft does not deform in a way that affects its cross-section).
  3. The angle of twist is small.
  4. The torque is applied gradually and is constant along the length of the shaft.
  5. The shaft is subjected to pure torsion, with no axial or bending loads.
  6. There is no friction or wear in the shaft's bearing supports.

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4.4 Problems on Design of Shaft Based on Strength and Rigidity

• Design of Shaft Based on Strength: The design of a shaft is based on its ability to withstand the applied torque without failing. The maximum shear stress $\tau_{\text{max}}$ in the shaft should not exceed the allowable shear stress $\tau_{\text{allow}}$. The formula for calculating the shear stress in a shaft under torsion is:

  $$\tau = \frac{T \cdot r}{J}$$

  where:

  • $T$ is the applied torque,
  • $r$ is the radius of the shaft,
  • $J$ is the polar moment of inertia of the shaft.

• Design of Shaft Based on Rigidity: The rigidity of a shaft relates to its ability to resist deformation under torsion. The angle of twist $\theta$ for a shaft of length $L$ is given by:

  $$\theta = \frac{T \cdot L}{J \cdot G}$$

  where:

  • $T$ is the applied torque,
  • $L$ is the length of the shaft,
  • $J$ is the polar moment of inertia,
  • $G$ is the modulus of rigidity (shear modulus).

  The shaft design ensures that the angle of twist does not exceed the permissible value based on the system's requirements.

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4.5 Numerical Problems Related to Comparison of Strength and Weight of Solid and Hollow Shafts

• Strength Comparison of Solid and Hollow Shafts:

  • For a given torque, compare the maximum shear stress in a solid and hollow shaft with the same material and dimensions. This involves calculating the shear stress using the formula:
  $$\tau = \frac{T \cdot r}{J}$$

• Weight Comparison of Solid and Hollow Shafts:

  • The weight of the shaft depends on its volume. For a solid shaft:
  $$W_{\text{solid}} = \rho \cdot V = \rho \cdot (\pi r^2 L)$$
  

  For a hollow shaft:

  $$W_{\text{hollow}} = \rho \cdot V = \rho \cdot (\pi (r_o^2 - r_i^2) L)$$
  

  where:

  • $\rho$ is the density of the material,
  • $r_o$ is the outer radius,
  • $r_i$ is the inner radius,
  • $L$ is the length of the shaft.

  After calculating the weight and strength for both shafts, the design decision can be made based on the required performance criteria.

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4.6 Classification of Springs

Springs are classified based on their application, design, and shape. The main types include:

1. Tension Springs: Used to resist pulling forces. These springs are wound to stretch under load.
2. Compression Springs: Used to resist compressive forces. These springs are designed to compress under load.
3. Torsion Springs: Used to resist twisting forces. These springs exert a rotational force when twisted.
4. Leaf Springs: Often used in vehicle suspension systems to provide flexibility and load support.
5. Helical Springs: Wound in a coil shape and can be tension, compression, or torsion types.

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4.7 Deflection Formula for Closed Coil Helical Spring (Without Derivation)

• The deflection $\delta$ for a closed coil helical spring under an axial load $F$ is given by: $$\delta = \frac{64 F D^3 n}{G d^4}$$where:
  • $F$ is the applied load,
  • $D$ is the diameter of the spring coil,
  • $n$ is the number of coils,
  • $G$ is the modulus of rigidity (shear modulus),
  • $d$ is the diameter of the wire used to make the spring.

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4.8 Stiffness of Spring

• The stiffness of a spring is defined as the force required to produce a unit deflection.
• The spring stiffness $k$ is given by: $$k = \frac{F}{\delta}$$ where:
  • $F$ is the applied load,
  • $\delta$ is the deflection produced by the load.

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4.10 Numerical Problems on Closed Coil Helical Spring to Find Safe Load, Deflection, Size of Coil, and Number of Coils

• Problem 1: Safe Load and Deflection

  • Given the spring dimensions and material properties, calculate the safe load and deflection for a closed coil helical spring.
  • Example:
    • Load $F = 200 \, \text{N}$,
    • Coil diameter $D = 50 \, \text{mm}$,
    • Wire diameter $d = 10 \, \text{mm}$,
    • Number of coils $n = 10$,
    • Modulus of rigidity $G=80\text{GPa.}$
    Find the deflection using the formula: $$\delta = \frac{64 F D^3 n}{G d^4}$$

• Problem 2: Size of Coil and Number of Coils

  • Given a specific deflection and load requirement, determine the size of the coil and the number of coils for a closed coil helical spring.
  • Rearranging the deflection formula, solve for $D$ and $n$.

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Conclusion:

Torsion in shafts and the design of springs are crucial topics in mechanical engineering. Understanding the principles of torsion and the behavior of shafts under loads is necessary for designing reliable and efficient machines. Similarly, the design of springs involves calculating deflection, stiffness, and load capacity to ensure performance and safety. Numerical problems in these areas help reinforce the application of formulas and design considerations.

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