Modulus of Rigidity of Shaft Given Total Angle of Twist for Shaft Calculator

Formula Used:

\[ G = \frac{32 \cdot \tau \cdot L \cdot \left(\frac{1}{D_1^3} - \frac{1}{D_2^3}\right)}{\pi \cdot \theta \cdot (D_2 - D_1)} \]

Torque Exerted on Wheel (τ):

N·m

Length of Shaft (L):

Diameter of Shaft at Left End (D₁):

Diameter of Shaft at Right End (D₂):

Angle of Twist (θ):

rad

Modulus of Rigidity of Shaft (G):

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1. What is the Modulus of Rigidity?

The Modulus of Rigidity, also known as the shear modulus, is a measure of a material's stiffness when subjected to shear stress. It quantifies the relationship between shear stress and shear strain in a material, representing its resistance to deformation under shear forces.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ G = \frac{32 \cdot \tau \cdot L \cdot \left(\frac{1}{D_1^3} - \frac{1}{D_2^3}\right)}{\pi \cdot \theta \cdot (D_2 - D_1)} \]

Where:

\( G \) — Modulus of rigidity of the shaft (Pa)
\( \tau \) — Torque exerted on the wheel (N·m)
\( L \) — Length of the shaft (m)
\( D_1 \) — Diameter of shaft at left end (m)
\( D_2 \) — Diameter of shaft at right end (m)
\( \theta \) — Angle of twist (rad)
\( \pi \) — Mathematical constant (approximately 3.14159)

Explanation: This formula calculates the modulus of rigidity for a tapered shaft by considering the torque applied, shaft dimensions, and the resulting angle of twist.

3. Importance of Modulus of Rigidity Calculation

Details: Calculating the modulus of rigidity is crucial for designing mechanical components, predicting material behavior under shear stress, ensuring structural integrity, and selecting appropriate materials for specific applications in engineering and manufacturing.

4. Using the Calculator

Tips: Enter all values in appropriate units (torque in N·m, lengths in meters, angle in radians). Ensure all values are positive and that D₂ > D₁ for a valid calculation. The angle of twist should be measured in radians.

5. Frequently Asked Questions (FAQ)

Q1: What is the typical range of modulus of rigidity for common materials?
A: The modulus of rigidity varies significantly: steel (~79 GPa), aluminum (~26 GPa), copper (~45 GPa), rubber (~0.0001-0.001 GPa).

Q2: How does modulus of rigidity differ from Young's modulus?
A: Young's modulus measures resistance to tensile/compressive deformation, while modulus of rigidity measures resistance to shear deformation.

Q3: Why is the shaft assumed to be tapered in this formula?
A: The formula specifically addresses tapered shafts where the diameter changes along the length, requiring integration to account for varying cross-sectional properties.

Q4: What happens if D₂ equals D₁ (uniform shaft)?
A: The formula becomes indeterminate (division by zero). For uniform shafts, use the standard formula: G = (32·τ·L)/(π·D⁴·θ).

Q5: How accurate is this calculation for real-world applications?
A: The formula provides theoretical values. Real-world accuracy depends on material homogeneity, measurement precision, and assumptions about linear elastic behavior.