Torque On Shaft Given Total Angle Of Twist For Shaft Calculator

Formula Used:

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

Angle of Twist (θ):

rad

Modulus of Rigidity (G):

Diameter at Left End (D₁):

Diameter at Right End (D₂):

Length of Shaft (L):

Torque Exerted on Wheel (τ):

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1. What is the Torque on Shaft Formula?

The torque on a tapered shaft formula calculates the torque exerted on a wheel based on the angle of twist, modulus of rigidity, shaft diameters at both ends, and the length of the shaft. This formula is essential in mechanical engineering for analyzing torsional stress in tapered shafts.

2. How Does the Calculator Work?

The calculator uses the following formula:

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

Where:

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

Explanation: This formula accounts for the varying cross-section of a tapered shaft and calculates the torque based on the torsional deformation characteristics.

3. Importance of Torque Calculation

Details: Accurate torque calculation is crucial for designing mechanical systems, ensuring shaft strength, preventing failure due to torsional stress, and optimizing power transmission efficiency.

4. Using the Calculator

Tips: Enter all values in appropriate units (radians for angle, Pascals for modulus, meters for dimensions). Ensure all values are positive and D₂ > D₁ for a properly tapered shaft.

5. Frequently Asked Questions (FAQ)

Q1: What is modulus of rigidity?
A: Modulus of rigidity (G) is a material property that measures the stiffness of a material under shear stress. It represents the ratio of shear stress to shear strain.

Q2: Why is the shaft tapered in this formula?
A: Tapered shafts are used in various mechanical applications to provide gradual stress distribution, reduce weight, or accommodate specific design requirements.

Q3: What are typical values for modulus of rigidity?
A: For steel: ~79.3 GPa, for aluminum: ~26 GPa, for brass: ~40 GPa. The exact value depends on the specific alloy and treatment.

Q4: When does this formula not apply?
A: This formula assumes linear elastic material behavior, homogeneous material properties, and pure torsion. It may not be accurate for very large deformations or non-uniform materials.

Q5: How does shaft taper affect torque capacity?
A: Tapered shafts typically have different stress distributions compared to uniform shafts. The smaller diameter end usually experiences higher stress for the same torque.