Understanding the Calculation of Torque Resistant to Rotation

Torque resistant to rotation quantifies the force opposing rotational motion in mechanical systems. It is essential for designing stable and efficient machinery.

This article explores detailed formulas, common values, and real-world applications for calculating torque resistance. Engineers will find comprehensive insights here.

• Calculate torque resistant to rotation for a steel shaft with given dimensions and load.
• Determine the torque resistance of a bolt under shear stress conditions.
• Analyze torque resistance in a rotating disc subjected to frictional forces.
• Compute torque resistance for a gear system with specified material properties.

Comprehensive Tables of Common Values for Torque Resistant to Rotation

Accurate torque resistance calculations depend on material properties, geometric dimensions, and loading conditions. The following tables summarize typical values used in engineering practice.

Additional geometric parameters commonly used in torque resistance calculations include:

Fundamental Formulas for Calculating Torque Resistant to Rotation

Torque resistance is primarily governed by the material’s shear strength and the geometry of the rotating element. Below are the essential formulas with detailed explanations.

1. Basic Torque Formula

The torque T required to resist rotation is related to the shear stress τ and the polar moment of inertia J by:

T = (τ × J) / r

• T: Torque resisting rotation (N·m)
• τ: Shear stress at the outer surface (Pa or N/m²)
• J: Polar moment of inertia of the cross-section (m⁴)
• r: Outer radius of the shaft (m)

The polar moment of inertia J for a solid circular shaft is calculated as:

J = (π × d⁴) / 32

• d: Diameter of the shaft (m)

For hollow shafts, the formula adjusts to:

J = (π × (d_o⁴ – d_i⁴)) / 32

• d_o: Outer diameter (m)
• d_i: Inner diameter (m)

2. Shear Stress from Applied Torque

Rearranging the basic torque formula, the shear stress at the outer surface is:

τ = (T × r) / J

This is critical for ensuring the material does not exceed its yield shear stress, preventing failure.

3. Maximum Torque Before Yielding

The maximum torque T_max a shaft can resist without yielding is:

T_max = (τ_y × J) / r

• τ_y: Yield shear stress of the material (Pa)

4. Torsional Angle of Twist

The angle of twist θ over a shaft length L under torque T is:

θ = (T × L) / (G × J)

• G: Shear modulus of the material (Pa)
• L: Length of the shaft (m)

This formula is essential for assessing deformation and ensuring operational limits are not exceeded.

5. Torque Due to Frictional Resistance

In systems where friction resists rotation, torque can be calculated as:

T = μ × N × r

• μ: Coefficient of friction (dimensionless)
• N: Normal force (N)
• r: Radius at which friction acts (m)

This is particularly relevant in brake systems, clutches, and bearing interfaces.

Real-World Applications and Detailed Examples

Example 1: Torque Resistance in a Steel Shaft for Industrial Machinery

An industrial steel shaft (AISI 1045) with a diameter of 50 mm and length 1.5 m is subjected to a torque. Determine the maximum torque it can resist without yielding. Use the following data:

• Yield shear stress, τ_y = 280 MPa
• Shear modulus, G = 79.3 GPa

Step 1: Calculate the polar moment of inertia J

d = 50 mm = 0.05 m

J = (π × d⁴) / 32 = (3.1416 × (0.05)⁴) / 32

J = (3.1416 × 6.25 × 10^-7) / 32 = 6.136 × 10^-8 m⁴

Step 2: Calculate the outer radius r

r = d / 2 = 0.025 m

Step 3: Calculate maximum torque T_max

T_max = (τ_y × J) / r = (280 × 10⁶ × 6.136 × 10^-8) / 0.025

T_max = (17.18) / 0.025 = 687.2 N·m

Interpretation: The shaft can resist up to approximately 687 N·m of torque before yielding occurs.

Example 2: Torque Resistance Due to Friction in a Brake Disc

A brake disc with radius 0.15 m experiences a normal force of 2000 N. The coefficient of friction between the brake pad and disc is 0.35. Calculate the torque resisting rotation due to friction.

Step 1: Apply the frictional torque formula

T = μ × N × r = 0.35 × 2000 × 0.15

T = 105 N·m

Interpretation: The brake system can resist 105 N·m of torque due to frictional forces, effectively slowing or stopping rotation.

Additional Considerations for Accurate Torque Resistance Calculations

Several factors influence the accuracy and reliability of torque resistance calculations:

• Material Anisotropy: Some materials exhibit direction-dependent properties affecting shear strength.
• Temperature Effects: Elevated temperatures can reduce shear modulus and yield stress.
• Surface Conditions: Roughness and lubrication impact frictional torque resistance.
• Stress Concentrations: Geometric discontinuities like keyways or shoulders increase local stresses.
• Dynamic Loading: Cyclic or impact loads require fatigue analysis beyond static torque calculations.

Incorporating these factors ensures safer and more efficient mechanical designs.

References and Further Reading

• ASME – American Society of Mechanical Engineers: Standards and guidelines on mechanical design.
• ASTM International: Material property standards and testing methods.
• Engineering Toolbox: Comprehensive engineering data and calculators.
• Machine Design: Articles and case studies on torque and mechanical design.

By mastering the calculation of torque resistant to rotation, engineers can optimize mechanical components for durability, safety, and performance across diverse applications.