19.4
Angle of Twist - Elastic Range
Consider a circular shaft of length L and a uniform cross-section of radius r subjected to a torque at its free end.
The maximum shearing strain in the shaft is proportional to both the angle of twist and the radial distance from the shaft's axis.
In the elastic range, this shearing strain can be expressed in terms of the applied torque, radial distance, polar moment of inertia, and the modulus of rigidity.
By equating these two equations, we derive an expression for the angle of twist within the elastic range. This equation applies to a homogeneous shaft with a uniform cross-section when torque is only applied at one of its ends.
If the shaft is subjected to torques at different locations, or if it consists of various parts with different cross sections or materials, the angle of twist must be considered separately for each part.
The total angle of twist is calculated by summing all individual values from each part of the shaft or by integrating along the length for shafts with non-uniform cross-sections.
Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By setting these two equations equal to each other, it is possible to formulate an expression for the twist angle within the elastic range.
This equation applies to a uniform homogeneous shaft, where torque is only exerted at its extremities. However, if the shaft is exposed to torques at varying points or composed of different parts with diverse cross-sections or materials, the twist angle must be evaluated distinctly for each section. The sum of all individual values from each shaft segment determines the total twist angle. Alternatively, it can be calculated by integrating along the length of shafts with non-uniform cross-sections. This approach presents a comprehensive understanding of the behavior of shafts under varying conditions.
Consider a circular shaft of length L and a uniform cross-section of radius r subjected to a torque at its free end.
The maximum shearing strain in the shaft is proportional to both the angle of twist and the radial distance from the shaft's axis.
In the elastic range, this shearing strain can be expressed in terms of the applied torque, radial distance, polar moment of inertia, and the modulus of rigidity.
By equating these two equations, we derive an expression for the angle of twist within the elastic range. This equation applies to a homogeneous shaft with a uniform cross-section when torque is only applied at one of its ends.
If the shaft is subjected to torques at different locations, or if it consists of various parts with different cross sections or materials, the angle of twist must be considered separately for each part.
The total angle of twist is calculated by summing all individual values from each part of the shaft or by integrating along the length for shafts with non-uniform cross-sections.
- Elastic Torsion - Twisting of a shaft under certain load within its elastic range.
- Shaft Twist - The rotation a shaft undergoes due to applied torque.
- Angle of Twist - The deformation angle resulting from twist in a shaft.
- Polar Moment of Inertia - A measure of a shaft's resistance to torsion.
- Non-Uniform Torsion - Torsion on a shaft with varying cross-sections or materials.
- Define Elastic Torsion – Understand deformation within the elastic range (e.g., shaft twist).
- Contrast Uniform vs Non-Uniform Torsion – Grasp differences under varying conditions (e.g., diverse material sections).
- Explore Angle of Twist Formulas – Understand how it can be computed (e.g., using polar moment of inertia).
- Explain Shaft Behavior – Understand how deformation occurs and is computed in real-world scenarios.
- Apply Twist Angle Calculation – Understand its relevance and implications in the field of engineering.
- What is Elastic Torsion and how to calculate Angle of Twist?
- How does Non-uniform Torsion affect the Angle of Twist?
- How is the Polar Moment of Inertia used in Twist computations?
From Chapter 19:
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19.4 : Angle of Twist - Elastic Range
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