Problem 17 - An Introduction to the Mechanics of Solids in SI Units

Question

For a hollow shaft whose outside diameter is twice its inside diameter, derive the relation between the horsepower that may be transmitted, the rpm, the maximum allowable shear stress, and the outside diameter \(d_{o}\).

Step-by-Step Solution

Answer

The derived relationship between the horsepower (\(P\)), the rpm (\(N\)), the maximum allowable shear stress (\(\tau\)), and the outside diameter (\(d_{o}\)) of a hollow shaft (whose outside diameter is twice its inside diameter) is given as \(P = \frac{\pi^2 N \tau d_o^3}{256}\)

Step 1: Analyze the Shaft Geometry

Assuming a hollow shaft where the outside diameter is twice the inside diameter, the inside diameter \(d_{i}\) can be represented as \(d_{i} = 0.5 \cdot d_{o}\). This will be needed to compute the polar moment of inertia (J) of the hollow shaft.

Step 2: Determine the Polar Moment of Inertia

The polar moment of inertia (J) for a hollow cylinder \(J = \pi \cdot \frac{(d_o^4 - d_{i}^4)}{32}\). Substitute the value of \(d_{i}\) into the equation: \(J = \pi \cdot \frac{(d_o^4 - (0.5d_{o})^4)}{32} = \frac{15 \pi d_o^4}{128}\).

Step 3: Define the Shear Stress Formula

In this scenario, the shear stress (\tau) in the material is given by the formula \(\tau = \frac{T}{J} \cdot \frac{d}{2}\) where T is the torque and d is the diameter.

Step 4: Compute the Torque

Substitute the values of \tau and J into the shear stress equation to yield the torque: \(T = \tau \frac{J}{ \frac{d_o}{2}} = \tau \frac{\frac{15 \pi d_o^4}{128}}{ \frac{d_o}{2}} = \tau \frac{30 \pi d_o^3}{128}\). This gives an expression for the torque in terms of the shear stress and outer diameter.

Step 5: Formulate Power Transmission and Substitute Torque

The power P transmitted by a rotating shaft is given by \(P = \frac{2\pi NT}{60}\). Substitute the expression obtained for T into the power equation: \(P = \frac{2\pi N \tau \frac{30 \pi d_o^3}{128}}{60}\). Simplifying this gives the final relationship: \(P = \frac{\pi^2 N \tau d_o^3}{256}\).

Key Concepts

Hollow Shaft Analysis

In the world of mechanics, understanding the geometry of a hollow shaft is crucial for analyzing its strength and efficiency. A hollow shaft is a cylindrical object that has a hole running through its center. The key aspect of this shaft is that the outside diameter is twice the inside diameter. This relationship helps determine the hollow shaft's mechanical properties. In this case, the inside diameter \(d_i\) can be expressed as half of the outside diameter \(d_o\), given by the formula \(d_i = 0.5 \cdot d_o\).
The geometry of the hollow shaft allows engineers to design strong and lightweight components. By optimizing the distribution of material, a hollow shaft can deliver the same performance as a solid shaft with much less weight. This is particularly advantageous in applications that require the transmission of torque, such as automotive axles or industrial machinery shafts.

Torque Calculation

Torque is the twisting force that causes rotation. When analyzing hollow shafts, calculating the torque is essential for determining its performance.
To find the torque \(T\) within a hollow shaft, we use the formula: \(T = \tau \frac{J}{\frac{d_o}{2}}\), where:

• \(\tau\) is the shear stress applied to the shaft
• \(J\) is the polar moment of inertia
• \(d_o\) is the outside diameter

This formula reveals how the torque is influenced by the maximum shear stress and the shaft's dimensions. In the exercise, substituting in the previously derived equation for \(J\) gives: \(T = \tau \frac{30 \pi d_o^3}{128}\). Understanding torque is vital because it determines how much rotational force the shaft can withstand before failing.

Shear Stress Formulation

Shear stress is a measure of how much force per unit area is exerted parallel to the face of a material. In hollow shaft analysis, shear stress is represented by \(\tau = \frac{T}{J} \cdot \frac{d}{2}\). This formula shows how shear stress is influenced by external torque \(T\), the shaft's structure, and its geometry.
The calculation of shear stress in hollow shafts is essential for design and safety. It helps engineers predict how materials will behave under various loads and conditions. A key goal is ensuring that the shear stress does not exceed the material's maximum allowable limits to prevent failure. Thus, practicing shear stress computation helps consolidate your understanding of material behavior in rotating machinery.

Power Transmission in Rotating Shafts

For rotating shafts like those found in machinery and vehicles, power transmission is a crucial function. Power \(P\) transmitted in a rotating shaft is determined by the formula: \(P = \frac{2\pi NT}{60}\), where:

• \(N\) is the rotational speed in revolutions per minute (rpm)
• \(T\) is the torque

This formula links the rotational speed and torque to the power output of the shaft. Using the exercise's torque calculation, the final expression for power transmission becomes \(P = \frac{\pi^2 N \tau d_o^3}{256}\). This equation ties together the mechanical and geometrical aspects of the shaft to understand its capability to transmit energy.Power transmission is integral to machinery efficiency. By understanding and applying this concept, you'll gain insights into optimizing machinery performance and ensuring robust design.