Problem 39 - An Introduction to the Mechanics of Solids in SI Units

Question

A 125 -mm-diameter \(20^{\circ}\) spur gear is attached to the end of a 125 -mm- long cantilevered shaft. A smaller gear (ratio \(3: 1\) ) transmits a \(130 \mathrm{~N} \cdot \mathrm{m}\) torque. Sketch the shear-force, bending-moment, and twisting-moment diagrams for the cantilevered shaft, labeling important values.

Step-by-Step Solution

Answer

Transmitted Torque: 43.33 N.m, Shear Force: 695.28 N, Bending Moment: 86.91 N.m, Twisting Moment: 43.33 N.m. The shear force diagram is a horizontal line at 695.28 N, the bending moment diagram is a slant line reaching 86.91 N.m at the fixed end, and the twisting moment diagram is a horizontal line at 43.33 N.m.

Step 1: Determine the Transmitted Torque

First, let's determine the torque transmitted by the smaller gear. Since the gear ratio is 3:1, the smaller gear transmits a torque that's one third of the original 130 N.m, i.e., \( \frac{1}{3} \times 130 = 43.33 \) N.m

Step 2: Calculate the Shear Force

The shear force is equal to the torque divided by the radius of the gear. The radius of the gear is half its diameter i.e., \( \frac{125}{2} = 62.5 \) mm. Therefore, the shear force is \( \frac{43.33}{0.0625} = 695.28 \) N.

Step 3: Calculate the Bending Moment and Twisting Moment

The bending moment at the fixed end of a cantilever due to a force acting at its free end is given by the product of the force and the length of the cantilever. The twisting moment is equal to the transmitted torque. Therefore, bending moment = shear force * length of cantilevered shaft = \( 695.28 \times 0.125 = 86.91 \) N.m and twisting moment = 43.33 N.m

Step 4: Draw the Diagrams

Sketch the shear force, bending moment and twisting moment diagrams, labeling important values. The shear force is constant along the length of the shaft, thus the shear force diagram is a horizontal line at 695.28 N. The bending moment linearly increases, reaching its maximum value of 86.91 N.m at the fixed end, thus the bending moment diagram is a slant line. The twisting moment is constant along the length and equal to the transmitted torque, thus the twisting moment diagram is a horizontal line at 43.33 N.m.

Key Concepts

Cantilevered Shaft

A cantilevered shaft is a type of structure that is rigidly supported at one end while the other end is free. This setup is commonly used in mechanical applications where a shaft needs to resist moments, torques, and forces.
Cantilevered shafts are integral in systems where loads are applied at the free end, such as in this scenario with gears.

• This sort of positioning allows for supporting maximum bending moments and shear forces at the fixed end.
• In mechanical systems, they are used to transmit loads while maintaining alignment and integrity.

Due to these characteristics, understanding the behavior of cantilevered shafts under various loadings is crucial in designing robust and effective machinery.

Bending Moment

In the context of a cantilevered shaft, a bending moment arises when a force is applied perpendicular to the shaft. This causes the shaft to bend.
The bending moment on a cantilevered shaft is greatest at the fixed support and reduces to zero at the free end.

• The magnitude of the bending moment is determined by multiplying the force applied by the distance from the point of interest to where the force is applied.
• In our discussed example, the bending moment is calculated as the product of the shear force and the length of the shaft.
• This linear distribution can be visualized with a slanted diagram, starting from zero at the free end to the maximum value at the support.

Understanding this allows engineers to create structures that can safely handle applied loads without failure.

Shear Force

Shear force in a cantilevered shaft is an internal force that acts parallel to the cross-section. It results from external loads applied along the shaft's axis.
This force typically acts to "slide" one part of the material relative to another, creating a shearing effect.

• The shear force in the shaft remains constant along its length.
• It can be calculated by dividing the torque by the radius of the gear involved.
• In the given problem, the shear force had a consistent value, resulting in a horizontal line in the shear force diagram.

Shear forces need to be carefully monitored since excessive shear can lead to structural failure, making it vital in the design and analysis of mechanical components.

Twisting Moment

A twisting moment in mechanics of solids refers to torque applied to a structure, causing it to twist around its axis. In shafts, this can influence the operation significantly if not managed appropriately.
Twisting moments are crucial in ensuring that the shaft can handle rotational forces without yielding or becoming deformed.

• Twisting moment is equivalent to the applied torque on the system.
• In cantilevered shafts, it remains constant along the shaft length.
• This constant value results in a flat line in a twisting moment diagram.

Accounting for twisting moments helps in selecting material and designing shafts that perform effectively under rotation-intensive operations.