Problem 37 - An Introduction to the Mechanics of Solids in SI Units

Question

The shaft \(A D\) is supported in bearings at \(A\) and \(D\) and has pulleys attached at \(B\) and \(C\). The pulley at \(B\) is \(20-\mathrm{cm}\) diameter while that at \(C\) is 30 -cm diameter. The shaft transmits a maximum of \(20 \mathrm{~kW}\) at \(1750 \mathrm{rpm}\). The belt tensions are adjusted so that $$ \frac{T_{1}}{T_{2}}=\frac{T_{3}}{T_{4}}=3 $$ Sketch the shear-force, bending-moment, and twisting-moment diagrams for \(A D\), labeling important values. (Note : A horsepower is \(745 \mathrm{~N}-\mathrm{m} / \mathrm{s}\). Rotational power is the product of torque times angular velocity in radians per unit time.)

Step-by-Step Solution

Answer

The diagrams illustrating the shear-force, bending-moment, and twisting-moment should indicate a constant effect between B and C and between A and B, and C and D. This is due to the constant values of tension and diameter along these parts.

Step 1: Conversion of Power

First, convert the power from kilowatts to newton meter per second. Given 1 kW = 1000 watts and 1 watt = 1 N.m/s. Hence, 20 kW = 20000 N.m/s.

Step 2: Calculation of Torque

Next, calculate the torque using formula \[ Torque = \frac{Power}{\omega} \]. Where \(\omega\) (Angular velocity) = 1750 rpm = 183.3 rad/s (Remember to convert rpm to rad/s). Hence, Torque = \(\frac{20000N.m/s}{183.3rad/s}\) = 109.1 N.m.

Step 3: Application of Belt Tension Relation

Apply the given relation between the tensions, \(\frac{T_{1}}{T_{2}} = \frac{T_{3}}{T_{4}} = 3\). From this, it is clear that \(T_{1} = 3T_{2}\) and \(T_{3} = 3T_{4}\). For belts on pulleys, the moment created by a tension is \(Moment = Tension × Radius\). Applying this, we find \(Moment_{B} = T_{1}R_{B} - T_{2}R_{B}\) and \(Moment_{C} = T_{3}R_{C} - T_{4}R_{C}\).

Step 4: Calculate the Tensions

Given that maximum moment that can be transmitted = 109.1 N.m, we can find the tensions using the effective radius for B and C, that is, 0.1 m and 0.15 m respectively. We can write the relation as \(109.1 = (3T_{2} - T_{2})× 0.1 - (3T_{4}-T_{4}) × 0.15\). Solving this equation will give us the values of \(T_{2}\) and \(T_{4}\). From which we can find \(T_{1} = 3T_{2}\) and \(T_{3} = 3T_{4}\).

Step 5: Creating the Diagrams

Using the tensions calculated and given diameters, we can now draw the shear-force, bending-moment, and twisting-moment diagrams. Note that the effects between B and C and between A and B, and C and D are constant as there are no changes along these shaft sections.

Key Concepts

Shaft Mechanics

In mechanics of solids, understanding shaft mechanics is essential for analyzing how shafts handle loads and transmit power. A shaft is a cylindrical object that rotates to perform a mechanical task. It is designed to transfer energy from one part of a system to another while experiencing forces such as tension, compression, shear, and torque.
This exercise involves a shaft supported by bearings at points A and D, with pulleys at points B and C. These pulleys have different diameters, capable of transmitting power via belts. The ability to efficiently transmit power from the motor to the tool or machine is controlled by the shaft's material, dimensions, and the forces applied to it. Both the material strength and design geometry must adequately support the operational loads.

• Material strength ensures the shaft does not permanently deform under load.
• Design geometry helps avoid excess stress concentrations that could lead to failure.

By analyzing the shaft through diagrams and calculations, we can ensure that it will perform effectively under its mechanical duties.

Shear Force Diagram

The shear force diagram is a graphical representation that shows how shear forces vary along the length of a shaft. Shear forces arise in the shaft due to transverse loads, such as the weights of the pulleys and the tensions in the belts.
It's crucial to draw a correct shear force diagram, as it helps identify points of maximum shear stress. This visualization is key to understanding where the shaft might experience issues such as shearing or fatigue over time.

• To create the diagram, start by calculating the shear forces at various points along the shaft, especially where pulleys and bearings are located.
• Plot these forces on a graph (y-axis representing shear force and x-axis representing the shaft length).
• Identify points where the shear force changes, often where external loads are applied or removed.

The shear force diagram assists in recognizing stress patterns within the shaft, providing insight into its structural integrity.

Bending Moment Diagram

The bending moment diagram illustrates how the bending moments differ along the shaft's length. A bending moment results from forces inducing bending, such as the reactions at the bearings and forces due to belt tension on the pulleys. Understanding this helps ensure the shaft's design minimizes the risk of bending-related failures.
To create the bending moment diagram, follow these steps:

• Determine the bending moments at critical points (e.g., where forces act).
• Plot the calculated moments on a graph (y-axis showing bending moments and x-axis for shaft length).
• Where the bending moment curve peaks, maximum stress or bending occurs, which might necessitate design adjustments if too high.

This diagram is invaluable for engineering applications that require determining the robustness of a shaft to withstand bending forces.

Twisting Moment Diagram

A twisting moment, or torque, is the force causing the shaft to rotate. The twisting moment diagram is used to display how torque varies along the shaft's length.
The torque in the exercise arises from the transmitted power and the belt tensions on the pulleys. This diagram helps engineers prevent shaft twisting failure by ensuring that the material and dimensions can handle the twisting forces.

• Calculate the torques at different sections of the shaft, typically where changes in radius or tension occur.
• Plot the torques on a graph to show how the shaft's capability to resist twisting varies.
• Find peaks and trends in the twisting moment diagram to assess if reinforcement is needed.

Proper understanding and interpretation of this diagram are crucial to ensuring the shaft's rotational integrity.

Belt Tension Analysis

Belt tension analysis involves understanding how the forces in belt-driven systems like this exercise are distributed and their sustaining effects on the shaft. Pulley diameters and belt tensions significantly influence performance and efficiency.
The given problem provides a tension ratio, indicating a designed need to manage power transmission efficiently. In this scenario, ensuring correct tension can prevent belts from slipping, ensuring consistent power delivery.

• Start by examining the given tension ratios and calculate absolute tensions for each side of the belt.
• Apply these tensions to determine the effective force transmission over the pulleys' radius.
• Use the calculated tensions to check if the shaft can handle these combined tensions without deforming.

Belt tension analysis is fundamental in designing shaft systems that can efficiently and reliably drive mechanical processes.