Question

A \(30 \mathrm{~cm}\) aluminum rod possessing a circular cross section of \(1.25 \mathrm{~cm}\) radius is inserted into a testing machine where it is fixed at one end and attached to a load cell at the other end. At some point during a torsion test the clamp at the load cell slips, releasing that end of the rod. If the \(20 \mathrm{~kg}\) clamp remains attached to the end of the rod, determine the frequency of the oscillations of the rod-clamp system. The radius of gyration of the clamp is \(5 \mathrm{~cm}\). Fig. P2.7 Fig. P2.7

Short answer

Answer: To determine the frequency of oscillations (f), follow the steps below: Step 1: Calculate the moment of inertia of the rod and clamp using the formulas \(I_{rod} = \frac{1}{4} \pi r_{rod}^4 \rho_{Al} L\) and \(I_{clamp} = M_{clamp} k^2\). Step 2: Calculate the torsional stiffness of the rod using the formula \(K = \frac{GJ}{L}\), where \(J = \pi r_{rod}^4 / 2\). Step 3: Determine the frequency of oscillations using the formula \(f = \frac{1}{2\pi} \sqrt{\frac{K}{I_{total}}}\), where \(I_{total} = I_{rod} + I_{clamp}\). Plug in the given values and solve for the frequency, \(f\).

Step-by-step solution

Calculate moment of inertia of the rod and the clamp

To calculate the moment of inertia of the rod and the clamp, we need to use the formulas: \(I_{rod} = \frac{1}{4} \pi r_{rod}^4 \rho_{Al} L\) for the rod and \(I_{clamp} = M_{clamp} k^2\) for the clamp, where \(I_{rod}\) and \(I_{clamp}\) are the moment of inertia of the rod and clamp, respectively, \(r_{rod}\) is the radius of the rod, \(\rho_{Al}\) is the density of aluminum, \(L\) is the length of the rod, \(M_{clamp}\) is the mass of the clamp, and \(k\) is the radius of gyration. Plug in the given values for the variables.

Calculate torsional stiffness of the rod

Next, we need to find the torsional stiffness of the rod, which can be calculated using the formula \(K = \frac{GJ}{L}\), where \(K\) is the torsional stiffness, \(G\) is the shear modulus of aluminum, \(J\) is the polar moment of inertia of the rod (can be calculated as \(J=\pi r_{rod}^4 / 2\)), and \(L\) is the length of the rod. Plug in the given values for the variables.

Determine the frequency of oscillations

Now, we have all the information needed to calculate the frequency of oscillations. The formula for the frequency of oscillations is \(f = \frac{1}{2\pi} \sqrt{\frac{K}{I_{total}}}\), where \(f\) is the frequency of oscillations, \(K\) is the torsional stiffness, and \(I_{total} = I_{rod} + I_{clamp}\) is the total moment of inertia of the rod-clamp system. Plug in the values obtained in Steps 1 and 2 and solve for the frequency, \(f\). Following these steps will help you determine the frequency of oscillations of the rod-clamp system after the clamp slips during a torsion test.

Key concepts

Moment of Inertia

Understanding the moment of inertia is crucial when studying mechanical vibrations, especially in systems like a rod-clamp. Moment of inertia, or rotational inertia, measures how much resistance an object has to change in its rotation. It's like the rotational equivalent of mass in linear motion. In the given problem, we calculate the moment of inertia for both the rod and the clamp.

For the **rod**, the formula used is:

• \(I_{rod} = \frac{1}{4} \pi r_{rod}^4 \rho_{Al} L\)

This formula takes into account the rod's geometric properties and material density.

• \(r_{rod}\) is the radius of the circular cross-section of the rod.
• \(\rho_{Al}\) is the density of aluminum.
• \(L\) is the length of the rod.

For the **clamp**, the calculation is slightly different:

• \(I_{clamp} = M_{clamp} k^2\)

Where:

• \(M_{clamp}\) is the mass of the clamp.
• \(k\) is the radius of gyration, which describes how mass is distributed around the axis of rotation.

Combining the inertia of the rod and clamp gives us the total rotational inertia, necessary for further steps in analyzing the system's behavior.

Torsional Stiffness

Torsional stiffness is a key concept when dealing with mechanical vibrations involving twisting or torsion. It describes the material's resistance to twist. Imagine trying to twist a rod; torsional stiffness tells us how hard it is to make it twist. It's crucial in calculating oscillation frequencies because a stiffer system will oscillate differently than a softer one.

The torsional stiffness of the rod is given by the formula:

• \(K = \frac{GJ}{L}\)

Where:

• \(G\) is the shear modulus, which describes how the material deforms under shear stress. For aluminum, this value is usually provided in tables.
• \(J\) is the polar moment of inertia, expressing how mass is distributed with respect to the axis. It's calculated as \(J = \frac{\pi r_{rod}^4}{2}\).
• \(L\) is the rod's length, influencing how stiffness translates into the system's vibrational characteristics.

This concept helps us understand how the rod's rigidity influences the dynamics when the clamp starts to oscillate.

Frequency of Oscillations

Once we know both the total moment of inertia and the torsional stiffness, we can determine how often the system oscillates. The frequency of oscillations gives insight into how many oscillations happen per second. It's essential for understanding how the system behaves dynamically after being disturbed.

The frequency of oscillations is determined using this formula:

• \(f = \frac{1}{2\pi} \sqrt{\frac{K}{I_{total}}}\)

Where:

• \(f\) represents the frequency of oscillations.
• \(K\) is the torsional stiffness calculated previously.
• \(I_{total}\) is the total moment of inertia, which is the sum of the rod’s and clamp’s moments of inertia.

This results in a frequency value showing how the system will behave in real-world conditions. Understanding this frequency helps engineers and designers ensure the system operates safely within mechanical and structural limits.