### Vibration Analysis of Connected Cylinders with Spring**Problem Statement:**Two identical circular cylinders, each with mass \( m \) and radius \( r \) according to the figure, are connected by a spring with spring stiffness \( k \). Determine the natural frequencies and illustrate the shape of the vibration modes. The moment of inertia of the cylinder is given by:\[ I_{\text{cylinder}} = \frac{1}{2} m r^2 \]**System Description:**The system consists of:- Two identical circular cylinders.- Each cylinder has a mass \( m \) and radius \( r \).- The cylinders are connected by a spring with a spring constant \( k \).- The cylinders are placed on a surface with friction modeled in the system dynamics.**Diagram Explanation:**The diagram included in the problem consists of:1. Two circular cylinders, labeled with radius \( r \) and mass \( m \).2. A spring with spring constant \( k \) connects the two cylinders horizontally.3. Below the system is a patterned ground line indicating that the cylinders rest on a horizontal surface.4. The spring is depicted as a typical mechanical spring connecting the centers of the two cylinders.5. Numerical value \( 36 \) over the length of the spring indicating a possible distance or labeling relevant to a specific problem context not detailed here.**Approach to Solution:**1. **Mathematical Modeling:** - We start by setting up the equations of motion for each cylinder considering both translational and rotational dynamics. - Let \( x_1 \) and \( x_2 \) be the displacements of cylinders 1 and 2 respectively.2. **Translational Motion:** - The force exerted by the spring is \( F_{\text{spring}} = k (x_2 - x_1) \).3. **Rotational Motion:** - The torque on each cylinder is \( \tau = I \alpha \), where \( \alpha \) is the angular acceleration. - Given \( I_{\text{cylinder}} = \frac{1}{2} m r^2 \).4. **Equations of Motion:** - Using Newton's second law for each cylinder’s center of mass and rotational dynamics.5. **Natural Frequencies:** - Solve the characteristic equation derived from the differential equations to find the natural frequencies.

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Two identical circular cylinders, each with mass m and radius r according to the figure, are connected by a spring with hardness k. Get the natural frequencies and draw the shape of the modes .(Teylinder =;mr?) 36 K www m m

Two identical circular cylinders, each with mass m and radius r according to the figure, are connected by a spring with hardness k. Get the natural frequencies and draw the shape of the modes .(Teylinder =;mr?) 36 K www m m

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

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Concept explainers

Combined Loading

Any functioning component, such as machine parts, structural members, pressure vessels, and so on, are all under the influence of more than one force. When anybody is under the influence of more than one force such as shear forces, bending moments, axial forces, torsion, and so on, then such body is said to be under combined loading. A very practical example of combined loading is the crankshaft of an internal combustion engine (IC engine). The crankshaft of the IC engine is under high rpm, which is caused due to the torsional forces. Due to the presence of piston, counterbalances, and internal imbalances, the crankshaft experiences lateral forces and due to the rise in temperature, the crankshaft undergoes thermal expansion, which pulls the crankshaft along the axial direction and lateral direction. The presence of unbalanced loads along the periphery of the crankshaft also induces a bending moment.

Question

### Vibration Analysis of Connected Cylinders with Spring

**Problem Statement:**
Two identical circular cylinders, each with mass \( m \) and radius \( r \) according to the figure, are connected by a spring with spring stiffness \( k \). Determine the natural frequencies and illustrate the shape of the vibration modes. The moment of inertia of the cylinder is given by:
\[ I_{\text{cylinder}} = \frac{1}{2} m r^2 \]

**System Description:**

The system consists of:
- Two identical circular cylinders.
- Each cylinder has a mass \( m \) and radius \( r \).
- The cylinders are connected by a spring with a spring constant \( k \).
- The cylinders are placed on a surface with friction modeled in the system dynamics.

**Diagram Explanation:**

The diagram included in the problem consists of:
1. Two circular cylinders, labeled with radius \( r \) and mass \( m \).
2. A spring with spring constant \( k \) connects the two cylinders horizontally.
3. Below the system is a patterned ground line indicating that the cylinders rest on a horizontal surface.
4. The spring is depicted as a typical mechanical spring connecting the centers of the two cylinders.
5. Numerical value \( 36 \) over the length of the spring indicating a possible distance or labeling relevant to a specific problem context not detailed here.

**Approach to Solution:**

1. **Mathematical Modeling:**
- We start by setting up the equations of motion for each cylinder considering both translational and rotational dynamics.
- Let \( x_1 \) and \( x_2 \) be the displacements of cylinders 1 and 2 respectively.

2. **Translational Motion:**
- The force exerted by the spring is \( F_{\text{spring}} = k (x_2 - x_1) \).

3. **Rotational Motion:**
- The torque on each cylinder is \( \tau = I \alpha \), where \( \alpha \) is the angular acceleration.
- Given \( I_{\text{cylinder}} = \frac{1}{2} m r^2 \).

4. **Equations of Motion:**
- Using Newton's second law for each cylinder’s center of mass and rotational dynamics.

5. **Natural Frequencies:**
- Solve the characteristic equation derived from the differential equations to find the natural frequencies.

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