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Observe the diagram properly. Solve the question accordingly

The problem you've provided involves a three-degree-of-freedom system subject to a harmonic force \(10\cos(t)\). The system includes masses \(m_1\), \(m_2\), and \(m_3\) connected by springs with stiffness \(k_1\), \(k_2\), and \(k_3\), and dampers with damping coefficients \(c_1\), \(c_2\), and \(c_3\). Here are the steps to solve the problem as per the instructions:

### Step 1: Derive the Equations of Motion
For a three-degree-of-freedom system, the equations of motion can be derived using Newton's second law:

\[
m_i \ddot{x}_i = \text{sum of forces on mass } i
\]

For each mass, the forces include contributions from the springs and dampers connected to it, and the external force for the first mass.

#### Mass 1 (\(m_1\)):
\[
m_1 \ddot{x}_1 = -k_1(x_1 - x_2) - c_1(\dot{x}_1 - \dot{x}_2) + 10\cos(t)
\]

#### Mass 2 (\(m_2\)):
\[
m_2 \ddot{x}_2 = -k_2(x_2 - x_3) - c_2(\dot{x}_2 - \dot{x}_3) + k_1(x_1 - x_2) + c_1(\dot{x}_1 - \dot{x}_2)
\]

#### Mass 3 (\(m_3\)):
\[
m_3 \ddot{x}_3 = k_2(x_2 - x_3) + c_2(\dot{x}_2 - \dot{x}_3) - k_3x_3 - c_3\dot{x}_3
\]

### Step 2: Find the Natural Frequencies and Mode Shapes
To find the natural frequencies and mode shapes, we would solve the eigenvalue problem associated with the system:

\[
[M]\{\ddot{x}\} + [K]\{x\} = \{0\}
\]

Where \([M]\) is the mass matrix and \([K]\) is the stiffness matrix. Given the values:

\[
[M] = \begin{bmatrix}
400 & 0 & 0 \\
0 & 7000 & 0 \\
0 & 0 & 1000
\end{bmatrix}
\]

\[
[K] = \begin{bmatrix}
k_1 + k_2 & -k_2 & 0 \\
-k_2 & k_2 + k_3 & -k_3 \\
0 & -k_3 & k_3
\end{bmatrix}
\]

Plugging in the provided values for \(k_1\), \(k_2\), and \(k_3\):

\[
[K] = \begin{bmatrix}
300000 + 1000000 & -1000000 & 0 \\
-1000000 & 1000000 + 800000 & -800000 \\
0 & -800000 & 800000
\end{bmatrix}
\]

You would then solve the eigenvalue problem \(det([K] - \omega^2[M]) = 0\) to find the natural frequencies \(\omega\) and corresponding mode shapes.

### Step 3: Calculate the Response
For forced vibration, assuming harmonic forcing and steady-state response, you would use the method of undetermined coefficients or modal analysis. The particular solution typically takes the form:

\[
x(t) = X\cos(\omega t + \phi)
\]

Where \(X\) is the amplitude vector found by solving the system for the specific forcing function, which is \(10\cos(t)\).

### Step 4: Plot the Response
Using the solutions obtained in the previous steps, plot the response \(x(t)\) for each mass \(m_1\), \(m_2\), and \(m_3\). This involves substituting the values back into the general solution obtained and plotting over a suitable time range.

Would you like a detailed breakdown of one of these steps, or do you have any specific questions about the solution?

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### Further Questions:
1. How would you solve for the eigenvalues and eigenvectors of the mass and stiffness matrices to determine the natural frequencies and mode shapes?
2. What is the significance of damping in the system and how would it affect the response?
3. How do you interpret the mode shapes physically in this context?
4. How does the harmonic force influence the response, particularly in relation to the natural frequencies?
5. What simplifications can be made if the system is assumed to be undamped?

### Tip:
When dealing with multi-degree-of-freedom systems, using MATLAB or any computational tool to solve for eigenvalues and eigenvectors can significantly simplify the process, especially for complex or large matrices.

Explore the detailed solution for a three-degree-of-freedom system subjected to a harmonic force. Learn about equations of motion, eigenvalue problem for natural frequencies, and response analysis. Suitable for advanced undergraduate or graduate studies.

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