Math Problem Statement

Observe the diagram properly. Solve the question accordingly

Solution

The problem you've provided involves a three-degree-of-freedom system subject to a harmonic force 10cos(t)10\cos(t). The system includes masses m1m_1, m2m_2, and m3m_3 connected by springs with stiffness k1k_1, k2k_2, and k3k_3, and dampers with damping coefficients c1c_1, c2c_2, and c3c_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:

mix¨i=sum of forces on mass im_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 (m1m_1):

m1x¨1=k1(x1x2)c1(x˙1x˙2)+10cos(t)m_1 \ddot{x}_1 = -k_1(x_1 - x_2) - c_1(\dot{x}_1 - \dot{x}_2) + 10\cos(t)

Mass 2 (m2m_2):

m2x¨2=k2(x2x3)c2(x˙2x˙3)+k1(x1x2)+c1(x˙1x˙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 (m3m_3):

m3x¨3=k2(x2x3)+c2(x˙2x˙3)k3x3c3x˙3m_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][M] is the mass matrix and [K][K] is the stiffness matrix. Given the values:

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? --- ### 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.

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Math Problem Analysis

Mathematical Concepts

Mechanical Vibrations
Multi-degree-of-freedom Systems
Eigenvalue Problem

Formulas

Equations of Motion for multi-DOF systems
Eigenvalue problem for natural frequencies

Theorems

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Suitable Grade Level

Advanced Undergraduate or Graduate