To find the mass and stiffness matrices for the system described in the problem, we need to analyze the system based on the provided diagram and details.

Analyzing the System:

From the figure, the system is a pipeline supported on soil and modeled as a spring-dashpot system with three degrees of freedom corresponding to three masses $m_1$, $m_2$, and $m_3$. The springs and dashpots are represented by their stiffness $k$ and damping $c$ values, respectively.

Degrees of Freedom:

• Degree of Freedom 1: Horizontal displacement of mass $m_1$.
• Degree of Freedom 2: Horizontal displacement of mass $m_2$.
• Degree of Freedom 3: Horizontal displacement of mass $m_3$.

Mass Matrix $\mathbf{M}$:

Since the system has three masses and the degrees of freedom are purely translational, the mass matrix is diagonal:

m_1 & 0 & 0 \\ 0 & m_2 & 0 \\ 0 & 0 & m_3 \end{bmatrix}$$ #### Stiffness Matrix $$\mathbf{K}$$: The stiffness matrix can be constructed based on the configuration of springs connecting the masses and the ground. Let's identify the springs: - **Spring $$k_1$$** between mass $$m_1$$ and ground. - **Spring $$k_2$$** between masses $$m_1$$ and $$m_2$$. - **Spring $$k_3$$** between masses $$m_2$$ and $$m_3$$. - **Spring $$k_4$$** between mass $$m_3$$ and ground. The stiffness matrix $$\mathbf{K}$$ for this system is symmetric and can be written as: $$\mathbf{K} = \begin{bmatrix} k_1 + k_2 & -k_2 & 0 \\ -k_2 & k_2 + k_3 & -k_3 \\ 0 & -k_3 & k_3 + k_4 \end{bmatrix}$$ ### Example Values from the Problem: - $$m_1 = m_2 = m_3 = m$$ - $$k_1 = k_4 = 48 \, \text{kN/m}$$ - $$k_2 = k_3 = 16 \, \text{kN/m}$$ Plugging in these values, the mass and stiffness matrices are: $$\mathbf{M} = \begin{bmatrix} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & m \end{bmatrix}$$ $$\mathbf{K} = \begin{bmatrix} 64 & -16 & 0 \\ -16 & 32 & -16 \\ 0 & -16 & 64 \end{bmatrix} \text{(in kN/m)}$$ These matrices represent the system's dynamics in terms of mass distribution and stiffness. Would you like more details on how to derive these matrices or any further questions about this problem? ### Further Questions for Exploration: 1. How would the damping matrix be formulated for this system considering the damping ratio? 2. What are the natural frequencies of this system using the mass and stiffness matrices? 3. How would the system response change if one of the springs' stiffness values were altered? 4. Can we derive the mode shapes for this system using the given matrices? 5. How does the damping ratio of 7.5% affect the dynamic response of the system? ### Tip: When forming the stiffness matrix, carefully consider the connections between each mass and the springs to ensure the correct values are assigned to each element of the matrix.