Let's break down the solution steps for the given mechanical system: --- ## **Step 1: Equation of Motion (Differential Equation)** ### **Free Body Diagram Analysis** - **Mass \( m \):** Subjected to external force \( p(t) \). - **Right Side:** Two parallel elements (\(k_1, c_1\)) in parallel with a series combination of \(k_2, c_2\). #### **Equivalent Stiffness and Damping** - **Parallel branch:** \(k_1, c_1\) - **Series branch:** \(k_2\) and \(c_2\) in series #### **Force on Mass:** \[ m\ddot{x}(t) = p(t) - F_{\text{parallel}} - F_{\text{series}} \] #### **Parallel Branch:** \[ F_{k_1+c_1} = k_1x + c_1\dot{x} \] #### **Series Branch (Spring-Damper in Series):** Let \( z \) be the displacement at the node between \( k_2 \) and \( c_2 \). - Force through both elements is the same: \( F_{series} \) - Displacement across both: \( x \) - \( x = x_{k_2} + x_{c_2} \) - \( F_{k_2} = k_2(x - z) \) - \( F_{c_2} = c_2(\dot{x} - \dot{z}) \) - \( F_{k_2} = F_{c_2} = F_{series} \) So, \[ k_2(x-z) = c_2(\dot{x} - \dot{z}) \] \[ \Rightarrow k_2(x-z) = c_2(\dot{x} - \dot{z}) \] \[ \Rightarrow k_2(x-z) + c_2(\dot{z} - \dot{x}) = 0 \] \[ k_2(x-z) - c_2(\dot{x} - \dot{z}) = 0 \] \[ k_2(x-z) = c_2(\dot{x} - \dot{z}) \] So, \[ z = x - \frac{c_2}{k_2}(\dot{x} - \dot{z}) \] But let's express \( F_{series} \) in terms of \( x \) and \( \dot{x} \): \[ F_{series} = k_2(x-z) = c_2(\dot{x} - \dot{z}) \] Let \( y = x-z \), so \( \dot{y} = \dot{x} - \dot{z} \): \[ y = x-z \] \[ F_{series} = k_2 y = c_2 \dot{y} \] \[ \dot{y} + \frac{k_2}{c_2} y = \dot{x} \] #### **Combined Equation of Motion:** \[ m\ddot{x} + c_1\dot{x} + k_1x + F_{series} = p(t) \] But \( F_{series} \) itself is a dynamic variable. So the system is of **second order in \( x \)** plus a first order equation for \( y \): --- ## **Step 2: State Equations** Let \( x_1 = x \) \( x_2 = \dot{x} \) \( x_3 = y \) The equations are: \[ \begin{cases} \dot{x}_1 = x_2 \\ \dot{x}_2 = \frac{1}{m}\left[p(t) - k_1 x_1 - c_1 x_2 - k_2 x_3 \right] \\ \dot{x}_3 = x_2 - \frac{k_2}{c_2} x_3 \end{cases} \] --- ## **Step 3: Eigenvalues and Eigenvectors** Write in matrix form assuming \( p(t) = 0 \): \[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \\ -\frac{k_1}{m} & -\frac{c_1}{m} & -\frac{k_2}{m} \\ 0 & 1 & -\frac{k_2}{c_2} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \] Find eigenvalues \( \lambda \) from: \[ \det(A - \lambda I) = 0 \] --- ## **Step 4: Decoupled State Equations** Diagonalize \( A \) if possible, or use modal coordinates. --- ## **Step 5: Initial Condition Response** Set \( x_1(0) = 0 \), \( x_2(0) = 1 \), \( x_3(0) = 0 \). Solve the state equations for \( x_1(t) \). --- ## **Step 6: Harmonic Response** For \( p(t) = P_0 \sin(\omega t) \), solve using Laplace or Fourier transform. --- ## **Step 7: Numerical Values** Adopt given values and use zero-order hold to verify results numerically. --- ### **Summary** The system forms a **third-order** system due to the series damper-spring branch. The main steps above outline how to model, analyze, and solve for responses both analytically and numerically. Would you like to see the detailed calculations for each step?