Let's break down the steps as requested: --- ## (a) Free Body Diagram and Equation Verification for \( m_2 \) ### **Free Body Diagram for \( m_2 \):** **Forces acting on \( m_2 \):** - **Spring \( k_2 \) and Damper \( c_2 \) (between \( m_1 \) and \( m_2 \)):** - Spring force: \( F_{k2} = k_2(x_1 - x_2) \) - Damper force: \( F_{c2} = c_2(\dot{x}_1 - \dot{x}_2) \) - **External Force:** \( F_s \) (applied to \( m_2 \)) **Total force on \( m_2 \):** \[ F_{m_2} = k_2(x_1 - x_2) + c_2(\dot{x}_1 - \dot{x}_2) + F_s \] **Newton's Second Law:** \[ m_2 \ddot{x}_2 = k_2(x_1 - x_2) + c_2(\dot{x}_1 - \dot{x}_2) + F_s \] **Expanding:** \[ m_2 \ddot{x}_2 = k_2 x_1 - k_2 x_2 + c_2 \dot{x}_1 - c_2 \dot{x}_2 + F_s \] **Divide by \( m_2 \):** \[ \ddot{x}_2 = \frac{k_2}{m_2} x_1 - \frac{k_2}{m_2} x_2 + \frac{c_2}{m_2} \dot{x}_1 - \frac{c_2}{m_2} \dot{x}_2 + \frac{F_s}{m_2} \] **Rearrange:** \[ \ddot{x}_2 = -\frac{c_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2} x_2 + \frac{c_2}{m_2} \dot{x}_1 + \frac{k_2}{m_2} x_1 + \frac{F_s}{m_2} \] **This matches the given equation.** --- ## (b) State Vector \( \vec{X} \) Let’s define the state vector: \[ \vec{X} = \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} \] If we only consider the \( m_2 \) mass, the state-space form is: \[ \frac{d}{dt} \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} \dot{x}_2 \\ -\frac{c_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2} x_2 + \frac{c_2}{m_2} \dot{x}_1 + \frac{k_2}{m_2} x_1 + \frac{F_s}{m_2} \end{bmatrix} \] But note: \( x_1, \dot{x}_1 \) are from the other mass and are not defined by this equation alone! --- ## (c) Is State Space Solution Possible? **No, it is _not_ possible to fully solve the system using only the given equation.** ### **Reason:** - The equation for \( m_2 \) depends on both \( x_1 \) and \( \dot{x}_1 \) (the position and velocity of \( m_1 \)). - We do **not** have the equation of motion for \( m_1 \). - Without the coupled equation for \( m_1 \), the system is **underdetermined**; we have two unknowns (\( x_1 \) and \( x_2 \)) and only one equation. - To construct the **full state-space model**, we need both equations (for \( m_1 \) and \( m_2 \)). --- ## **Summary Table** | Part | Solution | |------|----------| | (a) | The free body diagram for \( m_2 \) leads to the given equation. Verified. | | (b) | State vector: \( \vec{X} = \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} \) but the equation is not self-contained. | | (c) | **No,** we cannot find a state-space solution from this equation alone because the dynamics of \( x_1 \), \( \dot{x}_1 \) are missing. | --- **If you need the state-space for the full two-mass system, you would need both equations (for \( m_1 \) and \( m_2 \)).**

# Step-by-Step Solution to the Problem ## (a) Free Body Diagram and Equation Verification for \( m_2 \) ### **Free Body Diagram for \( m_2 \)** **Forces acting on \( m_2 \):** - **Spring \( k_2 \)**: - Force due to spring: \( F_{k2} = k_2(x_1 - x_2) \) - **Damper \( c_2 \)**: - Force due to damper: \( F_{c2} = c_2(\dot{x}_1 - \dot{x}_2) \) - **External Force**: - Applied force: \( F_s \) ### **Total Force on \( m_2 \)** Using Newton's Second Law: \[ F_{m_2} = m_2 \ddot{x}_2 \] The total force equation becomes: \[ m_2 \ddot{x}_2 = k_2(x_1 - x_2) + c_2(\dot{x}_1 - \dot{x}_2) + F_s \] ### **Expanding and Rearranging** Rearranging gives: \[ m_2 \ddot{x}_2 = k_2 x_1 - k_2 x_2 + c_2 \dot{x}_1 - c_2 \dot{x}_2 + F_s \] Dividing by \( m_2 \): \[ \ddot{x}_2 = \frac{k_2}{m_2} x_1 - \frac{k_2}{m_2} x_2 + \frac{c_2}{m_2} \dot{x}_1 - \frac{c_2}{m_2} \dot{x}_2 + \frac{F_s}{m_2} \] This matches the given equation, confirming its validity for mass \( m_2 \). --- ## (b) State Vector \( \vec{X} \) ### **Defining the State Vector** We define the state vector as: \[ \vec{X} = \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} \] ### **State-Space Formulation** The state-space representation can be expressed as: \[ \frac{d}{dt} \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} \dot{x}_2 \\ -\frac{c_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2} x_2 + \frac{c_2}{m_2} \dot{x}_1 + \frac{k_2}{m_2} x_1 + \frac{F_s}{m_2} \end{bmatrix} \] ### **Note on Dependencies** The variables \( x_1 \) and \( \dot{x}_1 \) are from mass \( m_1 \), which are not defined by this equation alone. --- ## (c) Is a State Space Solution Possible? ### **Conclusion: No, it is not possible.** ### **Reasoning:** - The motion equation for \( m_2 \) depends on \( x_1 \) and \( \dot{x}_1 \), which are not provided. - The system is **underdetermined** because we have two unknowns (\( x_1 \) and \( x_2 \)) but only one equation. - To fully derive a state-space model, we need the equations of motion for both \( m_1 \) and \( m_2 \). --- ## **Summary Table** | Part | Solution | |------|----------| | (a) | Verified the equation for \( m_2 \) using a free body diagram. | | (b) | State vector: \( \vec{X} = \begin{bmatrix} x_2 \\ \dot{x}_2 \end{bmatrix} \) with a dependent equation. | | (c) | **No,** the solution cannot be found with the given information alone. | --- This structured approach clarifies the verification process, state vector definition, and the limitations in deriving a complete state-space solution.