# Mechanical System Analysis ## **Step 1: Draw the Free Body Diagram (FBD)** Below is a free body diagram for the system, where: - \( m_1 \) and \( m_2 \): masses - \( k_1, k_2, k_3 \): spring constants - \( b_1 \): damping/friction coefficient (applies only to \( m_2 \)) - \( x_1(t) \), \( x_2(t) \): displacements of \( m_1 \) and \( m_2 \) **Free Body Diagram:** ``` |---/\/\/\---[m1]---/\/\/\---[m2]---/\/\/\---| k1 k2 k3 ↑ ↑ x1(t) x2(t) | Friction b1 ``` - **For \( m_1 \):** - Left: Spring \( k_1 \) - Right: Spring \( k_2 \) (connected to \( m_2 \)) - **For \( m_2 \):** - Left: Spring \( k_2 \) (connected to \( m_1 \)) - Right: Spring \( k_3 \) - Friction \( b_1 \) (acts on \( m_2 \) only) --- ## **Step 2: Write the Differential Equations** Let \( x_1 \) and \( x_2 \) be the displacements of \( m_1 \) and \( m_2 \) from equilibrium. ### **For Mass \( m_1 \):** - **Forces on \( m_1 \):** - Left: \( -k_1 x_1 \) - Right: \( k_2 (x_2 - x_1) \) - No friction \[ m_1 \ddot{x}_1 = -k_1 x_1 + k_2 (x_2 - x_1) \] \[ m_1 \ddot{x}_1 + (k_1 + k_2)x_1 - k_2 x_2 = \] --- ### **For Mass \( m_2 \):** - **Forces on \( m_2 \):** - Left: \( -k_2 (x_2 - x_1) \) - Right: \( -k_3 x_2 \) - Friction: \( -b_1 \dot{x}_2 \) \[ m_2 \ddot{x}_2 = -k_2 (x_2 - x_1) - k_3 x_2 - b_1 \dot{x}_2 \] \[ m_2 \ddot{x}_2 + b_1 \dot{x}_2 + (k_2 + k_3)x_2 - k_2 x_1 = \] --- ### **Final Answer** #### **System of Differential Equations:** \[ \boxed{ \begin{aligned} &1.\quad m_1 \ddot{x}_1 + (k_1 + k_2)x_1 - k_2 x_2 = \\ 2.\quad m_2 \ddot{x}_2 + b_1 \dot{x}_2 + (k_2 + k_3)x_2 - k_2 x_1 = \end{aligned} } \] --- ### **Example Values (for clarity):** Suppose \( m_1 = 1 \), \( m_2 = 2 \), \( k_1 = 3 \), \( k_2 = 4 \), \( k_3 = 5 \), \( b_1 = 6 \): \[ \begin{aligned} &1.\quad 1 \ddot{x}_1 + (3+4)x_1 - 4x_2 = \\ &\Rightarrow \ddot{x}_1 + 7x_1 - 4x_2 = \\ &2.\quad 2 \ddot{x}_2 + 6 \dot{x}_2 + (4+5)x_2 - 4x_1 = \\ &\Rightarrow 2 \ddot{x}_2 + 6 \dot{x}_2 + 9x_2 - 4x_1 = \end{aligned} \] --- **Summary:** 1. **Draw the FBD** (see diagram above). 2. **Write the equations** for each mass as shown. **Final answer:** The two coupled differential equations above describe the motion of the mechanical system.

# Mechanical System Analysis ## **Step 1: Draw the Free Body Diagram (FBD)** The Free Body Diagram (FBD) of the system includes the following components: - **\( m_1 \)** and **\( m_2 \)**: masses - **\( k_1, k_2, k_3 \)**: spring constants - **\( b_1 \)**: damping/friction coefficient (applies to \( m_2 \)) - **\( x_1(t) \)** and **\( x_2(t) \)**: displacements of \( m_1 \) and \( m_2 \) ### **Free Body Diagram Representation:** ``` |---/\/\/\---[m1]---/\/\/\---[m2]---/\/\/\---| k1 k2 k3 ↑ ↑ x1(t) x2(t) | Friction b1 ``` - **For \( m_1 \) (left mass):** - Spring \( k_1 \) acting left - Spring \( k_2 \) acting right (connected to \( m_2 \)) - **For \( m_2 \) (right mass):** - Spring \( k_2 \) acting left (connected to \( m_1 \)) - Spring \( k_3 \) acting right - Friction \( b_1 \) acting to the left --- ## **Step 2: Write the Differential Equations** Let \( x_1 \) and \( x_2 \) represent the displacements of \( m_1 \) and \( m_2 \) from their equilibrium positions. ### **For Mass \( m_1 \):** - **Forces on \( m_1 \):** - Left force from spring \( k_1 \): \( -k_1 x_1 \) - Right force from spring \( k_2 \): \( k_2 (x_2 - x_1) \) The equation of motion for \( m_1 \) is: \[ m_1 \ddot{x}_1 = -k_1 x_1 + k_2 (x_2 - x_1) \] Rearranging gives: \[ m_1 \ddot{x}_1 + (k_1 + k_2)x_1 - k_2 x_2 = 0 \] --- ### **For Mass \( m_2 \):** - **Forces on \( m_2 \):** - Left force from spring \( k_2 \): \( -k_2 (x_2 - x_1) \) - Right force from spring \( k_3 \): \( -k_3 x_2 \) - Damping force: \( -b_1 \dot{x}_2 \) The equation of motion for \( m_2 \) is: \[ m_2 \ddot{x}_2 = -k_2 (x_2 - x_1) - k_3 x_2 - b_1 \dot{x}_2 \] Rearranging gives: \[ m_2 \ddot{x}_2 + b_1 \dot{x}_2 + (k_2 + k_3)x_2 - k_2 x_1 = 0 \] --- ### **Final Answer** #### **System of Differential Equations:** The resulting set of differential equations for the mechanical system is: \[ \boxed{ \begin{aligned} &1.\quad m_1 \ddot{x}_1 + (k_1 + k_2)x_1 - k_2 x_2 = 0 \\ &2.\quad m_2 \ddot{x}_2 + b_1 \dot{x}_2 + (k_2 + k_3)x_2 - k_2 x_1 = 0 \end{aligned} } \] --- ### **Example Values** Assume the following values for the parameters: - \( m_1 = 1 \, \text{kg} \) - \( m_2 = 2 \, \text{kg} \) - \( k_1 = 3 \, \text{N/m} \) - \( k_2 = 4 \, \text{N/m} \) - \( k_3 = 5 \, \text{N/m} \) - \( b_1 = 6 \, \text{Ns/m} \) Substituting these values into the equations gives: 1. For \( m_1 \): \[ 1 \ddot{x}_1 + (3 + 4)x_1 - 4x_2 = 0 \implies \ddot{x}_1 + 7x_1 - 4x_2 = 0 \] 2. For \( m_2 \): \[ 2 \ddot{x}_2 + 6 \dot{x}_2 + (4 + 5)x_2 - 4x_1 = 0 \implies 2 \ddot{x}_2 + 6 \dot{x}_2 + 9x_2 - 4x_1 = 0 \] --- ### **Summary** 1. **Draw the Free Body Diagram** to visualize the forces acting on the masses. 2. **Write the differential equations** for each mass based on the forces. **Final Answer:** The two coupled differential equations describe the motion of the mechanical system.