## Step 1: Number of Degrees of Freedom **Explanation:** Each block can move independently along the horizontal axis. Let \( x_1 \) and \( x_2 \) be their respective positions. Since both positions can change independently (constrained only by springs/damper, not by each other), **the system has 2 degrees of freedom**. > **Circle:** 2 > *Explanation: Each mass can move independently along the x-axis, giving us 2 independent coordinates.* --- ## Step 2: Number of States for State Space Model **Explanation:** For a state space model, each degree of freedom requires two states: position and velocity. - For \( x_1 \): states are \( x_1 \) and \( \dot{x}_1 \) - For \( x_2 \): states are \( x_2 \) and \( \dot{x}_2 \) So, total states = \( 2 \times 2 = 4 \). > **Circle:** 4 > *Explanation: Each of the 2 masses has a position and velocity, making 4 states total.* --- ## Step 3: Complete the Free-Body Diagram Let's draw forces on each block: ### Block 1 (\( m \), position \( x_1 \)): - **Spring force (left):** \( -k x_1 \) (restoring force from wall) - **Spring force (right):** \( k(x_2 - x_1) \) (from middle spring, pulls toward \( x_2 \)) - **Damping force (damper):** \( c(\dot{x}_2 - \dot{x}_1) \) (from damper, proportional to relative velocity) ### Block 2 (\( m \), position \( x_2 \)): - **Spring force (left):** \( -k(x_2 - x_1) \) (from middle spring, opposite to force on block 1) - **Damping force (damper):** \( -c(\dot{x}_2 - \dot{x}_1) \) (opposite to force on block 1) - **Spring force (right):** None (not attached to wall) - **External force:** \( F \) (applied rightwards) **Free-body diagram:** ``` Block 1 (x1): Block 2 (x2): <--- [ -k x1 ] <--- [ -k(x2-x1) ] <--- [ -c(ẋ2-ẋ1) ] <--- [ -c(ẋ2-ẋ1) ] ---> [ k(x2-x1) ] <--- [ -k(x2-x1) ] ---> [ c(ẋ2-ẋ1) ] <--- [ -c(ẋ2-ẋ1) ] ---> [ F ] (external) ``` Or, as vectors on the diagram: - For Block 1 (\( x_1 \)): - Left: \(-k x_1\) - Right: \(k(x_2 - x_1) + c(\dot{x}_2 - \dot{x}_1)\) - For Block 2 (\( x_2 \)): - Left: \(-k(x_2 - x_1) - c(\dot{x}_2 - \dot{x}_1)\) - Right: \(F\) --- ## **Final Answers** 1. **Degrees of freedom:** **2** 2. **Number of states:** **4** 3. **Free-body diagram:** - **Block 1:** Forces from wall spring, middle spring, and damper. - **Block 2:** Forces from middle spring, damper, and external force \( F \). *(See explanation above for force directions and expressions.)*