Let's analyze the mechanism step-by-step to find its degrees of freedom (DOF). --- ## Step 1 Identify the links and joints -Links:** - Link : Fixed frame (attached to the wall) - Link 2: Connecting link (the link between the fixed frame and the crank) - Link 3: Crank (connected to the piston) - Link 4: Connecting rod (between the crank and the piston) - Link 5: Piston (moving vertically) - **Joints:** - Joint 1: Fixed pivot at the wall (for link 2) - Joint 2: Revolute joint connecting link 2 and link 3 - Joint 3: Revolute joint between link 3 and link 4 - Joint 4: Revolute joint between link 4 and the piston (link 5) - Joint 5: Slider (piston moving vertically) --- ## Step 2: Count the total number of links and joints - **Number of links (including the fixed frame):** 5 - **Number of joints:** 5 --- ## Step 3: Determine the type of joints - Joints 1 to 4 are **revolute joints** (each providing 1 DOF) - Joint 5 is a **sliding joint** (prismatic joint, providing 1 DOF) --- ## Step 4: Apply Gruebler's equation for planar mechanisms The general formula for DOF is: \[ \text{DOF} = 3(n - 1) - 2j_1 - j_2 \] Where: - \( n \): total number of links (including the fixed frame) - \( j_1 \): number of **revolute joints** - \( j_2 \): number of **prismatic joints** For this mechanism: - \( n = 5 \) - \( j_1 = 4 \) (joints 1 to 4) - \( j_2 = 1 \) (joint 5) --- ## Step 5: Calculate the degrees of freedom \[ \text{DOF} = 3(5 - 1) - 2(4) - 1 = 3 \times 4 - 8 - 1 = 12 - 8 - 1 = 3 \] --- ## **Final Answer:** **The mechanism has 3 degrees of freedom.** This indicates it can move in three independent ways, which is typical for a mechanism controlling piston motion with additional possible movements.