Let's solve the given problem **step by step** using the **mobility (Gruebler's) equation**. --- ## **Step 1: Understanding the Problem** - We have a planar mechanism shown in Fig. 6.1. - We are to: 1. Use the mobility equation to determine the degrees of freedom (DOF). 2. If two lower pairs on the right side are connected, how is mobility affected? 3. If link \( G \) is removed, is the mechanism still constrained? 4. Are there any links that can be removed and still preserve a single DOF mechanism? --- ## **Step 2: The Mobility Equation (Gruebler’s Equation)** For planar mechanisms: \[ M = 3(n-1) - 2j - h \] Where: - \( M \) = Mobility (Degrees of Freedom) - \( n \) = Number of links (including frame) - \( j \) = Number of lower pairs (single DOF joints, e.g., revolute or prismatic) - \( h \) = Number of higher pairs (multiple DOF joints) --- ## **Step 3: Analyzing the Mechanism** ### **Count Links (\( n \))** From the figure (left side), label the links: - Ground (fixed frame): 1 - A, B, C, D, E, F, G (moving links): 7 Thus, \[ n = 7 + 1 = 8 \] ### **Count Lower Pairs (\( j \))** Count the revolute joints: - From the diagram, the joints are at each pin connection. - Let's count the number of pin joints: - Between frame and A - Between A and B - Between B and C - Between C and frame - Between B and E - Between E and D - Between D and C - Between E and F - Between F and D - Between E and G - Between G and F Let's count them one by one: 1. Frame-A 2. A-B 3. B-C 4. C-Frame 5. B-E 6. E-D 7. D-C 8. E-F 9. F-D 10. E-G 11. G-F So, \[ j = 11 \] ### **Count Higher Pairs (\( h \))** From the diagram, **all joints are lower pairs (revolute joints)**. Thus, \[ h = \] --- ## **Step 4: Calculate Mobility (\( M \))** Plug in the values: \[ M = 3(n-1) - 2j - h \] \[ M = 3(8-1) - 2 \times 11 - \] \[ M = 3 \times 7 - 22 \] \[ M = 21 - 22 \] \[ M = -1 \] --- ## **Step 5: Answering Each Question** ### **(a) Degrees of Freedom** \[ \boxed{M = -1} \] This means the mechanism is **overconstrained** (structure, not a mechanism). --- ### **(b) If two lower pairs on the right side of Fig. 6.1 are connected, how is mobility affected?** - If you connect two lower pairs (e.g., adding a link between two joints), you **reduce the number of links by 1** and **reduce the number of joints by 1**. Now, - \( n = 8 - 1 = 7 \) - \( j = 11 - 1 = 10 \) Plug in: \[ M = 3(7-1) - 2 \times 10 \] \[ M = 3 \times 6 - 20 \] \[ M = 18 - 20 = -2 \] \[ \boxed{M = -2} \] Mobility **decreases further** (even more overconstrained). --- ### **(c) If the link marked G is removed from Fig. 6.1, how is it mobility affected?** Removing link G: - \( n = 8 - 1 = 7 \) - Count the joints G is connected to: E-G and G-F (2 joints), so \( j = 11 - 2 = 9 \) Plug in: \[ M = 3(7-1) - 2 \times 9 \] \[ M = 18 - 18 = \] \[ \boxed{M = } \] So, **it becomes a rigid structure (no motion possible)**. --- ### **(d) Are there any links that can be removed and still preserve a single degree of freedom mechanism?** - Removing any link **reduces both n and j**. - Try removing another link (other than G), for example, F: - \( n = 7 \) - F is connected to E-F and F-D and G-F (3 joints), so \( j = 11 - 3 = 8 \) Plug in: \[ M = 3(7-1) - 2 \times 8 \] \[ M = 18 - 16 = 2 \] So, \( M = 2 \) Try another link, say E: - E is connected to B-E, E-D, E-F, E-G (4 joints), so \( n = 7 \), \( j = 11 - 4 = 7 \) \[ M = 18 - 14 = 4 \] **In general, this mechanism is overconstrained. Removing a single link either leaves it rigid or increases DOF above 1. No single link can be removed to create a mechanism with a single DOF.** \[ \boxed{\text{No, there are no links that can be removed to give a single degree of freedom mechanism.}} \] --- ## **Summary Table** | Case | \( n \) | \( j \) | \( M \) | Description | |------|-------|-------|------|-------------| | Original | 8 | 11 | -1 | Overconstrained | | Two right pairs connected | 7 | 10 | -2 | More overconstrained | | Remove G | 7 | 9 | | Rigid, no motion | | Remove other links | 7 | <9 | > | More mobility than 1 | --- ## **Final Answers** ### **(a)** \( \boxed{-1} \) (overconstrained) ### **(b)** \( \boxed{-2} \) (even more overconstrained) ### **(c)** \( \boxed{} \) (rigid, no motion) ### **(d)** \( \boxed{\text{No, removing any single link does not result in a single degree of freedom mechanism.}} \)