# Step-by-Step Solution: Degree of Freedom (DoF) of the Given Linkage We will use **Gruebler’s Equation** (or Kutzbach Criterion) for planar mechanisms to calculate the DoF: \[ \text{DoF} = 3(n-1) - 2j_1 - j_2 \] Where: - \( n \) = Number of links (including the frame) - \( j_1 \) = Number of lower pairs (single DoF joints: revolute or prismatic, called "full joints") - \( j_2 \) = Number of higher pairs (two DoF joints, called "half joints" – uncommon in basic mechanisms; typically, j2 = 0 unless cams/gears/etc.) --- ## 1. **Count the Number of Links (\( n \))** - **Frame (Fixed Link):** 1 - **Moving Links:** Count each rigid body that moves independently. Let's count visually: 1. Ground (fixed) 2. Left vertical slider 3. Link connecting left slider to triangle 4. Triangle link (central one) 5. Right vertical slider 6. Link connecting triangle to right slider 7. Link connecting triangle to the bottom slider 8. Bottom slider 9. Link from triangle to right ground (pivot) **Total links, \( n \):** **9** --- ## 2. **Count the Number of Joints** ### **Full Joints (\( j_1 \))** Full joints are: - **Revolute (pin) joints** (allow rotation) - **Prismatic (slider) joints** (allow translation) Let's count: - **Pin Joints** (circle marker): - 1 (left slider to ground) - 2 (left slider to link) - 3 (link to triangle) - 4 (triangle to link to right slider) - 5 (right slider to ground) - 6 (bottom slider to ground) - 7 (triangle to link to bottom slider) - 8 (triangle to link to right ground) **Total pin joints: 8** - **Slider Joints** (prismatic): - 1 (left slider) - 2 (right slider) - 3 (bottom slider) But note: Each slider joint counts as a full joint, and we've already included the pin joints at their ends. So each slider has one prismatic joint. **Total prismatic joints: 3** **Total full joints (\( j_1 \)):** \[ j_1 = 8 \text{ (pin)} + 3 \text{ (prismatic)} = 11 \] ### **Half Joints (\( j_2 \))** No cams, gears, or other higher pairs visible. **So, \( j_2 = 0 \)** --- ## 3. **Plug Values into Formula** \[ \text{DoF} = 3(n-1) - 2j_1 - j_2 \] \[ \text{DoF} = 3(9-1) - 2(11) - 0 \] \[ \text{DoF} = 3 \times 8 - 22 \] \[ \text{DoF} = 24 - 22 \] \[ \text{DoF} = 2 \] --- ## 4. **Interpretation** - If **DoF = 0**: Structure (does not move) - If **DoF > 0**: Mechanism (has mobility) **This is a mechanism with 2 degrees of freedom.** --- ## **Summary Table** | Quantity | Value | |-------------------|-------| | Number of links (\( n \)) | 9 | | Number of full joints (\( j_1 \)) | 11 | | Number of half joints (\( j_2 \)) | 0 | | Degree of Freedom (DoF) | 2 | | Structure or Mechanism | Mechanism | --- ## **Final Answer** - **Number of links:** 9 - **Number of full joints:** 11 - **Number of half joints:** 0 - **Degree of Freedom (\( \text{DoF} \)):** 2 - **Type:** **Mechanism** (because DoF > 0)