Let's solve **Problem 1** step by step. --- ### **Given Data** - Bearing size: \( 32 \times 32 \) mm (diameter \( D = 32 \) mm, length \( L = 32 \) mm) - Load \( W = 800 \) N - Journal speed \( N = 3600 \) rev/min - Oil: SAE 40, temperature = \( 70^\circ C \) - Assume **average viscosity** for SAE 40 at \( 70^\circ C \) is \( \mu \approx 0.018 \) Pa·s (you can refine this from tables). --- ### **1. Is this a thick film lubrication (stable lubrication)? Why?** #### **Step 1.1: Calculate Sommerfeld Number** The **Sommerfeld number** (\( S \)) is given by: \[ S = \frac{r}{c} \left( \frac{\mu N}{p} \right) \] Where: - \( r \) = radius = \( \frac{D}{2} = 16 \) mm = 0.016 m - \( c \) = radial clearance (unknown, but typically \( c \approx 0.001 \times D \)) - \( \mu \) = dynamic viscosity = 0.018 Pa·s - \( N \) = rev/s = \( \frac{3600}{60} = 60 \) rev/s - \( p \) = bearing pressure = \( \frac{W}{L D} \) Assume \( c = 0.032 \) mm = 0.000032 m (will verify in next part). \[ p = \frac{800}{0.032 \times 0.032} = \frac{800}{0.001024} = 781,250 \ \text{Pa} \] \[ \frac{\mu N}{p} = \frac{0.018 \times 60}{781,250} = \frac{1.08}{781,250} = 0.000001382 \] \[ \frac{r}{c} = \frac{0.016}{0.000032} = 500 \] \[ S = 500 \times 0.000001382 = 0.000691 \] **Typical thick film lubrication (hydrodynamic) occurs for \( S > 0.001 \).** - Our \( S \) is less than 0.001, so this suggests it may **not** be thick film. - But if we choose a slightly larger clearance, \( S \) could increase. For now, **not thick film**. #### **Conclusion:** **Not thick film lubrication, because the Sommerfeld number is less than the typical threshold for hydrodynamic lubrication.** --- ### **2. Assume minimum coefficient of friction.** #### **Step 2a: Estimate the radial clearance (\( c \))** Rule of thumb: \[ c = 0.001 \times D = 0.001 \times 32 = 0.032 \ \text{mm} = 0.000032 \ \text{m} \] --- #### **Step 2b: Estimate the maximum temperature** Max temperature is given as **70°C** (from problem statement). --- #### **Step 2c: Check the 4 Trumpler’s criteria** Trumpler's criteria are: 1. \( p < 10 \) MPa 2. \( v < 10 \) m/s 3. \( pv < 0.7 \) MPa·m/s 4. \( \frac{L}{D} < 2 \) ##### **1. Maximum Pressure** \[ p = \frac{W}{L D} = \frac{800}{0.032 \times 0.032} = 781,250 \ \text{Pa} = 0.781 \ \text{MPa} \] - **OK** (less than 10 MPa) ##### **2. Maximum velocity** \[ v = \pi D N = \pi \times 0.032 \times 60 = 6.03 \ \text{m/s} \] - **OK** (less than 10 m/s) ##### **3. \( pv \) value** \[ pv = 0.781 \times 6.03 = 4.71 \ \text{MPa·m/s} \] - **NOT OK** (should be < 0.7 MPa·m/s) ##### **4. \( L / D \) ratio** \[ \frac{L}{D} = \frac{32}{32} = 1 \] - **OK** (less than 2) --- ### **3. When Load is Increased to 3600 N** - All other dimensions remain the same. #### **Step 3a: Maximum Pressure** \[ p = \frac{3600}{0.032 \times 0.032} = \frac{3600}{0.001024} = 3,515,625 \ \text{Pa} = 3.52 \ \text{MPa} \] - **OK** (less than 10 MPa) #### **Step 3b: Maximum Temperature** Still 70°C (assuming oil temperature unchanged). #### **Step 3c: Maximum Speed** Still \( v = 6.03 \) m/s. ##### **Check new \( pv \) value** \[ pv = 3.52 \times 6.03 = 21.24 \ \text{MPa·m/s} \] - **NOT OK** (should be < 0.7 MPa·m/s) --- ## **Summary Table** | Check | At 800 N | At 3600 N | Criteria | Satisfied? | |--------------------------|----------|-----------|------------------|----------------| | Max Pressure (MPa) | 0.78 | 3.52 | < 10 | Yes | | Max Velocity (m/s) | 6.03 | 6.03 | < 10 | Yes | | \( pv \) (MPa·m/s) | 4.71 | 21.24 | < 0.7 | **No** | | \( L/D \) | 1 | 1 | < 2 | Yes | --- ## **Final Answers** ### 1. **Is this a thick film lubrication?** No, because the Sommerfeld number is below the typical threshold for stable hydrodynamic lubrication. ### 2. **Assuming minimum coefficient of friction:** - **Radial clearance \( c \):** \( 0.032 \) mm (or \( 0.001 \times D \)) - **Maximum temperature:** \( 70^\circ C \) - **Trumpler’s criteria:** - Max pressure: OK - Max velocity: OK - \( pv \): **NOT OK** - \( L/D \): OK ### 3. **When Load increases to 3600 N:** - **Maximum pressure:** \( 3.52 \) MPa (**OK**) - **Maximum temperature:** \( 70^\circ C \) - **Maximum speed:** \( 6.03 \) m/s (**OK**) - **\( pv \): \( 21.24 \) MPa·m/s (**NOT OK**)** --- **If you need more detailed calculations or have tables for oil viscosity, let me know!**