Let's break down **Problem 1** step by step, referencing **Table 12-5** where needed. ### **Given Data:** - **Bearing size:** 32 mm (diameter) × 32 mm (length) - **Load \( W \):** 800 N - **Journal speed \( n \):** 3600 rpm - **Oil:** SAE 40 at 70°C --- ## **1. Is this a thick film lubrication (stable lubrication)? And why?** (6 pts) **Step 1: Calculate the unit load (pressure).** \[ \text{Unit Load (P)} = \frac{W}{d \cdot L} \] Where: - \( W = 800\, \text{N} \) - \( d = 32\, \text{mm} = 0.032\, \text{m} \) - \( L = 32\, \text{mm} = 0.032\, \text{m} \) \[ P = \frac{800}{0.032 \times 0.032} = \frac{800}{0.001024} = 781250\, \text{N/m}^2 = 0.781\, \text{MPa} \] **Step 2: Compare with Table 12-5** For electric motors, steam turbines, gear reducers, the typical unit load is \( 0.8 - 1.5\, \text{MPa} \). For automotive main bearings, \( 4-5\, \text{MPa} \). Since \( 0.781\, \text{MPa} \) is on the lower end, it's within the range for light-duty applications. At this load, hydrodynamic (thick film) lubrication is possible if the speed and clearance are adequate. **Step 3: Calculate bearing characteristic number (\( S \))** We need oil viscosity (\( \eta \)). SAE 40 at 70°C ≈ \( 0.026\, \text{Pa·s} \) (from tables). \[ n' = \text{revolutions per second} = \frac{3600}{60} = 60\, \text{Hz} \] \[ S = \frac{\eta n'}{P} \] \[ S = \frac{0.026 \times 60}{0.781 \times 10^6} = \frac{1.56}{781000} \approx 2 \times 10^{-6} \] Typically, \( S > 1 \times 10^{-6} \) is indicative of thick film lubrication. **Conclusion:** **Yes, this is a thick film (hydrodynamic) lubrication condition, as the bearing operates at a reasonable load and high speed, and the viscosity is typical for stable lubrication.** --- ## **2. Assume the bearing operates at minimum coefficient of friction (Min \( f \)).** ### **a. Estimate the radial clearance (4 pts)** **Standard design: \( \frac{c}{d} \approx 0.001 \) to \( 0.002 \) for stable operation.** \[ c = (0.0015) \times d = 0.0015 \times 32\, \text{mm} = 0.048\, \text{mm} \] --- ### **b. Estimate the maximum temperature (4 pts)** **For a rough estimate (since detailed calculation needs heat dissipation info):** \[ \Delta T_{\text{max}} = \frac{P \cdot v \cdot L}{\text{Heat Dissipation Rate}} \] But usually, with good oil flow and normal operation, oil temperature rise is \( 10-30^\circ C \) above ambient. Given data: - Oil at 70°C (already stated) So, **maximum oil temperature = 70°C** (as per problem statement). --- ### **c. Check the 4 Trumpler’s criteria (4 pts)** **Trumpler’s criteria (for stable lubrication):** 1. \( \frac{L}{d} \approx 1 \) (here, \( L/d = 1 \), so OK) 2. \( \frac{c}{d} = 0.001 - 0.002 \) (here, \( 0.0015 \), so OK) 3. Unit load: \( 0.8 - 1.5\, \text{MPa} \) (here, \( 0.781\, \text{MPa} \), slightly low but acceptable) 4. Shaft surface velocity \( v > 1\, \text{m/s} \): \[ v = \pi d n' \] \[ = \pi \times 0.032 \times 60 = 6.03\, \text{m/s} \] **Conclusion: All criteria are met.** --- ## **3. When the load is increased to 3600 N:** ### **a. Maximum Pressure (4 pts)** \[ P = \frac{3600}{0.032 \times 0.032} = \frac{3600}{0.001024} = 3,515,625\, \text{N/m}^2 = 3.52\, \text{MPa} \] --- ### **b. Maximum temperature (4 pts)** Higher load increases friction/heat. Temperature rise will be higher, but with same oil and speed, typical design rise is 10–30°C above ambient, so up to **80–90°C** may be expected. If not enough oil cooling, further analysis is needed. --- ### **c. Maximum speed (4 pts)** The surface speed: \[ v = \pi d n' \] \[ = \pi \times 0.032 \times 60 = 6.03\, \text{m/s} \] This is already calculated; with increased load, the speed does not change. --- ### **Summary of Answers** #### **1. Is this a thick film lubrication?** - **Yes, thick film lubrication is achieved due to suitable viscosity, load, and speed.** #### **2. At minimum coefficient of friction:** - **a. Radial clearance:** \( c = 0.048\, \text{mm} \) - **b. Maximum temperature:** \( 70^\circ C \) (given) - **c. Trumpler’s criteria:** **All satisfied** #### **3. Increased load to 3600 N:** - **a. Maximum Pressure:** \( 3.52\, \text{MPa} \) - **b. Maximum temperature:** **80–90°C** (estimated) - **c. Maximum speed:** \( 6.03\, \text{m/s} \) --- **If you need more detailed calculations (such as frictional heat or coefficient of friction), let me know!**