Let's break down the solution for Problem 11-31 as requested, step by step. --- ### **Step 1: Conceptual Introduction** This problem involves determining the **Basic Load Rating (C₁₀)** for a ball bearing located at the right end of a shaft. The shaft rotates at 900 revolutions per minute (rev/min), and the desired bearing life is 12,000 hours with a reliability of 98%. An application factor of 1.2 is to be considered. The goal is also to select a bearing from an online catalog that meets the calculated requirements and fits the shaft geometry. #### **Explanation Block** Bearings must be selected based on the expected life and load they will encounter. The Basic Load Rating, C₁₀, is a standardized measure of bearing capacity, ensuring that the bearing will last for the desired period under given conditions. Reliability and application factors adjust the requirements to account for real-world differences. --- ### **Step 2: Formulae** #### **Bearing Life Equation:** \[ L_{10} = \left(\frac{C_{10}}{F_e}\right)^a \times 10^6 \text{ revolutions} \] or, rearranged for \( C_{10} \): \[ C_{10} = F_e \left(\frac{L_{10}}{10^6}\right)^{1/a} \] Where: - \( L_{10} \) = rating life in millions of revolutions - \( C_{10} \) = Basic Dynamic Load Rating (N) - \( F_e \) = Equivalent dynamic bearing load (N) - \( a \) = 3 for ball bearings #### **Converting Life to Revolutions:** \[ L_{10} (\text{rev}) = (\text{Life in hours}) \times (\text{speed in rev/min}) \times 60 \] #### **Adjusting for Reliability:** \[ a_1 = \text{Reliability factor (from tables)} \] \[ L_{nm} = a_1 \times L_{10} \] #### **Application Factor:** \[ F_e = F_{applied} \times f_{application} \] --- #### **Explanation Block** The bearing life equation relates the bearing’s rated load capacity to its life expectancy under load. For ball bearings, the exponent \( a \) is 3. The required life is converted from hours to revolutions. Reliability and application factors account for non-standard operating conditions and increased reliability requirements, ensuring the selected bearing is suitable for the intended use. --- ### **Step 3: Step-by-Step Calculations for (a)** #### 1. **Calculate Required Life in Revolutions** \[ \text{Life (h)} = 12,000~h \] \[ \text{Speed} = 900~\text{rev/min} \] \[ L_{10} = 12,000 \times 900 \times 60 = 648,000,000~\text{revolutions} \] \[ L_{10} = \frac{648,000,000}{1,000,000} = 648~\text{million revolutions} \] #### 2. **Adjust for Reliability (98%)** From bearing tables (e.g., Table 11-6), the reliability factor \( a_1 \) for 98% reliability is approximately 0.62: \[ L_{nm} = a_1 \times L_{10} = 0.62 \times 648 = 402~\text{million rev} \] #### 3. **Application Factor** Given \( f_{application} = 1.2 \): \[ F_e = F_{applied} \times 1.2 \] #### 4. **Basic Load Rating Equation** Rearrange for \( C_{10} \): \[ C_{10} = F_e \left(\frac{L_{nm}}{10^6}\right)^{1/3} \] --- #### **Explanation Block** First, the required bearing life is calculated in revolutions. The reliability factor decreases the expected life to account for the higher reliability requirement. The application factor increases the load to account for working conditions. Finally, the basic load rating formula is rearranged to solve for the required \( C_{10} \), ensuring the bearing can handle the load for the desired period. --- ### **Step 4: Step-by-Step Calculations for (b)** #### 1. **Choose a Ball Bearing** - Use the calculated \( C_{10} \) value. - Using an online catalog, find a bearing with a \( C_{10} \) value **equal to or greater** than the calculated value. - Ensure the bearing fits the shaft diameter and available space (from the drawing). #### 2. **Adjust Dimensions if Needed** - If no standard bearing fits, consider adjusting the shaft or housing dimensions to suit available bearings. --- #### **Explanation Block** This step connects engineering calculations to practical product selection. After determining the required load rating, the next step is to find a real bearing that matches these requirements. Catalogs provide load ratings and dimensions, making it possible to select or adapt to a suitable bearing. --- ### **Final Answer (Summary):** **(a)** The required Basic Load Rating (\( C_{10} \)) for the bearing at the right end is: \[ C_{10} = F_e \left(\frac{402}{1}\right)^{1/3} \] where \( F_e = F_{applied} \times 1.2 \). Substitute the actual applied load to get the numerical answer. **(b)** Use a bearing catalog to select a bearing with a \( C_{10} \) rating **equal to or greater than** the calculated value and ensure it fits the shaft's dimensions. --- *If you provide the applied load value (\( F_{applied} \)), I can compute the exact \( C_{10} \) value for you!*