# a) Find the Radial Bearing Loads at A and B ## Step 1: Draw the Free-Body Diagram and Write Equilibrium Equations - Shaft is supported at bearings A and B. - Loads \( F_1 = 300\,\mathrm{lbf} \) (at distance \( a = 8\,\mathrm{in} \) from A) and \( F_2 = 175\,\mathrm{lbf} \) (at distance \( b = 7\,\mathrm{in} \) from B). Let \( R_A \) and \( R_B \) be the radial reactions at A and B, respectively. The distance between A and B is \( L = a + b = 8 + 7 = 15\,\mathrm{in} \). Take moments about A: \[ \sum M_A = = F_1 \cdot a + F_2 \cdot (L-b) - R_B \cdot L \] \[ 300 \times 8 + 175 \times (15-7) = R_B \times 15 \] \[ 240 + 175 \times 8 = 15R_B \] \[ 240 + 140 = 15R_B \] \[ 380 = 15R_B \] \[ R_B = \frac{380}{15} = 253.\,\mathrm{lbf} \] **Explanation:** By taking moments about A, we calculate the reaction at B. --- ## Step 2: Use the Vertical Force Equilibrium to Find \( R_A \) \[ \sum F_y = = R_A + R_B - F_1 - F_2 \] \[ R_A + 253.33 - 300 - 175 = \] \[ R_A = 300 + 175 - 253.33 = 221.67\,\mathrm{lbf} \] **Explanation:** The sum of the upward reactions at the bearings equals the sum of the applied downward forces. --- **Final Answers:** - **Radial bearing load at A:** \( R_A = 221.67\,\mathrm{lbf} \) - **Radial bearing load at B:** \( R_B = 253.33\,\mathrm{lbf} \)

# Complete Solution to Bearing Loads and Design ## a) Find the Radial Bearing Loads at A and B ### Step 1: Draw the Free-Body Diagram and Write Equilibrium Equations Let the radial reactions at bearings A and B be \( R_A \) and \( R_B \) respectively. The total distance between A and B is: \[ L = a + b = 8\,\mathrm{in} + 7\,\mathrm{in} = 15\,\mathrm{in} \] Taking moments about A, we have: \[ \sum M_A = F_1 \cdot a + F_2 \cdot (L-b) - R_B \cdot L \] Substituting the values: \[ 300 \times 8 + 175 \times (15-7) = R_B \times 15 \] \[ 2400 + 1400 = 15R_B \] \[ 3800 = 15R_B \quad \Rightarrow \quad R_B = \frac{3800}{15} = 253.33\,\mathrm{lbf} \] **Explanation:** By taking moments about A, we determine the reaction force at B due to the applied loads. ### Step 2: Use the Vertical Force Equilibrium to Find \( R_A \) Using the vertical force equilibrium: \[ \sum F_y = R_A + R_B - F_1 - F_2 \] Substituting the known values: \[ R_A + 253.33 - 300 - 175 = 0 \] \[ R_A = 300 + 175 - 253.33 = 221.67\,\mathrm{lbf} \] **Explanation:** The sum of the forces must equal zero, allowing us to solve for the reaction at A. --- **Final Answers for Part a:** - **Radial bearing load at A:** \( R_A = 221.67\,\mathrm{lbf} \) - **Radial bearing load at B:** \( R_B = 253.33\,\mathrm{lbf} \) --- ## b) Calculate the Viscosity of Lubricant Required at A ### Step 1: Use the Bearing Equation For a hydrodynamically lubricated bearing, the viscosity \( \mu \) can be estimated using: \[ \mu = \frac{R_A \cdot L^2}{\eta \cdot N} \] Where: - \( \eta = 0.0015 \) (clearance ratio) - \( N = 2500\,\mathrm{rpm} \) Substituting the values: \[ \mu = \frac{221.67 \cdot (15)^2}{0.0015 \cdot 2500} \] ### Step 2: Solve for \( \mu \) Calculating: \[ \mu = \frac{221.67 \cdot 225}{3.75} = \frac{49875.75}{3.75} \approx 13380.2\,\mathrm{cP} \] **Explanation:** The viscosity is determined by the load, bearing dimensions, and operating conditions. --- **Final Answer for Part b:** - **Viscosity required at A:** \( \mu \approx 13380.2\,\mathrm{cP} \) --- ## c) Find the Angle of Maximum Pressure at A The angle at which the pressure is maximum in a hydrodynamic bearing can be calculated as: \[ \theta_{max} = \tan^{-1}\left(\frac{R_A}{R_B}\right) \] Substituting the values: \[ \theta_{max} = \tan^{-1}\left(\frac{221.67}{253.33}\right) \approx 0.707 \text{ radians} \approx 40.6^\circ \] **Explanation:** The angle is derived from the ratio of the radial loads, indicating where pressure concentration occurs. --- **Final Answer for Part c:** - **Angle of maximum pressure at A:** \( \theta_{max} \approx 40.6^\circ \) --- ## d) Find the Maximum Pressure in the Bearing at A Using the formula for maximum pressure in a hydrodynamic bearing: \[ P_{max} = \frac{R_A}{\pi \cdot d \cdot \eta} \] Where \( d = 1.2\,\mathrm{in} \). Thus, \[ P_{max} = \frac{221.67}{\pi \cdot 1.2 \cdot 0.0015} \] Calculating: \[ P_{max} \approx \frac{221.67}{0.005656} \approx 39141.0\,\mathrm{psi} \] **Explanation:** The maximum pressure is calculated based on the radial load and bearing dimensions. --- **Final Answer for Part d:** - **Maximum pressure in the bearing at A:** \( P_{max} \approx 39141.0\,\mathrm{psi} \) --- ## e) Find the Power Loss in the Bearing at A The power loss \( P \) in the bearing can be calculated as: \[ P = \frac{P_{max} \cdot Q}{33000} \] Where \( Q \) is the flow rate. For hydrodynamic bearings, \( Q \) can be approximated as: \[ Q = \frac{\pi \cdot d^2 \cdot (R_A / \mu)}{N} \] Substituting known values, we can calculate \( P \). **Explanation:** Power loss in a bearing is influenced by the maximum pressure and lubricant viscosity. --- **Final Answer for Part e:** - **Power loss in the bearing A:** (calculation to be completed based on flow rate) --- ## f) Find the Equivalent Radial Load at B The equivalent radial load \( P_B \) for B can be calculated as: \[ P_B = \sqrt{R_B^2 + \left(\frac{F_a}{C_o}\right)^2} \] Where \( F_a = 175\,\mathrm{lbf} \) and \( C_o \) is the dynamic load rating of the bearing. **Explanation:** The equivalent load accounts for both radial and axial forces acting on the bearing. --- **Final Answer for Part f:** - **Equivalent radial load at B:** (requires \( C_o \) value) --- ## g) Select the Suitable Bearing for B Using the L10 life formula: \[ L_{10} = \left(\frac{C}{P_B}\right)^3 \] Where \( L_{10} = 5700\,\mathrm{million\,revolutions} \) and \( Fa/C_o = 0.056 \) leads to determining the appropriate bearing that meets these specifications. **Explanation:** Selecting the correct bearing requires ensuring it meets both load ratings and life expectations. --- **Final Answer for Part g:** - **Suitable bearing selection for B:** (requires specific bearing details) --- ## h) Determine the Expected Life Time of the Ball Bearing at B The expected life can be calculated using: \[ L = \frac{C}{P_B}^3 \] Where \( C \) is the dynamic load rating and the failure rate is taken into account. **Explanation:** Expected lifetime is calculated based on the load and bearing rating to evaluate reliability. --- **Final Answer for Part h:** - **Expected lifetime of ball bearing at B:** (requires specific \( C \) value and calculations) --- **Note:** For parts e, f, g, and h, specific bearing values and calculations should be completed with appropriate data.