Let's break down the given **problem 37** and the provided **figure** step by step: --- ### **Given Data:** - **Vertical shaft speed:** 600 rpm - **Power input:** 4.0 hp (entering through bevel gear) - **Each chain sprocket delivers:** 2.0 hp (to the side for mixer blades) - **Bevel Gear:** - **Diametral pitch:** 5 - **Pitch diameter:** 9.000 in - **Face width:** 1.31 in - **Pressure angle:** 20° - **Material:** SAE 4140 OQT 1000 steel for the shaft - **Safety Factor:** N = 2 - **Reliability:** 0.99 --- ### **What You Need to Do:** 1. **Compute the forces on the bevel gear** (using Chapter 10 methods). 2. **Analyze the shaft** for strength and reliability with the given factor of safety. 3. **Design/check the shaft** for the loading from the bevel gear and sprockets. --- ## **Step 1: Power and Torque Calculation** **Power (P):** \( 4.0 \text{ hp} \) **Speed (n):** \( 600 \text{ rpm} \) Convert power to watts: \[ P = 4.0 \text{ hp} \times 746 = 2984 \text{ W} \] **Torque (T):** \[ T = \frac{63025 \times P_{\text{hp}}}{n} \] \[ T = \frac{63025 \times 4}{600} = 420.2 \text{ in-lb} \] --- ## **Step 2: Forces on Bevel Gear** You need to find the **tangential (W_T), radial (W_R), and axial (W_A) forces** on the bevel gear. - **Pitch diameter (d):** 9.0 in \[ W_T = \frac{2T}{d} \] \[ W_T = \frac{2 \times 420.2}{9} = 93.4 \text{ lb} \] - **Pressure angle (\(\phi\)):** 20° - **Pitch angle (\(\theta\)):** For a 90° bevel gear set, pinion and gear have pitch angles of 45° - **Axial force:** \( W_A = W_T \tan(\theta) \) - **Radial force:** \( W_R = W_T \tan(\phi) \cos(\theta) \) \[ W_A = W_T \tan(45°) = 93.4 \times 1 = 93.4 \text{ lb} \] \[ W_R = W_T \tan(20°) \cos(45°) \] \[ = 93.4 \times 0.3640 \times 0.7071 = 24.0 \text{ lb} \] --- ## **Step 3: Sprocket Loads** Each sprocket delivers 2.0 hp. \[ T_{\text{sprocket}} = \frac{63025 \times 2}{600} = 210.1 \text{ in-lb} \] \[ W_{\text{chain}} = \frac{2 \times 210.1}{10} = 42.0 \text{ lb} \] --- ## **Step 4: Load Distribution and Shaft Analysis** - The figure shows locations for bearings and points where loads are applied (bevel gear, chain sprockets). - Use the calculated forces at locations A, C, D (where gear and sprocket loads are applied). - Draw the free body diagram and calculate reaction forces at the bearings. - Use the **shaft design criteria** (bending, torsion, combined stress) and apply the safety factor \( N = 2 \) and reliability correction. --- ## **Step 5: Material Strength** - SAE 4140 OQT 1000 steel properties (approximate): - Yield strength, \( S_y \) ≈ 90 ksi - Ultimate strength, \( S_u \) ≈ 115 ksi --- ### **Summary of Calculated Forces:** - **Bevel Gear:** - Tangential: **93.4 lb** - Axial: **93.4 lb** - Radial: **24.0 lb** - **Each Sprocket:** **42.0 lb** tangential chain force --- ### **Next Steps** 1. **Draw the complete force diagram on the shaft.** 2. **Calculate the bending moment and torsional moment at each critical section.** 3. **Check the shaft for combined stresses using the selected factor of safety and reliability.** If you need the full shaft design and stress calculations, let me know and I can walk you through each step in detail!