Certainly! To address the given engineering design problem step by step, each important parameter will be calculated and explained clearly. The solution will be structured for maximum clarity and completeness. --- ### **Step 1: Data Extraction and Problem Restatement** The problem provides the following information: - **Motor Power:** 20 hp electric motor - **Motor Speed (N₁):** 1400 rpm - **Gear Pump Speed (N₂):** 560 rpm - **Nominal Center Distance (C):** 22.00 in - **Adjustment Range:** ±0.375 in - **Motor Shaft Diameter:** 1.425 in - **Operation:** 8–16 h/day, 5 days/week **Tasks:** - Specify the complete drive system. - Determine the minimum sprocket size on the pump to avoid interference with the base plate. --- ### **Step 2: Speed Ratio Calculation** The **speed ratio** (reduction ratio) is calculated as: \[ \text{Speed Ratio} = \frac{N_1}{N_2} = \frac{1400}{560} = 2.5 \] **Interpretation:** The driven sprocket (pump) must rotate at 2.5 times slower than the driver sprocket (motor). --- ### **Step 3: Sprocket Teeth Selection** Let \(N_M\) be the number of teeth on the motor sprocket and \(N_P\) be the number of teeth on the pump sprocket. \[ \frac{N_P}{N_M} = \frac{N_1}{N_2} = 2.5 \] Assume a practical minimum number of teeth for the small sprocket (to avoid excessive wear and interference) is **18 teeth** (standard minimum for synchronous belts). \[ N_M = 18 \] \[ N_P = 2.5 \times N_M = 2.5 \times 18 = 45 \] So, the closest standard number of teeth for \(N_P\) is **45**. --- ### **Step 4: Pitch Diameter Calculation** For synchronous belts, the **pitch diameter (PD)** of a sprocket is given by: \[ \text{Pitch Diameter (in)} = \frac{\text{Number of Teeth} \times \text{Pitch}}{\pi} \] Assume a standard pitch, such as **0.5 in** (typical for this power and speed): - **Motor Sprocket:** \[ \text{PD}_M = \frac{18 \times 0.5}{\pi} \approx 2.87 \text{ in} \] - **Pump Sprocket:** \[ \text{PD}_P = \frac{45 \times 0.5}{\pi} \approx 7.16 \text{ in} \] --- ### **Step 5: Center Distance Check** The actual **center distance** using the selected sprockets and belt length should be close to 22.00 in ± 0.375 in. The belt length can be verified using the standard formula for two-pulley systems: \[ L = \pi \left(\frac{D_1 + D_2}{2}\right) + 2C + \frac{(D_2 - D_1)^2}{4C} \] Where: - \(L\) = belt length (in) - \(D_1\) = small sprocket PD (motor) - \(D_2\) = large sprocket PD (pump) - \(C\) = center distance (in) Plug values: \[ L = \pi\left(\frac{2.87 + 7.16}{2}\right) + 2(22) + \frac{(7.16 - 2.87)^2}{4 \times 22} \] \[ L = \pi(5.015) + 44 + \frac{(4.29)^2}{88} \] \[ L = 15.75 + 44 + 0.21 = 59.96 \text{ in} \] Choose a **standard belt length** close to this value (e.g., 60 in). --- ### **Step 6: Minimum Sprocket Size to Avoid Base Plate Interference** The **minimum clearance** is determined by the sprocket radius plus the shaft radius, ensuring the bottom of the sprocket does not touch the base plate. - **Pump shaft radius:** \( \frac{1.425}{2} = 0.7125 \) in - **Radius of pump sprocket:** \( \frac{7.16}{2} \approx 3.58 \) in **Total minimum clearance:** \[ \text{Clearance} = 3.58 \text{ in (sprocket)} + 0.7125 \text{ in (shaft)} = 4.2925 \text{ in} \] The mounting must ensure at least this much vertical distance between the base plate and the pump shaft center to prevent interference. --- ### **Step 7: Complete Drive System Specification** - **Motor:** 20 hp, 1400 rpm, 1.425 in shaft - **Belt:** Synchronous, 0.5 in pitch, nominal length 60 in - **Motor Sprocket:** 18 teeth, 2.87 in PD - **Pump Sprocket:** 45 teeth, 7.16 in PD - **Center Distance:** 22 in (adjustable ±0.375 in) - **Minimum Pump Sprocket Clearance:** 4.29 in from base plate center to bottom of sprocket --- ### **Short Explanation** **The sprocket sizes are chosen based on the required speed reduction, standard pitch, and practical tooth counts. The minimum clearance ensures the sprocket will not interfere with the base plate during operation or adjustment.** --- ### **Overall Short Final Answer** The complete drive system requires an 18-tooth motor sprocket (2.87 in pitch diameter), a 45-tooth pump sprocket (7.16 in pitch diameter), a 0.5 in pitch synchronous belt of approximately 60 in length, and a minimum vertical clearance of 4.29 in from the pump shaft center to the base plate to avoid interference.