# Gearset Rating Using AGMA Method This problem involves calculating the horsepower capacity of a spur gear set based on given parameters using the **AGMA (American Gear Manufacturers Association)** standards. --- ## Given Data | Parameter | Value | Description | |---|---|---| | Number teeth on pinion | 24 | \( N_p = 24 \) | | Number of teeth on gear | 72 | \( N_g = 72 \) | | Pressure angle | 20° | \( \phi = 20^\circ \) | | Overload factor | \( K_O = 1 \) | Load factor | | Diametral pitch | 6 teeth/in | \( P = 6 \) | | Face width | 2 in | \( F = 2 \text{ in} \) | | Pinion speed | 120 rpm | \( N_p = 120 \text{ rpm} \) | | Mission life | \( 10^8 \) revolutions | \( L_{10} \) | | Reliability | .99 | \( R = .99 \) | | Quality number | 5 | \( Q = 5 \) | | Material | Through hardened steel | Grade 1 | | Brinell hardness | 232 | \( HB = 232 \) | | Design factor | 2 | \( F_{d} = 2 \) | --- ## Step 1: Calculate the Gear Ratio and Output Speed \[ i = \frac{N_g}{N_p} = \frac{72}{24} = 3 \] The gear ratio is 3:1, with the gear turning at 1/3 the pinion speed: \[ N_g = \frac{N_p}{i} = \frac{120}{3} = 400 \text{ rpm} \] --- ## Step 2: Determine the Transmitted Power Capacity ### 2.1 Basic AGMA Formula for Power The **AGMA** provides a contact stress-based capacity: \[ P_{rated} = \frac{C \times S_{H} \times F \times \text{Number of teeth} \times \text{Efficiency}}{Q} \] But more accurate is the **AGMA tooth strength equation**: \[ P_{t} = \frac{W_t \times v}{550} \] Where: - \( W_t \): Transmitted load (lb) - \( v \): Pitch line velocity (ft/min) ### 2.2 Calculate pitch line velocity: \[ v = \frac{\pi \times d_p \times N_p}{12} \] The pitch diameter: \[ d_p = \frac{N_p}{P} = \frac{24}{6} = 4 \text{ in} \] Convert to ft/min: \[ v = \frac{\pi \times 4 \times 120}{12} = \pi \times 4 \times 100 = 1256.64 \text{ ft/min} \] --- ## Step 3: Determine the AGMA Contact Stress Capacity ### 3.1 Contact stress \( \sigma_c \): AGMA's basic formula for allowable contact stress: \[ \sigma_c = C \times \text{Material factor} \times \text{Quality factor} \] Where: - \( C \) depends on material and surface finish. From AGMA tables for grade 1, through hardened steel, the **design factor \( C \)** is approximately **25000 psi**. ### 3.2 Calculate the dynamic factor \( K_v \): For high speeds (~120 rpm), the dynamic factor \( K_v \) accounts for velocity effects. Approximate \( K_v \): \[ K_v \approx 1. \quad \text{(assuming good condition and moderate speed)} \] ### 3.3 Load capacity: \[ W_t = \frac{\sigma_c \times F \times N_{teeth}}{Q} \] Where: - \( F = 2 \text{ in} \) - Number of teeth on pinion \( N_p = 24 \) - \( Q \) is the gear quality number: 5 --- ## Step 4: Calculate the Power Rating ### 4.1 Find the allowable contact stress: Using AGMA Tables for grade 1, \( \sigma_c \approx 25000 \text{ psi} \). ### 4.2 Calculate the transmitting force: \[ W_t = \frac{\sigma_c \times F \times N_p}{Q} \approx \frac{25000 \times 2 \times 24}{5} = 25000 \times 9.6 = 240,000 \text{ lb} \] (Note: This is a simplified approximation; detailed AGMA calculations involve more factors, but this provides a reasonable estimate.) ### 4.3 Convert force to horsepower: \[ P = \frac{W_t \times v}{550} = \frac{240,000 \times 1256.64}{550} \approx 548,746 \text{ ft-lb/min} \] Convert to horsepower: \[ P_{hp} = \frac{548,746}{33000} \approx 16.61 \text{ hp} \] --- ## **Final Step: Apply Design Factor** Given a design factor \( F_d = 2 \): \[ P_{rated} = \frac{16.61}{2} \approx 8.3 \text{ hp} \] --- # **Final Answer** > **The gearset can reliably transmit approximately **8.3 horsepower** under the given conditions, considering the AGMA design method and safety factors.**