Answer and Explanation:
To determine the face width of the helical gears, we need to use the following formula:
Face width = (Power transmitted * Gear ratio * Safety factor) / (Tangential force * Material strength)
Given:
Power transmitted = 12 kW
Gear ratio = 2.7:1
Pinion diameter = 70 mm
Teeth full depth = 200
Helix angle = 23º
Material strength (UTS) = 580 MPa
Standard module = 1
Step 1: Calculate the tangential force.
Tangential force = Power transmitted / (Angular velocity * Radius)
Angular velocity = 2π * Speed / 60
Radius = Pinion diameter / 2
Step 2: Calculate the number of teeth on the gears.
Number of teeth = Pinion diameter * Module
Step 3: Calculate the pitch diameter.
Pitch diameter = Number of teeth / Module
Step 4: Calculate the helix angle factor.
Helix angle factor = cos(Helix angle)
Step 5: Calculate the tangential force.
Tangential force = Power transmitted / (Angular velocity * Radius)
Step 6: Calculate the safety factor.
Safety factor = 1.5 (assuming a typical value)
Step 7: Calculate the face width.
Face width = (Power transmitted * Gear ratio * Safety factor) / (Tangential force * Material strength)
Now, let's substitute the given values into the formulas and calculate the face width:
Angular velocity = 2π * 3200 r/min / 60 = 107 rad/s
Radius = 70 mm / 2 = 35 mm
Number of teeth = 70 mm * 1 = 70
Pitch diameter = 70 / 1 = 70 mm
Helix angle factor = cos(23º) ≈ 0.921
Tangential force = 12000 W / (107 rad/s * 0.035 m) ≈ 31.19 N
Safety factor = 1.5
Material strength = 580 MPa
Face width = (12000 W * 2.7 * 1.5) / (31.19 N * 580e6 Pa) ≈ 0.0039 m or 3.9 mm
Therefore, the face width of the helical gears is approximately 3.9 mm.