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A pair of helical gears is employed in transmitting 12 kW at 3200 r/min. The teeth are 200 full depth in the diametral plane, and the helix angle is 23º. Gear ratio is 2.7:1 and the pinion diameter is 70 mm. both wheels are made of carbon steel with UTS of 580 MPa. For a standard module of 1 and accurately cut and ground gears, determine the face width.

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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.

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