Calculate shaft diameter.

Steps to Solve

1. Convert Power to Watts

First, convert the power from horsepower to watts because the other parameters are in metric units.

[ P = 25 , \text{HP} \times 745.7 , \text{W/HP} = 18642.5 , \text{W} ]

2. Calculate Torque

Use the relationship between power, torque, and speed. The formula is:

[ P = \frac{T \times N}{9.549} ]

Where:

• (P) is in watts,
• (T) is torque in Nm,
• (N) is speed in rpm.

Rearranging gives us:

[ T = \frac{P \times 9.549}{N} ]

Substituting the values:

[ T = \frac{18642.5 , \text{W} \times 9.549}{125 , \text{rpm}} = 1415.028 , \text{Nm} ]

3. Calculate Design Torque with Safety Factors

Using the given gear ratio and safety factors:

[ T_{\text{design}} = T \times \left( \frac{T_1}{T_2} \times k_f \times k_m \right) ]

Substituting the values:

[ T_{\text{design}} = 1415.028 , \text{Nm} \times \left( 2.5 \times 1.5 \times 2 \right) = 1415.028 , \text{Nm} \times 7.5 = 10613.71 , \text{Nm} ]

4. Calculate Shaft Diameter

To find the diameter, use the formula for shear stress. First, rearrange for the diameter:

[ \sigma = \frac{T}{\frac{\pi}{16} \times d^3} ]

Where:

• $\sigma$ = allowable shear stress = 63 MPa = $63 \times 10^6$ Pa,
• $T$ = design torque.

Rearranging gives:

[ d^3 = \frac{16T}{\pi \sigma} ]

Substituting the values into the formula:

[ d^3 = \frac{16 \times 10613.71 , \text{Nm}}{\pi \times (63 \times 10^6 , \text{Pa})} ]

Calculating this yields:

[ d^3 = 8.4984 \times 10^{-4} , \text{m}^3 ]

Taking the cube root gives:

[ d = \sqrt[3]{8.4984 \times 10^{-4}} \approx 0.094 , \text{m} \approx 94 , \text{mm} ]