Two spur gears of 24 teeth and 36 teeth of 8 mm module and 20° pressure angle are in mesh. Addendum of each gear is 7.5 mm. The teeth are of involute form. Determine: 1. The angle... Two spur gears of 24 teeth and 36 teeth of 8 mm module and 20° pressure angle are in mesh. Addendum of each gear is 7.5 mm. The teeth are of involute form. Determine: 1. The angle through which the pinion turns while any pair of teeth are in contact, and 2. The velocity of sliding between the teeth when the contact on the pinion is at a radius of 102 mm. The speed of the pinion is 450 r.p.m. Read more

Steps to Solve

1. Calculate the Pitch Circle Diameters (PCDs)

For spur gears, the Pitch Circle Diameter (PCD) can be calculated using the formula:

$$ \text{PCD} = \text{Teeth} \times \text{Module} $$

For the pinion (24 teeth):

$$ \text{PCD}_{\text{pinion}} = 24 \times 8 , \text{mm} = 192 , \text{mm} $$

For the gear (36 teeth):

$$ \text{PCD}_{\text{gear}} = 36 \times 8 , \text{mm} = 288 , \text{mm} $$

2. Calculate the Contact Ratio

The contact ratio is defined as the ratio of the length of the line of action to the base pitch. It can be calculated using:

$$ \text{Contact Ratio} = \frac{\text{PCD}_{\text{pinion}} \cdot \sin(\phi)}{\text{Base Pitch}} $$

Where $\phi$ is the pressure angle (20°). The Base Pitch can be found from:

$$ \text{Base Pitch} = \frac{\pi \cdot \text{Module}}{\cos(\phi)} $$

Calculating the Base Pitch:

$$ \text{Base Pitch} = \frac{\pi \cdot 8 , \text{mm}}{\cos(20^\circ)} \approx 16.49 , \text{mm} $$

Now, substitute into the Contact Ratio formula:

$$ \text{Contact Ratio} = \frac{192 \cdot \sin(20^\circ)}{16.49} $$

3. Compute the Angle of Contact

The angle through which the pinion turns while teeth are in contact is given by:

$$ \text{Angle of Contact} = \text{Contact Ratio} \times \frac{360}{\text{Teeth}_{\text{pinion}}} $$

Calculating this:

$$ \text{Angle of Contact} = \text{Contact Ratio} \times \frac{360}{24} $$

4. Calculate the Velocity of Sliding

The velocity of sliding can be calculated using the formula:

$$ v = r \cdot \omega $$

Where $r$ is the radius at contact (102 mm) and $\omega$ is the angular velocity in rad/s.

Convert the pinion speed from RPM to rad/s:

$$ \omega = \frac{450 \times 2\pi}{60} \approx 47.1 , \text{rad/s} $$

Now, calculate the velocity of sliding at the specified radius:

$$ v = 102 \times 10^{-3} \cdot 47.1 $$

5. Final Computations

Substitute values and compute both the angle and velocity.