Question

The minimum number of teeth on the pinion in order to avoid interference for $20^\circ$ full depth involute system is

• A. $12$
• B. $14$
• C. $18$
• D. $32$

Hint:

This is a standard reference value in mechanical design: - For a $14.5^\circ$ full-depth system $\rightarrow$ Minimum teeth = 32 - For a $20^\circ$ full-depth system $\rightarrow$ Minimum teeth = 18 - For a $20^\circ$ stub-tooth system $\rightarrow$ Minimum teeth = 14

Answer 1

The Correct Option is C

Concept: In involute gear systems, interference occurs if the teeth are designed such that the tip of the mating gear tooth penetrates into the non-involute flank region near the base of the pinion. To ensure smooth conjugate gear action, we limit the addendum height or enforce a minimum number of teeth on the smaller gear (the pinion). For a pinion meshing with a standard rack, the minimum number of teeth $T_{\text{min}}$ required to completely avoid interference is given by the formula: \[ T_{\text{min}} = \frac{2 a_w}{\sin^2\phi} \] where $a_w$ is the addendum coefficient factor (expressed as a fraction of the module), and $\phi$ is the pressure angle of the gear system.

Step 1: Extracting standard parameters for the gear system.
From the problem statement, we identify the following standard system specifications:
• Gear system profile type = $20^\circ$ full-depth involute system.
• For any standard full-depth gear system, the addendum is equal to one module, which means the addendum coefficient is: $a_w = 1.0$
• The operating pressure angle is given as: $\phi = 20^\circ$

Step 2: Substituting values into the minimum teeth equation.
Substitute $a_w = 1$ and $\phi = 20^\circ$ into the formula: \[ T_{\text{min}} = \frac{2 \times 1}{\sin^2(20^\circ)} \]

Step 3: Calculating the numerical value.
First, find the sine of $20^\circ$: \[ \sin(20^\circ) \approx 0.3420 \] Squaring this value: \[ \sin^2(20^\circ) \approx (0.3420)^2 \approx 0.1170 \] Now, divide 2 by this result to find the minimum number of teeth: \[ T_{\text{min}} = \frac{2}{0.1170} \approx 17.09 \]

Step 4: Rounding up to the nearest integer.
Since you cannot have a fraction of a gear tooth, we must round up to the next whole number to guarantee that no interference occurs: \[ T_{\text{min}} = 18 \text{ teeth} \] This calculated value matches Option (C).