--- title: "Interference and minimum number of teeth" subject: "Theory of Mechanism and Machine I" chapter: 4 key: "semester_6/theory_of_mechanism_and_machine_i:interference-minimum-teeth-pinion" source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/interference-minimum-teeth-pinion site: BCT Engineering --- # Interference and minimum number of teeth Theory of Mechanism and Machine I · Chapter 4: Spur Gears - Asked in: year not given, 8 marks - Video: [MINIMUM NUMBER OF TEETH ON PINION TO AVOID INTERFERENCE](https://www.youtube.com/watch?v=KD_JCxvdvuE) — TECHNICAL CLASSES - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/interference-minimum-teeth-pinion - Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-4 ## Question What is interference in involute gears? Derive an expression for the minimum number of teeth on a pinion meshing with a rack to avoid interference. Calculate this number for 14.5, 20 and 25 degree full-depth systems, and the minimum pinion teeth for a 20 degree full-depth pinion meshing with a gear having three times as many teeth. ## Short answer Interference is when a gear tooth tip digs into the mating flank below the base circle, causing undercutting. Minimum pinion teeth with rack: z_min = 2/sin²φ; for 14.5°, 20°, 25°: 32, 18, 12. For a 20° pinion driving a gear with three times teeth, minimum pinion teeth = 15. ## Answer #### Interference Involute action exists only outside the base circle. If the tip of the driven gear tooth goes below the base circle of the pinion (that is, beyond the tangent point $N$ of the line of action), the tip digs into the flank of the pinion below its base circle. This mismatch is called **interference**. It causes undercutting, weakens the tooth and gives noise and wear. ```text pinion base circle ____ / \----- N (tangent point) | O | \ \____/ \ line of action \ K <- gear tip beyond N: interference ``` #### Minimum teeth with a rack Let pinion pitch radius be $R = mz/2$ and the rack addendum $= a_w m$ ($a_w = 1$ for full depth). The rack tip should not go beyond the point $N$ (the tangent point on the pinion base circle). The distance from the pitch point $P$ to the rack's tip along the line of action is $\dfrac{a_w m}{\sin\phi}$. The distance $PN = R\sin\phi$. For no interference: $$ \frac{a_w m}{\sin\phi} \le R\sin\phi = \frac{mz}{2}\sin\phi $$ $$ z_{min} = \frac{2a_w}{\sin^2\phi} $$ #### Values for full depth ($a_w = 1$), $\sin^2 20^\circ = 0.1170$ | Pressure angle | $z_{min}$ (calculated) | Used in practice | |---|---|---| | 14.5 degrees | 31.90 | 32 | | 20 degrees | 17.10 | 18 (or 17) | | 25 degrees | 11.20 | 12 | #### Pinion meshing with a gear (ratio $G = 3$) Interference is avoided when the gear addendum circle does not pass the pinion's base-circle tangent point $N_1$: $$ (R_G + a_wm)^2 \le R_G^2\cos^2\phi + (R + R_G)^2\sin^2\phi $$ with $R_G = GR$. Solving the resulting quadratic for the pinion pitch radius $R$: $$ z_{min} = \frac{2a_w\left[G + \sqrt{G^2 + (2G+1)\sin^2\phi}\right]}{(2G+1)\sin^2\phi} $$ For $G = 3$, $\phi = 20^\circ$, $a_w = 1$: $z_{min} = 14.98$, so a **pinion of at least 15 teeth** is needed (the value rises to 17.1 as $G \to \infty$, which is the rack case, and drops to 12.32 for $G = 1$). #### Methods to avoid interference - Use more teeth on the pinion than $z_{min}$. - Use stub teeth (shorter addendum) or larger pressure angle. - Use profile (addendum) correction, that is, a long-addendum pinion and short-addendum gear (X-gears). - Increase the centre distance (gives a larger working pressure angle). **Answer: $z_{min} = 2a_w/\sin^2\phi = 31.90$ (14.5 deg), 17.10 (20 deg), 11.20 (25 deg); for $G = 3$, 20 deg: 14.98, take 15 teeth.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is interference in involute gears? A. Two teeth touching on both flanks B. Teeth failing due to fatigue C. The tip of the mating tooth digging into the flank below the base circle of the other gear D. Backlash being too large Answer: C. Involute action exists only outside the base circle. If a tip goes beyond the tangent point N, it cuts into the flank below the base circle, causing undercutting, weakness, noise and wear. (Key point: Meaning of interference) 2. The minimum number of pinion teeth meshing with a rack to avoid interference is: A. 2a_w / sin^2(phi) B. 2a_w / cos^2(phi) C. a_w / sin(phi) D. 2a_w sin^2(phi) Answer: A. Requiring a_w*m/sin(phi) <= (mz/2) sin(phi) gives z_min = 2a_w/sin^2(phi). (Key point: Minimum teeth with rack) 3. The calculated z_min for a 20° full-depth pinion meshing with a rack is: A. 31.90 B. 11.20 C. 17.10 D. 14.98 Answer: C. z_min = 2/sin^2(20°) = 2/0.1170 = 17.10, so 18 teeth (or 17) in practice. (Key point: Value for 20° system) 4. For a 20° pinion meshing with a gear of 3 times the teeth, how does z_min compare with the rack case? A. Higher, about 17.1 B. Lower, about 15 teeth needed C. Same, 17.1 D. Zero, no interference possible Answer: B. The calculated value is 14.98, so at least 15 teeth. It rises to 17.1 for a rack (G to infinity) and falls to 12.32 for G = 1. 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