--- title: "Straight bevel gear geometry and formative teeth" subject: "Theory of Mechanism and Machine I" chapter: 5 key: "semester_6/theory_of_mechanism_and_machine_i:straight-bevel-gear-geometry-formative-teeth" source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/straight-bevel-gear-geometry-formative-teeth site: BCT Engineering --- # Straight bevel gear geometry and formative teeth Theory of Mechanism and Machine I · Chapter 5: Bevel, Helical and Worm Gears - Asked in: year not given, 6 marks - Video: [Bevel Gears Explained, Calculated & Modeled](https://www.youtube.com/watch?v=0N1SVoDFThM) — INTEGRAL PHYSICS - 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/straight-bevel-gear-geometry-formative-teeth - Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-5 ## Question Define pitch cone angle, cone distance, back cone and formative (virtual) number of teeth for straight bevel gears. Explain Tredgold's approximation. A pair of straight bevel gears with 20 and 40 teeth, module 6 mm at the large end, connects shafts at 90 degrees. Find the pitch cone angles, pitch diameters, cone distance, the maximum recommended face width and the formative numbers of teeth. ## Short answer Pitch cone angle is angle between axis and pitch cone; cone distance is apex to large-end pitch radius; back cone is perpendicular to pitch cone at large end; formative teeth z'=z/cosγ (Tredgold). For z1=20, z2=40, m=6 mm, 90° shafts: γ1=26.57°, γ2=63.43°, d1=120 mm, d2=240 mm, A0=134.16 mm, maximum face width b_max=44.72 mm, z1'=22.36, z2'=89.44. ## Answer #### Definitions - **Pitch cone angle ($\gamma$)**: the angle between the gear axis and the pitch cone element. - **Cone distance ($A_0$)**: the length of a pitch cone element from the apex to the large-end pitch circle. - **Back cone**: a cone whose elements are perpendicular to the pitch cone elements at the large end of the tooth. When developed (laid flat), it gives a spur gear. - **Formative (virtual) number of teeth**: the number of teeth on the spur gear whose pitch radius equals the back cone distance. #### Tredgold's approximation The tooth profile of a bevel gear is taken as that of a spur gear lying on the back cone surface. The back cone radius is $r_v = \dfrac{d}{2\cos\gamma}$, so the virtual spur gear has pitch diameter $d/\cos\gamma$ and $$ z' = \frac{z}{\cos\gamma} $$ This $z'$ is used for tooth strength design and to check interference ($z'$ need not be an integer). ```text apex /|\ / | \ A0 = cone distance / | \ / | \ /_(gamma)_\ gear pitch dia d ``` #### Shafts at 90 degrees For shaft angle $\Sigma = 90^\circ$: $$ \tan\gamma_1 = \frac{z_1}{z_2}, \qquad \gamma_2 = 90^\circ - \gamma_1 $$ #### Calculation $$ \gamma_1 = \tan^{-1}\frac{20}{40} = 26.57^\circ, \qquad \gamma_2 = 90^\circ - 26.57^\circ = 63.43^\circ $$ Pitch diameters (large end): $$ d_1 = mz_1 = 6 \times 20 = 120\ \text{mm}, \qquad d_2 = mz_2 = 6 \times 40 = 240\ \text{mm} $$ Cone distance: $$ A_0 = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} = \sqrt{60^2 + 120^2} = 134.16\ \text{mm} $$ Face width limit (to avoid weak tooth tips and cutting problems): $b \le A_0/3$ or $10m$, whichever is less. Here $10m = 60$ mm, so $$ b_{max} = \frac{134.16}{3} = 44.72\ \text{mm} $$ Formative numbers of teeth: $$ z_1' = \frac{20}{\cos 26.57^\circ} = 22.36, \qquad z_2' = \frac{40}{\cos 63.43^\circ} = 89.44 $$ Tooth proportions (large end): addendum $= m = 6$ mm, dedendum $= 1.2m = 7.2$ mm. **Answer: $\gamma_1 = 26.57^\circ$, $\gamma_2 = 63.43^\circ$, $d_1 = 120$ mm, $d_2 = 240$ mm, $A_0 = 134.16$ mm, $b_{max} = 44.72$ mm, $z_1' = 22.36$, $z_2' = 89.44$.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Tredgold's approximation treats a bevel gear tooth profile as that of: A. A helical gear on the pitch cone B. A rack on the apex C. A worm on the shaft D. A spur gear on the back cone, giving z' = z/cos(gamma) Answer: D. The profile is taken on the back cone of radius d/(2cos gamma), so the virtual spur gear has z' = z/cos(gamma) teeth. (Key point: Tredgold's approximation) 2. For shafts at 90° with 20 and 40 teeth, the pitch cone angles are: A. 45° and 45° B. 26.57° and 63.43° C. 30° and 60° D. 63.43° and 26.57° for pinion and gear Answer: B. tan(gamma1) = z1/z2 = 0.5, so gamma1 = 26.57° (pinion) and gamma2 = 90° - 26.57° = 63.43° (gear). (Key point: Pitch cone angles at 90°) 3. For d1 = 120 mm and d2 = 240 mm at 90°, the cone distance A0 is: A. 180 mm B. 134.16 mm C. 268.3 mm D. 60 mm Answer: B. A0 = sqrt(60^2 + 120^2) = 134.16 mm. (Key point: Cone distance) 4. Using b <= A0/3 and z' = z/cos(gamma), the results for this pair are: A. b_max = 44.72 mm; z1' = 22.36, z2' = 89.44 B. b_max = 134.16 mm; z1' = 20, z2' = 40 C. b_max = 44.72 mm; z1' = 17.9, z2' = 17.9 D. b_max = 60 mm; z1' = 22.36, z2' = 44.72 Answer: A. b_max = 134.16/3 = 44.72 mm; z1' = 20/cos26.57° = 22.36 and z2' = 40/cos63.43° = 89.44. 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