External Spur Gear Nomenclature Quiz

External Spur Gear Nomenclature

• Pitch Circle (D) is the circle with a radius equal to the distance from the gear axis to the pitch point. It represents the diameter of a cylinder if the gear pair is replaced by rolling cylinders
• Addendum (a): The height of the tooth above the pitch circle
• Dedendum (d): The radial distance from the pitch circle to the root circle (bottom of the tooth space). The formula for dedendum is: 𝑑 = 𝑎 + 𝑐
• Clearance (c): The distance separating the outside diameter of a gear and the root diameter of its mating gear, preventing tooth interference
• Addendum Circle (Da): The circle that bounds the outer ends of the teeth. The formula is: 𝐷𝑎 = 𝐷 + 2𝑎 (where D is the pitch circle diameter)
• Dedendum Circle (Dd): The circle coinciding with or tangent to the bottoms of the tooth spaces. Formula: 𝐷𝑑 = 𝐷 − 2𝑑 or 𝐷𝑑 = 𝐷 − 2(𝑎 + 𝑐)
• Root Diameter: Diameter of the dedendum circle
• Base Circle (Db): The circle from which the involute of a gear is drawn. Formula: 𝐷𝐵 = 𝐷cos 𝜃 (where 𝜃 is the pressure angle)
• Working Depth (WR): The depth of engagement of two gears, calculated by summing their addenda. Formula: 𝑊𝑅 = 2𝑎
• Whole Depth (WL): The total depth of a tooth space, equal to the addendum plus the dedendum. Alternatively, it is equal to working depth plus clearance. Formula: 𝑊𝐿 = 2𝑎 + 𝑐 or 𝑊𝐿 = 𝑎 + 𝑑
• Diametral Pitch (PD): Ratio of the number of teeth (T) to the pitch diameter (D): 𝑃𝐷 = 𝑇/𝐷
• Circular Pitch (PC): Distance measured along the pitch circle between corresponding points on adjacent teeth. Formula: 𝑃𝐶 = 𝜋𝐷/𝑇. Important relationship: 𝑃𝐶 𝑥 𝑃𝐷 = 𝜋. Gears in mesh must have the same circular pitch. The term "pitch" typically refers to diametral pitch unless otherwise specified.
• Pitch Point: Point of tangency between two pitch circles, located on the line of centers
• Pitch Angle (α): Angle subtended by an arc on the pitch circle equal in length to the circular pitch. Formula: 𝛼° = 360/𝑇
• Pitch Line: Line passing through the pitch point, perpendicular to the line of centers
• Tooth Flank: Surface of the tooth between the pitch circle and the root
• Tooth Face: Surface of the tooth between the pitch circle and the addendum circle
• Face Width: Length of the teeth in the axial plane
• Tooth Thickness (Tt): Width of the tooth measured along the pitch circle
• Tooth Space (Ts): Space between teeth measured along the pitch circle
• Backlash (b): Excess thickness of the tooth space over the thickness of the mating tooth. Formula: 𝑏 = 𝑇𝑠 − 𝑇𝑡. Relationship: 𝑃𝑐 = 𝑇𝑡 + 𝑇𝑠
• Chordal Thickness: Tooth width measured along the chord at the pitch point
• Gear Ratio: Number of teeth on the gear divided by the number of teeth on the pinion
• Speed Ratio: Angular speed of the driver divided by the angular speed of the driven gear

Standard Gear Tooth Proportions

• System: 14 1/2°, Brown and Sharpe, 14 1/2°, Composite, Cycloidal, Full-Depth, 200 Full Depth, 200 Stub, 200 Fellows
• Addendum: 1/Pd, 1/Pd, 1/Pd, 0.8/Pd, 1/P2
• Dedendum: 1.157/Pd, 1.157/Pd, 1.157/Pd, 1/Pd, 1.25/P2
• Clearance: 0.157/Pd, 0.157/Pd, 0.157/Pd, 0.2/Pd, 0.25/P2
• Working Depth: 2/Pd, 2/Pd, 2/Pd, 1.6/Pd, 2/P2
• Total Depth: 2.157/Pd, 2.157/Pd, 2.157/Pd, 1.8/Pd, 2.25/P2
• Outside Diameter: T + 2 / Pd, T + 2 / Pd, T + 2 / Pd, T + 1.6 / Pd, T/P1 + 2/P2
• Tooth Thickness: 1.5708/Pd, 1.5708/Pd, 1.5708/Pd, 1.5708/Pd, 1.5708/P1
• Tooth Space: 1.5708/Pd, 1.5708/Pd, 1.5708/Pd, 1.5708/Pd, 1.5708/P1
• Fillet Radius: 0.209/Pd, 0.209/Pd, 0.236/Pd, 0.3/Pd, 0.25/P1

Law Governing the Shape of Teeth

• The shape of gear teeth must follow a fundamental law ensuring smooth operation: The line drawn from the pitch point to the point of contact between teeth must be perpendicular to the tangent line drawn through the point of contact

The Involute of a Circle

• The involute is a curve generated by a point on a taut string unwinding from a circle, creating the tooth profile.
• To draw an involute:
  • Start with a circle and a point on its circumference.
  • Use dividers to mark equal distances along the circle's circumference.
  • Draw radial lines from the center of the circle through each mark.
  • Construct perpendicular lines to these radial lines at each mark.
  • Lay off equal distances from the marked point along the perpendicular lines, increasing the number of spaces for each consecutive perpendicular line.
  • A smooth curve drawn through these points approximates the involute.

Example 1: Finding the Center Distance between Gears

• Given: T1 (number of teeth on gear 1) = 12, T2 (number of teeth on gear 2) = 37, PD (diametral pitch) = 7
• Required: Center distance (C)
• Solution: 𝐷1 (pitch diameter of gear 1) = 𝑇1/𝑃𝐷 = 12/7 = 1.714 in, 𝐷2 (pitch diameter of gear 2) = 𝑇2/𝑃𝐷 = 37/7 = 5.286 in
• Final Answer: 𝐶 = (𝐷1 + 𝐷2)/2 = (1.714 + 5.286)/2 = 3.5 in

Example 2: Finding the Speed of a Gear

• Given: Center distance (C) = 15 in, T1 (number of teeth of gear 1) = 40, PD (diametral pitch) = 2, N2 (speed of gear 2) = 150 rpm
• Required: N1 (speed of gear 1)
• Solution: 𝐷1 (pitch diameter of gear 1) = 𝑇1/𝑃𝐷 = 40/2 = 20 in, 𝐷2 (pitch diameter of gear 2) = 2C - D1 = 2(15) - 20 = 10 in
• Final Answer: N1 (speed of gear 1) = N2 * 𝐷2/𝐷1 = 150 * 10 / 20 = 75 rpm

Example 3: Gearset Calculations

• Given: T1 (number of teeth on pinion) = 16, T2 (number of teeth on gear) = 40, PD (diametral pitch) = 2, addendum = 1/P, dedendum = 1.25/P, pressure angle (𝜃) = 20°
• Required: Circular pitch (PC), center distance (C), radii of base circles (Db)
• Solution:
  • PC (Circular Pitch) = π/PD = π/2 = 1.571 in
  • D1 (pitch diameter of pinion) = T1/PD = 16/2 = 8 in
  • D2 (pitch diameter of gear) = T2/PD = 40/2 = 20 in
  • C (center distance) = (D1 + D2)/2 = (8 + 20)/2 = 14 in
  • Db1 (base circle radius of pinion) = D1 * cos 𝜃 = 8 * cos 20° = 7.52 in
  • Db2 (base circle radius of gear) = D2 * cos 𝜃 = 20 * cos 20° = 18.79 in