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Gear Terminology and Equations

Gears Engineering and Design

Gear Terminology and Equations  

• Active Profile – is that part a gear tooth that actually comes into contact with its mating tooth along the line of action. 
• Addendum (a)  – is the radial or perpendicular distance between the pitch circle and the top of the teeth. 

Spur Gears - a = 1 / Pd            (Pd = Diametral Pitch)Helical Gears - a = 1 / Pnd            (Pnd = Normal Diametral Pitch) 

• Angular Bevel Gears – are bevel gears in which the axes are not at a right angle.
• Angular Pitch (θN) – is the angle subtended by the circular pitch. 
• Arc of Approach (Qt) – is the arc of the pitch circle through which a tooth travels from the time it first makes contact with a mating tooth until it is in contact at the pitch point.
• Arc of Recession (Qr) – is the arc of pitch circle through which a tooth travels from the time it is in contact with a mating tooth at the pitch point until contact ceases.
• Axial Direction – A direction parallel to an axes.
• Axial Pitch (Px) – is the distance in an axial plane surface between corresponding adjacent tooth profiles.

Px = Pt / Tan Ψ         (Pt = Transverse Circular Pitch, Ψ = Helix Angle)

• Axial Pressure Angle (Φx) – is the angle between the tangent to the tooth profile in an axial plane at the pitch surface and a line perpendicular to the axis.
• Axial Runout  – is the runout measured in a direction parallel to the axes of rotation.
• Backlash (B) – is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth on the pitch circles.
• Backlash Variation – is the difference between the maximum and minimum backlash occurring in one revolution of the larger of a pair of mating gears.
• Base Circular Thickness (tb) – is the length of arc on the base circle between two involute curves forming the profiles of a tooth.
• Base Cylinder – is the cylinder of the same diameter as the base circle.
• Base Lead Angle (λb) – is the lead angle at the base cylinder.
• Base Diameter (Db) – is the diameter of the circle from which the involute form is generated.

Db = D · Cos Φ         (D = Pitch Diameter, Φ = Pressure Angle

• Base Helix Angle (Ψb) – is the helix angle at the base cylinder.
• Base Pitch (Pb) – is the circular pitch in the plane of rotation at the base circle.

Pb = Cos Φ · π / Pd      (Φ = Pressure Angle, Pd = Diametral Pitch)

• Base Radius (rb) – is the radius of the circle from which the involute form is generated.
• Base Tooth Thickness (Tb) – is the distance on the base circle in the plane of rotation between involutes of the same pitch.
• Bevel Gears – are gears of conical form designed to operate on intersecting axes.
• Bore Diameter – is the diameter of the hole in a gear.
• Bottom Land – is the surface of a gear between the flanks of two adjacent teeth. For external gears it is measured in the root and for internal gears it is measured at the minor tip of the tooth.
• Center Distance (C) – is the radial distance from the center of a gear to the center of its mating gear.
• Chordal Addendum (ac) – is the radial distance from the circular tooth thickness chord to the top of the tooth.
• Chordal Thickness (tc) – is the length of the chord subtended by the circular tooth thickness arc.
• Circular Pitch (P) – is the distance on the circumference of the pitch circle between corresponding points of adjacent teeth.

P = π / Pd            (Pd = Diametral Pitch)

• Circular thickness (t) – is the thickness of the tooth on the pitch circle.

t (Basic) = .5 · P          (P = Circular Pitch)

• Clearance (c) – is the radial distance between the top of a tooth and the bottom of the mating tooth space.

c = .157 / Pd                (Pd = Diametral Pitch)

• Composite Action – is the variation in center distance when two gears are rolled in tight mesh.
• Contact Ratio (mc) – is the ratio of the arc of action to the circular pitch.

mc = sqrt(Ro1² - Rb1²) + sqrt(Ro2² - Rb2²-C · SIN Φ )

                             P · COS Φ(Ro1 and Ro2 = Outside Radius of gear 1 and Gear 2)(Rb1 and Rb2 = Base Radius of Gear 1 and Gear 2)(C = Center Distance, P = Circular Pitch, Φ = Pressure Angle)

•  Contact Ratio Total (mf) – is the ratio of the sum of the arc of action and the face advance to the circular pitch.
• Crowned Teeth  – are teeth having modified surfaces in the lengthwise direction to prevent contact at their ends.
• Dedendum (b) – is the radial or perpendicular distance between the pitch circle and the bottom of the tooth space.

b = 1.157 / Pd              (Pd = Diametral Pitch) Diametral Pitch (Pd) – is the ratio of the number of teeth per inches of pitch diameter.Pd = π / P                      (P = Circular Pitch)Pd = N / D                     (N = Number of Teeth, D = Pitch Diameter)Pd = N + 2 / Do         (Do = Outside Diameter)Pd = (Pb / Cos Φ) / π     (Pb = Base Pitch, Φ = Pressure Angle)Pd = 25.4 / m                 (m = Module)

•  Double Helical Gear – is a gear of cylindrical form with two sections of teeth, one right hand and the other left hand, that engage simultaneously with teeth of a similarly designed mating gear.
• Effective Face Width (Fe) – is the width of face that actually comes into contact with a mating gear.
• External Gear – is a gear with teeth formed on the outer surface of a cylinder or cone.
• Face Width (F) – is the width of the pitch surface.
• Fillet Curve – is the concave portion of the tooth profile where it joins the bottom of the tooth space.
• Fillet Radius (rt) – is the radius of a circular arc approximating the fillet curve.
• Form Diameter (TIF) - is the diameter of the circle beyond which the tooth profile must conform to the specified involute curve.
• Full Depth Teeth – are teeth in which the working depth equals 2.0 divided by the Diametral pitch.
• Gear(s) – are machine elements that transmit motion by means of successively engaging teeth.
• Gears Center – is the center of the pitch circle.
• Gear Ratio (mG  – is the ratio of the number of teeth in a mating set of gears. 
• Helical Gear – is a cylindrical gear with helical teeth.
• Helix Angle (Ψ) – is the angle between a tangent to the helix and an element of the cylinder. Usually the pitch helix is referred to.
• Hub Diameter – is the diameter of the central part of the gear body surrounding the bore and extending beyond the web, spokes or body.
• Hub Extension – is the distance that the hub extends beyond the face of the gear body..
• Interference – is the contact between mating teeth at some other point than along the line of action.
• Inside Diameter (Di) – is the diameter of the circle which contains the tops of the teeth of an internal gear.
• Involute – is the curve that is described by the end of a line which is unwound from the circumference of a circle.
• Involute function of an angle – is the difference between its Tangent and its angular value in radians.

Inv.a=tan(a)-(a · (π / 180))            (a = An angle in degrees)

•  Involute Polar Angle (θ) – is the angle between a radius vector to a point on an involute curve and a radial line to the point where the curve touches the base circle.
• Involute Roll Angle (ε) – is an angle whose arc on the base circle of radius unity equals the tangent of the pressure angle at a selected point on the involute.
• K Factor (K) – Is the ratio of tooth thickness change to the measurement over/between pins change.
• Lead (L) – is the axial advance of a helix for one complete turn, as in the threads of cylindrical worms and teeth of helical gears.

L =  π · D / Tan Ψ         (D = Pitch Diameter, Ψ = Helix Angle)

•  Lead Angle (λ) – is the angle between a tangent to the pitch helix and a plane of rotation.
• Length of Action (Z) – is the distance on an involute line of action through which the point of contact moves during the action of the tooth profiles.
• Line of Action – is that portion of the common tangent to the base circles along which contact between the mating involute teeth occurs.
• Line of Centers – is the straight line through the center of tangent pitch circles.
• Line of Contact – is the line or curve along which two tooth surfaces are tangent to each other.
• Long and Short addendum teeth – are the teeth of engaging gears (on a standard designed center distance), one of which has a long addendum and the other has a short addendum.
• Module (Metric) (m) – is the ratio of the pitch diameter in millimeters to the number of teeth.

m = 25.4 / Pd       (Pd = Diametral Pitch)

•  Normal Base Pitch (Pnb)  – is the circular pitch taken along the base circle normal to the gear helix angle.

Pnb = Pn · Cos Φn        (Pn = Normal Circular Pitch, Φn = Normal Pressure Angle)

•  Normal Chordal Addendum (anc) – is the chordal addendum the normal plane.
• Normal Chordal Thickness (tnc)  – is the length of the normal thickness chord between pitch line elements of a tooth.
• Normal Circular Thickness (tn) – is the circular thickness in a plane normal to the helix angle.

tn = tt · Cos Ψ             (tt = Transverse Circular Tooth Thickness, Ψ = Helix Angle)tn = .5 · Pn                   (Pn = Normal Circular Pitch)

•  Normal Circular Pitch (Pn) – (Normal to the helix angle) is the shortest distance on the pitch surface between corresponding pitch line elements of adjacent teeth.

Pn = Pt · Cos Ψ         (Pt = Transverse Circular Pitch, Ψ = Helix Angle)Pn = π / Pnd                 (Pnd = Normal Diametral Pitch)

•  Normal Diametral Pitch (Pnd) – is the diametral pitch corresponding to the normal circular pitch and calculated in the normal plane.

Pnd = Pd / CosΨ          (Pd = Transverse Diametral Pitch, Ψ = Helix Angle)

•  Normal Helix – is the helix on a pitch cylinder normal to the pitch helix.
• Normal Plane (n) – is the plane perpendicular to a given straight line or to a tangent to a curved line.
• Normal Pressure Angle (Φn) – is the pressure angle in a plane normal to the pitch line element.

Tan Φn = Tan Φt / Cos Ψ            (Φn = Normal Pressure Angle, Ψ = Helix Angle) Normal Profile Angle – is the profile angle in a normal plane of a helical or spiral tooth.

•  Number of teeth – Number of gear teeth.

N = Pd · D            (Pd = Diametral Pitch, D = Pitch Diameter)

•  Operating Pitch Diameters (dp) – are the pitch diameters determined from the numbers of teeth and the center distance at which gears operate.

dp = Db / Cos Φ         (Db = Base Circle Diameter, Φ = Pressure Angle)

•  Outside Diameter (Do) – is the diameter of the circle that contains the tops of the teeth of an external gear.

Do = N + 2 / Pd        (N = Number Of Teeth, Pd = Diametral Pitch)Do = D + 2 / Pd        (D = Pitch Diameter, Pd = Diametral Pitch)

•  Outside Cylinder – is the surface which coincides with the tops of the teeth of an external cylindrical gear.
• Outside helix Angle (Ψo) – is the helix angle at the outside Cylinder.
• Outside Lead Angle (λo) – is the lead angle at the outside cylinder.
• Outside Radius (Ro) – is the radius of the circle which contains the tops of the teeth of external gears.
• Pinion – is a gear with a small number of teeth. Of two mating gears, the one with the smaller number of teeth is called the pinion.
• Pitch Circle – is the circle through the pitch point having its center at the axis of the gear.
• Pitch Cylinder – is the imaginary cylinder in a gear that rolls without slipping on a pitch cylinder or pitch plane of another gear.
• Pitch Diameter (D) – is the diameter of the pitch circle.

D = N / Pd                   (N=Number of teeth, Pd = Diametral Pitch)D = N · .3183 · P          (P = Circular Pitch)D = N / Pd                   (Pd = Diametral Pitch or Transverse Diametral Pitch for helicals)

•  Pitch Radius (R) – is the radius of the pitch circle.

R = .5 · D            (D = Pitch Diameter)

•  Pitch Point – is the intersection between the axes of the line of centers and the line of action.
• Pitch Range – is the difference between the longest and the shortest pitches on a gear.
• Point of Contact – is the point at which two profiles touch each other.
•  Plane of Rotation – is any plane perpendicular to a gear axis.
• Pressure Angle (Φ) – is the angle between a tangent to the tooth profile and a line perpendicular to the pitch surface.

Cos Φ = Db / D         (Db = Base Diameter, D = Pitch Diameter)Cos Φ = Pb / (π / Pd)            (Pb = Base Pitch, Pd = Diametral Pitch)

•  Profile Control Diameter – is the diameter of the circle beyond which the tooth profile must conform to the specified involute curve.
• Rack – is a gear with teeth spaced along a straight line, and suitable for straight-line motion. It is also described as a gear with an infinite number of teeth.
• Right Hand Helical Gear – is a gear in which the teeth twist clockwise as they recede from an observer looking along the axis.
• Roll Angle (Involute) (ε) – is an angle whose arc on the base circle of radius unity equals the tangent of the pressure angle at a selected point on the involute.
• Roll Angle to Form Diameter – Is the inactive angle of involute or the amount of roll in degrees from the base circle to the start of active profile.
• Root Diameter (DR) – is the diameter of the circle which contains the roots of the teeth.
• Root Radius (Rr) – is the radius of the root circle containing the bottoms of the tooth spaces.
• Tooth Face – is the surface between the pitch line element and the top of the tooth.
• Tooth Fillet – is the curved surface of the tooth flank joining it to the bottom land.
• Tooth Flank – is the surface between the pitch line and the bottom land including the fillet.
• Tooth Surface – is the total area including the tooth face and the tooth flank.
• Tooth to Tooth Composite Action – is the amount of composite action from one tooth to the next.
• Top Land – is the surface of the top of the tooth.
• Total Composite Action – is the total amount of composite action for an entire gear.
• Transverse Circular Pitch (Pt)  – is the circular pitch in the transverse plane.

Pt = Pn / Cos Ψ         (Pn = Normal Circular Pitch, Ψ = Helix Angle)Pt = π / Ptd                  (Ptd = Transverse Diametral Pitch)

•  Transverse Circular Thickness (tt) – is the circular thickness in the transverse plane.

tt = tn / Cos Ψ             (tn = Normal Circular tooth Thickness, Ψ = Helix Angle)

•  Transverse Contact Ratio (mp) – is the ratio of the arc of action to the transverse circular pitch.

mp = sqrt(Ro1² - Rb1²) + sqrt(Ro2² - Rb2²-C · SIN Φt )                             Pt · COS Φt(Ro1 and Ro2 = Outside Radius of gear 1 and Gear 2)(Rb1 and Rb2 = Base Radius of Gear 1 and Gear 2)(C = Center Distance, Pt = Transverse Circular Pitch, Φt = Transverse Pressure Angle)

•  Transverse Diametral Pitch (Pd) – is the ratio of the number of teeth to inches of transverse pitch diameter.

Pd = Pnd · Cos Ψ         (Pnd = Normal Diametral Pitch, Ψ = Helix Angle)Pd = π / Pt                   (Pt = Transverse Circular Pitch)

•  Transverse Plane (t) – is perpendicular to the axial plane and to the pitch plane. In parallel axis gears, the transverse plane and the plane of rotation coincide. 
• Transverse Pressure Angle (Φt) – is the pressure angle in the plane of rotation.

Tan Φt = Tan Φn / Cos Ψ            (Φn = Normal Pressure Angle, Ψ = Helix Angle)Cos Φt = Db / D                     (Db = Base Diameter, D = Pitch Diameter)

•  True Involute Form Diameter (TIF) - is the diameter of the circle beyond which the tooth profile must conform to the specified involute curve.
• Undercut – is a condition in generated gear teeth when any part of the fillet curve lies inside of the line drawn tangent to the true involute form and its lowest point. Undercut may be deliberately introduced to facilitate finishing operation.
• Whole Depth (ht)  – is the radial distance between the outside circle and the root circle.

ht = 2.157 / Pd             (Pd = Diametral Pitch)

•  Working Depth (hk)  – is the greatest depth to which a tooth of one gear extends into the tooth space of the mating gear.

hk = 2 / Pd                   (Pd = Diametral Pitch)

Contributed by: Samir Harsh, Pune India Topal Machinery