Spur Gears(1)

Introduction

Attach a string to a point on a curve. Extend the string so that it is tangent to curve at point of attachment. Then wind string up and keeping it always taut. The locus of points traced out by end of the string is called involute of original curve and the original curve is called evolute of its involute. In an involute gear the profiles of teeth are involutes of a circle. The involute of a circle is the spiraling curve traced by the end of an imaginary taut string unwinding itself from that stationary circle called the base circle.

Gears Terms

Fig: Terms used in gear

• Pitch circle: It is the imaginary circle which by pure rolling action and would give the same motion as given actual gear.
• Pitch circle diameter: It is the diameter of the pitch circle. The size of the gear is mostly specified by pitch circle diameter. It is also known as pitch diameter.
• Pitch point: It is a common point of contact between the two pitch circles.
• Pitch surface: It is the surface of the rolling discs which the meshing gears have transferred at the pitch circle.
• Pressure angle or angle of obliquity: It is the angle between the common normal to two gear teeth at point of contact with common tangent at pitch point. It is usually denoted by φ. The standard pressure angles are 14 °and 20°.
• Addendum: It is radial distance of a tooth from the pitch circle to the top of tooth.
• Dedendum: It is radial distance of a tooth from the pitch circle to the bottom of the tooth.
• Addendum circle: It is the circle drawn through top of the teeth and is concentric with the pitch circle.
• Dedendum circle: It is the circle drawn through bottom of the teeth. It is also called root circle. Noted that Root diameter = Pitch diameter × cos φ

Where φ is the pressure angle.

• Circular pitch: It is the distance measured on circumference of pitch circle from a point of one tooth to same corresponding point on next tooth. It is generally denoted by P_c.

Circular pitch (P_c) = π D/T

Where,

D = Diameter of the pitch circle, and

T = Number of teeth on the wheel.

A little supposition will show that two gears will mesh correctly if that two wheels have same circular pitch. If D₁ and D₂ are diameters of two meshing gears having teeth T₁ and T₂ respectively then for them to mesh effectively.

P_c = π D₁/T₁ = π D₂/T₂

Or, D₁/T₁ = D₂/T₂

• Diametral pitch: It is the ratio of number of teeth to pitch circle diameter. It is denoted by P_d.

Diametral pitch (P_d) = T/D = π/ P_c

• Module: It is the ratio of the pitch circle diameter in millimeters to the number of teeth. It is generally denoted by m

Module (m) = D /T

• Clearance: It is the radial distance from top of tooth to the bottom of the tooth in a meshing gear. A circle passing through the top of the meshing gear is clearance circle.
• Total depth: It is the radial distance between the addendum and the dedendum circles of a gear. It is equal to sum of the addendum and dedendum.
• Working depth: It is the radial distance from the addendum circle to clearance circle. It is equal to the sum of the addendum of the two meshing type gears.
• Tooth thickness: It is width of tooth measured along pitch circle.
• Tooth space: It is the width of space between two nearly teeth measured along pitch circle.
• Backlash: It is difference between the tooth space and the tooth thickness as measured along pitch circle. Theoretically backlash must be zero but in actual practice some backlash must be to prevent jamming of teeth due to tooth errors and thermal expansion.
• Face of tooth: It is surface of gear tooth above pitch surface.
• Flank of tooth: It is surface of gear tooth below pitch surface.
• Top land: It is surface of top of tooth.
• Face width: It is width of gear tooth measured parallel to its axis.
• Profile: It is curve formed by the face and flank of tooth.
• Fillet radius: It is radius that connects root circle to profile of tooth.
• Path of contact: It is path traced by point of contact of two teeth from beginning to end of engagement.
• Length of path of contact: It is length of common normal cut-off by addendum circles of wheel and pinion.
• Arc of contact: It is path traced by a point on pitch circle from beginning to end of engagement of a given pair of teeth. The arc of contact consists of two parts
• Arc of approach: It is the portion of path of contact from the beginning of the engagement to pitch point.
• Arc of recess: It is the portion of path of contact from pitch point to end of engagement of a pair of teeth. The ratio of length of arc of contact to the circular pitch is contact ratio i.e. number of pairs of teeth in contact.

Condition for Constant Velocity Ratio of Toothed Wheels–Law of Gearing

Consider the portions of the two teeth, one on the wheel 1 and other on the wheel 2 as shown by thick line curves in fig. Let the two teeth come in contact at point Q and the wheels rotate in that directions as shown in the figure.

Fig: Law of gearing

Let TT be common tangent and MN be the common normal to curves at the point of contact Q. From the centres O₁ and O₂, draw O₁M and O₂N perpendicular to MN. A little supposition will show that point Q moves in the direction QC, when considered as a point on wheel 1 and in the direction QD when supposed as a point on wheel 2. Let v1 and v2 be velocities of the point Q on wheels 1 and 2 respectively. If the teeth are to remain in contact then the components of velocities along common normal MN must be equal.

∴ v₁cos α = v₂cos β

Or (ω₁ × O₁Q )cos α = (ω₂ × O₂Q) cos

Or (ω₁ × O₁Q )O₁M/O₁Q = (ω₂ × O₂Q) O₂N/ O₂Q

Or (ω₁ ×O₁M) = (ω₂ × O₂N)

∴ (ω₁/ ω₂ )= (O₂N/ O₁M)

The angular velocity ratio is inversely proportional to the ratio of the distances of the point P from centres O₁ and O₂ or common normal to two surfaces at point of contact Q intersects the line of centres at point P that divide that centre distance inversely as the ratio of angular velocities.

Therefor to have a constant angular velocity ratio for all positions of the wheels the point P must be the fixed point for two wheels. The common normal at the point of contact between a pair of teeth must always pass through pitch point. This is the most fundamental condition which must be satisfied when designing the profiles for the teeth of gear wheels. It is also known as law of gearing.

∴ (ω₁/ ω₂ )= (O₂P/ O₁P) = D₁/D₂ = T₂/T₁

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.