Spur Gears(3)

Interference in Involute Gears

A pinion with centre O₁, in mesh with wheel or gear with centre O₂. MN is the common tangent to base circles and KL is path of contact between the two mating teeth.

Fig: Interference in involute gears.

A small consideration will show that if radius of addendum circle of pinion is increased to O₁N then the point of contact L will shift from L to N. When this radius is further increased the point of contact L will be on inside of base circle of wheel and not on involute profile of tooth on wheel. The tip of tooth on the pinion will undercut the tooth on wheel at root and remove part of the involute profile of tooth on the wheel. This effect is known as interference and its occurs when teeth are being cut. In brief the phenomenon when tip of tooth undercuts root on its mating gear is known as interference.

So if radius of addendum circle of wheel increases beyond O₂M then tip of tooth on wheel will cause interference with tooth on pinion. The points that M and N are called interference points. Obviously interference can be avoided if path of contact does not extend to beyond that interference points. Limiting value of the radius of the addendum circle of pinion is O₁N and wheel is O₂M. Interference may only be avoided if the point of contact between two teeth is always on involute profiles of both the teeth or we can say interference may only be prevented, if the addendum circles of two mating gears cut common tangent to base circles between the points of tangency.

When interference is just avoided then the maximum length of path of contact is MN when the maximum addendum circles for pinion and wheel pass through points of tangency N and M as shown in Fig. In such a case

Maximum length of path of approach,

MP = r sin φ

And maximum length of path of recess,

PN = R sin φ

∴ Maximum length of path of contact,

MN = MP + PN = r sin φ + R sin φ = (r + R) sin φ

And maximum length of arc of contact = (r + R) tanφ

Minimum Number of Teeth on Pinion to Avoid Interference

To avoid interference the addendum circles for the two mating gears type must cut common tangent to base circles between the points of tangency. The limiting condition reaches when addendum circles of pinion and wheel pass through points N and M respectively.

Let

t = Number of teeth on the pinion,

T = Number of teeth on the wheel,

m = Module of the teeth,

r = Pitch circle radius of pinion = m.t / 2

G = Gear ratio = T / t = R / r

φ = Pressure angle or angle of obliquity.

From² triangle O₁NP,

(O₁N)² = (O₁P)² + (PN)² − 2 × O₁P × PN cosO₁PN

= r² + R² sin² φ − 2r.R sinφ cos (90° + φ)

= r² + R² sin² φ + 2r.R sin²φ∴ ( PN = O₂ P sin φ = Rsin φ)

\[ = {r^2}[1 + \frac{{{R^2}{{\sin }^2}\varphi }}{{{r^2}}} + \frac{{2R{{\sin }^2}\varphi }}{r}]\]

∴ Limiting radius of the pinion addendum circle,

\[{O_1}N = r\sqrt {[1 + \frac{{{R^2}{{\sin }^2}\varphi }}{{{r^2}}} + \frac{{2R{{\sin }^2}\varphi }}{r}]} = m\frac{t}{2}\sqrt {1 + \frac{T}{t}[\frac{T}{t} + 2]{{\sin }^2}\varphi } \]

Let

A_p.m = Addendum of the pinion, where A_p is a fraction by which standard addendum of one module for pinion should be multiplied in order to avoid interference.

addendum of the pinion = O₁N - O₁P
\[{A_P}m = m\frac{t}{2}\sqrt {1 + \frac{T}{t}[\frac{T}{t} + 2]{{\sin }^2}\varphi } - m\frac{t}{2}\]

\[{A_P}m = m\frac{t}{2}[\sqrt {1 + \frac{T}{t}[\frac{T}{t} + 2]{{\sin }^2}\varphi } - 1]\]

\[t = \frac{{2{A_P}}}{{\sqrt {1 + \frac{T}{t}\{ \frac{T}{t} + 2\} {{\sin }^2}\varphi } - 1}} = \frac{{2{A_P}}}{{\sqrt {1 + G\{ G + 2\} {{\sin }^2}\varphi } - 1}}\]

This equation gives the minimum number of teeth required on the pinion to avoid interference.

Minimum Number of Teeth on the Wheel to Avoid Interference

Let

T= Minimum number of teeth need on the wheel to avoid interference,

A_W.m = Addendum of the wheel, where A_W is a fraction by which the standard addendum for the wheel should be multiplied.

We have from triangle O₂MP

(O₂M)² = (O₂P)² + (PM)² − 2 × O₂P × PM cosO₂PM

= R² + r² sin²φ − 2 R.r sinφ cos (90° + φ) ...(PM = O₁P sinφ= r)

= R² + r² sin² φ + 2R.r sin² φ

\[ = {R^2}[1 + \frac{{{r^2}{{\sin }^2}\varphi }}{{{R^2}}} + \frac{{2r{{\sin }^2}\varphi }}{R}] = {R^2}[1 + \frac{r}{R}\{ \frac{r}{R} + 2\} {\sin ^2}\varphi ]\]

∴ Limiting radius of wheel addendum circle,

\[{O_2}M = R\sqrt {1 + \frac{r}{R}\{ \frac{r}{R} + 2\} {{\sin }^2}\varphi } = m\frac{T}{2}\sqrt {1 + \frac{t}{T}\{ \frac{t}{T} + 2\} {{\sin }^2}\varphi } \]

We know that the addendum of the wheel = O₂M - O₂P

\[\therefore {A_W}.m = m\frac{T}{2}\sqrt {1 + \frac{t}{T}\{ \frac{t}{T} + 2\} {{\sin }^2}\varphi } - - m\frac{T}{2}\]

\[\therefore T = \frac{{2{A_W}}}{{\sqrt {1 + \frac{t}{T}(\frac{t}{T} + 2){{\sin }^2}\varphi } - 1}} = \frac{{2{A_W}}}{{\sqrt {1 + \frac{1}{G}(\frac{1}{G} + 2){{\sin }^2}\varphi } - 1}}\]

Standardization of Gears; Metric system

The following table shows the standard proportions in module (m) for the four gear systems.

Table: Standard proportions of gear systems

When gear dimensions are in metric system the pitch specification is mostly in terms of module or modulus which is effectively a length measurement across pitch diameter. Module is mean the pitch diameter in millimeters divided by number of teeth. When module is based on inch measurements it is known as English module to avoid confusion with metric module. Module is a direct dimension unlike diametral pitch which is an inverse dimension. So if pitch diameter of a gear is 40 mm and number of teeth 20 with the module is 2 which means that there are 2 mm of pitch diameter for each tooth. The preferred standard module values are 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 1.25, 1.5, 2.0, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 32, 40 and 50.

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.