Subject: Kinematics of Machinery

Topic: Gears and Gear Trains

Difficulty: Medium

​The addendum circles for two matting gears must cut the common tangent to the base circles between the points of tangency.The limiting condition reaches, when the addendum circles of pinion and wheel pass through points N and M as shown in fig 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

$\phi$ = Pressure angle or angle of obliquity

From triangle​ $O_{1}NP$​,​

$(O_{1}N)^{2}=(O_{1}P)^{2}+(PN)^{2}-2\times O_{1}P\times PN\cos O_{1}PN$

$=r^{2}+R^{2}\sin^{2}\phi-2r.R\sin\phi\cos(90^{\circ}+\phi)$.............$(\because PN=O_{2}P\sin\phi=R\sin\phi)$

$=r^{2}+R^{2}\sin^{2}\phi-2r.R\sin^{2}\phi$

$=r^{2}[1+\frac{R^{2}\sin^{2}\phi}{r^{2}}+\frac{2.R\sin^{2}\phi}{r}]=r^{2}[1+\frac{R}{r}(\frac{R}{r}+2)\sin^{2}\phi]$

Limiting radius of the pinion addendum circle,

​$ O_{1}N=r\sqrt{1+\frac{R}{r}(\frac{R}{r}+2)\sin^{2}\phi}=\frac{mt}{2}\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)\sin^{2}\phi} $​

Let $A_{P}m$= Addendum of the pinion, where $A_{P}$ is a fraction by which the standard addendum of one module for the pinion should be multiplied in order to avoid interference.

We know that the addendum of the pinion

​$=O_{1}N-O_{1}P$

$A_{P}m=\frac{mt}{2}\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)\sin^{2}\phi}-\frac{mt}{2}$.......$(\because O_{1}P=r=\frac{mt}{2})$

$=\frac{mt}{2}[\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)\sin^{2}\phi}-1]$

$A_{P}=\frac{t}{2}[\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)\sin^{2}\phi}-1]$

$t=\frac{2A_{P}}{\sqrt{1+\frac{T}{t}(\frac{T}{t}+2)\sin^{2}\phi}-1}=\frac{2A_{P}}{\sqrt{1+G(G+2)\sin^{2}\phi}-1} $​