Minimum Number of Teeth on the Pinion in Order to Avoid Interference

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written 5.5 years ago by

yashbeer ★ 11k

The phenomenon when the tip of a tooth undercuts the root on its mating gear is known as interference. The minimum number of teeth on the pinion which will mesh with any gear (also rack) without interference is 17 or it may be determined by,

$Z_{1}=\frac{2}{(\sin \alpha)^{2}}$ where $\alpha$ is pressure angle of gear and $Z_{1}=$ no. of teeth on the pinion

Design Criteria for Gears (Also refer PSG Design Data Book)

Parameter	Spur Gear	Helical Gear
Strength Criteria - Design	$\mathrm{m} \geq 1.26 \sqrt[3]{\frac{\left[M_{t}\right]}{y \left[ \sigma_{b} ] \varphi_{m} \cdot Z_{1}\right.}}$ PSG 8.13A	$m_{n} \geq 1.15 \cos \beta^{3} \sqrt{\frac{\left[M_{t}\right]}{y_{v}\left[\sigma_{b} \varphi_{m} \cdot Z_{1}\right.}}$ PSG 8.13A
Strength Criteria - Checking	$F_{s}=\pi y b m\left[\sigma_{b}\right]$ $F_{s} \geq F_{L D}$ or $F_{d}$ $F_{L D}=F_{t} \times C_{v}$ PSG 8.50 OR $\sigma_{b}=\frac{i \pm 1}{a m b y}\left[M_{t}\right] \leq\left[\sigma_{b}\right]$ PSG 8.13A	$F_{s}=\pi y b m_{n}\left[\sigma_{b}\right]$ $F_{s} \geq F_{L D}$ or $F_{d}$ $F_{L D}=F_{t} \times C_{v}$ PSG 8.50 OR $\sigma_{b}=0.7 \frac{i \pm 1}{a b m_{n} y_{v}}\left[M_{t}\right] \leq\left[\sigma_{b}\right]$ PSG 8.13A
Wear Criteria - Design	Find module using $F_{w}=F_{L D} \quad$ Or $d_{1} Q k b=F_{t} \times C_{v}$ PSG 8.51	Find module using $F_{w}=F_{L D} \quad$ Or $\frac{d_{1} Q k b}{\cos \beta^{2}}=F_{t} \times C_{v}$ PSG 8.51
Wear Critera - Checking	$F w=d_{1} Q k b$ $F w \geq F_{L D}$ PSG 8.51	$F w=\frac{d_{1} Q k b}{\cos \beta^{2}}, F w \geq F_{L D}$ PSG 8.51
Surface / Contact Stresses - Design	$a \geq(i \pm 1)^{3} \sqrt{\left(\frac{0.74}{\left[\sigma_{c}\right]}\right)^{2} \frac{E\left[M_{t}\right]}{i \varphi}}$ Find module using $a=\frac{m}{2}\left(Z_{1}+Z_{2}\right)$ PSG 8.13	$a \geq(i \pm 1)^{3} \sqrt{\left(\frac{0.7}{\left[\sigma_{c}\right]}\right)^{2} \frac{E\left[M_{t}\right]}{i \varphi}}$ Find a. Find module using $ a=\frac{m_{n}}{\cos \beta} \frac{\left(Z_{1}+Z_{2}\right)}{2}$ PSG 8.13
Surface / Contact Stresses - Checking	$\sigma_{c}= 0.74 \frac{i \pm 1}{a} \sqrt{\frac{i \pm 1}{i b} E\left[M_{t}\right]} \leq \left[\sigma_{c}\right]$ PSG 8.13	$ \sigma_c = 0.7 \frac{i \pm 1}{a} \sqrt{\frac{i \pm 1}{i b} E\left[M_{t}\right]}$ $\leq\left[\sigma_{c}\right]$, PSG 8.13

Parameter	Bevel Gear	Worm Gear
Strength Criteria - Design	$m_{a v} \geq 1.28$ $\sqrt[3]{\frac{\left[M_{t}\right]}{y_{v}\left[\sigma_{b ]} \varphi_{m} \cdot Z_{1}\right.}}$ PSG 8.13A	$m_{x} \geq 1.24^{3} \sqrt{\frac{\left[M_{t}\right]}{z q y_{v}\left[\sigma_{b ]}\right.}}$ PSG 8.44
Strength Criteria - Checking	$F_{s}=\left[\sigma_{b}\right] b \pi y_{v}\left(1-\frac{b}{R}\right) m$ $F_{s} \geq F_{L D}$ or $F_{d}$ $F_{L D}=C_{v} N_{s f} K_{m} F_{t}$ PSG 8.52 OR $\sigma_{b}=\frac{R \sqrt{i^{2}+1}\left[M_{t}\right]}{(R-0.5 b)^{2} b m y_{v}} \frac{1}{\cos \alpha} \leq \left[\sigma_{b}\right]$ PSG 8.13A	$F_{s}=\pi y b m\left[\sigma_{b}\right]$ $F_{s} \geq F_{L D}$ or $F_{d}$ $F_{L D}=F_{t} \times C_{v}, C_{v}=$ $\left(\frac{6+v_{m g}}{6}\right)$ PSG 8.25 OR $\sigma_{b}=\frac{1.9\left[M_{t}\right]}{m_{x}^{3} q z y_{v}} \leq\left[\sigma_{b}\right]$ PSG 8.44
Wear - Design	Find module using $F_{w}=F_{L D} \quad$ Or $\frac{d_{1} Q k b}{\cos \delta_{1}}=F_{t} \times C_{v}$ PSG 8. 52	Find module using $F_{w}=F_{L D}(\text { Fd }) \quad$ Or $D_{g} b k_{w}=F_{t} \times \left(\frac{6+v_{mg}}{6}\right)$ PSG 8.52
Wear - Checking	$F w=\frac{d_{1} Q k b}{\cos \delta_{1}}, F w \geq F_{L D}$ PSG 8.52	$F w=D_{g} b k_{w}$ $F w \geq F_{L D}(F d)$ PSG 8.52
Surface / Contact Stresses - Design	$R \geq \varphi_{y} \sqrt{i^{2}+1}^{3} \sqrt{\left(\frac{0.72}{\left(\varphi_{y}-0.5\right)\left[\sigma_{c}\right]}\right)}$ Find $^{\prime} \mathrm{R}^{\prime}$. Find module using formulae in Table 31, PSG 8.38	$a = \left(\frac{z}{q}+1\right)^{3} \sqrt{\left(\frac{540}{q}\left[\sigma_{c}\right]\right.} )^{2}\left[M_{t}\right] $ -- 8.44 Find 'a'. Find module using formulae on PSG 8.43
Surface / Contact Stresses - Checking	$\sigma_{C}=\frac{0.72}{(R-0.5 b)} \sqrt{\frac{\sqrt{\left(i^{2} \pm 1\right)^{3}}}{i b} E\left[M_{t}\right]} \leq \left[\sigma_{c}\right]$ PSG 8.13	$\sigma_c =\left(\frac{540}{\frac{z}{q}}\right) \sqrt{\frac{\sqrt{\left(\frac{z}{q}+1\right)^{3}}}{a}\left[M_{t}\right]} \leq {\left[\sigma_{C}\right]}$ PSG 8.44

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