# Friction wheel
## Power
$$P = Fv = fv$$

$$f = \mu N$$

$$P = \mu Nv = \mu F \frac{\pi DN}{60}$$

$$\text{PS} = \mu F \frac{\pi DN}{735\times60}$$
## Transmission distance
| type | Internal contact | External contact |
|:--------------------- |:----------------:|:----------------:|
| Transmission distance | $$R-r$$ | $$R+r$$ |
## speed ratio
$$v_1 = v_2$$

$$v_1 = r_1\omega_1 = v_2 = r_r\omega_2$$

$$\frac{\omega_1}{\omega_2} = \frac{N_1}{N_2} = \frac{R_2}{R_1} = \frac{D_2}{D_1}$$
## Conical friction wheel
### External contact
$$\frac{N_1}{N_2} = \frac{\sin\beta}{\sin\alpha} = \frac{\sin(\theta-\alpha)}{\sin\alpha}$$

$$= \frac{\sin\theta\cos\alpha-\cos\theta\sin\alpha}{\sin\alpha}$$

$$= \frac{\sin\theta-\cos\theta(\frac{\sin\alpha}{\cos\alpha})}{\frac{\sin\alpha}{\cos\alpha}}$$

$$= \frac{\sin\theta-cos\theta\tan\alpha}{\tan\alpha}$$

$$\frac{N_1}{N_2}\tan\alpha = \sin\theta-\cos\theta\tan\alpha$$
:::success
$$\tan\alpha = \frac{\sin\theta}{\frac{N_1}{N_2}+\cos\theta}$$

$$\tan\beta = \frac{\sin\theta}{\frac{N_2}{N_1}+\cos\theta}$$
:::
* if $\theta = \alpha+\beta = 90^o$
:::success
$$\tan\alpha = \frac{N_2}{N_1}$$

$$\tan\beta = \frac{N_1}{N_2}$$
:::
### Internal contact
$$\frac{N_1}{N_2} = \frac{\sin\beta}{\sin\alpha} = \frac{\sin(\alpha-\theta)}{\sin\alpha}$$

$$\frac{\sin\alpha\cos\theta-\cos\alpha\sin\theta}{\sin\alpha}$$

$$\frac{\cos\theta(\frac{\sin\alpha}{\cos\alpha})-\sin\theta}{\frac{\sin\alpha}{\cos\alpha}}$$

$$\frac{\cos\theta\tan\alpha-\sin\theta}{\tan\alpha}$$

$$\frac{N_1}{N_2}\tan\alpha = \cos\theta\tan\alpha-\sin\theta$$
:::success
$$\tan\alpha = \frac{\sin\theta}{\cos\theta-\frac{N_1}{N_2}}$$

$$\tan\beta = \frac{\sin\theta}{\frac{N_2}{N_1}-\cos\theta}$$
:::
## Perpendicular disks
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<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/TEZ-Reibradgetriebe_7271.JPG/1024px-TEZ-Reibradgetriebe_7271.JPG" alt="image"width="300">
</div>

$$v_A = v_B$$

$$2\pi r_AN_A = 2\pi r_BN_B$$

$$\frac{N_A}{N_B} = \frac{r_B}{r_A}$$