# Cylinder on a cylinder; axes at right angles

## Values for calculation

$P$ $\mathrm{N}$
$D_1$ $\mathrm{mm}$
$D_2$ $\mathrm{mm}$
$ν_1$
$ν_2$
$E_1$ $\mathrm{MPa}$
$E_2$ $\mathrm{MPa}$

## Calculation

### Coefficient $α$

$\text{if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 1.5$
$$α=0.908+\cfrac{1.045-0.908}{1.5-1}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 2$
$$α=1.045+\cfrac{1.158-1.045}{2-1.5}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1.5\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 3$
$$α=1.158+\cfrac{1.35-1.158}{3-2}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-2\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 4$
$$α=1.35+\cfrac{1.505-1.35}{4-3}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-3\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 6$
$$α=1.505+\cfrac{1.767-1.505}{6-4}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-4\right)$$
$\text{else}$
$$α=1.767+\cfrac{2.175-1.767}{10-6}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-6\right)$$

### Coefficient $β$

$\text{if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 1.5$
$$β=0.908+\cfrac{0.799-0.908}{1.5-1}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 2$
$$β=0.799+\cfrac{0.732-0.799}{2-1.5}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1.5\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 3$
$$β=0.732+\cfrac{0.651-0.732}{3-2}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-2\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 4$
$$β=0.651+\cfrac{0.602-0.651}{4-3}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-3\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 6$
$$β=0.602+\cfrac{0.544-0.602}{6-4}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-4\right)$$
$\text{else}$
$$β=0.544+\cfrac{0.481-0.544}{10-6}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-6\right)$$

### Coefficient $γ$

$\text{if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 1.5$
$$γ=0.825+\cfrac{0.818-0.825}{1.5-1}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 2$
$$γ=0.818+\cfrac{0.804-0.818}{2-1.5}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-1.5\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 3$
$$γ=0.804+\cfrac{0.774-0.804}{3-2}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-2\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 4$
$$γ=0.774+\cfrac{0.747-0.774}{4-3}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-3\right)$$
$\text{else if }\ \cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}\le 6$
$$γ=0.747+\cfrac{0.702-0.747}{6-4}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-4\right)$$
$\text{else}$
$$γ=0.702+\cfrac{0.641-0.702}{10-6}\cdot\left(\cfrac{\max\left(D_1, D_2\right)}{\min\left(D_1, D_2\right)}-6\right)$$

### Dimensional coefficient

$$K_D=\cfrac{D_1\cdot D_2}{D_1+D_2}$$

### Material coefficient

$$C_E=\cfrac{1-ν_1^2}{E_1}+\cfrac{1-ν_2^2}{E_2}$$

### Major semiaxis

$$c=α\cdot\sqrt[3]{P\cdot K_D\cdot C_E}$$

### Minor semiaxis of elliptical contact area

$$d=β\cdot\sqrt[3]{P\cdot K_D\cdot C_E}$$

### Contact stress

$$σ_c=\cfrac{1.5\cdot P}{π\cdot c\cdot d}$$

### Relative motion of approach along the axis of loading of two points, one in each of the two contact bodies, remote from the contact zone

$$y=γ\cdot\sqrt[3]{\cfrac{P^2\cdot C_E^2}{K_D}}$$