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Cylinder on a cylinder; axes at right angles

Cylinder on a cylinder; axes at right angles P P D 2 D 1
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 $

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

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

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

$$K_D=\cfrac{D_1\cdot D_2}{D_1+D_2}$$
$K_D=\cfrac{D_1\cdot D_2}{D_1+D_2}$
Equations in LaTeX code
K_D=\cfrac{D_1\cdot D_2}{D_1+D_2}

Material coefficient

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

Major semiaxis

$$c=α\cdot\sqrt[3]{P\cdot K_D\cdot C_E} $$
$c=α\cdot\sqrt[3]{P\cdot K_D\cdot C_E} $
Equations in LaTeX code
c=α\cdot\sqrt[3]{P\cdot K_D\cdot C_E}

Minor semiaxis of elliptical contact area

$$d=β\cdot\sqrt[3]{P\cdot K_D\cdot C_E}$$
$d=β\cdot\sqrt[3]{P\cdot K_D\cdot C_E}$
Equations in LaTeX code
d=β\cdot\sqrt[3]{P\cdot K_D\cdot C_E}

Contact stress

$$σ_c=\cfrac{1.5\cdot P}{π\cdot c\cdot d}$$
$σ_c=\cfrac{1.5\cdot P}{π\cdot c\cdot d}$
Equations in LaTeX code
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

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