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	\end{equation}		\end{equation}
−	The coordinate $x$ is measured perpendicular to that of the cylinder axes.	+	The coordinate $x$ is measured perpendicular to that of the cylinder axes. For the case of nominal contact between cylinders closed form analytical solutions are available.
−	For the case of nominal contact between cylinders closed from analytical solutions are available.	+	The surfaces stresses are given by the equations:
−	\end{equation}	+
		+	[[File:Screenshot_2016-11-16_16-05-58.png|500px]]
		+
		+	[[File:Screenshot_2016-11-16_16-07-37.png|700px]]
		+
		+	[[File:Screenshot_2016-11-16_16-08-50.png|700px]]
		+
		+	The surface stresses and stresses along the line of symmetry are shown in the following two graphs. The $x$ and $z$ coordinates are normalized with the contact width $b$.
		+
		+	[[File:Screenshot_2016-11-16_16-13-26.png|300px]]            [[File:Screenshot_2016-11-16_16-12-32.png|300px]]
		+
		+
		+	= References =
	= FreeFem++ numerical solution =		= FreeFem++ numerical solution =

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Revision as of 17:19, 16 November 2016

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Contact of two cylinders with axes parallel

Normal contact

If two circular cylinders with radii $R_1$ and $R_2$ are pressed together by a force per unit length of magnitude $P$ with their axes parallel, then the contact patch will be of half-width $b$ such that

$$\begin{equation} b = \sqrt{\frac{2PR}{\pi E^*}} \end{equation}$$ where $R$ and $E^*$ are the reduced radius of contact and the contact modulus defined by $$\begin{equation} \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}, \end{equation}$$ $$\begin{equation} \frac{1}{E^*} = \frac{1-{\nu_1}^2}{E_1} + \frac{1-{\nu_2}^2}{E_2}. \end{equation}$$

The resulting pressure distribution $p(x)$ is semielliptical, i.e., of the form

$$\begin{equation} p(x) = p_0 \sqrt{1-\frac{x^2}{b^2}} \end{equation}$$ where the peak pressure $$\begin{equation} p_0 = \sqrt{\frac{PE^*}{\pi R}}. \end{equation}$$

The coordinate $x$ is measured perpendicular to that of the cylinder axes. For the case of nominal contact between cylinders closed form analytical solutions are available.

The surfaces stresses are given by the equations:

500px

700px

700px

The surface stresses and stresses along the line of symmetry are shown in the following two graphs. The $x$ and $z$ coordinates are normalized with the contact width $b$.

300px 300px

References

FreeFem++ numerical solution