# 悬链线

## 维基百科，自由的百科全书

${\displaystyle y=a\cosh {\frac {x}{a))}$或者简单地表示为${\displaystyle y={\frac {a\left(e^{\frac {x}{a))+e^{-{\frac {x}{a))}\right)}{2))}$

${\displaystyle {\frac {L}{a))=\sinh {\frac {d}{a))}$

## 方程的推导

${\displaystyle T\sin \theta =mg}$

${\displaystyle T\cos \theta =H}$

${\displaystyle \tan \theta ={\frac {\mathrm {d} y}{\mathrm {d} x))={\frac {mg}{H))}$

${\displaystyle mg=\rho s}$，　其中${\displaystyle s}$是右段${\displaystyle AB}$绳子的长度，${\displaystyle \rho }$是绳子线重量密度，${\displaystyle \tan \theta }$为切线方向，记${\displaystyle a={\frac {\rho }{H))}$, 代入得微分方程${\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} x))=as}$;

${\displaystyle p'={\frac {\rho }{H)){\sqrt {1+p^{2))}\ \cdots \cdots \ (2)}$

${\displaystyle \int {\frac {dp}{\sqrt {1+p^{2))))=\int adx}$

${\displaystyle ln(p+{\sqrt {1+p^{2))})=ax+C}$，即${\displaystyle \mathrm {arsinh} p=ax+C}$

${\displaystyle x=0}$时，${\displaystyle {\frac {dy}{dx))=p=0}$

## 工程中的应用

${\displaystyle y=a\ \left(\cosh {\frac {x}{a))-1\right)}$

${\displaystyle L=a\ \sinh {\frac {x}{a))}$
${\displaystyle \tan \alpha =\sinh {\frac {x}{a))}$
${\displaystyle F_{0}=a\ \gamma }$