悬链线

不同的悬链线
鐵鏈形式的悬链线。
蜘蛛絲形成多個(近似的)悬链线。

悬链线(Catenary)是一种常用曲线,物理上用于描绘質量均勻分佈而不可延伸的長鏈悬掛在两支点间,因均勻引力作用下而形成向下彎曲之曲線,因此而得名。

雖然彎曲的形狀看似二次方的拋物線,但是1638年在伽利略的《Two New Sciences》中證明因為繩子的張力會隨著吊掛重量的不同,在底端為最小、愈高的地方愈大,如此一來,它所形成的形狀就不是拋物線。

隨後在1670年胡克根據力學推導出懸鏈線的數學特性。1691年萊布尼茲惠更斯約翰·白努利近一步推导出數學模型。

它的公式为:

或者简单地表示为

其中cosh是雙曲余弦函数, 是一个由绳子本身性质和悬挂方式决定的常数軸為其準線。具体来说,,其中是重力加速度,是线密度(假设绳子密度均匀),而是绳子上每一点处张力的水平分量,它取决于绳子的悬挂方式;若绳子两端在同一水平面上,则下面的方程决定了

其中L是绳子总长的一半,d是端点距离的一半。

方程的推导

表达式的证明

如右图,设最低点处受水平向左的拉力,右悬挂点处表示为点,在弧线区段任意取一段设为点,则受一个斜向上的拉力,设和水平方向夹角为,绳子的质量为,受力分析有:

, 其中是右段绳子的长度,是绳子线重量密度,为切线方向,记, 代入得微分方程;

利用弧长公式;

所以;

再把代入微分方程得

对于微分处理

其中;

对(2)分离常量求积分

,即

其中反双曲函数;

时,

带入得

整理得

工程中的应用

悬索桥双曲拱桥架空电缆都用到悬链线的原理。 在工程中有一种应用,称作悬链系数。如果我们改变公式的写法,会给工程应用带来很大帮助,公式及图像如下:

还有以下几个公式,可能也有用:

其中是曲线中某点到0点的链索长度,是该点的正切角,是0点处的水平张力,是链索的单位重量。利用上述公式即能计算出任意点的张力。

參考資料

外部連結

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