Name: catenary cable length in power lines
Uploaded: 2026-01-12T12:24:12.000Z
Duration: 1 min 21 s
Channel: JoVE

Une ligne électrique haute tension suspendue entre deux tours de transmission couvre un espace de 40 mètres avec un affaissement vertical significatif de 10 mètres dû à la dilatation thermique.L’objectif est de trouver la longueur exacte de la ligne électrique nécessaire pour connecter ces aiguillages en toute sécurité.Les ingénieurs modélisent cela à l’aide d’une caténaire, la courbe naturelle d’un câble suspendu décrite mathématiquement par la fonction cosinus hyperbolique.Cette courbe est contrôlée par un paramètre a, qui relie le poids du câble par mètre à la tension horizontale. Pour cette installation spécifique, la valeur du paramètre a est supposée être de 20 mètres.Pour trouver la longueur exacte, le calcul utilise la fonction longueur d’arc. Ici, la dérivée de la caténaire donne le sinus hyperbolique.Une identité standard impliquant des fonctions hyperboliques simplifie encore l’expression en supprimant la racine carrée.L’intégrale du cosinus hyperbolique résultant est simplement le sinus hyperbolique. L’évaluation de la fonction sinusoïdal hyperbolique aux limites de moins vingt à vingt positif, on obtient une longueur totale d’environ 47 mètres.

longueur d’arc d’une courbe : résolution de problèmes

Arc Length of a Curve: Problem Solving

Une ligne &#xE0; haute tension s&#x2019;&#xE9;tend sur une distance horizontale de 40 m entre deux pyl&#xF4;nes, ce qui g&#xE9;n&#xE8;re une fl&#xE8;che verticale de 10 m due &#xE0; la gravit&#xE9; et &#xE0; la dilatation thermique. La courbe form&#xE9;e par le c&#xE2;ble suspendu est une cha&#xEE;nette, qui repr&#xE9;sente pr&#xE9;cis&#xE9;ment le comportement d&#x2019;un c&#xE2;ble uniforme et flexible sous son propre poids. Contrairement &#xE0; une courbe parabolique, la cha&#xEE;nette est d&#xE9;crite par la fonction cosinus hyperbolique et offre une repr&#xE9;sentation pr&#xE9;cise de la forme du c&#xE2;ble.Dans cette configuration, les ing&#xE9;nieurs donnent au param&#xE8;tre a une valeur de 20 m. Ce param&#xE8;tre correspond au rapport entre la tension horizontale du c&#xE2;ble et son poids lin&#xE9;ique. En pla&#xE7;ant l&#x2019;origine au point le plus bas de la courbe, les pyl&#xF4;nes sont sym&#xE9;triquement situ&#xE9;s &#xE0; 20 m de part et d&#x2019;autre de la courbe.Pour d&#xE9;terminer la longueur exacte du c&#xE2;ble n&#xE9;cessaire pour l&#x2019;installation, on calcule la longueur d&#x2019;arc de la cha&#xEE;nette entre les deux pyl&#xF4;nes. La formule utilis&#xE9;e est la suivante :\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Le r&#xE9;sultat de cette int&#xE9;gration est :\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Cette valeur donne la longueur totale du c&#xE2;ble n&#xE9;cessaire pour l&#x2019;installation, garantissant une mise en place correcte tout en tenant compte de la fl&#xE8;che caus&#xE9;e par le poids propre du c&#xE2;ble et des effets thermiques.

Longueur d’arc d’une courbe : résolution de problèmes

Une ligne &#xE0; haute tension s&#x2019;&#xE9;tend sur une distance horizontale de 40 m entre deux pyl&#xF4;nes, ce qui g&#xE9;n&#xE8;re une fl&#xE8;che ...

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Calculus

Calcul

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Application of Techniques of Integration

Application des techniques d'intégration

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catenary cable length in power lines

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight.

A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the parameter a = 20 meters. This value represents the ratio between the horizontal tension in the cable and its weight per unit length.

To make the calculation easier, the origin is placed at the lowest point of the curve. That puts the two towers 20 meters to either side of the center. The arc length of the catenary between those towers is then found with an integration formula. The result gives the total cable length needed for installation, including the sag caused by the cable’s own weight and thermal effects.

Catenary Cable Length in Power Lines

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arc length of a curve: problem solving

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight. A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the...

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key terms and concepts

Find the cable length of a sagging power line by modeling it as a catenary. Learn how arc length, tension, and sag work together in this example.

Catenary Cable Length in Power Lines | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Catenary Cable Length in Power Lines

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