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

Eine Hochspannungsleitung, die zwischen zwei Sendetürmen hängt, überspannt einen 40-Meter-Abstand mit einer signifikanten 10-Meter-Höhensenkung aufgrund der thermischen Ausdehnung.Das Ziel ist es, die genaue Länge der Stromleitung zu bestimmen, die erforderlich ist, um diese Punkte sicher zu verbinden.Ingenieure modellieren dies mit einer Oberleitung, der natürlichen Kurve eines hängenden Kabels, die mathematisch durch die hyperbolische Kosinusfunktion beschrieben wird.Diese Kurve wird von einem Parameter a gesteuert, der das Gewicht des Kabels pro Meter mit der horizontalen Spannung in Beziehung setzt. Für diese spezielle Installation wird der Wert des Parameters a mit 20 Metern angenommen.Um die genaue Länge zu bestimmen, verwendet die Berechnung die Bogenlängenfunktion. Hier ergibt die Ableitung der Leitung den hyperbolischen Sinus.Eine Standardidentität mit hyperbolischen Funktionen vereinfacht den Ausdruck weiter, indem sie die Quadratwurzel entfernt.Das Integral des resultierenden hyperbolischen Kosinus ist einfach der hyperbolische Sinus. Die Berechnung der hyperbolischen Sinusfunktion an den Grenzen von minus zwanzig bis positiv zwanzig ergibt eine Gesamtlänge von etwa 47 Metern.

bogenlänge einer kurve: problemlösung

Arc Length of a Curve: Problem Solving

Eine Hochspannungsleitung &#xFC;berspannt eine horizontale Distanz von 40 m zwischen zwei &#xDC;bertragungsmasten. Durch die Wirkung von Schwerkraft und thermischer Ausdehnung entsteht ein vertikaler Durchhang von 10 m. Die von dem h&#xE4;ngenden Kabel gebildete Kurve stellt eine Kettenlinie dar, welche das Verhalten eines gleichm&#xE4;&#xDF;igen, flexiblen Kabels unter seinem Eigengewicht pr&#xE4;zise modelliert. Im Unterschied zu einer Parabel wird die Kettenlinie durch die hyperbolische Kosinusfunktion beschrieben und liefert eine exakte Darstellung der Form des Kabels.In diesem Szenario setzen die Ingenieure den Parameter a auf einen Wert von 20 m. Dieser Parameter entspricht dem Verh&#xE4;ltnis zwischen der horizontalen Zugkraft im Kabel und seinem Gewicht pro L&#xE4;ngeneinheit. Wird der Ursprung am tiefsten Punkt der Kurve festgelegt, befinden sich die Masten symmetrisch bei jeweils 20 m links und rechts des Ursprungs.Zur Bestimmung der f&#xFC;r die Installation erforderlichen Kabell&#xE4;nge wird die Bogenl&#xE4;nge der Kettenlinie zwischen den beiden Masten berechnet. Die verwendete Formel lautet:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Das Ergebnis dieser Integration lautet:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Dieser Wert gibt die gesamte ben&#xF6;tigte L&#xE4;nge der Hochspannungsleitung an und stellt sicher, dass die Installation fachgerecht erfolgt, wobei der Durchhang durch das Eigengewicht sowie thermische Effekte des Kabels ber&#xFC;cksichtigt werden.

Bogenlänge einer Kurve: Problemlösung

Eine Hochspannungsleitung &#xFC;berspannt eine horizontale Distanz von 40 m zwischen zwei &#xDC;bertragungsmasten. Durch die Wirkung von Schwerkraft und ...

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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

arc length of a curve

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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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