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

Linia wysokonapięciowa wisząca między dwoma wieżami przesyłowymi zajmuje 40-metrową szczelinę z wyraźnym pionowym zapadnięciem o wysokości 10 metrów spowodowanym rozszerzaniem termicznym.Celem jest ustalenie dokładnej długości linii energetycznej potrzebnej do bezpiecznego połączenia tych punktów.Inżynierowie modelują to za pomocą sieci kontaktowej, naturalnej krzywej wiszącego kabla opisanej matematycznie przez hiperboliczną funkcję cosinusową.Krzywa ta jest kontrolowana parametrem a, który odnosi wagę liny na metr do poziomego napięcia. Dla tej konkretnej instalacji wartość parametru a zakłada się na 20 metrów.Aby wyznaczyć dokładną długość, obliczenia używają funkcji długości łuku. Tutaj pochodna sieci trakcyjnej daje sinus hiperboliczny.Standardowa tożsamość obejmująca funkcje hiperboliczne upraszcza wyrażenie przez usunięcie pierwiastka kwadratowego.Całka powstałego kosinusa hiperbolicznego to po prostu sinus hiperboliczny. Ocena funkcji sinusoidalnej hiperbolicznej na granicach od minus dwadzieścia do plus dwadzieścia daje całkowitą długość około 47 metrów.

długość łuku krzywej: rozwiązywanie problemów

Arc Length of a Curve: Problem Solving

Linia wysokiego napi&#x119;cia rozci&#x105;ga si&#x119; na 40 m w poziomie mi&#x119;dzy dwoma s&#x142;upami energetycznymi, co powoduje 10 m pionowe ugi&#x119;cie spowodowane dzia&#x142;aniem grawitacji i rozszerzalno&#x15B;ci cieplnej. Krzywa utworzona przez zawieszony przew&#xF3;d jest krzyw&#x105; katenarn&#x105;, kt&#xF3;ra dok&#x142;adnie modeluje zachowanie jednorodnego, elastycznego kabla pod wp&#x142;ywem jego w&#x142;asnego ci&#x119;&#x17C;aru. W przeciwie&#x144;stwie do kszta&#x142;tu parabolicznego, krzywa katenarna jest opisana funkcj&#x105; cosinusa hiperbolicznego i oferuje precyzyjne odwzorowanie kszta&#x142;tu kabla.W tym przypadku in&#x17C;ynierowie przyjmuj&#x105; dla parametru aaa warto&#x15B;&#x107; 20 m. Parametr ten odpowiada stosunkowi mi&#x119;dzy napr&#x119;&#x17C;eniem poziomym kabla a jego ci&#x119;&#x17C;arem na jednostk&#x119; d&#x142;ugo&#x15B;ci. Umieszczaj&#x105;c pocz&#x105;tek uk&#x142;adu wsp&#xF3;&#x142;rz&#x119;dnych w najni&#x17C;szym punkcie krzywej, s&#x142;upy s&#x105; rozmieszczone symetrycznie w odleg&#x142;o&#x15B;ci 20 m po obu stronach krzywej.Aby okre&#x15B;li&#x107; dok&#x142;adn&#x105; d&#x142;ugo&#x15B;&#x107; kabla wymagan&#x105; do instalacji, oblicza si&#x119; d&#x142;ugo&#x15B;&#x107; &#x142;uku krzywej katenarnej mi&#x119;dzy dwoma s&#x142;upami. Zastosowany wz&#xF3;r to:\begin{equation*}L = \int_{-20}^{20} \cosh\!\liparens{\jfrac{x}{20}}\, dx\end{equation*}Wynik ca&#x142;kowania to:\begin{equation*}L = 40 \sinh(1) \approx 47.01 \,\text{meters}\end{equation*}Ta warto&#x15B;&#x107; okre&#x15B;la ca&#x142;kowit&#x105; d&#x142;ugo&#x15B;&#x107; linii energetycznej, zapewniaj&#x105;c prawid&#x142;owy monta&#x17C;, z uwzgl&#x119;dnieniem ugi&#x119;&#x107; spowodowanych ci&#x119;&#x17C;arem w&#x142;asnym kabla i efektami termicznymi.

Długość łuku krzywej: rozwiązywanie problemów

Linia wysokiego napi&#x119;cia rozci&#x105;ga si&#x119; na 40 m w poziomie mi&#x119;dzy dwoma s&#x142;upami energetycznymi, co powoduje 10 m pionowe ...

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Application of Techniques of Integration

Zastosowanie technik całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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...

arc length function

Measuring Distance Along a Curve

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Hydrostatic Force on Dam Walls with Integration

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applications of integration to find centers of mass

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Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

theorem of pappus

Pappus Theorem for Volume and Surface Area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

applications of integration to find consumer surplus

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Consumer Surplus from Demand and Integration

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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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