Name: modeling hanging cables with hyperbolic curves
Uploaded: 2026-03-30T09:05:59.000Z
Duration: 1 min 26 s
Channel: JoVE

Kabel zawieszony między dwoma punktami na tej samej wysokości tworzy sieć kontaktową.Ta krzywa jest opisywana funkcjami hiperbolicznymi, które są specjalnymi kombinacjami funkcji wykładniczych. Nazywają się tak, ponieważ odnoszą się do hiperboli w taki sam sposób, w jaki standardowe funkcje trygonometryczne odnoszą się do okręgu.Katenara opisywana jest przez hiperboliczną funkcję kosinusową, która jest średnią dwóch funkcji wykładniczych o odwrotności. Jego graf tworzy krzywą w kształcie litery U z minimum w x równym zero i symetrią wokół osi y.Nachylenie w dowolnym punkcie na linie wyraża się za pomocą funkcji sinusoidalnej hiperbolicznej. Funkcja ta definiowana jest jako połowa różnicy dwóch funkcji wykładniczych. Wykres dla tego to gładka krzywa przechodząca przez początek układu i gwałtownie wznosząca się zarówno dla wartości dodatniej, jak i ujemnej x.Styczna hiperboliczna to stosunek sinusa hiperbolicznego do kosinusu hiperbolicznego. Jego graf przechodzi przez początek układu i zbliża się do poziomych asymptot w punkcie y równym się jeden, a y równa się minus jeden, tworząc krzywą w kształcie litery S.Te hiperboliczne funkcje cosinus i sinus pomagają inżynierom opisać i obliczyć kształt oraz napięcie kabla.

funkcje hiperboliczne

Elastyczny kabel zawieszony mi&#x119;dzy dwoma punktami na tej samej wysoko&#x15B;ci naturalnie tworzy krzyw&#x105; zwan&#x105; krzyw&#x105; &#x142;a&#x144;cuchow&#x105;. Kszta&#x142;t ten wynika z r&#xF3;wnowagi mi&#x119;dzy ci&#x119;&#x17C;arem kabla a si&#x142;&#x105; naci&#x105;gu dzia&#x142;aj&#x105;c&#x105; wzd&#x142;u&#x17C; jego d&#x142;ugo&#x15B;ci, co reprezentuje stan r&#xF3;wnowagi mechanicznej. W przeciwie&#x144;stwie do prostszych przybli&#x17C;e&#x144;, rzeczywisty kszta&#x142;t wisz&#x105;cego kabla jest opisywany za pomoc&#x105; funkcji hiperbolicznych.Funkcje hiperboliczne s&#x105; &#x15B;ci&#x15B;le powi&#x105;zane z funkcjami wyk&#x142;adniczymi i zosta&#x142;y nazwane ze wzgl&#x119;du na ich zwi&#x105;zek z geometri&#x105; hiperboli, tak jak funkcje trygonometryczne s&#x105; powi&#x105;zane z okr&#x119;giem. Kszta&#x142;tem krzywej &#x142;a&#x144;cuchowej rz&#x105;dzi funkcja cosinusa hiperbolicznego, kt&#xF3;ra okre&#x15B;la, jak kabel wygina si&#x119; pod wp&#x142;ywem w&#x142;asnego ci&#x119;&#x17C;aru. Posta&#x107; matematyczna krzywej &#x142;a&#x144;cuchowej jest podana przez\begin{equation*} L(P) = h(P_i) + h&#x27;(P_i)\,(P - P_i) \end{equation*}gdzie sta&#x142;a okre&#x15B;la krzywizn&#x119; kabla i jest zwi&#x105;zana z si&#x142;&#x105; naci&#x105;gu dzia&#x142;aj&#x105;c&#x105; wzd&#x142;u&#x17C; niego. Powsta&#x142;y wykres jest g&#x142;adk&#x105; krzyw&#x105; w kszta&#x142;cie litery U z najni&#x17C;szym punktem w &#x15B;rodku prz&#x119;s&#x142;a i symetri&#x105; wzgl&#x119;dem osi pionowej, odzwierciedlaj&#x105;c&#x105; r&#xF3;wnomierny rozk&#x142;ad si&#x142; po obu stronach.Nachylenie kabla w dowolnym punkcie krzywej opisane jest funkcj&#x105; sinusa hiperbolicznego. Funkcja ta pokazuje, jak stromo kabel wznosi si&#x119; od najni&#x17C;szego punktu i stale ro&#x15B;nie w miar&#x119; zbli&#x17C;ania si&#x119; do podp&#xF3;r. Stosunek sinusa hiperbolicznego do cosinusa hiperbolicznego definiuje tangens hiperboliczny, kt&#xF3;ry opisuje zachowanie si&#x119; nachylenia wzgl&#x119;dem og&#xF3;lnego kszta&#x142;tu krzywej.Funkcje cosinusa hiperbolicznego i sinusa hiperbolicznego zapewniaj&#x105; in&#x17C;ynierom precyzyjne narz&#x119;dzia matematyczne do modelowania kszta&#x142;tu, nachylenia i si&#x142;y naci&#x105;gu podwieszonych kabli. Zale&#x17C;no&#x15B;ci te s&#x105; niezb&#x119;dne w projektowaniu i analizie konstrukcji takich jak mosty wisz&#x105;ce, napowietrzne linie energetyczne i inne systemy podparte kablami.

Funkcje hiperboliczne

Elastyczny kabel zawieszony mi&#x119;dzy dwoma punktami na tej samej wysoko&#x15B;ci naturalnie tworzy krzyw&#x105; zwan&#x105; krzyw&#x105; ...

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Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Differentiation Rules

Zasady różniczkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Hyperbolic Functions

modeling hanging cables with hyperbolic curves

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium.

The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions. They are named for their connection to the hyperbola, much like trigonometric functions are linked to the circle.

The catenary is governed by the hyperbolic cosine function. This function sets the curve of the cable as it hangs under its own weight. In the mathematical form of the catenary, a constant controls the curvature and is tied to the tension in the cable. The graph is smooth and U-shaped. Its lowest point is at the center of the span, and it is symmetric about the vertical axis.

The slope of the cable is described by the hyperbolic sine function. This function shows how steeply the cable rises away from the lowest point. The slope increases steadily as the cable moves toward the supports. The hyperbolic tangent is the ratio of hyperbolic sine to hyperbolic cosine, and it describes how the slope changes relative to the overall curve.

Together, hyperbolic cosine and hyperbolic sine give engineers precise tools for modeling shape, slope, and tension in suspended cables. These ideas matter in the design and analysis of suspension bridges, overhead power lines, and other cable-supported systems.

Modeling Hanging Cables with Hyperbolic Curves

modeling-hanging-cables-with-hyperbolic-curves

derivatives of simple functions

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limits on trigonometric functions

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the chain rule

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implicit differentiation

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the number e as a limit

e from Natural Log Derivatives

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application of rates of change

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related rates

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linearization and approximation

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hyperbolic functions

Hyperbolic functions help describe the shape of a hanging cable. When a flexible cable is suspended between two points at the same height, it naturally forms a catenary. That curve comes from the balance between the cable’s weight and the tension along its length. It shows a state of mechanical equilibrium. The true shape of this hanging cable is not captured well by a simple approximation. Instead, it is described with hyperbolic functions. These functions are related to exponential functions.

inverse hyperbolic functions and their derivatives

inverse-hyperbolic-functions-and-their-derivatives

Deriving the Slope of a Catenary Curve

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hyperbolic-and-inverse-hyperbolic-functions-problem-solving

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key terms and concepts

Learn how hyperbolic functions describe catenary curves, cable slope, and tension in hanging cables used in bridges and power lines.

Modeling Hanging Cables with Hyperbolic Curves | 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

Modeling Hanging Cables with Hyperbolic Curves

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