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

Un cable que cuelga entre dos puntos a la misma altura forma una catenaria.Esta curva se describe mediante funciones hiperbólicas, que son combinaciones especiales de funciones exponenciales. Se llaman así porque se relacionan con la hipérbole de la misma manera que las funciones trigonométricas estándar se relacionan con el círculo.La catenaria se describe mediante la función coseno hiperbólica, que es la media de dos funciones exponenciales recíprocas. Su gráfico forma una curva en forma de U con un mínimo en x igual a cero y simetría respecto al eje y.La pendiente en cualquier punto del cable se expresa usando la función seno hiperbólica. Esta función se define como la mitad de la diferencia de dos funciones exponenciales. El gráfico para esto es una curva suave que pasa por el origen y sube bruscamente tanto para valores positivos como negativos de x.La tangente hiperbólica es la proporción entre el seno hiperbólico y el coseno hiperbólico. Su gráfica pasa por el origen y se aproxima a asíntotas horizontales en y igual a uno y igual a menos uno, formando una curva en forma de S.Estas funciones hiperbólicas de coseno y seno ayudan a los ingenieros a describir y calcular la forma y la tensión del cable.

funciones hiperbólicas

Un cable flexible suspendido entre dos puntos a la misma altura adopta naturalmente una curva conocida como catenaria. Esta forma resulta del equilibrio entre el peso del cable y la tensi&#xF3;n que act&#xFA;a a lo largo de su longitud, lo que constituye un estado de equilibrio mec&#xE1;nico. A diferencia de aproximaciones m&#xE1;s simples, la forma real de un cable suspendido se describe mediante funciones hiperb&#xF3;licas.Las funciones hiperb&#xF3;licas est&#xE1;n estrechamente relacionadas con las funciones exponenciales y se denominan as&#xED; por su conexi&#xF3;n con la geometr&#xED;a de la hip&#xE9;rbola, de la misma manera que las funciones trigonom&#xE9;tricas se asocian con la circunferencia. La forma de una catenaria est&#xE1; determinada por la funci&#xF3;n coseno hiperb&#xF3;lico, que establece c&#xF3;mo se curva el cable bajo su propio peso. La forma matem&#xE1;tica de la catenaria se expresa mediante:\begin{equation*} L(P) = h(P_i) + h&#x27;(P_i)\,(P - P_i) \end{equation*}donde la constante controla la curvatura del cable y est&#xE1; relacionada con la tensi&#xF3;n en su interior. La gr&#xE1;fica resultante es una curva suave en forma de U, con su punto m&#xE1;s bajo en el centro del tramo y simetr&#xED;a respecto al eje vertical, lo que refleja la distribuci&#xF3;n uniforme de fuerzas en ambos lados.La pendiente del cable en cualquier punto de la curva est&#xE1; determinada por la funci&#xF3;n seno hiperb&#xF3;lico. Esta funci&#xF3;n indica la inclinaci&#xF3;n con la que el cable asciende desde su punto m&#xE1;s bajo y aumenta progresivamente a medida que se acerca a los soportes. El cociente entre el seno hiperb&#xF3;lico y el coseno hiperb&#xF3;lico define la tangente hiperb&#xF3;lica, que indica c&#xF3;mo var&#xED;a la pendiente en relaci&#xF3;n con la forma general de la curva.Juntas, las funciones coseno hiperb&#xF3;lico y seno hiperb&#xF3;lico proporcionan a los ingenieros herramientas matem&#xE1;ticas precisas para modelar la forma, la pendiente y la tensi&#xF3;n de cables suspendidos. Estas relaciones son esenciales en el dise&#xF1;o y an&#xE1;lisis de estructuras como puentes colgantes, l&#xED;neas el&#xE9;ctricas a&#xE9;reas y otros sistemas soportados por cables.

Funciones hiperbólicas

Un cable flexible suspendido entre dos puntos a la misma altura adopta naturalmente una curva conocida como catenaria. Esta forma resulta del equilibrio ...

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Calculus

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Differentiation Rules

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KEY TERMS AND CONCEPTS

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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

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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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