7.2
Longueur d’arc d’une courbe : résolution de problèmes
Une ligne à haute tension s’étend sur une distance horizontale de 40 m entre deux pylônes, ce qui génère une flèche verticale de 10 m due à la gravité…
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.
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Q1: Why is a catenary curve used to model a hanging power line instead of a parabola?
A catenary accurately describes a uniform, flexible cable hanging under its own weight, unlike a parabola which is merely an approximation. The catenary is defined by the hyperbolic cosine function and precisely represents how gravity and the cable's weight distribution affect its shape. This mathematical model ensures engineers can calculate exact cable lengths needed for safe installation between transmission towers.
Q2: What does the parameter 'a' represent in the catenary equation for a power line?
The parameter a represents the ratio between the horizontal tension in the cable and its weight per unit length. In the power line example, a equals 20 meters. This parameter controls the catenary's shape and determines how much the cable sags under its own weight and thermal expansion effects.
Q3: How does the arc length function calculate the exact length of a suspended cable?
The arc length function integrates the derivative of the catenary curve, which yields the hyperbolic sine function. A standard hyperbolic identity simplifies the expression by eliminating the square root. Integrating the resulting hyperbolic cosine and evaluating at the tower positions gives the total cable length needed for the installation.
Q4: Why is the origin placed at the lowest point of the catenary curve in this power line problem?
Placing the origin at the lowest point creates a symmetric coordinate system where the two transmission towers are positioned equidistantly on either side. For this 40-meter span with parameter a of 20 meters, the towers are located 20 meters to the left and right of the curve's minimum. This symmetry simplifies the integration limits and calculations.
Q5: What is the relationship between hyperbolic sine and hyperbolic cosine in arc length calculations?
The derivative of the catenary's hyperbolic cosine function produces the hyperbolic sine. When computing arc length, a standard hyperbolic identity involving these functions eliminates the square root from the integrand. The integral of the simplified hyperbolic cosine expression yields the hyperbolic sine, which is then evaluated at the integration limits.
Q6: How much cable length is required for a 40-meter power line span with 10-meter sag?
For a 40-meter horizontal span between towers with a 10-meter vertical sag, the catenary model with parameter a of 20 meters yields approximately 47 meters of cable length. This additional 7 meters accounts for the curve's natural shape under gravity and thermal expansion, ensuring the cable hangs safely without excessive tension.
Q7: What role do hyperbolic functions play in modeling real-world engineering problems?
Hyperbolic functions like hyperbolic cosine and sine naturally describe physical phenomena involving hanging cables and structures under their own weight. These functions appear in applications of integration across engineering and physics. Their mathematical properties, including standard identities, enable engineers to solve complex real-world problems such as determining exact cable lengths for power line installations.