Hanging cable - catenary
April 3, 2022
These things make a curve, which is rather similar to a parabola, a graph of a quadratic function. But actually it is not a parabola, but a special curve called catenary. We shall show that such curve is given by a hyperbolic function with two ways: 1. equilibrium of the force and 2. using a calculus of variations. 1. Equilibrium of the force:We shall see a small part of the hanging cable, and study the total force executed to this part.
Since the cable is in stable condition, the result of all force is cancelled. Horizontal compornents of the force: Vertical compornents of the force: From (a), the horizontal compornents of the force is constant. Then From (b), let we have Substitute (c) in this expression. Suppose that the density of the cable is ρ, the area of the section is A and the length of the cable is s. Then Since we have Therefore Let For integration of the right hand side of the equation, we substitute Then Hence We use a boundary condition such that Hence which is our equation of the catenary (hanging cable). 2. Using calculus of variations:The total energy of the hanging cable is given by As a hanging cable does not move, it has no velocity energy, but only has a potential energy. Let s be a length of the cable and let ρ be its density and A be the area of its section. Then the total energy of the cable is given by Since we have Our curve is given such that the total energy E is the minimum. When the energy E is the minimum, we have a Beltrami identity: Then For intagration of the right hand side of the equation, substituting y = acosht, then dy = asinhtdt. Hence We suppose that t = 0, when x = 0. Then b = 0, and Therefore which is the equation of a catenary.
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