Catenary
A chain line (also rope curve catenoid or catenary, catenary or funicular curve english ) is a mathematical curve that describes the slack of a suspended at their ends chain under the influence of gravity. It is an elementary mathematical function, the hyperbolic cosine, cosh shortly.
- 2.1 parabola
- 2.2 catenoid
- 2.3 tractrix
Mathematical Description
The calculation of the catenary is a classical problem of the calculus of variations. People imagine a rope of certain mass and length, which is suspended at its ends. The catenary curve is the result of the lowest possible potential energy of the cable. The attempts to mathematically understand.
This requires the mathematical expression for the potential energy. He is a refinement of the well-known " weight by height ." The refinement is that the energy is " all parts" of the rope evaluated separately and added at the end. This is necessary because the parts of the rope are at different heights. The imaginary decomposition of the rope into smaller and smaller parts makes the sum of an integral. The height of is replaced by the unknown function, the mass by the mass of the rope piece over the interval; according to Pythagoras is this:
The mass per meter is. When the rope is suspended at the points accordingly, the energy is obtained ( " weight times height " ) as
A similar reasoning leads to the expression for the length of the cable:
The energy to be minimized, however, the length is specified. Man this brings under one roof by a Lagrange multiplier, ie minimizing the expression now
The variation results in the equation ( Euler-Lagrange equation):
Interestingly, fell out in this step both the mass size and the acceleration of gravity. A heavy rope thus takes on like a light the same form, and on the moon arises despite other acceleration due to gravity the same shape as on Earth.
The solutions of the equations, the functions
It is enlarged and shifted cosine hyperbolic functions. is the radius of curvature at the vertex (as shown), while the magnification factor. is the displacement in the direction, the displacement in the direction.
The actual form in which the rope ultimately to be calculated by, and thus adapts to the curve passing through the points of suspension and has the predetermined length.
Example
As an example, one between two posts (distance) suspended rope length is given ( see figure above). The posts are the same height and are with him, and, it is so.
To calculate the radius of curvature, we write the rope length as a function of:
This relation defines a function of and unambiguous. Since you can not specify a closed expression for the value of a numerical method for solving nonlinear equations must be calculated approximate.
Recently we read from the figure from the condition from which it is obtained. Furthermore, the relationships are
Where the " slack " is.
Is the potential energy of this system
More specifically it is the difference in energy compared to the case that the cable is fully on the level of the suspension ().
With the help of energy one can calculate the force in the suspension points. For this purpose, it is envisioned that the rope runs in a suspension over a pulley, which deflects the force in the horizontal direction. To ready to pull out the rope around a very small distance, you have to expend the energy. These can be calculated and thus obtains the force. For the calculation of comparing the energy of the original cable with the order shortened rope. The result is surprisingly simple, namely
The same formula can also be applied to sections of the rope. Since the sections all have the same radius of curvature, but for small parts ( in the valley below), the sag is negligible, there is the rope tension in the valley of the rope.
If one together the posts close, then dominated the slack quite accurate which is then half the rope length. The force is then expected to be half the weight of the rope, (note that two suspension points share the load ).
The formula also shows how the force at increasing cable tension half the weight by a factor greater than. The factor is practically 1 for very small radii of curvature, but about or even for very large radii of curvature.
In everyday life, the factor about 2 to 4 in suspension then affects the whole or twice the weight of the rope is.
Relations to other functions
Parabola
Joachim Jung dismissed in 1669 after that the catenary is not a parabola. Gottfried Leibniz, Christiaan Huygens and Johann Bernoulli found 1690/91 out how the chain curve is appropriate. The parable sets in at a line load, eg evenly distributed across the span of x a suspension bridge, wherein the weight of the cables with respect to which the road surface may be neglected. The figure on the right compares the curve of a chain line (red) with a normal parabola ( green).
Catenoid
The surface of revolution generated by the rotation of catenoid the x -axis is referred to as a catenoid, and is a minimal area.
Tractrix
The chain line is the evolute of the tractrix to (following curve).
Examples
For m = 100 and a pole spacing of 200 m ( ie a special case ) is a 2117, 5 m long rope needed. The sag is 54 m. For a steel cable with 100 cm2 cross section weighs a half rope 9,2 t. The corresponding weight force of 9,104 N is the vertical force on a hanger. The horizontal force to a suspension of 7.7 × 104 N.
Is about 20.2 % of the total width, the sag is equal to the width ( square-shaped overall dimension ). This is the case for example in front of the Gateway Arch ( see below in the architecture section ) which is 630 feet wide and high as well. The exact formula
With a = 127.7 feet and w / 2 = 315 feet is on display inside the monument.
Applications in architecture
- One of the catenary -like support line follows the shear force-free arc of the 192 meter high Gateway Arch in St. Louis. Due to the different thickness of the sheet, however there is no real catenary.
- Antoni Gaudí frequently used on the foot end design principle, among others, the Sagrada Familia in Barcelona.
- Another example is the built according to plans by Christopher Wren in 1666 after St Paul's Cathedral in London. Between an outer and inner wooden hemisphere he had put a catenoid, which began the severity of the lantern, but even allowed a lower structural weight.
- Cross-section of the roof of the Keleti railway station (Budapest, Hungary ) forms a catenary. Built from 1881 to 1884 Designer:. János Feketeházy.
- The Nubian ton, a barrel vault, is a variant of the Nubian vault, a vault in earth building construction without formwork and often without lessons that takes its name from the traditional designs in Nubia. To achieve the greatest possible stability, it follows the line of pressure, as a rule of the chain line.
Gallery
Gateway Arch in St. Louis
Casa Milà by Antoni Gaudí
Cross-section of the roof of the Keleti railway station (Budapest, Hungary)
Variation of the parameter a, or different from each remote attachment points
Drawing by Christiaan Huygens
Architectural model of Gaudí
Free-hanging power lines follow the chain line
Spinnenfäden approximately follow the chain line, emphasized here by dewdrop