Chain

Etymology and history

The problem of the form taken by a flexible heavy wire interested very early the mathematicians. Galileo thought that this form was to be an arc of Parabole, but the proof of the opposite was brought in 1669 by Jungius, after a first questioning by Huygens in 1646.

In 1691, Leibniz, Jean Bernoulli, and Huygens, under the impulse of a challenge launched by Jacques Bernoulli show quasi simultaneously that the exact form is a chain. It is besides Huygens which baptizes it thus, in a letter addressed to Leibniz. Forsaking the Latin term of the problema funicularium , (problem relating to the cord), used by Bernoulli, it uses the word catenaria , curve relating to the chain ( catena ), then passes to French chain , thus joining again with the term catenella used by Galileo (whereas the anglophone mathematicians preserve the designation of Huygens to name it catenary , the same English word being translated into French by Caténaire with the same Latin origin, word used also in French for certain constructions autoportées in the shape of chain).

Certain French-speaking authors thus give him also the name of overhead line although it catenary indicates rather the association of a autoporté cable (in the shape of chain) supporting a second cable linaire in its lower part, the two cables being subjected to a force of longitudinal traction balanced by a series of pendulums, the system of bearing deforming the carrying chain to give him a form closer in fact to the Parabole, the chain being present virtually only in the central axis between the two cables where the pendulums variable length are articulated.

Mathematical definition

The Cartesian equation of the shape of the chain is:

y (X) = has \ cdot \ operatorname {CH} \ left ({X \ over has} \ right) = {has \ over 2} \ cdot \ left (e^ {X \ over has} + e^ {- {X \ over has}} \ right)

in which where

ch

indicates the hyperbolic Cosinus and the constant:

a = \ frac {T_o} {dP}. {DLL}

of the horizontal component of the force of tension mechanical

T_o

is the report/ratio and linear weight

P/L = dP/dL

of the chain (the relationship between the weight and the length of the chain or any of its elementary links).

To note that this equation depends on only one parameter $a$ (a constant, which with the dimension a length in its physical interpretation). A curve of equation:

y (X) =b \ cdot \ operatorname {CH} \ left ({X \ over has} \ right)

is generally not a chain in a strict sense. However, the shape of the curve does not vary except for an additive constant (determining its height of range), and the following curve will be also regarded as a generalized chain:

y (X) =a \ cdot \ operatorname {CH} \ left ({X \ over has} \ right) + c

One can also see it in the form of a parametric equation:

\ left \ {\ begin {matrix}

X (T) & = & has \ cdot \ ln \ left (T \ right) \ \ there (T) & = & {have \ over 2} \ left (T + {1 \ over T} \ right) + C \ \ T > 0 && \end{matrix} \right.

One can note that even if the chain undergoes a lengthening, because of the mechanical tension, it keeps the shape in chain because the total force of tension resulting from the horizontal mechanical tension and of the vertical weight remains constant throughout the chain and induces an identical lengthening and a antiproportionelle reduction of the identical linear weight over the entire length of the chain (in condition however that elasticity is perfect, i.e. that its matérieux is homogeneous and at temperature and constant pressure). It will not be the case if the constitution of the chain varies according to its length or if this one is subjected to environments of different pressure and temperature over its length (because that modifies its elongation in an unequal way in consequence of different dilation producing a variable linear density along the chain).

Mechanical calculation

In Mechanical, this equation can be calculated in two manners.

With the law of the balance of the forces: the sum of the force S being exerted on a link is null since one is with balance. The link is subjected to its Poids, like with traction on behalf of the close links. The direction of traction varies from one link to another.

One can also apply the Lagrangian formalism. The equation of Euler-Lagrange which one obtains is that of the vibrating Corde

\ frac {d^2 F} {dt^2} + \ frac {d^2 F} {dx^2} = f

The stationary solution checks

\ frac {df} {dt} = 0

and

\ frac {d^2 F} {dt^2} = 0

that is to say

\ frac {d^2 F} {dx^2} = f

what gives the hyperbolic Cosinus.

Properties and applications

the y-axis is axis of symmetry of the curve. For the x-axis, one speaks about bases .
the chain is a particular case of Alysoïde and Courbe of Ribaucour.
the chain is almost vertical close to the points of suspension, because it is there that the most important weight downwards draws more the chain. On the other hand, to the bottom of the curve, the slope decreases little by little since the chain supports weight less and less. It is besides one of the differences between the chain and the parabola: for an equal length, the parabola is more “pointed” in its lower part.
the chain does not appear only in the shape of a suspended wire. It is also found:

reversed, for an arc holding by its own weight (see on this subject the architectural tests of Gaudi).
vertical, in the profile of a rectangular sail attached to 2 horizontal bars, swollen by a blowing wind perpendicular to these bars, by neglecting the actual weight of the veil compared to the force of the wind. It is this property which justifies the name of “velar” (veil) given by Jacques Bernoulli.

Practical aspects

the theory of the chain follows the curve of balance of a line (chain or cable) suspended between two points, homogeneous, inextensible, without rigidity in inflection, subjected to its only weight. The solution of the problem is simple if the characteristics of the line (length and linear weight) are given and forces it applied at an end to calculate its extension (distances horizontal and vertical) and forces it at the other end. The corresponding formulas define the basic module of any calculation.
If the elasticity of the line cannot be neglected any more, it is taken into account by applying the Loi of Hooke, which involves simply a complication of the calculation of the extension in the basic module.
If the lines are consisted a succession of segments of different characteristics, the repeated call of the basic module makes it possible to obtain for the line a result similar to that of the segment by transmitting the forces of a segment to another and by adding up the extensions.
If there exists a bottom on which part of the line rests, the vertical force applied at an end makes it possible to determine the suspended length not deformed to add to the length posed on the bottom.
In all these cases, it is thus possible to obtain for the line a module which transforms the characteristics of the line and the force at an end into the extension and the force at the other end. Unfortunately, these relatively simple calculations are not adapted to the concrete problems in which one generally wishes to calculate the forces at the two ends according to the characteristics and of the extension. Two loops of Dichotomy, unconditionally convergent, solve the problem.

See too

For the historical context to see Mathematical in Europe at the XVIIe century
For more information on the hyperbolic function cosine, to see the hyperbolic article Function.
a cord subjected to a force can take other forms: to see the Courbe of the skipping rope, which undergoes not only one force distributed equitably to its own weight (which gives him the shape of a chain when it is not in rotation), but also a centrifugal force more important in the center of the skipping rope (in rotation) that at its ends, which deforms the chain at the point to make even take with this one a more pointed form in extreme cases triangular with a number of revolutions tending towards the infinite one where the centrifugal forces make negligible the forces related to the actual weight of the cord). The more quickly the cord turns, the more it becomes deformed and the differential of tension between the upper part and the lower part of the cord increases, this différenciel being maximum in the middle of the length of cord (which will be thus the point of rupture of this one if it too quickly is turned).
When one draws aside two initially jointed circles just left a soapy solution, the tubular surface which is created between these two profiles has a profile of chain: it is about a Caténoïde, of which the central axis of the tube to the shape of a chain: the tension on the upper surface of the tube catenoid (exerted longitudinally in the direction of the axis of the tube) is lower than that of lower surface and explains why the water tube romp always by bottom when this tension of spacing becomes higher than the tension of bringing together exerted between the soapy molecules.
the catenoid is carrying the ideal form to adopt for a autoportée structure adopting a profile of chain because it is possible to compensate for the compressive forces exerted on the upper surface by a precompression of this surface, and to compensate for the forces of spacing on the lower surface of the tube by allowing him a greater elasticity. This form is thus adopted for the tubes of carrying arches.
a bridge not suspended made up of only one quasi plane arch and only carried at its two ends, adopts the shape of chain naturally if it is subjected to no other constraint vertical or horizontal but its own weight or if the load which it supports is distributed equitably along its length. It is the same for a horizontal horse frame posed between two load-bearing walls.
the strongest constraint exerted on a structure in chain is that of a maximum weight carried in its center: it is the case of the skipping rope in rotation, where the centrifugal force related to its rotation is maximum in the center of the cord (where the spacing compared to the axis of rotation is maximum), or if the cord supports a weight suspended in its center supports (as on a clothes line if one does not fix the clothing worn on the wire to prevent that they do not slip all worms the medium of the cord).
Of other systems exists in the construction of autoportés bridges (built like an arch in reversed chain), enabling them to resist other exerted constraints horizontally either perpendicular to the arches or cables carrying in the axis the bridge (primarily by the wind) or vertically on surface the bridge (or by the wind or by the vehicles which circulate there: this constraint is easier to control because it has an effect identical to a variation of its weight actual and led to shorten the length of range); this requires that the ends of the bridge can move horizontally, in order to avoid the rupture in the center of the bridge by increase in the longitudinal tension if one prevents this longitudinal displacement which makes it possible to preserve the ideal profile of chain), and can be carried out by zones at each end sliding freely one in the other in the axis of the bridge; the permanent monitoring of the spacing or the bringing together of these sliding zones makes it possible to measure instantaneously the longitudinal tension of the autoportée structure and thus to prevent the ruptures (or to close circulation as soon as safety thresholds are exceeded for example because of winds too violent one).
the same system is employed for the horizontal frames, slightly longer than the spacing of the walls or ram vertical carriers, and sometimes carried by a rotary arm articulated at the top of rams carrying allowing to balance the spacing at each end.

External bonds

On the site Mathcurve.com

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